Research ArticleNoncommutative Wormhole Solutions in EinsteinGauss-Bonnet Gravity

Shamaila Rani and Abdul Jawad

Department of Mathematics COMSATS Institute of Information Technology Lahore 54000 Pakistan

Correspondence should be addressed to Abdul Jawad jawadab181yahoocom

Received 2 November 2015 Revised 25 January 2016 Accepted 3 February 2016

Academic Editor Rong-Gen Cai

Copyright copy 2016 S Rani and A JawadThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

We explore static spherically symmetric wormhole solutions in the framework of 119899-dimensional Einstein Gauss-Bonnet gravityOur objective is to find out wormhole solutions that satisfy energy conditions For this purpose we consider two frameworks suchas Gaussian distributed and Lorentzian distributed noncommutative geometry Taking into account constant redshift function weobtain solutions in the form of shape function The fifth and sixth dimensional solutions with positive as well as negative Gauss-Bonnet coefficient are discussed Also we check the equilibrium condition for the wormhole solutions with the help of generalizedTolman-Oppenheimer-Volkoff equation It is interesting tomention here that we obtain fifth dimensional stablewormhole solutionsin both distributions that satisfy the energy conditions

1 Introduction

The study of wormhole solution becomes a prime focus ofinterest in the modern cosmology as it connects differentdistant parts of the universe as a shortcut The wormholeis like a tunnel or bridge with two ends which are openin distant parts of the universe to join To develop themathematical structure of wormhole in general relativity [1]the basic ingredient is an energy-momentum tensor whichconstitutes exotic matter This is hypothetical form of matterresults of the violation of energy conditions For two-waytravel the traversable condition must be fulfilled that is thethroat of wormhole must remains open due to violation ofthe null energy condition Since normal matter satisfies theenergy conditions the matter violating the energy conditionis called exotic The phantom dark energy violates the energyconditions which may be a reasonable source of wormholeconstruction [2 3]The inclusion of some scalar fieldmodelselectromagnetic field Gaussian and Lorentzian distributionsof noncommutative geometry thin-shell formalism and soforth demonstratesmore interesting and useful results [4ndash11]

In order to minimize the usage of exotic matter orto find another source of violation while normal matter

satisfies the energy conditions many directions are adoptedso far Modified theories of gravity are the most appeal-ing direction which contribute to using effective energy-momentum tensor For instance in 119891(119877) [12] as well as119891(119879) [13] theories it has been proved that the effectiveenergy-momentum tensor which consists of higher ordercurvature terms or some torsion terms is responsible forthe necessary violation to traverse through the wormholeNowadays higher dimensional wormhole solutions are alsounder discussion Many theories point out the existence ofextra dimensions in the universe which leads to explorewormhole solutions for higher dimensions

Rahaman with his collaborators have done a lot ofwork taking noncommutative geometry in four-dimensionalspacetime as well as higher dimensional cases Rahaman etal [14] studied the higher dimensional static spherically sym-metricwormhole solutions in general relativitywithGaussiandistribution They found these solutions up to four dimen-sionswhile they found these solutions in a very restrictivewayfor fifth dimensional case Bhar and Rahaman [15] obtainedthe same result by taking Lorentzian distribution Rahamanet al [16] worked for viable physical properties of newwormhole solutions inspired by noncommutative geometry

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2016 Article ID 7815242 11 pageshttpdxdoiorg10115520167815242

2 Advances in High Energy Physics

with conformal killing vectors In119891(119877) gravity with Gaussian[17] and Lorentzian distributions [8] wormhole solutions areconstructed but with violation of energy conditions Sharifand Rani [13] explored the noncommutative wormhole solu-tions in 119891(119879) gravity and found some physically acceptablesolutions

In a recent paper Jawad and Rani [18] studied Lorentzdistributed wormhole solutions in 119891(119879) gravity and foundsome stable wormhole solutions satisfying energy conditionsIn the higher dimensional gravity theories 119899-dimensionalEinstein Gauss-Bonnet gravity is widely used The mod-ern string theory established its natural appearance in thelow energy effective action Bhawal and Kar [19] studiedLorentzian wormhole solutions in 119863-dimensional EinsteinGauss-Bonnet gravity which depend on the dimensionalityof the spacetime and coupling coefficient of Gauss-Bonnetcombination Taking traceless fluid Mehdizadeh et al [20]extended this work and found wormhole solutions whichsatisfy energy conditions

We explore wormhole solutions taking spacetime of(119899 minus 2) sphere in the framework of 119899-dimensional EinsteinGauss-Bonnet gravity We consider two frameworks non-commutative geometry having Gaussian distributed energydensity and Lorentzian distributed energy density For 119899 =

5 and 6-dimensions we take positive as well as negativeGauss-Bonnet coefficient Also we check the stability ofwormhole solutions with the help of generalized Tolman-Oppenheimer-Volkoff equation The paper is organized asfollows in Section 2 we construct the field equation forhigher dimensional wormhole solutions in 119899-dimensionalGauss-Bonnet gravity Section 3 is devoted to the construc-tion of wormhole solutions in Gaussian and Lorentziannoncommutative frameworks In Section 4 we check theequilibrium condition of the wormhole solutions Section 5summarizes the discussions and results

2 Field Equations

In this section we provide some basic and brief reviewsabout 119899-dimensional Einstein Gauss-Bonnet gravity as wellas wormhole geometry and construct field equations in theunderlying scenario

21 119899-Dimensional EinsteinGauss-BonnetGravity Theactionfor 119899-dimensional Einstein-Gauss-Bonnet gravity is an out-come of string theory in low energy limit It is given by

119878119899GB = intradicminus119892 [119877 minus 1205981LGB] 119889

119899

119909 (1)

where 119877 is the 119899-dimensional Ricci scalar 1205981is the Gauss-

Bonnet coefficient and LGB is the Gauss-Bonnet termdefined as

LGB = 1198772

minus 4119877120572120573119877120572120573

+ 119877120572120573120574120575

119877120572120573120574120575

(2)

Varying the action with respect to metric tensor the fieldequations becomes

119866120572120573+ 1205981G120572120573= T120572120573 (3)

where 119866120572120573

andT120572120573

are the Einstein and energy-momentumtensors respectively while Gauss-Bonnet tensor is defined as

G120572120573= 2 (119877119877

120572120573minus 2119877120572120574119877120574

120573minus 2119877120572120574120573120575

119877120574120575

minus 119877120572120575120590120591

119877120590120591120575

120573)

minus

1

2

LGB119892120572120573

(4)

It is noted that we assume 8120587119866119899= 1 where 119866

119899is the 119899-

dimensional gravitational constant

22Wormhole Geometry Thewormhole spacetime for (119899minus2)sphere is given by [14 15]

1198891199042

= minus1198902120582(119903)

1198891199052

+

1198891199032

1 minus 119904 (119903) 119903

+ 1199032

119889Ω2

119899minus2 (5)

where 120582(119903) is the redshift function and 119904(119903) is the shapefunction For traversable wormhole scenario we have tochoose 120582 to be finite to satisfy the no-horizon conditionUsually it is taken as zero for the sake of simplicity whichgives 1198902120582(119903) rarr 1 The reason behind finite redshift function isas followsThe redshift function defines that part of themetricresponsible for finding the magnitude of the gravitationalredshift The gravitational redshift is the reduction in thefrequency that a photon will experience when it climbs outfrom gravitational potential well in order to escape to infinityIn doing so the photon uses energy Its energy is proportionalto its frequency A reduction in energy then is equivalent toa reduction in frequency which is also known as redshiftfunction If the wormhole has an event horizon it means thata photon emitted outwardly from the horizon cannot escapeto infinity In other words it would take an infinite amountof energy for the photon to escape Its frequency wouldbe infinity reduced that is its redshift would be negativelyinfinite A negatively infinite value of the redshift functionat a particular value of the radial coordinate indicates thepresence of an event horizon there Thus to be traversablewormhole solution the magnitude of its redshift functionmust be finite

The shape of the wormhole is such that a spherical holein space with increasing length of diameter as moving farfrom throat (theminimumnonzero value of radial coordinatedenoted as 119903

0) and combines two asymptomatically flat

regions In order to have a proper shape of the wormhole theshape function must attain the ratio to radial coordinate as1 and represents increasing behavior with respect to radialcoordinate which is 1 minus 119904(119903)119903 ge 0 This condition of ratio isknown as flare-out condition In addition the value of shapefunction and radial coordinate must be same at throat thatis 119904(119903

0) = 1199030 There are also some other constraints applied

on the derivative of shape functions which must be satisfiedThese are 1199031199041015840 minus 119904 lt 0 and 119904

1015840

(1199030) lt 1 Also the proper

distance119863(119903) = plusmn int1199031199030

(1 minus 119904(119903)119903)minus12

119889119903must meet the criteriaas decreasing behavior from upper region 119863 = +infin towardsthroat where 119889 = 0 and then towards lower region where119863 = minusinfin

In order to make the wormhole traversable the throatmust remain open To prevent shrinking of wormhole

Advances in High Energy Physics 3

throat there must exist such form of energy-momentumtensor which provides the corresponding matter contentThis matter content violates the energy conditions in orderto keep throat open and thus is named as exotic matterThis implies that violation of these conditions is the basickey ingredient to construct traversable wormhole solutionsSince the usual energy-momentum tensor satisfies the energyconditions therefore the search for wormhole solutions forwhich violation may come from other sources while mattercontent satisfying energy conditions becomes one of themostchallenging problems in astrophysics

The higher dimensional gravity theories and modifiedtheories may play positive role by providing violation fromhigher order Lagrangian terms and effective form of energy-momentum tensor The relationship between Raychaudhuriequation and attractiveness of gravity yields the weak energycondition (WEC) asT

120572120573120583120572

120583120573

ge 0 for any time-like vector120583120572In terms of components of the energy-momentum tensor thisinequality yields120588 ge 0 and120588+119901 ge 0Thenull energy condition(NEC) is developed by continuity through WEC that is theNEC isT

120572120573120594120572

120594120573

ge 0 for any null vector 120594120572 This inequalitygives 120588 + 119901 ge 0 Also it is noted that WEC keeps NEC

The anisotropic energy-momentum is given by

T120572

120573= (120588 + 119901

119903) 119906120572

119906120573minus 119901119903119892120572

120573+ (119901119905minus 119901119903) 120578120572

120578120573 (6)

where 119901119903and 119901

119905are the radial and tangential pressure

components with 120588 = 120588(119903) 119901 = 119901(119903) and satisfy 119906120572119906120573=

minus120578120572

120578120573= 1 Using (3) the field equations become

120588 (119903) =

(119899 minus 2)

21199032

[(1199041015840

minus

119904

119903

) (1 +

2120598119904

1199033)

+

119904

119903

(119899 minus 3) + (119899 minus 5)

120598119904

1199033]

(7)

119901119903(119903) =

(119899 minus 2)

2119903

[2 (1 minus

119904

119903

) (1 +

2120598119904

1199033) 1205821015840

minus

119904

1199032(119899 minus 3) + (119899 minus 5)

120598119904

1199033]

(8)

119901119905(119903) = (1 minus

119904

119903

) (1 +

2120598119904

1199033)[12058210158401015840

+ 12058210158402

+

(119904 minus 1199031199041015840

) 1205821015840

2119903 (119903 minus 119904)

]

+ (1 minus

119904

119903

)(

1205821015840

119903

+

119904 minus 1199031199041015840

21199032(119903 minus 119904)

) (119899 minus 3)

+ (119899 minus 5)

2120598119904

1199033 minus

119904

21199033(119899 minus 3) (119899 minus 4)

+ (119899 minus 5) (119899 minus 6)

120598119904

1199033 minus

21205821015840

120598

1199034(1 minus

119904

119903

) (119904 minus 1199031199041015840

) (119899

minus 5)

(9)

where prime refers derivative with respect to 119903 and 120598 = (119899 minus

3)(119899 minus 4)1205981for the sake of notational simplicity

3 Wormhole Solutions

In order to discuss the wormhole geometry and solutionsthere are several frameworks and strategies used to findunknown functions For instance we have five unknownfunctions 120588(119903) 119901

119903(119903) 119901

119905(119903) 120582(119903) and 119904(119903) in the under-

lying case One may choose some kind of equation ofstate representing accelerated expansion of the universe ordifferent forms of energy density such as energy densityof static spherically symmetric object with noncommutativegeometry having Gaussian or Lorentzian distributions andgalactic halo region and so forth The traceless energy-momentum tensor is also usedwhich is related to the Casimireffect In order to construct viable wormhole solutions in 119899-dimensional Einstein-Gauss-Bonnet gravity we assume dif-ferent forms of energy density of noncommutative geometryin the following

Nicolini et al [21] have improved the short distancebehavior of point-like structures in a new conceptualapproach based on coordinate coherent state formalism tononcommutative gravity In their method curvature singu-larities which appear in general relativity can be eliminatedThey have demonstrated that black hole evaporation processshould be stopped when a black hole reaches a minimalmass This minimal mass named black hole remnant is aresult of the existence of a minimal observable length Thisapproach which is the so-called noncommutative geometryinspired model via a minimal length caused by averagingnoncommutative coordinate fluctuations cures the curvaturesingularity in black holes In fact the curvature singularity atthe origin of black holes is substituted for a regular de Sittercore Accordingly the ultimate phase of the Hawking evapo-ration as a novel thermodynamically steady state comprisinga nonsingular behavior is concluded

It must be noted that generally it is not required toconsider the length scale of the coordinate noncommutativityto be the same as the Planck length Since the noncom-mutativity influences appear on a length scale connectedto that region they can behave as an adjustable parametercorresponding to that pertinent scale The presence of auniversal short distance cut-off leads to the effects such as inquantum field theory it curves UV divergences while it curescurvature singularities in general relativity In the specificcase of the gravity field equations the only modificationoccurs at the level of the energy-momentum tensor while119866120583] is formally left unchanged In nonommutative space the

usual definition of mass density in the form of Dirac deltafunction does not hold So in this space the usual form ofthe energy density of the static spherically symmetry smearedand particle-like gravitational source requires some otherforms of distribution

In view of the above explanations we are going to discusswormhole solutions with the help of two well-known energydistributions such as Gaussian and Lorentzian in noncom-mutative scenario As an important remark the essentialaspects of the noncommutativity approach are not specificallysensitive to any of these distributions of the smearing effects[22] rather only distribution parameter is different TheGaussian source has also been used by Sushkov [23] to model

4 Advances in High Energy Physics

0

2

4

6

8

minus2

00 10 30252005 15r

1minuss +r

(a)

0

2

4

minus2

86420r

1minuss minusr

(b)

Figure 1 Plots of (a) 1 minus 119904+119903 and (b) 1 minus 119904

minus119903 versus 119903 in Gaussian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed) 119899 = 6 120598 = 3

(purple) and 119899 = 6 120598 = minus3 (purple dashed)

phantom-energy supported wormholes as well as by Nicoliniand Spallucci [24] for the purpose of modeling the physicaleffects of short distance fluctuations of noncommutativecoordinates in the study of black holes Galactic rotationcurves inspired by a noncommutative geometry backgroundare discussed [25] The stability of a particular class of thin-shell wormholes in noncommutative geometry is analyzedelsewhere [26]

31 Gaussian Distributed Noncommutative Framework Anintrinsic characteristic of spacetime is the noncommuta-tivity which plays an effective role in several areas Itis an interesting consequence of string theory where thecoordinates of spacetime become noncommutative operatorson 119863-brane [27] The noncommutativity of spacetime canbe converted in the commutator [119909120572 119909120573] = 119894120579

120572120573 where120579120572120573 is an antisymmetric matrix describing discretization ofspacetime and has dimension (length)2 This discretizationprocess is similar to the discretization of phase space byPlanck constant Replacing the point-like structures withsmeared objects the energy density of the particle-like staticspherically symmetric gravitational source having mass 119872takes the following form [21]

120588nc =119872

(4120587120579)(119899minus1)2

119890minus11990324120579

(10)

where 120579 is the noncommutative parameter in Gaussiandistribution The mass 119872 could be a diffused centralizedobject such as a wormhole [28] It is mentioned here that thesmearing effect is achieved by replacing the Gaussian distri-bution of minimal length radic120579 with the Dirac delta functionIn order to construct 119899-dimensional Einstein Gauss-Bonnetwormhole geometry we equate the energy density given in(7) and 120588nc in (10) yields the following differential equation

1199041015840

=

1

1 + 21199041205981199033[(4 minus 119899 + (7 minus 119899)

119904120598

1199033)

+

21199032

119872

(119899 minus 2)

(4120587120579)(1minus119899)2

119890minus11990324120579

]

(11)

The solution of this equation is

119904 (119903) = minus

1

2

[

1199033

120598

plusmn

119903(7minus119899)2

120587minus1198994

radic120598 (119899 minus 1)

(119899 minus 2) 1205871198992

(119903119899minus1

+ 41205982

1198881)

minus 4119872radic120587120598Gamma[119899 minus 12

1199032

4120579

]

12

]

(12)

where 1198881is an integration constant This solution has two

roots with plus and minus signs and we assign these rootsas 119904+(119903) and 119904

minus(119903) solutions In order to plot the quantities

1 minus 119904119903 120588 120588 + 119901119903 and 120588 + 119901

119905to obtain wormhole solutions

for both of these solutions we restrict ourselves to five- andsix-dimensional cases with constant redshift function 120582 = 0The expressions of NEC takes the form

120588 + 119901119903=

(119899 minus 2)

21199032

(1199041015840

minus

119904

119903

) (1 +

2120598119904

1199033)

120588 + 119901119905=

1199031199041015840

minus 119904

21199033

(1 +

6120598119904

1199033)

+

119904

1199033[(119899 minus 3) + (119899 minus 5)

2119904120598

1199033]

(13)

In this regard we assume values of some constants as119872 = 0008 120579 = 0002 while 120598

1= 05 minus05 result in 120598 =

1 minus1 for five-dimensional and 120598 = 3 minus3 for six-dimensionalcase Figure 1(a) represents the plot of 1 minus 119904

+119903 versus 119903

for five and six-dimensional wormhole solutions The graphrepresents positively increasing behavior for positive 120598 forboth curves For negative 120598 we examine that graph initiallyrepresents positive behavior for both dimensions in the range119903 lt 1 and 119903 lt 18 for 119899 = 5 and 6 respectively and thendecreases towards negative values In Figure 1(b) the graphof 1 minus 119904

minus119903 versus 119903 shows the same behavior for all curves as

in Figure 1(a) with positive behavior of dashed curves in therange 119903 lt 4 and 119903 lt 8 for 119899 = 5 ad 6 However we have less

Advances in High Energy Physics 5

025 030 035 040020r

000

002

120588

004

006

008

010

(a) Evolution of 120588+versus 119903

000

002

004

006

008

010

120588

025 030 035 040020r

(b) Evolution of 120588minusversus 119903

Figure 2 Plots of (a) 120588+and (b) 120588

minusversus 119903 in Gaussian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed) 119899 = 6 120598 = 3 (purple)

and 119899 = 6 120598 = minus3 (purple dashed)

possibility of wormhole scenario with respect to 119903 for positiveroot solution as compared to negative root solution of shapefunction

Figure 2 shows the behavior of energy density versus119903 as positively decreasing behavior for both dimensionscorresponding to 119899 = 5 and 6 dimensions For 119904

+and positive

120598 the plot of 120588+119901119903represents positive behavior in decreasing

manner for 119899 = 5 and 6 while representing negative behaviorfor negative 120598 as shown in Figure 3(a) In Figure 3(b) weexamine same behavior for negative root solution Figure 4(a)depicts the opposite behavior to Figure 3(a) that is for 119899 =

5 6 dimensions and positive 120598 120588 + 119901119905demonstrates negative

behavior Thus these plots express the violation of WECincorporating the case of 119904

+ For 119904

minus we obtain the same

behavior as in Figure 3(b) which indicates positive behaviorfor positive 120598 and 119899 = 5 6 and negative behavior for negative120598 and 119899 = 5 6 This implies that WEC is satisfied fornegative root solutionwith positiveGauss-Bonnet coefficientThus we obtain physically acceptable wormhole solutionssatisfying WEC for both dimensions

32 Lorentzian Distributed Non-Commutative FrameworkNow we consider the case of noncommutative geometrywith Lorentzian distributionThe energy density of point-likesource under this distribution becomes [8 15]

120588Lnc =119872radic120601

1205872(1199032+ 120601)1198992

(14)

where 120601 is the noncommutative parameter in Lorentziandistribution Inserting 120588Lnc in (7) the differential equationtakes the following form

1199041015840

=

1

1 + 21199041205981199033[

21199032

119872radic120601

1205872(119899 minus 2) (119903

2+ 120601)1198992

+

119887

119903

4 minus 119899 + (7 minus 119899)

119904120598

1199033]

(15)

The solution of this equation is given by

119904 (119903) =

1199033

2120598120587

[minus120587 plusmn

1

(119899 minus 2) (119899 minus 1)

1205872

(1198992

minus 3119899 + 2) (119903119899

+ 41199031205982

1198882)

+ 8119872120598119903119899

120601(1minus119899)2Hypergeometric 2F1 [1

2

(119899 minus 1)

119899

2

119899 + 1

2

minus

1199032

120601

]

12

]

(16)

where 1198882is an integration constant We again assign both

solutions as 119904+and 119904minusand explore the wormhole solutions

In this case we choose constants as 120601 = 2 119872 = 2 and1198882= 05 for same dimensions and Gauss-Bonnet coefficient

as for noncommutative backgroundIn Figure 5(a) we plot 1 minus 119904

+119903 versus 119903 which represents

that the condition for wormhole geometry that is 1minus119904119903 gt 0holds for 5 and 6 dimensions with negative 120598 For positive 120598the positivity of this expression depends on some ranges sothat it remains positive for 119903 lt 19 for 119899 = 5 while 119899 = 6

observes the range 119903 lt 21 Figure 5(b) corresponds to theplot of 1 minus 119904

minus119903 with respect to 119903 showing positive behavior

for positive 120598 For 119899 = 5 it expresses a very short range119903 lt 05 for positivity whereas it remains negative for 6-dimensional solutionThat is we have no wormhole solutionfor 119899 = 6 120598 = minus3 taking negative root solution So we skipthis case in further discussion The behavior of 120588

+and 120588

minus

remains positive for all cases except 119899 = 6 120598 = minus3 for whichit describes negative behavior as shown in Figures 6(a) and6(b) This implies that we have no wormhole solutions for 6-dimensional case with negative Gauss-Bonnet coefficient forboth solutions

Figure 7(a) expresses the behavior of 120588 + 119901119903for positive

root solution versus 119903 which remains positive for the range119903 lt 1 for 119899 = 5 and 6 with positive 120598 and then turns towardsnegative behavior It demonstrates negative behavior for 119903 lt 1and then moves to positive region Incorporating 119904

minussolution

120588 + 119901119903shows positive behavior only for the case 119899 = 5

6 Advances in High Energy Physics

pr+120588

minus06

minus04

minus02

00

02

04

06

0402 06 08 10 12 1400r

(a) Evolution of (120588 + 119901119903)+ versus 119903

pr+120588

minus3

minus2

minus1

0

1

2

3

04 06 08 10 12 1402r

(b) Evolution of (120588 + 119901119903)minus versus 119903

Figure 3 Plots of (a) 120588 + 119901119903for 119904+and (b) 120588 + 119901

119903for 119904minusversus 119903 in Gaussian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

119899 = 6 120598 = 3 (purple) and 119899 = 6 120598 = minus3 (purple dashed)

pt+120588

pt+120588

n = 5 120598 = minus1

minus04

minus02

00

02

04

05 10 15 20 25 3000r

000001500000200000250000030000035

17 18 19 2016r

(a) Evolution of (120588 + 119901119905)+ versus 119903

pt+120588

minus10

minus5

0

5

10

05 10 15 20 25 3000r

(b) Evolution of (120588 + 119901119905)minus versus 119903

Figure 4 Plots of (a) 120588 + 119901119905for 119904+and (b) 120588 + 119901

119905for 119904minusversus 119903 for in Gaussian distribution 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

119899 = 6 120598 = 3 (purple) and 119899 = 6 120598 = minus3 (purple dashed)

n = 5 120598 = minus1

minus2

minus1

0

1

2

05 10 15 20 25 3000r

0985

0990

0995

1000

005 010 015 020000

r

1minuss +r

1minuss +r

(a)

05 10 15 20 25 3000r

minus4

minus2

0

2

4

1minuss minusr

(b)

Figure 5 Plots of (a) 1minus 119904+119903 and (b) 1minus 119904

minus119903 versus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed) and 119899 = 6

120598 = 3 (purple)

Advances in High Energy Physics 7

04 05 06 07 08 09 10050100150200

r

05 10 15 20

0

05 10 15 20r

n = 6 120598 = minus3

n = 6 120598 = 3

minus1000minus2000minus3000minus4000minus5000minus6000

001

002

120588

120588

120588

003

004

005

(a) Evolution of 120588+versus 119903

05 10 15 20000

001

002

003

004

005

r

120588

(b) Evolution of 120588minusversus 119903

Figure 6 Plots of (a) 120588+and (b) 120588

minusversus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed) and 119899 = 6 120598 = 3

(purple) For (a) 120588+ 119899 = 6 and 120598 = minus3 (purple dashed)

05 10 15 20 25 30

0

5

10

15

r

minus10

minus5

pr+120588

(a) Evolution of (120588 + 119901119903)+versus 119903

02 04 06 08 101020304050

02 04 06 08 10

0

n = 5 120598 = 1

r

rminus8000

minus6000

minus4000

minus2000

pr+120588

pr+120588

(b) Evolution of (120588 + 119901119903)minusversus 119903

Figure 7 Plots of (a) 120588 + 119901119903for 119904+and (b) 120588 + 119901

119903for 119904minusversus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

and 119899 = 6 120598 = 3 (purple)

120598 gt 0 and negative behavior for the remaining two casesas shown in Figure 7(b) In Figures 8(a) and 8(b) we draw120588 + 119901119905for both solutions versus 119903 This expression represents

the negative behavior for (119899 = 5 120598 = 1) (119899 = 5 120598 = minus1) with119904+and (119899 = 5 120598 = minus1) with 119904

minusand preserves positivity for

(119899 = 6 120598 = 3) with 119904+ (119899 = 5 120598 = 1) (119899 = 6 120598 = 3) with 119904

minus

solution

4 Equilibrium Condition

In order to find the equilibrium configuration of thewormhole solutions in Gaussian as well as Lorentzian dis-tributed noncommutative backgrounds we use the general-ized Tolman-Oppenheimer-Volkoff equation This equationis derived by solving the Einstein equations for a general time-invariant spherically symmetric metric having metric tensor119892120572120573

= (119890120591(119903)

minus119890120596(119903)

minus1199032

minus1199032sin2120579) where 120572120573 represents

only diagonal entries and 120591 120596 are general metric functionsdependent on 119903 The generalized Tolman-Oppenheimer-Volkoff equation is

119889119901119903

119889119903

+

1205911015840

2

(120588 + 119901119903) +

2

119903

(119901119903minus 119901119905) = 0 (17)

Keeping in mind the above equation de Leon [28] proposedan equation for anisotropicmass distributionwhich naturallygives the equilibrium for the wormhole subject It is given by

2

119903

(119901119905minus 119901119903) minus

119890(120596minus120591)2

119898eff1199032

(120588 + 119901119903) minus

119889119901119903

119889119903

= 0 (18)

where effective gravitational mass 119898eff = (12)1199032

119890(120591minus120596)2

1205911015840 is

measured from throat to some arbitrary radius 119903 Accord-ingly the gravitational hydrostatic and anisotropic force dueto anisotropic matter distribution are defied as follows

119891119892= minus

1205911015840

(120588 + 119901119903)

2

119891ℎ= minus

119889119901119903

119889119903

119891119886=

2 (119901119905minus 119901119903)

119903

(19)

It is required that119891119892+119891ℎ+119891119886= 0must hold for the wormhole

solutions to be in equilibrium In the underlying cases we

8 Advances in High Energy Physics

05 10 15 20 25 30

0

r

r

150 152 154 156 158 160

r

05 10 15 200

500

1000

1500

2000

pt+120588

pt+120588

pt+120588

minus0058

minus15

minus10

minus5

minus0056

minus0054

minus0052

minus0050n = 5 120598 = minus1

n = 5 120598 = 1

(a) Evolution of (120588 + 119901119905)+versus 119903

r

05 10 15 20

r

00 05 10 15 200

5

10

15

20

pt+120588 p

t+120588

minus70

minus60

minus50

minus40

minus30

minus20

minus10

n = 5 120598 = minus1

(b) Evolution of (120588 + 119901119905)minusversus 119903

Figure 8 Plots of (a) 120588 + 119901119905for 119904+and (b) 120588 + 119901

119905for 119904minusversus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

and 119899 = 6 120598 = 3 (purple)

proceedwith constant redshift function 120582 = 0which vanishesthe gravitational contribution119891

119892in the equilibrium equation

that is 120591 = 2120582 leads to 1205911015840 = 0 for constant 120582 Thus we are leftwith hydrostatic and anisotropic forces with correspondingequilibrium condition as

119891119886+ 119891ℎ= 0 (20)

Using (8) and (9) we obtain the following expressions for thehydrostatic and anisotropic forces

119891119886=

1

1199033(119899 minus 1)

119904

119903

minus 1199041015840

(119899 minus 3 + (119899 minus 5)

119904120598

1199033)

minus

119904

1199034[(119899 minus 3) (119899 minus 4) + (119899 minus 5) (119899 minus 6)

119904120598

1199033]

119891ℎ=

(119899 minus 2)

21199033

[119899 minus 3 + (119899 minus 5)

119904120598

1199033+

119904120598 (119899 minus 5)

1199033

]

sdot (1199041015840

minus

3119904

119903

)

(21)

Figure 9 represents the plots of anisotropic as well ashydrostatic forces for the wormhole solutions in Gaussiandistributed framework Figures 9(a) and 9(c) describe theequilibrium for positive root solution for 119899 = 5 120598 =

1 minus1 and 119899 = 6 120598 = 3 minus3 through opposite behaviorand hence they cancel each other in order to satisfy (20)For negative root solution 119904

minus Figures 9(b) and 9(d) show

the stable configuration of wormhole solutions for fifthdimensional case only The behaviors of both forces (whichare not in opposite manner) do not cancel each other sothere is no equilibrium configuration examined for sixthdimensional wormhole solutions In the case of Lorentziannoncommutative background Figure 10 expresses that allwormhole solutions are in equilibrium by satisfying theequilibrium condition for both dimensions

Table 1 Wormhole solutions with Gaussian distributed noncom-mutative framework for 119904

+

Expressions 119899 = 5 120598 =

1

119899 = 5 120598 =

minus1

119899 = 6 120598 =

3

119899 = 6 120598 =

minus3

1 minus 119904+119903 Positive Positive

for 119903 lt 1 Positive Positivefor 119903 lt 18

120588 Positive Positive Positive Positive120588 + 119901

119903 Positive Negative Positive Negative120588 + 119901

119905 Negative Positive Negative PositiveWEC Violates Violates Violates ViolatesEC Holds Holds Holds Holds

5 Conclusion

It is a well-known fact that the existence of wormhole solu-tions is based on violation of NEC Since the normal mattersatisfies the energy conditions this violation is associatedwith an energy-momentum tensor which provides exoticmatter a hypothetical form of matter To explore realisticmodel or physically acceptable wormhole solutions it isnecessary to find such a source which gives the violationof NEC while normal matter meets the energy conditionsHere in this paper we have explored wormhole solutionsin 119899-dimensional Einstein Gauss-Bonnet gravity with Gaus-sian and Lorentzian noncommutative backgrounds We haverestricted ourselves to fifth and sixth dimensional cases withpositive as well as negativeGauss-Bonnet coefficient Also wehave checked the condition of equilibrium for the wormholesolutions

The results are summarized in Tables 1 and 2 It is notedthat the 119904

+(Tables 1 and 3) and 119904

minus(Tables 2 and 4) represent

the two roots of the solutions in both backgrounds and ECdenotes the equilibrium condition

In the Lorentzan distributed noncommutative frame-work we have found negative energy density for 119904

+while

Advances in High Energy Physics 9

02 04 06 1008

0

2

4

r

fa

minus2

minus4

(a) Evolution of 119891119886versus 119903

r

02 04 06 08 10

0

2

4

fa

minus2

minus4

(b) Evolution of 119891119886versus 119903

02 04 06 08 10

0

2

4

r

fh

minus2

minus4

(c) Evolution of 119891ℎversus 119903

02 04 06 08 10

0

2

4

r

fh

minus2

minus4

(d) Evolution of 119891ℎversus 119903

Figure 9 Plots of (a) 119891119886for 119904+ (b) 119891

119886for 119904minus (c) 119891

ℎfor 119904+ (d) 119891

ℎfor 119904minusversus 119903 with Gaussian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1

(red dashed) and 119899 = 6 120598 = 3 (purple)

Table 2 Wormhole solutions with Gaussian distributed noncom-mutative framework for 119904

minus

Expressions 119899 = 5 120598 =

1

119899 = 5 120598 =

minus1

119899 = 6 120598 =

3

119899 = 6 120598 =

minus3

1 minus 119904minus119903 Positive Positive

for 119903 lt 4 Positive Positivefor 119903 lt 8

120588 Positive Positive Positive Positive120588 + 119901

119903 Positive Negative Positive Negative120588 + 119901

119905 Positive Negative Positive NegativeWEC Holds Violates Holds Violates

EC Holds Holds Does nothold

Does nothold

violation of 1 minus 119904119903 gt 0 incorporating 119904minusfor the case 119899 = 6

120598 = minus3 so we skipped this case The remaining results aresummarized in Tables 3 and 4

In the paper [20] higher dimensional asymptotically flatwormhole solutions have been explored in the frameworkof Gauss-Bonnet gravity by considering a specific choicefor a radial dependent redshift function and by imposinga particular equation of state The WEC is satisfied at thethroat by considering a negative Gauss-Bonnet coupling

Table 3 Wormhole solutions with Lorentzian noncommutativebackground for 119904

+

Expressions 119899 = 5 120598 = 1 119899 = 5 120598 = minus1 119899 = 6 120598 = 3

1 minus 119904+119903

Positive for119903 lt 19

Positive Positive for119903 lt 21

120588 Positive Positive Positive

120588 + 119901119903

Positive for119903 lt 1

Positive for119903 gt 1

Positive for119903 lt 1

120588 + 119901119905 Negative Negative Positive

WEC Does nothold

Does nothold

Holds for119903 lt 1

EC Holds Holds Holds

constant Furthermore they have considered a constantredshift function and shown specifically that for negativeGauss-Bonnet coupling constant one may have normalmatter in a determined radial region and that the increaseof coupling constant enlarges the normal matter regionIn the present paper we have taken energy density undernoncommutative geometry distributions instead of particularequation of stateWe have obtained results for positiveGauss-Bonnet coefficient satisfying energy conditions It contains

10 Advances in High Energy Physics

02 04 06 08 10

0

200

400

r

fa

minus400

minus200

(a) Evolution of 119891119886versus 119903

02 04 06 08 10

0

200

400

r

fa

minus400

minus200

(b) Evolution of 119891119886versus 119903

02 04 06 08 10r

minus400

minus200

0

200

400

fh

(c) Evolution of 119891ℎversus 119903

02 04 06 08 10

0

200

400

r

fh

minus400

minus200

(d) Evolution of 119891ℎversus 119903

Figure 10 Plots of (a) 119891119886for 119904+ (b) 119891

119886for 119904minus (c) 119891

ℎfor 119904+ (d) 119891

ℎfor 119904minusversus 119903 with Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5

120598 = minus1 (red dashed) and 119899 = 6 120598 = 3 (purple)

Table 4 Wormhole solutions with Lorentzian noncommutativebackground for 119904

minus

Expressions 119899 = 5 120598 = 1 119899 = 5 120598 = minus1 119899 = 6 120598 = 3

1 minus 119904minus119903 Positive Positive for

119903 lt 05

Positive

120588 Positive Positive Positive120588 + 119901

119903 Positive Negative Negative120588 + 119901

119905 Positive Negative Positive

WEC Holds Does nothold

Does nothold

EC Holds Holds Holds

fifth dimensional wormhole solutions in both backgroundssatisfying equilibrium condition and sixth dimensional withdisequilibrium in noncommutative background with 119904

minussolu-

tion Also there is possibility for the existence of wormholein equilibrium satisfying WEC for 119899 = 6 120598 = 3 taking intoaccount 119904

+solution for the range 119903 lt 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M S Morris and K S Thorne ldquoWormholes in spacetimeand their use for interstellar travel a tool for teaching generalrelativityrdquo American Journal of Physics vol 56 no 5 pp 395ndash412 1988

[2] M Sharif and A Jawad ldquoPhantom-like generalized cosmicchaplygin gas and traversable wormhole solutionsrdquo The Euro-pean Physical Journal Plus vol 129 article 15 2014

[3] F S N Lobo F Parsaei and N Riazi ldquoNew asymptotically flatphantom wormhole solutionsrdquo Physical Review D vol 87 no 8Article ID 084030 2013

[4] S-W Kim andH Lee ldquoExact solutions of a charged wormholerdquoPhysical Review D vol 63 no 6 Article ID 064014 5 pages2001

[5] M Jamil and M U Farooq ldquoPhantom wormholes in (2 + 1)-dimensionsrdquo International Journal ofTheoretical Physics vol 49no 4 pp 835ndash841 2010

[6] M U Farooq M Akbar and M Jamil ldquoDynamics and ther-modynamics of (2+1)minusdimensional evolving lorentzian worm-holerdquo in Proceedings of the 3rd AlgerianWorkshop on Astronomyand Astrophysics vol 1295 of AIP Conference Proceedings pp176ndash190 Constantine Algeria June 2010

[7] P K Kuhfittig ldquoGravitational lensing of wormholes in thegalactic halo regionrdquo The European Physical Journal C vol 74no 3 article 2818 2014

Advances in High Energy Physics 11

[8] F Rahaman A Banerjee M Jamil A K Yadav and H IdrisldquoNoncommutative wormholes in f(R) gravity with lorentziandistributionrdquo International Journal ofTheoretical Physics vol 53no 6 pp 1910ndash1919 2014

[9] M Sharif and S Rani ldquoDynamical wormhole solutions in f(T)gravityrdquo General Relativity and Gravitation vol 45 no 11 pp2389ndash2402 2013

[10] M Sharif and S Rani ldquoCharged noncommutative wormholesolutions in f (T) gravityrdquo The European Physical Journal Plusvol 129 no 10 article 237 2014

[11] M Sharif and S Rani ldquoGalactic halo wormhole solutions in119891(119879) gravityrdquoAdvances in High Energy Physics vol 2014 ArticleID 691497 9 pages 2014

[12] F S N Lobo andM A Oliveira ldquoWormhole geometries in f (R)modified theories of gravityrdquo Physical Review D vol 80 ArticleID 104012 2009

[13] M Sharif and S Rani ldquoWormhole solutions in f(T) gravity withnoncommutativerdquo Physical Review D vol 88 no 12 Article ID123501 2013

[14] F Rahaman S Islam P K F Kuhfittig and S Ray ldquoSearch-ing for higher-dimensional wormholes with noncommutativegeometryrdquo Physical Review D vol 86 no 10 Article ID 1060102012

[15] P Bhar and F Rahaman ldquoSearch of wormholes in differ-ent dimensional non-commutative inspired space-times withLorentzian distributionrdquo The European Physical Journal C vol74 no 12 article 3213 2014

[16] F Rahaman S Karmakar I Karar and S Ray ldquoWormholeinspired by non-commutative geometryrdquo Physics Letters B vol746 pp 73ndash78 2015

[17] M Jamil F Rahaman R Myrzakulov P Kuhfittig N Ahmedand U Mondal ldquoNonommutative wormholes in f(R) gravityrdquoJournal of the Korean Physical Society vol 65 no 6 pp 917ndash9252014

[18] A Jawad and S Rani ldquoLorentz distributed noncommutativewormhole solutions in extended teleparallel gravityrdquoThe Euro-pean Physical Journal C vol 75 article 173 2015

[19] B Bhawal and S Kar ldquoLorentzian wormholes in Einstein-Gauss-Bonnet theoryrdquo Physical Review D vol 46 no 6 pp2464ndash2468 1992

[20] M R Mehdizadeh M K Zangeneh and F S N LoboldquoEinstein-Gauss-Bonnet traversable wormholes satisfying theweak energy conditionrdquo Physical ReviewD vol 91 no 8 ArticleID 084004 2015

[21] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[22] S H Mehdipour ldquoEntropic force approach to noncommutativeSchwarzschild black holes signals a failure of current physicalideasrdquo The European Physical Journal Plus vol 127 article 802012

[23] S Sushkov ldquoWormholes supported by a phantom energyrdquoPhysical Review D vol 71 no 4 Article ID 043520 2005

[24] P Nicolini and E Spallucci ldquoNoncommutative geometry-inspired dirty black holesrdquo Classical and Quantum Gravity vol27 no 1 Article ID 015010 2010

[25] F Rahaman P K Kuhfittig K Chakraborty A A Usmani andS Ray ldquoGalactic rotation curves inspired by a noncommutative-geometry backgroundrdquo General Relativity and Gravitation vol44 no 4 pp 905ndash916 2012

[26] P K F Kuhfittig ldquoOn the stability of thin-shell wormholes innoncommutative geometryrdquo Advances in High Energy Physicsvol 2012 Article ID 462493 12 pages 2012

[27] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquo Journal of High Energy Physics vol 1999 no 9 article032 1999

[28] J P de Leon ldquoMass and charge in brane-world and non-compact Kaluza-KLEin theories in 5 dimrdquo General Relativityand Gravitation vol 35 no 8 pp 1365ndash1384 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

2 Advances in High Energy Physics

with conformal killing vectors In119891(119877) gravity with Gaussian[17] and Lorentzian distributions [8] wormhole solutions areconstructed but with violation of energy conditions Sharifand Rani [13] explored the noncommutative wormhole solu-tions in 119891(119879) gravity and found some physically acceptablesolutions

In a recent paper Jawad and Rani [18] studied Lorentzdistributed wormhole solutions in 119891(119879) gravity and foundsome stable wormhole solutions satisfying energy conditionsIn the higher dimensional gravity theories 119899-dimensionalEinstein Gauss-Bonnet gravity is widely used The mod-ern string theory established its natural appearance in thelow energy effective action Bhawal and Kar [19] studiedLorentzian wormhole solutions in 119863-dimensional EinsteinGauss-Bonnet gravity which depend on the dimensionalityof the spacetime and coupling coefficient of Gauss-Bonnetcombination Taking traceless fluid Mehdizadeh et al [20]extended this work and found wormhole solutions whichsatisfy energy conditions

We explore wormhole solutions taking spacetime of(119899 minus 2) sphere in the framework of 119899-dimensional EinsteinGauss-Bonnet gravity We consider two frameworks non-commutative geometry having Gaussian distributed energydensity and Lorentzian distributed energy density For 119899 =

5 and 6-dimensions we take positive as well as negativeGauss-Bonnet coefficient Also we check the stability ofwormhole solutions with the help of generalized Tolman-Oppenheimer-Volkoff equation The paper is organized asfollows in Section 2 we construct the field equation forhigher dimensional wormhole solutions in 119899-dimensionalGauss-Bonnet gravity Section 3 is devoted to the construc-tion of wormhole solutions in Gaussian and Lorentziannoncommutative frameworks In Section 4 we check theequilibrium condition of the wormhole solutions Section 5summarizes the discussions and results

2 Field Equations

In this section we provide some basic and brief reviewsabout 119899-dimensional Einstein Gauss-Bonnet gravity as wellas wormhole geometry and construct field equations in theunderlying scenario

21 119899-Dimensional EinsteinGauss-BonnetGravity Theactionfor 119899-dimensional Einstein-Gauss-Bonnet gravity is an out-come of string theory in low energy limit It is given by

119878119899GB = intradicminus119892 [119877 minus 1205981LGB] 119889

119899

119909 (1)

where 119877 is the 119899-dimensional Ricci scalar 1205981is the Gauss-

Bonnet coefficient and LGB is the Gauss-Bonnet termdefined as

LGB = 1198772

minus 4119877120572120573119877120572120573

+ 119877120572120573120574120575

119877120572120573120574120575

(2)

Varying the action with respect to metric tensor the fieldequations becomes

119866120572120573+ 1205981G120572120573= T120572120573 (3)

where 119866120572120573

andT120572120573

are the Einstein and energy-momentumtensors respectively while Gauss-Bonnet tensor is defined as

G120572120573= 2 (119877119877

120572120573minus 2119877120572120574119877120574

120573minus 2119877120572120574120573120575

119877120574120575

minus 119877120572120575120590120591

119877120590120591120575

120573)

minus

1

2

LGB119892120572120573

(4)

It is noted that we assume 8120587119866119899= 1 where 119866

119899is the 119899-

dimensional gravitational constant

22Wormhole Geometry Thewormhole spacetime for (119899minus2)sphere is given by [14 15]

1198891199042

= minus1198902120582(119903)

1198891199052

+

1198891199032

1 minus 119904 (119903) 119903

+ 1199032

119889Ω2

119899minus2 (5)

where 120582(119903) is the redshift function and 119904(119903) is the shapefunction For traversable wormhole scenario we have tochoose 120582 to be finite to satisfy the no-horizon conditionUsually it is taken as zero for the sake of simplicity whichgives 1198902120582(119903) rarr 1 The reason behind finite redshift function isas followsThe redshift function defines that part of themetricresponsible for finding the magnitude of the gravitationalredshift The gravitational redshift is the reduction in thefrequency that a photon will experience when it climbs outfrom gravitational potential well in order to escape to infinityIn doing so the photon uses energy Its energy is proportionalto its frequency A reduction in energy then is equivalent toa reduction in frequency which is also known as redshiftfunction If the wormhole has an event horizon it means thata photon emitted outwardly from the horizon cannot escapeto infinity In other words it would take an infinite amountof energy for the photon to escape Its frequency wouldbe infinity reduced that is its redshift would be negativelyinfinite A negatively infinite value of the redshift functionat a particular value of the radial coordinate indicates thepresence of an event horizon there Thus to be traversablewormhole solution the magnitude of its redshift functionmust be finite

The shape of the wormhole is such that a spherical holein space with increasing length of diameter as moving farfrom throat (theminimumnonzero value of radial coordinatedenoted as 119903

0) and combines two asymptomatically flat

regions In order to have a proper shape of the wormhole theshape function must attain the ratio to radial coordinate as1 and represents increasing behavior with respect to radialcoordinate which is 1 minus 119904(119903)119903 ge 0 This condition of ratio isknown as flare-out condition In addition the value of shapefunction and radial coordinate must be same at throat thatis 119904(119903

0) = 1199030 There are also some other constraints applied

on the derivative of shape functions which must be satisfiedThese are 1199031199041015840 minus 119904 lt 0 and 119904

1015840

(1199030) lt 1 Also the proper

distance119863(119903) = plusmn int1199031199030

(1 minus 119904(119903)119903)minus12

119889119903must meet the criteriaas decreasing behavior from upper region 119863 = +infin towardsthroat where 119889 = 0 and then towards lower region where119863 = minusinfin

In order to make the wormhole traversable the throatmust remain open To prevent shrinking of wormhole

Advances in High Energy Physics 3

throat there must exist such form of energy-momentumtensor which provides the corresponding matter contentThis matter content violates the energy conditions in orderto keep throat open and thus is named as exotic matterThis implies that violation of these conditions is the basickey ingredient to construct traversable wormhole solutionsSince the usual energy-momentum tensor satisfies the energyconditions therefore the search for wormhole solutions forwhich violation may come from other sources while mattercontent satisfying energy conditions becomes one of themostchallenging problems in astrophysics

The higher dimensional gravity theories and modifiedtheories may play positive role by providing violation fromhigher order Lagrangian terms and effective form of energy-momentum tensor The relationship between Raychaudhuriequation and attractiveness of gravity yields the weak energycondition (WEC) asT

120572120573120583120572

120583120573

ge 0 for any time-like vector120583120572In terms of components of the energy-momentum tensor thisinequality yields120588 ge 0 and120588+119901 ge 0Thenull energy condition(NEC) is developed by continuity through WEC that is theNEC isT

120572120573120594120572

120594120573

ge 0 for any null vector 120594120572 This inequalitygives 120588 + 119901 ge 0 Also it is noted that WEC keeps NEC

The anisotropic energy-momentum is given by

T120572

120573= (120588 + 119901

119903) 119906120572

119906120573minus 119901119903119892120572

120573+ (119901119905minus 119901119903) 120578120572

120578120573 (6)

where 119901119903and 119901

119905are the radial and tangential pressure

components with 120588 = 120588(119903) 119901 = 119901(119903) and satisfy 119906120572119906120573=

minus120578120572

120578120573= 1 Using (3) the field equations become

120588 (119903) =

(119899 minus 2)

21199032

[(1199041015840

minus

119904

119903

) (1 +

2120598119904

1199033)

+

119904

119903

(119899 minus 3) + (119899 minus 5)

120598119904

1199033]

(7)

119901119903(119903) =

(119899 minus 2)

2119903

[2 (1 minus

119904

119903

) (1 +

2120598119904

1199033) 1205821015840

minus

119904

1199032(119899 minus 3) + (119899 minus 5)

120598119904

1199033]

(8)

119901119905(119903) = (1 minus

119904

119903

) (1 +

2120598119904

1199033)[12058210158401015840

+ 12058210158402

+

(119904 minus 1199031199041015840

) 1205821015840

2119903 (119903 minus 119904)

]

+ (1 minus

119904

119903

)(

1205821015840

119903

+

119904 minus 1199031199041015840

21199032(119903 minus 119904)

) (119899 minus 3)

+ (119899 minus 5)

2120598119904

1199033 minus

119904

21199033(119899 minus 3) (119899 minus 4)

+ (119899 minus 5) (119899 minus 6)

120598119904

1199033 minus

21205821015840

120598

1199034(1 minus

119904

119903

) (119904 minus 1199031199041015840

) (119899

minus 5)

(9)

where prime refers derivative with respect to 119903 and 120598 = (119899 minus

3)(119899 minus 4)1205981for the sake of notational simplicity

3 Wormhole Solutions

In order to discuss the wormhole geometry and solutionsthere are several frameworks and strategies used to findunknown functions For instance we have five unknownfunctions 120588(119903) 119901

119903(119903) 119901

119905(119903) 120582(119903) and 119904(119903) in the under-

lying case One may choose some kind of equation ofstate representing accelerated expansion of the universe ordifferent forms of energy density such as energy densityof static spherically symmetric object with noncommutativegeometry having Gaussian or Lorentzian distributions andgalactic halo region and so forth The traceless energy-momentum tensor is also usedwhich is related to the Casimireffect In order to construct viable wormhole solutions in 119899-dimensional Einstein-Gauss-Bonnet gravity we assume dif-ferent forms of energy density of noncommutative geometryin the following

Nicolini et al [21] have improved the short distancebehavior of point-like structures in a new conceptualapproach based on coordinate coherent state formalism tononcommutative gravity In their method curvature singu-larities which appear in general relativity can be eliminatedThey have demonstrated that black hole evaporation processshould be stopped when a black hole reaches a minimalmass This minimal mass named black hole remnant is aresult of the existence of a minimal observable length Thisapproach which is the so-called noncommutative geometryinspired model via a minimal length caused by averagingnoncommutative coordinate fluctuations cures the curvaturesingularity in black holes In fact the curvature singularity atthe origin of black holes is substituted for a regular de Sittercore Accordingly the ultimate phase of the Hawking evapo-ration as a novel thermodynamically steady state comprisinga nonsingular behavior is concluded

It must be noted that generally it is not required toconsider the length scale of the coordinate noncommutativityto be the same as the Planck length Since the noncom-mutativity influences appear on a length scale connectedto that region they can behave as an adjustable parametercorresponding to that pertinent scale The presence of auniversal short distance cut-off leads to the effects such as inquantum field theory it curves UV divergences while it curescurvature singularities in general relativity In the specificcase of the gravity field equations the only modificationoccurs at the level of the energy-momentum tensor while119866120583] is formally left unchanged In nonommutative space the

usual definition of mass density in the form of Dirac deltafunction does not hold So in this space the usual form ofthe energy density of the static spherically symmetry smearedand particle-like gravitational source requires some otherforms of distribution

In view of the above explanations we are going to discusswormhole solutions with the help of two well-known energydistributions such as Gaussian and Lorentzian in noncom-mutative scenario As an important remark the essentialaspects of the noncommutativity approach are not specificallysensitive to any of these distributions of the smearing effects[22] rather only distribution parameter is different TheGaussian source has also been used by Sushkov [23] to model

4 Advances in High Energy Physics

0

2

4

6

8

minus2

00 10 30252005 15r

1minuss +r

(a)

0

2

4

minus2

86420r

1minuss minusr

(b)

Figure 1 Plots of (a) 1 minus 119904+119903 and (b) 1 minus 119904

minus119903 versus 119903 in Gaussian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed) 119899 = 6 120598 = 3

(purple) and 119899 = 6 120598 = minus3 (purple dashed)

phantom-energy supported wormholes as well as by Nicoliniand Spallucci [24] for the purpose of modeling the physicaleffects of short distance fluctuations of noncommutativecoordinates in the study of black holes Galactic rotationcurves inspired by a noncommutative geometry backgroundare discussed [25] The stability of a particular class of thin-shell wormholes in noncommutative geometry is analyzedelsewhere [26]

31 Gaussian Distributed Noncommutative Framework Anintrinsic characteristic of spacetime is the noncommuta-tivity which plays an effective role in several areas Itis an interesting consequence of string theory where thecoordinates of spacetime become noncommutative operatorson 119863-brane [27] The noncommutativity of spacetime canbe converted in the commutator [119909120572 119909120573] = 119894120579

120572120573 where120579120572120573 is an antisymmetric matrix describing discretization ofspacetime and has dimension (length)2 This discretizationprocess is similar to the discretization of phase space byPlanck constant Replacing the point-like structures withsmeared objects the energy density of the particle-like staticspherically symmetric gravitational source having mass 119872takes the following form [21]

120588nc =119872

(4120587120579)(119899minus1)2

119890minus11990324120579

(10)

where 120579 is the noncommutative parameter in Gaussiandistribution The mass 119872 could be a diffused centralizedobject such as a wormhole [28] It is mentioned here that thesmearing effect is achieved by replacing the Gaussian distri-bution of minimal length radic120579 with the Dirac delta functionIn order to construct 119899-dimensional Einstein Gauss-Bonnetwormhole geometry we equate the energy density given in(7) and 120588nc in (10) yields the following differential equation

1199041015840

=

1

1 + 21199041205981199033[(4 minus 119899 + (7 minus 119899)

119904120598

1199033)

+

21199032

119872

(119899 minus 2)

(4120587120579)(1minus119899)2

119890minus11990324120579

]

(11)

The solution of this equation is

119904 (119903) = minus

1

2

[

1199033

120598

plusmn

119903(7minus119899)2

120587minus1198994

radic120598 (119899 minus 1)

(119899 minus 2) 1205871198992

(119903119899minus1

+ 41205982

1198881)

minus 4119872radic120587120598Gamma[119899 minus 12

1199032

4120579

]

12

]

(12)

where 1198881is an integration constant This solution has two

roots with plus and minus signs and we assign these rootsas 119904+(119903) and 119904

minus(119903) solutions In order to plot the quantities

1 minus 119904119903 120588 120588 + 119901119903 and 120588 + 119901

119905to obtain wormhole solutions

for both of these solutions we restrict ourselves to five- andsix-dimensional cases with constant redshift function 120582 = 0The expressions of NEC takes the form

120588 + 119901119903=

(119899 minus 2)

21199032

(1199041015840

minus

119904

119903

) (1 +

2120598119904

1199033)

120588 + 119901119905=

1199031199041015840

minus 119904

21199033

(1 +

6120598119904

1199033)

+

119904

1199033[(119899 minus 3) + (119899 minus 5)

2119904120598

1199033]

(13)

In this regard we assume values of some constants as119872 = 0008 120579 = 0002 while 120598

1= 05 minus05 result in 120598 =

1 minus1 for five-dimensional and 120598 = 3 minus3 for six-dimensionalcase Figure 1(a) represents the plot of 1 minus 119904

+119903 versus 119903

for five and six-dimensional wormhole solutions The graphrepresents positively increasing behavior for positive 120598 forboth curves For negative 120598 we examine that graph initiallyrepresents positive behavior for both dimensions in the range119903 lt 1 and 119903 lt 18 for 119899 = 5 and 6 respectively and thendecreases towards negative values In Figure 1(b) the graphof 1 minus 119904

minus119903 versus 119903 shows the same behavior for all curves as

in Figure 1(a) with positive behavior of dashed curves in therange 119903 lt 4 and 119903 lt 8 for 119899 = 5 ad 6 However we have less

Advances in High Energy Physics 5

025 030 035 040020r

000

002

120588

004

006

008

010

(a) Evolution of 120588+versus 119903

000

002

004

006

008

010

120588

025 030 035 040020r

(b) Evolution of 120588minusversus 119903

Figure 2 Plots of (a) 120588+and (b) 120588

minusversus 119903 in Gaussian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed) 119899 = 6 120598 = 3 (purple)

and 119899 = 6 120598 = minus3 (purple dashed)

possibility of wormhole scenario with respect to 119903 for positiveroot solution as compared to negative root solution of shapefunction

Figure 2 shows the behavior of energy density versus119903 as positively decreasing behavior for both dimensionscorresponding to 119899 = 5 and 6 dimensions For 119904

+and positive

120598 the plot of 120588+119901119903represents positive behavior in decreasing

manner for 119899 = 5 and 6 while representing negative behaviorfor negative 120598 as shown in Figure 3(a) In Figure 3(b) weexamine same behavior for negative root solution Figure 4(a)depicts the opposite behavior to Figure 3(a) that is for 119899 =

5 6 dimensions and positive 120598 120588 + 119901119905demonstrates negative

behavior Thus these plots express the violation of WECincorporating the case of 119904

+ For 119904

minus we obtain the same

behavior as in Figure 3(b) which indicates positive behaviorfor positive 120598 and 119899 = 5 6 and negative behavior for negative120598 and 119899 = 5 6 This implies that WEC is satisfied fornegative root solutionwith positiveGauss-Bonnet coefficientThus we obtain physically acceptable wormhole solutionssatisfying WEC for both dimensions

32 Lorentzian Distributed Non-Commutative FrameworkNow we consider the case of noncommutative geometrywith Lorentzian distributionThe energy density of point-likesource under this distribution becomes [8 15]

120588Lnc =119872radic120601

1205872(1199032+ 120601)1198992

(14)

where 120601 is the noncommutative parameter in Lorentziandistribution Inserting 120588Lnc in (7) the differential equationtakes the following form

1199041015840

=

1

1 + 21199041205981199033[

21199032

119872radic120601

1205872(119899 minus 2) (119903

2+ 120601)1198992

+

119887

119903

4 minus 119899 + (7 minus 119899)

119904120598

1199033]

(15)

The solution of this equation is given by

119904 (119903) =

1199033

2120598120587

[minus120587 plusmn

1

(119899 minus 2) (119899 minus 1)

1205872

(1198992

minus 3119899 + 2) (119903119899

+ 41199031205982

1198882)

+ 8119872120598119903119899

120601(1minus119899)2Hypergeometric 2F1 [1

2

(119899 minus 1)

119899

2

119899 + 1

2

minus

1199032

120601

]

12

]

(16)

where 1198882is an integration constant We again assign both

solutions as 119904+and 119904minusand explore the wormhole solutions

In this case we choose constants as 120601 = 2 119872 = 2 and1198882= 05 for same dimensions and Gauss-Bonnet coefficient

as for noncommutative backgroundIn Figure 5(a) we plot 1 minus 119904

+119903 versus 119903 which represents

that the condition for wormhole geometry that is 1minus119904119903 gt 0holds for 5 and 6 dimensions with negative 120598 For positive 120598the positivity of this expression depends on some ranges sothat it remains positive for 119903 lt 19 for 119899 = 5 while 119899 = 6

observes the range 119903 lt 21 Figure 5(b) corresponds to theplot of 1 minus 119904

minus119903 with respect to 119903 showing positive behavior

for positive 120598 For 119899 = 5 it expresses a very short range119903 lt 05 for positivity whereas it remains negative for 6-dimensional solutionThat is we have no wormhole solutionfor 119899 = 6 120598 = minus3 taking negative root solution So we skipthis case in further discussion The behavior of 120588

+and 120588

minus

remains positive for all cases except 119899 = 6 120598 = minus3 for whichit describes negative behavior as shown in Figures 6(a) and6(b) This implies that we have no wormhole solutions for 6-dimensional case with negative Gauss-Bonnet coefficient forboth solutions

Figure 7(a) expresses the behavior of 120588 + 119901119903for positive

root solution versus 119903 which remains positive for the range119903 lt 1 for 119899 = 5 and 6 with positive 120598 and then turns towardsnegative behavior It demonstrates negative behavior for 119903 lt 1and then moves to positive region Incorporating 119904

minussolution

120588 + 119901119903shows positive behavior only for the case 119899 = 5

6 Advances in High Energy Physics

pr+120588

minus06

minus04

minus02

00

02

04

06

0402 06 08 10 12 1400r

(a) Evolution of (120588 + 119901119903)+ versus 119903

pr+120588

minus3

minus2

minus1

0

1

2

3

04 06 08 10 12 1402r

(b) Evolution of (120588 + 119901119903)minus versus 119903

Figure 3 Plots of (a) 120588 + 119901119903for 119904+and (b) 120588 + 119901

119903for 119904minusversus 119903 in Gaussian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

119899 = 6 120598 = 3 (purple) and 119899 = 6 120598 = minus3 (purple dashed)

pt+120588

pt+120588

n = 5 120598 = minus1

minus04

minus02

00

02

04

05 10 15 20 25 3000r

000001500000200000250000030000035

17 18 19 2016r

(a) Evolution of (120588 + 119901119905)+ versus 119903

pt+120588

minus10

minus5

0

5

10

05 10 15 20 25 3000r

(b) Evolution of (120588 + 119901119905)minus versus 119903

Figure 4 Plots of (a) 120588 + 119901119905for 119904+and (b) 120588 + 119901

119905for 119904minusversus 119903 for in Gaussian distribution 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

119899 = 6 120598 = 3 (purple) and 119899 = 6 120598 = minus3 (purple dashed)

n = 5 120598 = minus1

minus2

minus1

0

1

2

05 10 15 20 25 3000r

0985

0990

0995

1000

005 010 015 020000

r

1minuss +r

1minuss +r

(a)

05 10 15 20 25 3000r

minus4

minus2

0

2

4

1minuss minusr

(b)

Figure 5 Plots of (a) 1minus 119904+119903 and (b) 1minus 119904

minus119903 versus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed) and 119899 = 6

120598 = 3 (purple)

Advances in High Energy Physics 7

04 05 06 07 08 09 10050100150200

r

05 10 15 20

0

05 10 15 20r

n = 6 120598 = minus3

n = 6 120598 = 3

minus1000minus2000minus3000minus4000minus5000minus6000

001

002

120588

120588

120588

003

004

005

(a) Evolution of 120588+versus 119903

05 10 15 20000

001

002

003

004

005

r

120588

(b) Evolution of 120588minusversus 119903

Figure 6 Plots of (a) 120588+and (b) 120588

minusversus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed) and 119899 = 6 120598 = 3

(purple) For (a) 120588+ 119899 = 6 and 120598 = minus3 (purple dashed)

05 10 15 20 25 30

0

5

10

15

r

minus10

minus5

pr+120588

(a) Evolution of (120588 + 119901119903)+versus 119903

02 04 06 08 101020304050

02 04 06 08 10

0

n = 5 120598 = 1

r

rminus8000

minus6000

minus4000

minus2000

pr+120588

pr+120588

(b) Evolution of (120588 + 119901119903)minusversus 119903

Figure 7 Plots of (a) 120588 + 119901119903for 119904+and (b) 120588 + 119901

119903for 119904minusversus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

and 119899 = 6 120598 = 3 (purple)

120598 gt 0 and negative behavior for the remaining two casesas shown in Figure 7(b) In Figures 8(a) and 8(b) we draw120588 + 119901119905for both solutions versus 119903 This expression represents

the negative behavior for (119899 = 5 120598 = 1) (119899 = 5 120598 = minus1) with119904+and (119899 = 5 120598 = minus1) with 119904

minusand preserves positivity for

(119899 = 6 120598 = 3) with 119904+ (119899 = 5 120598 = 1) (119899 = 6 120598 = 3) with 119904

minus

solution

4 Equilibrium Condition

In order to find the equilibrium configuration of thewormhole solutions in Gaussian as well as Lorentzian dis-tributed noncommutative backgrounds we use the general-ized Tolman-Oppenheimer-Volkoff equation This equationis derived by solving the Einstein equations for a general time-invariant spherically symmetric metric having metric tensor119892120572120573

= (119890120591(119903)

minus119890120596(119903)

minus1199032

minus1199032sin2120579) where 120572120573 represents

only diagonal entries and 120591 120596 are general metric functionsdependent on 119903 The generalized Tolman-Oppenheimer-Volkoff equation is

119889119901119903

119889119903

+

1205911015840

2

(120588 + 119901119903) +

2

119903

(119901119903minus 119901119905) = 0 (17)

Keeping in mind the above equation de Leon [28] proposedan equation for anisotropicmass distributionwhich naturallygives the equilibrium for the wormhole subject It is given by

2

119903

(119901119905minus 119901119903) minus

119890(120596minus120591)2

119898eff1199032

(120588 + 119901119903) minus

119889119901119903

119889119903

= 0 (18)

where effective gravitational mass 119898eff = (12)1199032

119890(120591minus120596)2

1205911015840 is

measured from throat to some arbitrary radius 119903 Accord-ingly the gravitational hydrostatic and anisotropic force dueto anisotropic matter distribution are defied as follows

119891119892= minus

1205911015840

(120588 + 119901119903)

2

119891ℎ= minus

119889119901119903

119889119903

119891119886=

2 (119901119905minus 119901119903)

119903

(19)

It is required that119891119892+119891ℎ+119891119886= 0must hold for the wormhole

solutions to be in equilibrium In the underlying cases we

8 Advances in High Energy Physics

05 10 15 20 25 30

0

r

r

150 152 154 156 158 160

r

05 10 15 200

500

1000

1500

2000

pt+120588

pt+120588

pt+120588

minus0058

minus15

minus10

minus5

minus0056

minus0054

minus0052

minus0050n = 5 120598 = minus1

n = 5 120598 = 1

(a) Evolution of (120588 + 119901119905)+versus 119903

r

05 10 15 20

r

00 05 10 15 200

5

10

15

20

pt+120588 p

t+120588

minus70

minus60

minus50

minus40

minus30

minus20

minus10

n = 5 120598 = minus1

(b) Evolution of (120588 + 119901119905)minusversus 119903

Figure 8 Plots of (a) 120588 + 119901119905for 119904+and (b) 120588 + 119901

119905for 119904minusversus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

and 119899 = 6 120598 = 3 (purple)

proceedwith constant redshift function 120582 = 0which vanishesthe gravitational contribution119891

119892in the equilibrium equation

that is 120591 = 2120582 leads to 1205911015840 = 0 for constant 120582 Thus we are leftwith hydrostatic and anisotropic forces with correspondingequilibrium condition as

119891119886+ 119891ℎ= 0 (20)

Using (8) and (9) we obtain the following expressions for thehydrostatic and anisotropic forces

119891119886=

1

1199033(119899 minus 1)

119904

119903

minus 1199041015840

(119899 minus 3 + (119899 minus 5)

119904120598

1199033)

minus

119904

1199034[(119899 minus 3) (119899 minus 4) + (119899 minus 5) (119899 minus 6)

119904120598

1199033]

119891ℎ=

(119899 minus 2)

21199033

[119899 minus 3 + (119899 minus 5)

119904120598

1199033+

119904120598 (119899 minus 5)

1199033

]

sdot (1199041015840

minus

3119904

119903

)

(21)

Figure 9 represents the plots of anisotropic as well ashydrostatic forces for the wormhole solutions in Gaussiandistributed framework Figures 9(a) and 9(c) describe theequilibrium for positive root solution for 119899 = 5 120598 =

1 minus1 and 119899 = 6 120598 = 3 minus3 through opposite behaviorand hence they cancel each other in order to satisfy (20)For negative root solution 119904

minus Figures 9(b) and 9(d) show

the stable configuration of wormhole solutions for fifthdimensional case only The behaviors of both forces (whichare not in opposite manner) do not cancel each other sothere is no equilibrium configuration examined for sixthdimensional wormhole solutions In the case of Lorentziannoncommutative background Figure 10 expresses that allwormhole solutions are in equilibrium by satisfying theequilibrium condition for both dimensions

Table 1 Wormhole solutions with Gaussian distributed noncom-mutative framework for 119904

+

Expressions 119899 = 5 120598 =

1

119899 = 5 120598 =

minus1

119899 = 6 120598 =

3

119899 = 6 120598 =

minus3

1 minus 119904+119903 Positive Positive

for 119903 lt 1 Positive Positivefor 119903 lt 18

120588 Positive Positive Positive Positive120588 + 119901

119903 Positive Negative Positive Negative120588 + 119901

119905 Negative Positive Negative PositiveWEC Violates Violates Violates ViolatesEC Holds Holds Holds Holds

5 Conclusion

It is a well-known fact that the existence of wormhole solu-tions is based on violation of NEC Since the normal mattersatisfies the energy conditions this violation is associatedwith an energy-momentum tensor which provides exoticmatter a hypothetical form of matter To explore realisticmodel or physically acceptable wormhole solutions it isnecessary to find such a source which gives the violationof NEC while normal matter meets the energy conditionsHere in this paper we have explored wormhole solutionsin 119899-dimensional Einstein Gauss-Bonnet gravity with Gaus-sian and Lorentzian noncommutative backgrounds We haverestricted ourselves to fifth and sixth dimensional cases withpositive as well as negativeGauss-Bonnet coefficient Also wehave checked the condition of equilibrium for the wormholesolutions

The results are summarized in Tables 1 and 2 It is notedthat the 119904

+(Tables 1 and 3) and 119904

minus(Tables 2 and 4) represent

the two roots of the solutions in both backgrounds and ECdenotes the equilibrium condition

In the Lorentzan distributed noncommutative frame-work we have found negative energy density for 119904

+while

Advances in High Energy Physics 9

02 04 06 1008

0

2

4

r

fa

minus2

minus4

(a) Evolution of 119891119886versus 119903

r

02 04 06 08 10

0

2

4

fa

minus2

minus4

(b) Evolution of 119891119886versus 119903

02 04 06 08 10

0

2

4

r

fh

minus2

minus4

(c) Evolution of 119891ℎversus 119903

02 04 06 08 10

0

2

4

r

fh

minus2

minus4

(d) Evolution of 119891ℎversus 119903

Figure 9 Plots of (a) 119891119886for 119904+ (b) 119891

119886for 119904minus (c) 119891

ℎfor 119904+ (d) 119891

ℎfor 119904minusversus 119903 with Gaussian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1

(red dashed) and 119899 = 6 120598 = 3 (purple)

Table 2 Wormhole solutions with Gaussian distributed noncom-mutative framework for 119904

minus

Expressions 119899 = 5 120598 =

1

119899 = 5 120598 =

minus1

119899 = 6 120598 =

3

119899 = 6 120598 =

minus3

1 minus 119904minus119903 Positive Positive

for 119903 lt 4 Positive Positivefor 119903 lt 8

120588 Positive Positive Positive Positive120588 + 119901

119903 Positive Negative Positive Negative120588 + 119901

119905 Positive Negative Positive NegativeWEC Holds Violates Holds Violates

EC Holds Holds Does nothold

Does nothold

violation of 1 minus 119904119903 gt 0 incorporating 119904minusfor the case 119899 = 6

120598 = minus3 so we skipped this case The remaining results aresummarized in Tables 3 and 4

In the paper [20] higher dimensional asymptotically flatwormhole solutions have been explored in the frameworkof Gauss-Bonnet gravity by considering a specific choicefor a radial dependent redshift function and by imposinga particular equation of state The WEC is satisfied at thethroat by considering a negative Gauss-Bonnet coupling

Table 3 Wormhole solutions with Lorentzian noncommutativebackground for 119904

+

Expressions 119899 = 5 120598 = 1 119899 = 5 120598 = minus1 119899 = 6 120598 = 3

1 minus 119904+119903

Positive for119903 lt 19

Positive Positive for119903 lt 21

120588 Positive Positive Positive

120588 + 119901119903

Positive for119903 lt 1

Positive for119903 gt 1

Positive for119903 lt 1

120588 + 119901119905 Negative Negative Positive

WEC Does nothold

Does nothold

Holds for119903 lt 1

EC Holds Holds Holds

constant Furthermore they have considered a constantredshift function and shown specifically that for negativeGauss-Bonnet coupling constant one may have normalmatter in a determined radial region and that the increaseof coupling constant enlarges the normal matter regionIn the present paper we have taken energy density undernoncommutative geometry distributions instead of particularequation of stateWe have obtained results for positiveGauss-Bonnet coefficient satisfying energy conditions It contains

10 Advances in High Energy Physics

02 04 06 08 10

0

200

400

r

fa

minus400

minus200

(a) Evolution of 119891119886versus 119903

02 04 06 08 10

0

200

400

r

fa

minus400

minus200

(b) Evolution of 119891119886versus 119903

02 04 06 08 10r

minus400

minus200

0

200

400

fh

(c) Evolution of 119891ℎversus 119903

02 04 06 08 10

0

200

400

r

fh

minus400

minus200

(d) Evolution of 119891ℎversus 119903

Figure 10 Plots of (a) 119891119886for 119904+ (b) 119891

119886for 119904minus (c) 119891

ℎfor 119904+ (d) 119891

ℎfor 119904minusversus 119903 with Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5

120598 = minus1 (red dashed) and 119899 = 6 120598 = 3 (purple)

Table 4 Wormhole solutions with Lorentzian noncommutativebackground for 119904

minus

Expressions 119899 = 5 120598 = 1 119899 = 5 120598 = minus1 119899 = 6 120598 = 3

1 minus 119904minus119903 Positive Positive for

119903 lt 05

Positive

120588 Positive Positive Positive120588 + 119901

119903 Positive Negative Negative120588 + 119901

119905 Positive Negative Positive

WEC Holds Does nothold

Does nothold

EC Holds Holds Holds

fifth dimensional wormhole solutions in both backgroundssatisfying equilibrium condition and sixth dimensional withdisequilibrium in noncommutative background with 119904

minussolu-

tion Also there is possibility for the existence of wormholein equilibrium satisfying WEC for 119899 = 6 120598 = 3 taking intoaccount 119904

+solution for the range 119903 lt 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M S Morris and K S Thorne ldquoWormholes in spacetimeand their use for interstellar travel a tool for teaching generalrelativityrdquo American Journal of Physics vol 56 no 5 pp 395ndash412 1988

[2] M Sharif and A Jawad ldquoPhantom-like generalized cosmicchaplygin gas and traversable wormhole solutionsrdquo The Euro-pean Physical Journal Plus vol 129 article 15 2014

[3] F S N Lobo F Parsaei and N Riazi ldquoNew asymptotically flatphantom wormhole solutionsrdquo Physical Review D vol 87 no 8Article ID 084030 2013

[4] S-W Kim andH Lee ldquoExact solutions of a charged wormholerdquoPhysical Review D vol 63 no 6 Article ID 064014 5 pages2001

[5] M Jamil and M U Farooq ldquoPhantom wormholes in (2 + 1)-dimensionsrdquo International Journal ofTheoretical Physics vol 49no 4 pp 835ndash841 2010

[6] M U Farooq M Akbar and M Jamil ldquoDynamics and ther-modynamics of (2+1)minusdimensional evolving lorentzian worm-holerdquo in Proceedings of the 3rd AlgerianWorkshop on Astronomyand Astrophysics vol 1295 of AIP Conference Proceedings pp176ndash190 Constantine Algeria June 2010

[7] P K Kuhfittig ldquoGravitational lensing of wormholes in thegalactic halo regionrdquo The European Physical Journal C vol 74no 3 article 2818 2014

Advances in High Energy Physics 11

[8] F Rahaman A Banerjee M Jamil A K Yadav and H IdrisldquoNoncommutative wormholes in f(R) gravity with lorentziandistributionrdquo International Journal ofTheoretical Physics vol 53no 6 pp 1910ndash1919 2014

[9] M Sharif and S Rani ldquoDynamical wormhole solutions in f(T)gravityrdquo General Relativity and Gravitation vol 45 no 11 pp2389ndash2402 2013

[10] M Sharif and S Rani ldquoCharged noncommutative wormholesolutions in f (T) gravityrdquo The European Physical Journal Plusvol 129 no 10 article 237 2014

[11] M Sharif and S Rani ldquoGalactic halo wormhole solutions in119891(119879) gravityrdquoAdvances in High Energy Physics vol 2014 ArticleID 691497 9 pages 2014

[12] F S N Lobo andM A Oliveira ldquoWormhole geometries in f (R)modified theories of gravityrdquo Physical Review D vol 80 ArticleID 104012 2009

[13] M Sharif and S Rani ldquoWormhole solutions in f(T) gravity withnoncommutativerdquo Physical Review D vol 88 no 12 Article ID123501 2013

[14] F Rahaman S Islam P K F Kuhfittig and S Ray ldquoSearch-ing for higher-dimensional wormholes with noncommutativegeometryrdquo Physical Review D vol 86 no 10 Article ID 1060102012

[15] P Bhar and F Rahaman ldquoSearch of wormholes in differ-ent dimensional non-commutative inspired space-times withLorentzian distributionrdquo The European Physical Journal C vol74 no 12 article 3213 2014

[16] F Rahaman S Karmakar I Karar and S Ray ldquoWormholeinspired by non-commutative geometryrdquo Physics Letters B vol746 pp 73ndash78 2015

[17] M Jamil F Rahaman R Myrzakulov P Kuhfittig N Ahmedand U Mondal ldquoNonommutative wormholes in f(R) gravityrdquoJournal of the Korean Physical Society vol 65 no 6 pp 917ndash9252014

[18] A Jawad and S Rani ldquoLorentz distributed noncommutativewormhole solutions in extended teleparallel gravityrdquoThe Euro-pean Physical Journal C vol 75 article 173 2015

[19] B Bhawal and S Kar ldquoLorentzian wormholes in Einstein-Gauss-Bonnet theoryrdquo Physical Review D vol 46 no 6 pp2464ndash2468 1992

[20] M R Mehdizadeh M K Zangeneh and F S N LoboldquoEinstein-Gauss-Bonnet traversable wormholes satisfying theweak energy conditionrdquo Physical ReviewD vol 91 no 8 ArticleID 084004 2015

[21] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[22] S H Mehdipour ldquoEntropic force approach to noncommutativeSchwarzschild black holes signals a failure of current physicalideasrdquo The European Physical Journal Plus vol 127 article 802012

[23] S Sushkov ldquoWormholes supported by a phantom energyrdquoPhysical Review D vol 71 no 4 Article ID 043520 2005

[24] P Nicolini and E Spallucci ldquoNoncommutative geometry-inspired dirty black holesrdquo Classical and Quantum Gravity vol27 no 1 Article ID 015010 2010

[25] F Rahaman P K Kuhfittig K Chakraborty A A Usmani andS Ray ldquoGalactic rotation curves inspired by a noncommutative-geometry backgroundrdquo General Relativity and Gravitation vol44 no 4 pp 905ndash916 2012

[26] P K F Kuhfittig ldquoOn the stability of thin-shell wormholes innoncommutative geometryrdquo Advances in High Energy Physicsvol 2012 Article ID 462493 12 pages 2012

[27] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquo Journal of High Energy Physics vol 1999 no 9 article032 1999

[28] J P de Leon ldquoMass and charge in brane-world and non-compact Kaluza-KLEin theories in 5 dimrdquo General Relativityand Gravitation vol 35 no 8 pp 1365ndash1384 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 3

throat there must exist such form of energy-momentumtensor which provides the corresponding matter contentThis matter content violates the energy conditions in orderto keep throat open and thus is named as exotic matterThis implies that violation of these conditions is the basickey ingredient to construct traversable wormhole solutionsSince the usual energy-momentum tensor satisfies the energyconditions therefore the search for wormhole solutions forwhich violation may come from other sources while mattercontent satisfying energy conditions becomes one of themostchallenging problems in astrophysics

The higher dimensional gravity theories and modifiedtheories may play positive role by providing violation fromhigher order Lagrangian terms and effective form of energy-momentum tensor The relationship between Raychaudhuriequation and attractiveness of gravity yields the weak energycondition (WEC) asT

120572120573120583120572

120583120573

ge 0 for any time-like vector120583120572In terms of components of the energy-momentum tensor thisinequality yields120588 ge 0 and120588+119901 ge 0Thenull energy condition(NEC) is developed by continuity through WEC that is theNEC isT

120572120573120594120572

120594120573

ge 0 for any null vector 120594120572 This inequalitygives 120588 + 119901 ge 0 Also it is noted that WEC keeps NEC

The anisotropic energy-momentum is given by

T120572

120573= (120588 + 119901

119903) 119906120572

119906120573minus 119901119903119892120572

120573+ (119901119905minus 119901119903) 120578120572

120578120573 (6)

where 119901119903and 119901

119905are the radial and tangential pressure

components with 120588 = 120588(119903) 119901 = 119901(119903) and satisfy 119906120572119906120573=

minus120578120572

120578120573= 1 Using (3) the field equations become

120588 (119903) =

(119899 minus 2)

21199032

[(1199041015840

minus

119904

119903

) (1 +

2120598119904

1199033)

+

119904

119903

(119899 minus 3) + (119899 minus 5)

120598119904

1199033]

(7)

119901119903(119903) =

(119899 minus 2)

2119903

[2 (1 minus

119904

119903

) (1 +

2120598119904

1199033) 1205821015840

minus

119904

1199032(119899 minus 3) + (119899 minus 5)

120598119904

1199033]

(8)

119901119905(119903) = (1 minus

119904

119903

) (1 +

2120598119904

1199033)[12058210158401015840

+ 12058210158402

+

(119904 minus 1199031199041015840

) 1205821015840

2119903 (119903 minus 119904)

]

+ (1 minus

119904

119903

)(

1205821015840

119903

+

119904 minus 1199031199041015840

21199032(119903 minus 119904)

) (119899 minus 3)

+ (119899 minus 5)

2120598119904

1199033 minus

119904

21199033(119899 minus 3) (119899 minus 4)

+ (119899 minus 5) (119899 minus 6)

120598119904

1199033 minus

21205821015840

120598

1199034(1 minus

119904

119903

) (119904 minus 1199031199041015840

) (119899

minus 5)

(9)

where prime refers derivative with respect to 119903 and 120598 = (119899 minus

3)(119899 minus 4)1205981for the sake of notational simplicity

3 Wormhole Solutions

In order to discuss the wormhole geometry and solutionsthere are several frameworks and strategies used to findunknown functions For instance we have five unknownfunctions 120588(119903) 119901

119903(119903) 119901

119905(119903) 120582(119903) and 119904(119903) in the under-

lying case One may choose some kind of equation ofstate representing accelerated expansion of the universe ordifferent forms of energy density such as energy densityof static spherically symmetric object with noncommutativegeometry having Gaussian or Lorentzian distributions andgalactic halo region and so forth The traceless energy-momentum tensor is also usedwhich is related to the Casimireffect In order to construct viable wormhole solutions in 119899-dimensional Einstein-Gauss-Bonnet gravity we assume dif-ferent forms of energy density of noncommutative geometryin the following

Nicolini et al [21] have improved the short distancebehavior of point-like structures in a new conceptualapproach based on coordinate coherent state formalism tononcommutative gravity In their method curvature singu-larities which appear in general relativity can be eliminatedThey have demonstrated that black hole evaporation processshould be stopped when a black hole reaches a minimalmass This minimal mass named black hole remnant is aresult of the existence of a minimal observable length Thisapproach which is the so-called noncommutative geometryinspired model via a minimal length caused by averagingnoncommutative coordinate fluctuations cures the curvaturesingularity in black holes In fact the curvature singularity atthe origin of black holes is substituted for a regular de Sittercore Accordingly the ultimate phase of the Hawking evapo-ration as a novel thermodynamically steady state comprisinga nonsingular behavior is concluded

It must be noted that generally it is not required toconsider the length scale of the coordinate noncommutativityto be the same as the Planck length Since the noncom-mutativity influences appear on a length scale connectedto that region they can behave as an adjustable parametercorresponding to that pertinent scale The presence of auniversal short distance cut-off leads to the effects such as inquantum field theory it curves UV divergences while it curescurvature singularities in general relativity In the specificcase of the gravity field equations the only modificationoccurs at the level of the energy-momentum tensor while119866120583] is formally left unchanged In nonommutative space the

usual definition of mass density in the form of Dirac deltafunction does not hold So in this space the usual form ofthe energy density of the static spherically symmetry smearedand particle-like gravitational source requires some otherforms of distribution

In view of the above explanations we are going to discusswormhole solutions with the help of two well-known energydistributions such as Gaussian and Lorentzian in noncom-mutative scenario As an important remark the essentialaspects of the noncommutativity approach are not specificallysensitive to any of these distributions of the smearing effects[22] rather only distribution parameter is different TheGaussian source has also been used by Sushkov [23] to model

4 Advances in High Energy Physics

0

2

4

6

8

minus2

00 10 30252005 15r

1minuss +r

(a)

0

2

4

minus2

86420r

1minuss minusr

(b)

Figure 1 Plots of (a) 1 minus 119904+119903 and (b) 1 minus 119904

minus119903 versus 119903 in Gaussian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed) 119899 = 6 120598 = 3

(purple) and 119899 = 6 120598 = minus3 (purple dashed)

phantom-energy supported wormholes as well as by Nicoliniand Spallucci [24] for the purpose of modeling the physicaleffects of short distance fluctuations of noncommutativecoordinates in the study of black holes Galactic rotationcurves inspired by a noncommutative geometry backgroundare discussed [25] The stability of a particular class of thin-shell wormholes in noncommutative geometry is analyzedelsewhere [26]

31 Gaussian Distributed Noncommutative Framework Anintrinsic characteristic of spacetime is the noncommuta-tivity which plays an effective role in several areas Itis an interesting consequence of string theory where thecoordinates of spacetime become noncommutative operatorson 119863-brane [27] The noncommutativity of spacetime canbe converted in the commutator [119909120572 119909120573] = 119894120579

120572120573 where120579120572120573 is an antisymmetric matrix describing discretization ofspacetime and has dimension (length)2 This discretizationprocess is similar to the discretization of phase space byPlanck constant Replacing the point-like structures withsmeared objects the energy density of the particle-like staticspherically symmetric gravitational source having mass 119872takes the following form [21]

120588nc =119872

(4120587120579)(119899minus1)2

119890minus11990324120579

(10)

where 120579 is the noncommutative parameter in Gaussiandistribution The mass 119872 could be a diffused centralizedobject such as a wormhole [28] It is mentioned here that thesmearing effect is achieved by replacing the Gaussian distri-bution of minimal length radic120579 with the Dirac delta functionIn order to construct 119899-dimensional Einstein Gauss-Bonnetwormhole geometry we equate the energy density given in(7) and 120588nc in (10) yields the following differential equation

1199041015840

=

1

1 + 21199041205981199033[(4 minus 119899 + (7 minus 119899)

119904120598

1199033)

+

21199032

119872

(119899 minus 2)

(4120587120579)(1minus119899)2

119890minus11990324120579

]

(11)

The solution of this equation is

119904 (119903) = minus

1

2

[

1199033

120598

plusmn

119903(7minus119899)2

120587minus1198994

radic120598 (119899 minus 1)

(119899 minus 2) 1205871198992

(119903119899minus1

+ 41205982

1198881)

minus 4119872radic120587120598Gamma[119899 minus 12

1199032

4120579

]

12

]

(12)

where 1198881is an integration constant This solution has two

roots with plus and minus signs and we assign these rootsas 119904+(119903) and 119904

minus(119903) solutions In order to plot the quantities

1 minus 119904119903 120588 120588 + 119901119903 and 120588 + 119901

119905to obtain wormhole solutions

for both of these solutions we restrict ourselves to five- andsix-dimensional cases with constant redshift function 120582 = 0The expressions of NEC takes the form

120588 + 119901119903=

(119899 minus 2)

21199032

(1199041015840

minus

119904

119903

) (1 +

2120598119904

1199033)

120588 + 119901119905=

1199031199041015840

minus 119904

21199033

(1 +

6120598119904

1199033)

+

119904

1199033[(119899 minus 3) + (119899 minus 5)

2119904120598

1199033]

(13)

In this regard we assume values of some constants as119872 = 0008 120579 = 0002 while 120598

1= 05 minus05 result in 120598 =

1 minus1 for five-dimensional and 120598 = 3 minus3 for six-dimensionalcase Figure 1(a) represents the plot of 1 minus 119904

+119903 versus 119903

for five and six-dimensional wormhole solutions The graphrepresents positively increasing behavior for positive 120598 forboth curves For negative 120598 we examine that graph initiallyrepresents positive behavior for both dimensions in the range119903 lt 1 and 119903 lt 18 for 119899 = 5 and 6 respectively and thendecreases towards negative values In Figure 1(b) the graphof 1 minus 119904

minus119903 versus 119903 shows the same behavior for all curves as

in Figure 1(a) with positive behavior of dashed curves in therange 119903 lt 4 and 119903 lt 8 for 119899 = 5 ad 6 However we have less

Advances in High Energy Physics 5

025 030 035 040020r

000

002

120588

004

006

008

010

(a) Evolution of 120588+versus 119903

000

002

004

006

008

010

120588

025 030 035 040020r

(b) Evolution of 120588minusversus 119903

Figure 2 Plots of (a) 120588+and (b) 120588

minusversus 119903 in Gaussian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed) 119899 = 6 120598 = 3 (purple)

and 119899 = 6 120598 = minus3 (purple dashed)

possibility of wormhole scenario with respect to 119903 for positiveroot solution as compared to negative root solution of shapefunction

Figure 2 shows the behavior of energy density versus119903 as positively decreasing behavior for both dimensionscorresponding to 119899 = 5 and 6 dimensions For 119904

+and positive

120598 the plot of 120588+119901119903represents positive behavior in decreasing

manner for 119899 = 5 and 6 while representing negative behaviorfor negative 120598 as shown in Figure 3(a) In Figure 3(b) weexamine same behavior for negative root solution Figure 4(a)depicts the opposite behavior to Figure 3(a) that is for 119899 =

5 6 dimensions and positive 120598 120588 + 119901119905demonstrates negative

behavior Thus these plots express the violation of WECincorporating the case of 119904

+ For 119904

minus we obtain the same

behavior as in Figure 3(b) which indicates positive behaviorfor positive 120598 and 119899 = 5 6 and negative behavior for negative120598 and 119899 = 5 6 This implies that WEC is satisfied fornegative root solutionwith positiveGauss-Bonnet coefficientThus we obtain physically acceptable wormhole solutionssatisfying WEC for both dimensions

32 Lorentzian Distributed Non-Commutative FrameworkNow we consider the case of noncommutative geometrywith Lorentzian distributionThe energy density of point-likesource under this distribution becomes [8 15]

120588Lnc =119872radic120601

1205872(1199032+ 120601)1198992

(14)

where 120601 is the noncommutative parameter in Lorentziandistribution Inserting 120588Lnc in (7) the differential equationtakes the following form

1199041015840

=

1

1 + 21199041205981199033[

21199032

119872radic120601

1205872(119899 minus 2) (119903

2+ 120601)1198992

+

119887

119903

4 minus 119899 + (7 minus 119899)

119904120598

1199033]

(15)

The solution of this equation is given by

119904 (119903) =

1199033

2120598120587

[minus120587 plusmn

1

(119899 minus 2) (119899 minus 1)

1205872

(1198992

minus 3119899 + 2) (119903119899

+ 41199031205982

1198882)

+ 8119872120598119903119899

120601(1minus119899)2Hypergeometric 2F1 [1

2

(119899 minus 1)

119899

2

119899 + 1

2

minus

1199032

120601

]

12

]

(16)

where 1198882is an integration constant We again assign both

solutions as 119904+and 119904minusand explore the wormhole solutions

In this case we choose constants as 120601 = 2 119872 = 2 and1198882= 05 for same dimensions and Gauss-Bonnet coefficient

as for noncommutative backgroundIn Figure 5(a) we plot 1 minus 119904

+119903 versus 119903 which represents

that the condition for wormhole geometry that is 1minus119904119903 gt 0holds for 5 and 6 dimensions with negative 120598 For positive 120598the positivity of this expression depends on some ranges sothat it remains positive for 119903 lt 19 for 119899 = 5 while 119899 = 6

observes the range 119903 lt 21 Figure 5(b) corresponds to theplot of 1 minus 119904

minus119903 with respect to 119903 showing positive behavior

for positive 120598 For 119899 = 5 it expresses a very short range119903 lt 05 for positivity whereas it remains negative for 6-dimensional solutionThat is we have no wormhole solutionfor 119899 = 6 120598 = minus3 taking negative root solution So we skipthis case in further discussion The behavior of 120588

+and 120588

minus

remains positive for all cases except 119899 = 6 120598 = minus3 for whichit describes negative behavior as shown in Figures 6(a) and6(b) This implies that we have no wormhole solutions for 6-dimensional case with negative Gauss-Bonnet coefficient forboth solutions

Figure 7(a) expresses the behavior of 120588 + 119901119903for positive

root solution versus 119903 which remains positive for the range119903 lt 1 for 119899 = 5 and 6 with positive 120598 and then turns towardsnegative behavior It demonstrates negative behavior for 119903 lt 1and then moves to positive region Incorporating 119904

minussolution

120588 + 119901119903shows positive behavior only for the case 119899 = 5

6 Advances in High Energy Physics

pr+120588

minus06

minus04

minus02

00

02

04

06

0402 06 08 10 12 1400r

(a) Evolution of (120588 + 119901119903)+ versus 119903

pr+120588

minus3

minus2

minus1

0

1

2

3

04 06 08 10 12 1402r

(b) Evolution of (120588 + 119901119903)minus versus 119903

Figure 3 Plots of (a) 120588 + 119901119903for 119904+and (b) 120588 + 119901

119903for 119904minusversus 119903 in Gaussian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

119899 = 6 120598 = 3 (purple) and 119899 = 6 120598 = minus3 (purple dashed)

pt+120588

pt+120588

n = 5 120598 = minus1

minus04

minus02

00

02

04

05 10 15 20 25 3000r

000001500000200000250000030000035

17 18 19 2016r

(a) Evolution of (120588 + 119901119905)+ versus 119903

pt+120588

minus10

minus5

0

5

10

05 10 15 20 25 3000r

(b) Evolution of (120588 + 119901119905)minus versus 119903

Figure 4 Plots of (a) 120588 + 119901119905for 119904+and (b) 120588 + 119901

119905for 119904minusversus 119903 for in Gaussian distribution 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

119899 = 6 120598 = 3 (purple) and 119899 = 6 120598 = minus3 (purple dashed)

n = 5 120598 = minus1

minus2

minus1

0

1

2

05 10 15 20 25 3000r

0985

0990

0995

1000

005 010 015 020000

r

1minuss +r

1minuss +r

(a)

05 10 15 20 25 3000r

minus4

minus2

0

2

4

1minuss minusr

(b)

Figure 5 Plots of (a) 1minus 119904+119903 and (b) 1minus 119904

minus119903 versus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed) and 119899 = 6

120598 = 3 (purple)

Advances in High Energy Physics 7

04 05 06 07 08 09 10050100150200

r

05 10 15 20

0

05 10 15 20r

n = 6 120598 = minus3

n = 6 120598 = 3

minus1000minus2000minus3000minus4000minus5000minus6000

001

002

120588

120588

120588

003

004

005

(a) Evolution of 120588+versus 119903

05 10 15 20000

001

002

003

004

005

r

120588

(b) Evolution of 120588minusversus 119903

Figure 6 Plots of (a) 120588+and (b) 120588

minusversus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed) and 119899 = 6 120598 = 3

(purple) For (a) 120588+ 119899 = 6 and 120598 = minus3 (purple dashed)

05 10 15 20 25 30

0

5

10

15

r

minus10

minus5

pr+120588

(a) Evolution of (120588 + 119901119903)+versus 119903

02 04 06 08 101020304050

02 04 06 08 10

0

n = 5 120598 = 1

r

rminus8000

minus6000

minus4000

minus2000

pr+120588

pr+120588

(b) Evolution of (120588 + 119901119903)minusversus 119903

Figure 7 Plots of (a) 120588 + 119901119903for 119904+and (b) 120588 + 119901

119903for 119904minusversus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

and 119899 = 6 120598 = 3 (purple)

120598 gt 0 and negative behavior for the remaining two casesas shown in Figure 7(b) In Figures 8(a) and 8(b) we draw120588 + 119901119905for both solutions versus 119903 This expression represents

the negative behavior for (119899 = 5 120598 = 1) (119899 = 5 120598 = minus1) with119904+and (119899 = 5 120598 = minus1) with 119904

minusand preserves positivity for

(119899 = 6 120598 = 3) with 119904+ (119899 = 5 120598 = 1) (119899 = 6 120598 = 3) with 119904

minus

solution

4 Equilibrium Condition

In order to find the equilibrium configuration of thewormhole solutions in Gaussian as well as Lorentzian dis-tributed noncommutative backgrounds we use the general-ized Tolman-Oppenheimer-Volkoff equation This equationis derived by solving the Einstein equations for a general time-invariant spherically symmetric metric having metric tensor119892120572120573

= (119890120591(119903)

minus119890120596(119903)

minus1199032

minus1199032sin2120579) where 120572120573 represents

only diagonal entries and 120591 120596 are general metric functionsdependent on 119903 The generalized Tolman-Oppenheimer-Volkoff equation is

119889119901119903

119889119903

+

1205911015840

2

(120588 + 119901119903) +

2

119903

(119901119903minus 119901119905) = 0 (17)

Keeping in mind the above equation de Leon [28] proposedan equation for anisotropicmass distributionwhich naturallygives the equilibrium for the wormhole subject It is given by

2

119903

(119901119905minus 119901119903) minus

119890(120596minus120591)2

119898eff1199032

(120588 + 119901119903) minus

119889119901119903

119889119903

= 0 (18)

where effective gravitational mass 119898eff = (12)1199032

119890(120591minus120596)2

1205911015840 is

measured from throat to some arbitrary radius 119903 Accord-ingly the gravitational hydrostatic and anisotropic force dueto anisotropic matter distribution are defied as follows

119891119892= minus

1205911015840

(120588 + 119901119903)

2

119891ℎ= minus

119889119901119903

119889119903

119891119886=

2 (119901119905minus 119901119903)

119903

(19)

It is required that119891119892+119891ℎ+119891119886= 0must hold for the wormhole

solutions to be in equilibrium In the underlying cases we

8 Advances in High Energy Physics

05 10 15 20 25 30

0

r

r

150 152 154 156 158 160

r

05 10 15 200

500

1000

1500

2000

pt+120588

pt+120588

pt+120588

minus0058

minus15

minus10

minus5

minus0056

minus0054

minus0052

minus0050n = 5 120598 = minus1

n = 5 120598 = 1

(a) Evolution of (120588 + 119901119905)+versus 119903

r

05 10 15 20

r

00 05 10 15 200

5

10

15

20

pt+120588 p

t+120588

minus70

minus60

minus50

minus40

minus30

minus20

minus10

n = 5 120598 = minus1

(b) Evolution of (120588 + 119901119905)minusversus 119903

Figure 8 Plots of (a) 120588 + 119901119905for 119904+and (b) 120588 + 119901

119905for 119904minusversus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

and 119899 = 6 120598 = 3 (purple)

proceedwith constant redshift function 120582 = 0which vanishesthe gravitational contribution119891

119892in the equilibrium equation

that is 120591 = 2120582 leads to 1205911015840 = 0 for constant 120582 Thus we are leftwith hydrostatic and anisotropic forces with correspondingequilibrium condition as

119891119886+ 119891ℎ= 0 (20)

Using (8) and (9) we obtain the following expressions for thehydrostatic and anisotropic forces

119891119886=

1

1199033(119899 minus 1)

119904

119903

minus 1199041015840

(119899 minus 3 + (119899 minus 5)

119904120598

1199033)

minus

119904

1199034[(119899 minus 3) (119899 minus 4) + (119899 minus 5) (119899 minus 6)

119904120598

1199033]

119891ℎ=

(119899 minus 2)

21199033

[119899 minus 3 + (119899 minus 5)

119904120598

1199033+

119904120598 (119899 minus 5)

1199033

]

sdot (1199041015840

minus

3119904

119903

)

(21)

Figure 9 represents the plots of anisotropic as well ashydrostatic forces for the wormhole solutions in Gaussiandistributed framework Figures 9(a) and 9(c) describe theequilibrium for positive root solution for 119899 = 5 120598 =

1 minus1 and 119899 = 6 120598 = 3 minus3 through opposite behaviorand hence they cancel each other in order to satisfy (20)For negative root solution 119904

minus Figures 9(b) and 9(d) show

the stable configuration of wormhole solutions for fifthdimensional case only The behaviors of both forces (whichare not in opposite manner) do not cancel each other sothere is no equilibrium configuration examined for sixthdimensional wormhole solutions In the case of Lorentziannoncommutative background Figure 10 expresses that allwormhole solutions are in equilibrium by satisfying theequilibrium condition for both dimensions

Table 1 Wormhole solutions with Gaussian distributed noncom-mutative framework for 119904

+

Expressions 119899 = 5 120598 =

1

119899 = 5 120598 =

minus1

119899 = 6 120598 =

3

119899 = 6 120598 =

minus3

1 minus 119904+119903 Positive Positive

for 119903 lt 1 Positive Positivefor 119903 lt 18

120588 Positive Positive Positive Positive120588 + 119901

119903 Positive Negative Positive Negative120588 + 119901

119905 Negative Positive Negative PositiveWEC Violates Violates Violates ViolatesEC Holds Holds Holds Holds

5 Conclusion

It is a well-known fact that the existence of wormhole solu-tions is based on violation of NEC Since the normal mattersatisfies the energy conditions this violation is associatedwith an energy-momentum tensor which provides exoticmatter a hypothetical form of matter To explore realisticmodel or physically acceptable wormhole solutions it isnecessary to find such a source which gives the violationof NEC while normal matter meets the energy conditionsHere in this paper we have explored wormhole solutionsin 119899-dimensional Einstein Gauss-Bonnet gravity with Gaus-sian and Lorentzian noncommutative backgrounds We haverestricted ourselves to fifth and sixth dimensional cases withpositive as well as negativeGauss-Bonnet coefficient Also wehave checked the condition of equilibrium for the wormholesolutions

The results are summarized in Tables 1 and 2 It is notedthat the 119904

+(Tables 1 and 3) and 119904

minus(Tables 2 and 4) represent

the two roots of the solutions in both backgrounds and ECdenotes the equilibrium condition

In the Lorentzan distributed noncommutative frame-work we have found negative energy density for 119904

+while

Advances in High Energy Physics 9

02 04 06 1008

0

2

4

r

fa

minus2

minus4

(a) Evolution of 119891119886versus 119903

r

02 04 06 08 10

0

2

4

fa

minus2

minus4

(b) Evolution of 119891119886versus 119903

02 04 06 08 10

0

2

4

r

fh

minus2

minus4

(c) Evolution of 119891ℎversus 119903

02 04 06 08 10

0

2

4

r

fh

minus2

minus4

(d) Evolution of 119891ℎversus 119903

Figure 9 Plots of (a) 119891119886for 119904+ (b) 119891

119886for 119904minus (c) 119891

ℎfor 119904+ (d) 119891

ℎfor 119904minusversus 119903 with Gaussian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1

(red dashed) and 119899 = 6 120598 = 3 (purple)

Table 2 Wormhole solutions with Gaussian distributed noncom-mutative framework for 119904

minus

Expressions 119899 = 5 120598 =

1

119899 = 5 120598 =

minus1

119899 = 6 120598 =

3

119899 = 6 120598 =

minus3

1 minus 119904minus119903 Positive Positive

for 119903 lt 4 Positive Positivefor 119903 lt 8

120588 Positive Positive Positive Positive120588 + 119901

119903 Positive Negative Positive Negative120588 + 119901

119905 Positive Negative Positive NegativeWEC Holds Violates Holds Violates

EC Holds Holds Does nothold

Does nothold

violation of 1 minus 119904119903 gt 0 incorporating 119904minusfor the case 119899 = 6

120598 = minus3 so we skipped this case The remaining results aresummarized in Tables 3 and 4

In the paper [20] higher dimensional asymptotically flatwormhole solutions have been explored in the frameworkof Gauss-Bonnet gravity by considering a specific choicefor a radial dependent redshift function and by imposinga particular equation of state The WEC is satisfied at thethroat by considering a negative Gauss-Bonnet coupling

Table 3 Wormhole solutions with Lorentzian noncommutativebackground for 119904

+

Expressions 119899 = 5 120598 = 1 119899 = 5 120598 = minus1 119899 = 6 120598 = 3

1 minus 119904+119903

Positive for119903 lt 19

Positive Positive for119903 lt 21

120588 Positive Positive Positive

120588 + 119901119903

Positive for119903 lt 1

Positive for119903 gt 1

Positive for119903 lt 1

120588 + 119901119905 Negative Negative Positive

WEC Does nothold

Does nothold

Holds for119903 lt 1

EC Holds Holds Holds

constant Furthermore they have considered a constantredshift function and shown specifically that for negativeGauss-Bonnet coupling constant one may have normalmatter in a determined radial region and that the increaseof coupling constant enlarges the normal matter regionIn the present paper we have taken energy density undernoncommutative geometry distributions instead of particularequation of stateWe have obtained results for positiveGauss-Bonnet coefficient satisfying energy conditions It contains

10 Advances in High Energy Physics

02 04 06 08 10

0

200

400

r

fa

minus400

minus200

(a) Evolution of 119891119886versus 119903

02 04 06 08 10

0

200

400

r

fa

minus400

minus200

(b) Evolution of 119891119886versus 119903

02 04 06 08 10r

minus400

minus200

0

200

400

fh

(c) Evolution of 119891ℎversus 119903

02 04 06 08 10

0

200

400

r

fh

minus400

minus200

(d) Evolution of 119891ℎversus 119903

Figure 10 Plots of (a) 119891119886for 119904+ (b) 119891

119886for 119904minus (c) 119891

ℎfor 119904+ (d) 119891

ℎfor 119904minusversus 119903 with Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5

120598 = minus1 (red dashed) and 119899 = 6 120598 = 3 (purple)

Table 4 Wormhole solutions with Lorentzian noncommutativebackground for 119904

minus

Expressions 119899 = 5 120598 = 1 119899 = 5 120598 = minus1 119899 = 6 120598 = 3

1 minus 119904minus119903 Positive Positive for

119903 lt 05

Positive

120588 Positive Positive Positive120588 + 119901

119903 Positive Negative Negative120588 + 119901

119905 Positive Negative Positive

WEC Holds Does nothold

Does nothold

EC Holds Holds Holds

fifth dimensional wormhole solutions in both backgroundssatisfying equilibrium condition and sixth dimensional withdisequilibrium in noncommutative background with 119904

minussolu-

tion Also there is possibility for the existence of wormholein equilibrium satisfying WEC for 119899 = 6 120598 = 3 taking intoaccount 119904

+solution for the range 119903 lt 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M S Morris and K S Thorne ldquoWormholes in spacetimeand their use for interstellar travel a tool for teaching generalrelativityrdquo American Journal of Physics vol 56 no 5 pp 395ndash412 1988

[2] M Sharif and A Jawad ldquoPhantom-like generalized cosmicchaplygin gas and traversable wormhole solutionsrdquo The Euro-pean Physical Journal Plus vol 129 article 15 2014

[3] F S N Lobo F Parsaei and N Riazi ldquoNew asymptotically flatphantom wormhole solutionsrdquo Physical Review D vol 87 no 8Article ID 084030 2013

[4] S-W Kim andH Lee ldquoExact solutions of a charged wormholerdquoPhysical Review D vol 63 no 6 Article ID 064014 5 pages2001

[5] M Jamil and M U Farooq ldquoPhantom wormholes in (2 + 1)-dimensionsrdquo International Journal ofTheoretical Physics vol 49no 4 pp 835ndash841 2010

[6] M U Farooq M Akbar and M Jamil ldquoDynamics and ther-modynamics of (2+1)minusdimensional evolving lorentzian worm-holerdquo in Proceedings of the 3rd AlgerianWorkshop on Astronomyand Astrophysics vol 1295 of AIP Conference Proceedings pp176ndash190 Constantine Algeria June 2010

[7] P K Kuhfittig ldquoGravitational lensing of wormholes in thegalactic halo regionrdquo The European Physical Journal C vol 74no 3 article 2818 2014

Advances in High Energy Physics 11

[8] F Rahaman A Banerjee M Jamil A K Yadav and H IdrisldquoNoncommutative wormholes in f(R) gravity with lorentziandistributionrdquo International Journal ofTheoretical Physics vol 53no 6 pp 1910ndash1919 2014

[9] M Sharif and S Rani ldquoDynamical wormhole solutions in f(T)gravityrdquo General Relativity and Gravitation vol 45 no 11 pp2389ndash2402 2013

[10] M Sharif and S Rani ldquoCharged noncommutative wormholesolutions in f (T) gravityrdquo The European Physical Journal Plusvol 129 no 10 article 237 2014

[11] M Sharif and S Rani ldquoGalactic halo wormhole solutions in119891(119879) gravityrdquoAdvances in High Energy Physics vol 2014 ArticleID 691497 9 pages 2014

[12] F S N Lobo andM A Oliveira ldquoWormhole geometries in f (R)modified theories of gravityrdquo Physical Review D vol 80 ArticleID 104012 2009

[13] M Sharif and S Rani ldquoWormhole solutions in f(T) gravity withnoncommutativerdquo Physical Review D vol 88 no 12 Article ID123501 2013

[14] F Rahaman S Islam P K F Kuhfittig and S Ray ldquoSearch-ing for higher-dimensional wormholes with noncommutativegeometryrdquo Physical Review D vol 86 no 10 Article ID 1060102012

[15] P Bhar and F Rahaman ldquoSearch of wormholes in differ-ent dimensional non-commutative inspired space-times withLorentzian distributionrdquo The European Physical Journal C vol74 no 12 article 3213 2014

[16] F Rahaman S Karmakar I Karar and S Ray ldquoWormholeinspired by non-commutative geometryrdquo Physics Letters B vol746 pp 73ndash78 2015

[17] M Jamil F Rahaman R Myrzakulov P Kuhfittig N Ahmedand U Mondal ldquoNonommutative wormholes in f(R) gravityrdquoJournal of the Korean Physical Society vol 65 no 6 pp 917ndash9252014

[18] A Jawad and S Rani ldquoLorentz distributed noncommutativewormhole solutions in extended teleparallel gravityrdquoThe Euro-pean Physical Journal C vol 75 article 173 2015

[19] B Bhawal and S Kar ldquoLorentzian wormholes in Einstein-Gauss-Bonnet theoryrdquo Physical Review D vol 46 no 6 pp2464ndash2468 1992

[20] M R Mehdizadeh M K Zangeneh and F S N LoboldquoEinstein-Gauss-Bonnet traversable wormholes satisfying theweak energy conditionrdquo Physical ReviewD vol 91 no 8 ArticleID 084004 2015

[21] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[22] S H Mehdipour ldquoEntropic force approach to noncommutativeSchwarzschild black holes signals a failure of current physicalideasrdquo The European Physical Journal Plus vol 127 article 802012

[23] S Sushkov ldquoWormholes supported by a phantom energyrdquoPhysical Review D vol 71 no 4 Article ID 043520 2005

[24] P Nicolini and E Spallucci ldquoNoncommutative geometry-inspired dirty black holesrdquo Classical and Quantum Gravity vol27 no 1 Article ID 015010 2010

[25] F Rahaman P K Kuhfittig K Chakraborty A A Usmani andS Ray ldquoGalactic rotation curves inspired by a noncommutative-geometry backgroundrdquo General Relativity and Gravitation vol44 no 4 pp 905ndash916 2012

[26] P K F Kuhfittig ldquoOn the stability of thin-shell wormholes innoncommutative geometryrdquo Advances in High Energy Physicsvol 2012 Article ID 462493 12 pages 2012

[27] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquo Journal of High Energy Physics vol 1999 no 9 article032 1999

[28] J P de Leon ldquoMass and charge in brane-world and non-compact Kaluza-KLEin theories in 5 dimrdquo General Relativityand Gravitation vol 35 no 8 pp 1365ndash1384 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

4 Advances in High Energy Physics

0

2

4

6

8

minus2

00 10 30252005 15r

1minuss +r

(a)

0

2

4

minus2

86420r

1minuss minusr

(b)

Figure 1 Plots of (a) 1 minus 119904+119903 and (b) 1 minus 119904

minus119903 versus 119903 in Gaussian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed) 119899 = 6 120598 = 3

(purple) and 119899 = 6 120598 = minus3 (purple dashed)

phantom-energy supported wormholes as well as by Nicoliniand Spallucci [24] for the purpose of modeling the physicaleffects of short distance fluctuations of noncommutativecoordinates in the study of black holes Galactic rotationcurves inspired by a noncommutative geometry backgroundare discussed [25] The stability of a particular class of thin-shell wormholes in noncommutative geometry is analyzedelsewhere [26]

31 Gaussian Distributed Noncommutative Framework Anintrinsic characteristic of spacetime is the noncommuta-tivity which plays an effective role in several areas Itis an interesting consequence of string theory where thecoordinates of spacetime become noncommutative operatorson 119863-brane [27] The noncommutativity of spacetime canbe converted in the commutator [119909120572 119909120573] = 119894120579

120572120573 where120579120572120573 is an antisymmetric matrix describing discretization ofspacetime and has dimension (length)2 This discretizationprocess is similar to the discretization of phase space byPlanck constant Replacing the point-like structures withsmeared objects the energy density of the particle-like staticspherically symmetric gravitational source having mass 119872takes the following form [21]

120588nc =119872

(4120587120579)(119899minus1)2

119890minus11990324120579

(10)

where 120579 is the noncommutative parameter in Gaussiandistribution The mass 119872 could be a diffused centralizedobject such as a wormhole [28] It is mentioned here that thesmearing effect is achieved by replacing the Gaussian distri-bution of minimal length radic120579 with the Dirac delta functionIn order to construct 119899-dimensional Einstein Gauss-Bonnetwormhole geometry we equate the energy density given in(7) and 120588nc in (10) yields the following differential equation

1199041015840

=

1

1 + 21199041205981199033[(4 minus 119899 + (7 minus 119899)

119904120598

1199033)

+

21199032

119872

(119899 minus 2)

(4120587120579)(1minus119899)2

119890minus11990324120579

]

(11)

The solution of this equation is

119904 (119903) = minus

1

2

[

1199033

120598

plusmn

119903(7minus119899)2

120587minus1198994

radic120598 (119899 minus 1)

(119899 minus 2) 1205871198992

(119903119899minus1

+ 41205982

1198881)

minus 4119872radic120587120598Gamma[119899 minus 12

1199032

4120579

]

12

]

(12)

where 1198881is an integration constant This solution has two

roots with plus and minus signs and we assign these rootsas 119904+(119903) and 119904

minus(119903) solutions In order to plot the quantities

1 minus 119904119903 120588 120588 + 119901119903 and 120588 + 119901

119905to obtain wormhole solutions

for both of these solutions we restrict ourselves to five- andsix-dimensional cases with constant redshift function 120582 = 0The expressions of NEC takes the form

120588 + 119901119903=

(119899 minus 2)

21199032

(1199041015840

minus

119904

119903

) (1 +

2120598119904

1199033)

120588 + 119901119905=

1199031199041015840

minus 119904

21199033

(1 +

6120598119904

1199033)

+

119904

1199033[(119899 minus 3) + (119899 minus 5)

2119904120598

1199033]

(13)

In this regard we assume values of some constants as119872 = 0008 120579 = 0002 while 120598

1= 05 minus05 result in 120598 =

1 minus1 for five-dimensional and 120598 = 3 minus3 for six-dimensionalcase Figure 1(a) represents the plot of 1 minus 119904

+119903 versus 119903

for five and six-dimensional wormhole solutions The graphrepresents positively increasing behavior for positive 120598 forboth curves For negative 120598 we examine that graph initiallyrepresents positive behavior for both dimensions in the range119903 lt 1 and 119903 lt 18 for 119899 = 5 and 6 respectively and thendecreases towards negative values In Figure 1(b) the graphof 1 minus 119904

minus119903 versus 119903 shows the same behavior for all curves as

in Figure 1(a) with positive behavior of dashed curves in therange 119903 lt 4 and 119903 lt 8 for 119899 = 5 ad 6 However we have less

Advances in High Energy Physics 5

025 030 035 040020r

000

002

120588

004

006

008

010

(a) Evolution of 120588+versus 119903

000

002

004

006

008

010

120588

025 030 035 040020r

(b) Evolution of 120588minusversus 119903

Figure 2 Plots of (a) 120588+and (b) 120588

minusversus 119903 in Gaussian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed) 119899 = 6 120598 = 3 (purple)

and 119899 = 6 120598 = minus3 (purple dashed)

possibility of wormhole scenario with respect to 119903 for positiveroot solution as compared to negative root solution of shapefunction

Figure 2 shows the behavior of energy density versus119903 as positively decreasing behavior for both dimensionscorresponding to 119899 = 5 and 6 dimensions For 119904

+and positive

120598 the plot of 120588+119901119903represents positive behavior in decreasing

manner for 119899 = 5 and 6 while representing negative behaviorfor negative 120598 as shown in Figure 3(a) In Figure 3(b) weexamine same behavior for negative root solution Figure 4(a)depicts the opposite behavior to Figure 3(a) that is for 119899 =

5 6 dimensions and positive 120598 120588 + 119901119905demonstrates negative

behavior Thus these plots express the violation of WECincorporating the case of 119904

+ For 119904

minus we obtain the same

behavior as in Figure 3(b) which indicates positive behaviorfor positive 120598 and 119899 = 5 6 and negative behavior for negative120598 and 119899 = 5 6 This implies that WEC is satisfied fornegative root solutionwith positiveGauss-Bonnet coefficientThus we obtain physically acceptable wormhole solutionssatisfying WEC for both dimensions

32 Lorentzian Distributed Non-Commutative FrameworkNow we consider the case of noncommutative geometrywith Lorentzian distributionThe energy density of point-likesource under this distribution becomes [8 15]

120588Lnc =119872radic120601

1205872(1199032+ 120601)1198992

(14)

where 120601 is the noncommutative parameter in Lorentziandistribution Inserting 120588Lnc in (7) the differential equationtakes the following form

1199041015840

=

1

1 + 21199041205981199033[

21199032

119872radic120601

1205872(119899 minus 2) (119903

2+ 120601)1198992

+

119887

119903

4 minus 119899 + (7 minus 119899)

119904120598

1199033]

(15)

The solution of this equation is given by

119904 (119903) =

1199033

2120598120587

[minus120587 plusmn

1

(119899 minus 2) (119899 minus 1)

1205872

(1198992

minus 3119899 + 2) (119903119899

+ 41199031205982

1198882)

+ 8119872120598119903119899

120601(1minus119899)2Hypergeometric 2F1 [1

2

(119899 minus 1)

119899

2

119899 + 1

2

minus

1199032

120601

]

12

]

(16)

where 1198882is an integration constant We again assign both

solutions as 119904+and 119904minusand explore the wormhole solutions

In this case we choose constants as 120601 = 2 119872 = 2 and1198882= 05 for same dimensions and Gauss-Bonnet coefficient

as for noncommutative backgroundIn Figure 5(a) we plot 1 minus 119904

+119903 versus 119903 which represents

that the condition for wormhole geometry that is 1minus119904119903 gt 0holds for 5 and 6 dimensions with negative 120598 For positive 120598the positivity of this expression depends on some ranges sothat it remains positive for 119903 lt 19 for 119899 = 5 while 119899 = 6

observes the range 119903 lt 21 Figure 5(b) corresponds to theplot of 1 minus 119904

minus119903 with respect to 119903 showing positive behavior

for positive 120598 For 119899 = 5 it expresses a very short range119903 lt 05 for positivity whereas it remains negative for 6-dimensional solutionThat is we have no wormhole solutionfor 119899 = 6 120598 = minus3 taking negative root solution So we skipthis case in further discussion The behavior of 120588

+and 120588

minus

remains positive for all cases except 119899 = 6 120598 = minus3 for whichit describes negative behavior as shown in Figures 6(a) and6(b) This implies that we have no wormhole solutions for 6-dimensional case with negative Gauss-Bonnet coefficient forboth solutions

Figure 7(a) expresses the behavior of 120588 + 119901119903for positive

root solution versus 119903 which remains positive for the range119903 lt 1 for 119899 = 5 and 6 with positive 120598 and then turns towardsnegative behavior It demonstrates negative behavior for 119903 lt 1and then moves to positive region Incorporating 119904

minussolution

120588 + 119901119903shows positive behavior only for the case 119899 = 5

6 Advances in High Energy Physics

pr+120588

minus06

minus04

minus02

00

02

04

06

0402 06 08 10 12 1400r

(a) Evolution of (120588 + 119901119903)+ versus 119903

pr+120588

minus3

minus2

minus1

0

1

2

3

04 06 08 10 12 1402r

(b) Evolution of (120588 + 119901119903)minus versus 119903

Figure 3 Plots of (a) 120588 + 119901119903for 119904+and (b) 120588 + 119901

119903for 119904minusversus 119903 in Gaussian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

119899 = 6 120598 = 3 (purple) and 119899 = 6 120598 = minus3 (purple dashed)

pt+120588

pt+120588

n = 5 120598 = minus1

minus04

minus02

00

02

04

05 10 15 20 25 3000r

000001500000200000250000030000035

17 18 19 2016r

(a) Evolution of (120588 + 119901119905)+ versus 119903

pt+120588

minus10

minus5

0

5

10

05 10 15 20 25 3000r

(b) Evolution of (120588 + 119901119905)minus versus 119903

Figure 4 Plots of (a) 120588 + 119901119905for 119904+and (b) 120588 + 119901

119905for 119904minusversus 119903 for in Gaussian distribution 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

119899 = 6 120598 = 3 (purple) and 119899 = 6 120598 = minus3 (purple dashed)

n = 5 120598 = minus1

minus2

minus1

0

1

2

05 10 15 20 25 3000r

0985

0990

0995

1000

005 010 015 020000

r

1minuss +r

1minuss +r

(a)

05 10 15 20 25 3000r

minus4

minus2

0

2

4

1minuss minusr

(b)

Figure 5 Plots of (a) 1minus 119904+119903 and (b) 1minus 119904

minus119903 versus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed) and 119899 = 6

120598 = 3 (purple)

Advances in High Energy Physics 7

04 05 06 07 08 09 10050100150200

r

05 10 15 20

0

05 10 15 20r

n = 6 120598 = minus3

n = 6 120598 = 3

minus1000minus2000minus3000minus4000minus5000minus6000

001

002

120588

120588

120588

003

004

005

(a) Evolution of 120588+versus 119903

05 10 15 20000

001

002

003

004

005

r

120588

(b) Evolution of 120588minusversus 119903

Figure 6 Plots of (a) 120588+and (b) 120588

minusversus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed) and 119899 = 6 120598 = 3

(purple) For (a) 120588+ 119899 = 6 and 120598 = minus3 (purple dashed)

05 10 15 20 25 30

0

5

10

15

r

minus10

minus5

pr+120588

(a) Evolution of (120588 + 119901119903)+versus 119903

02 04 06 08 101020304050

02 04 06 08 10

0

n = 5 120598 = 1

r

rminus8000

minus6000

minus4000

minus2000

pr+120588

pr+120588

(b) Evolution of (120588 + 119901119903)minusversus 119903

Figure 7 Plots of (a) 120588 + 119901119903for 119904+and (b) 120588 + 119901

119903for 119904minusversus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

and 119899 = 6 120598 = 3 (purple)

120598 gt 0 and negative behavior for the remaining two casesas shown in Figure 7(b) In Figures 8(a) and 8(b) we draw120588 + 119901119905for both solutions versus 119903 This expression represents

the negative behavior for (119899 = 5 120598 = 1) (119899 = 5 120598 = minus1) with119904+and (119899 = 5 120598 = minus1) with 119904

minusand preserves positivity for

(119899 = 6 120598 = 3) with 119904+ (119899 = 5 120598 = 1) (119899 = 6 120598 = 3) with 119904

minus

solution

4 Equilibrium Condition

In order to find the equilibrium configuration of thewormhole solutions in Gaussian as well as Lorentzian dis-tributed noncommutative backgrounds we use the general-ized Tolman-Oppenheimer-Volkoff equation This equationis derived by solving the Einstein equations for a general time-invariant spherically symmetric metric having metric tensor119892120572120573

= (119890120591(119903)

minus119890120596(119903)

minus1199032

minus1199032sin2120579) where 120572120573 represents

only diagonal entries and 120591 120596 are general metric functionsdependent on 119903 The generalized Tolman-Oppenheimer-Volkoff equation is

119889119901119903

119889119903

+

1205911015840

2

(120588 + 119901119903) +

2

119903

(119901119903minus 119901119905) = 0 (17)

Keeping in mind the above equation de Leon [28] proposedan equation for anisotropicmass distributionwhich naturallygives the equilibrium for the wormhole subject It is given by

2

119903

(119901119905minus 119901119903) minus

119890(120596minus120591)2

119898eff1199032

(120588 + 119901119903) minus

119889119901119903

119889119903

= 0 (18)

where effective gravitational mass 119898eff = (12)1199032

119890(120591minus120596)2

1205911015840 is

measured from throat to some arbitrary radius 119903 Accord-ingly the gravitational hydrostatic and anisotropic force dueto anisotropic matter distribution are defied as follows

119891119892= minus

1205911015840

(120588 + 119901119903)

2

119891ℎ= minus

119889119901119903

119889119903

119891119886=

2 (119901119905minus 119901119903)

119903

(19)

It is required that119891119892+119891ℎ+119891119886= 0must hold for the wormhole

solutions to be in equilibrium In the underlying cases we

8 Advances in High Energy Physics

05 10 15 20 25 30

0

r

r

150 152 154 156 158 160

r

05 10 15 200

500

1000

1500

2000

pt+120588

pt+120588

pt+120588

minus0058

minus15

minus10

minus5

minus0056

minus0054

minus0052

minus0050n = 5 120598 = minus1

n = 5 120598 = 1

(a) Evolution of (120588 + 119901119905)+versus 119903

r

05 10 15 20

r

00 05 10 15 200

5

10

15

20

pt+120588 p

t+120588

minus70

minus60

minus50

minus40

minus30

minus20

minus10

n = 5 120598 = minus1

(b) Evolution of (120588 + 119901119905)minusversus 119903

Figure 8 Plots of (a) 120588 + 119901119905for 119904+and (b) 120588 + 119901

119905for 119904minusversus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

and 119899 = 6 120598 = 3 (purple)

proceedwith constant redshift function 120582 = 0which vanishesthe gravitational contribution119891

119892in the equilibrium equation

that is 120591 = 2120582 leads to 1205911015840 = 0 for constant 120582 Thus we are leftwith hydrostatic and anisotropic forces with correspondingequilibrium condition as

119891119886+ 119891ℎ= 0 (20)

Using (8) and (9) we obtain the following expressions for thehydrostatic and anisotropic forces

119891119886=

1

1199033(119899 minus 1)

119904

119903

minus 1199041015840

(119899 minus 3 + (119899 minus 5)

119904120598

1199033)

minus

119904

1199034[(119899 minus 3) (119899 minus 4) + (119899 minus 5) (119899 minus 6)

119904120598

1199033]

119891ℎ=

(119899 minus 2)

21199033

[119899 minus 3 + (119899 minus 5)

119904120598

1199033+

119904120598 (119899 minus 5)

1199033

]

sdot (1199041015840

minus

3119904

119903

)

(21)

Figure 9 represents the plots of anisotropic as well ashydrostatic forces for the wormhole solutions in Gaussiandistributed framework Figures 9(a) and 9(c) describe theequilibrium for positive root solution for 119899 = 5 120598 =

1 minus1 and 119899 = 6 120598 = 3 minus3 through opposite behaviorand hence they cancel each other in order to satisfy (20)For negative root solution 119904

minus Figures 9(b) and 9(d) show

the stable configuration of wormhole solutions for fifthdimensional case only The behaviors of both forces (whichare not in opposite manner) do not cancel each other sothere is no equilibrium configuration examined for sixthdimensional wormhole solutions In the case of Lorentziannoncommutative background Figure 10 expresses that allwormhole solutions are in equilibrium by satisfying theequilibrium condition for both dimensions

Table 1 Wormhole solutions with Gaussian distributed noncom-mutative framework for 119904

+

Expressions 119899 = 5 120598 =

1

119899 = 5 120598 =

minus1

119899 = 6 120598 =

3

119899 = 6 120598 =

minus3

1 minus 119904+119903 Positive Positive

for 119903 lt 1 Positive Positivefor 119903 lt 18

120588 Positive Positive Positive Positive120588 + 119901

119903 Positive Negative Positive Negative120588 + 119901

119905 Negative Positive Negative PositiveWEC Violates Violates Violates ViolatesEC Holds Holds Holds Holds

5 Conclusion

It is a well-known fact that the existence of wormhole solu-tions is based on violation of NEC Since the normal mattersatisfies the energy conditions this violation is associatedwith an energy-momentum tensor which provides exoticmatter a hypothetical form of matter To explore realisticmodel or physically acceptable wormhole solutions it isnecessary to find such a source which gives the violationof NEC while normal matter meets the energy conditionsHere in this paper we have explored wormhole solutionsin 119899-dimensional Einstein Gauss-Bonnet gravity with Gaus-sian and Lorentzian noncommutative backgrounds We haverestricted ourselves to fifth and sixth dimensional cases withpositive as well as negativeGauss-Bonnet coefficient Also wehave checked the condition of equilibrium for the wormholesolutions

The results are summarized in Tables 1 and 2 It is notedthat the 119904

+(Tables 1 and 3) and 119904

minus(Tables 2 and 4) represent

the two roots of the solutions in both backgrounds and ECdenotes the equilibrium condition

In the Lorentzan distributed noncommutative frame-work we have found negative energy density for 119904

+while

Advances in High Energy Physics 9

02 04 06 1008

0

2

4

r

fa

minus2

minus4

(a) Evolution of 119891119886versus 119903

r

02 04 06 08 10

0

2

4

fa

minus2

minus4

(b) Evolution of 119891119886versus 119903

02 04 06 08 10

0

2

4

r

fh

minus2

minus4

(c) Evolution of 119891ℎversus 119903

02 04 06 08 10

0

2

4

r

fh

minus2

minus4

(d) Evolution of 119891ℎversus 119903

Figure 9 Plots of (a) 119891119886for 119904+ (b) 119891

119886for 119904minus (c) 119891

ℎfor 119904+ (d) 119891

ℎfor 119904minusversus 119903 with Gaussian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1

(red dashed) and 119899 = 6 120598 = 3 (purple)

Table 2 Wormhole solutions with Gaussian distributed noncom-mutative framework for 119904

minus

Expressions 119899 = 5 120598 =

1

119899 = 5 120598 =

minus1

119899 = 6 120598 =

3

119899 = 6 120598 =

minus3

1 minus 119904minus119903 Positive Positive

for 119903 lt 4 Positive Positivefor 119903 lt 8

120588 Positive Positive Positive Positive120588 + 119901

119903 Positive Negative Positive Negative120588 + 119901

119905 Positive Negative Positive NegativeWEC Holds Violates Holds Violates

EC Holds Holds Does nothold

Does nothold

violation of 1 minus 119904119903 gt 0 incorporating 119904minusfor the case 119899 = 6

120598 = minus3 so we skipped this case The remaining results aresummarized in Tables 3 and 4

In the paper [20] higher dimensional asymptotically flatwormhole solutions have been explored in the frameworkof Gauss-Bonnet gravity by considering a specific choicefor a radial dependent redshift function and by imposinga particular equation of state The WEC is satisfied at thethroat by considering a negative Gauss-Bonnet coupling

Table 3 Wormhole solutions with Lorentzian noncommutativebackground for 119904

+

Expressions 119899 = 5 120598 = 1 119899 = 5 120598 = minus1 119899 = 6 120598 = 3

1 minus 119904+119903

Positive for119903 lt 19

Positive Positive for119903 lt 21

120588 Positive Positive Positive

120588 + 119901119903

Positive for119903 lt 1

Positive for119903 gt 1

Positive for119903 lt 1

120588 + 119901119905 Negative Negative Positive

WEC Does nothold

Does nothold

Holds for119903 lt 1

EC Holds Holds Holds

constant Furthermore they have considered a constantredshift function and shown specifically that for negativeGauss-Bonnet coupling constant one may have normalmatter in a determined radial region and that the increaseof coupling constant enlarges the normal matter regionIn the present paper we have taken energy density undernoncommutative geometry distributions instead of particularequation of stateWe have obtained results for positiveGauss-Bonnet coefficient satisfying energy conditions It contains

10 Advances in High Energy Physics

02 04 06 08 10

0

200

400

r

fa

minus400

minus200

(a) Evolution of 119891119886versus 119903

02 04 06 08 10

0

200

400

r

fa

minus400

minus200

(b) Evolution of 119891119886versus 119903

02 04 06 08 10r

minus400

minus200

0

200

400

fh

(c) Evolution of 119891ℎversus 119903

02 04 06 08 10

0

200

400

r

fh

minus400

minus200

(d) Evolution of 119891ℎversus 119903

Figure 10 Plots of (a) 119891119886for 119904+ (b) 119891

119886for 119904minus (c) 119891

ℎfor 119904+ (d) 119891

ℎfor 119904minusversus 119903 with Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5

120598 = minus1 (red dashed) and 119899 = 6 120598 = 3 (purple)

Table 4 Wormhole solutions with Lorentzian noncommutativebackground for 119904

minus

Expressions 119899 = 5 120598 = 1 119899 = 5 120598 = minus1 119899 = 6 120598 = 3

1 minus 119904minus119903 Positive Positive for

119903 lt 05

Positive

120588 Positive Positive Positive120588 + 119901

119903 Positive Negative Negative120588 + 119901

119905 Positive Negative Positive

WEC Holds Does nothold

Does nothold

EC Holds Holds Holds

fifth dimensional wormhole solutions in both backgroundssatisfying equilibrium condition and sixth dimensional withdisequilibrium in noncommutative background with 119904

minussolu-

tion Also there is possibility for the existence of wormholein equilibrium satisfying WEC for 119899 = 6 120598 = 3 taking intoaccount 119904

+solution for the range 119903 lt 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M S Morris and K S Thorne ldquoWormholes in spacetimeand their use for interstellar travel a tool for teaching generalrelativityrdquo American Journal of Physics vol 56 no 5 pp 395ndash412 1988

[2] M Sharif and A Jawad ldquoPhantom-like generalized cosmicchaplygin gas and traversable wormhole solutionsrdquo The Euro-pean Physical Journal Plus vol 129 article 15 2014

[3] F S N Lobo F Parsaei and N Riazi ldquoNew asymptotically flatphantom wormhole solutionsrdquo Physical Review D vol 87 no 8Article ID 084030 2013

[4] S-W Kim andH Lee ldquoExact solutions of a charged wormholerdquoPhysical Review D vol 63 no 6 Article ID 064014 5 pages2001

[5] M Jamil and M U Farooq ldquoPhantom wormholes in (2 + 1)-dimensionsrdquo International Journal ofTheoretical Physics vol 49no 4 pp 835ndash841 2010

[6] M U Farooq M Akbar and M Jamil ldquoDynamics and ther-modynamics of (2+1)minusdimensional evolving lorentzian worm-holerdquo in Proceedings of the 3rd AlgerianWorkshop on Astronomyand Astrophysics vol 1295 of AIP Conference Proceedings pp176ndash190 Constantine Algeria June 2010

[7] P K Kuhfittig ldquoGravitational lensing of wormholes in thegalactic halo regionrdquo The European Physical Journal C vol 74no 3 article 2818 2014

Advances in High Energy Physics 11

[8] F Rahaman A Banerjee M Jamil A K Yadav and H IdrisldquoNoncommutative wormholes in f(R) gravity with lorentziandistributionrdquo International Journal ofTheoretical Physics vol 53no 6 pp 1910ndash1919 2014

[9] M Sharif and S Rani ldquoDynamical wormhole solutions in f(T)gravityrdquo General Relativity and Gravitation vol 45 no 11 pp2389ndash2402 2013

[10] M Sharif and S Rani ldquoCharged noncommutative wormholesolutions in f (T) gravityrdquo The European Physical Journal Plusvol 129 no 10 article 237 2014

[11] M Sharif and S Rani ldquoGalactic halo wormhole solutions in119891(119879) gravityrdquoAdvances in High Energy Physics vol 2014 ArticleID 691497 9 pages 2014

[12] F S N Lobo andM A Oliveira ldquoWormhole geometries in f (R)modified theories of gravityrdquo Physical Review D vol 80 ArticleID 104012 2009

[13] M Sharif and S Rani ldquoWormhole solutions in f(T) gravity withnoncommutativerdquo Physical Review D vol 88 no 12 Article ID123501 2013

[14] F Rahaman S Islam P K F Kuhfittig and S Ray ldquoSearch-ing for higher-dimensional wormholes with noncommutativegeometryrdquo Physical Review D vol 86 no 10 Article ID 1060102012

[15] P Bhar and F Rahaman ldquoSearch of wormholes in differ-ent dimensional non-commutative inspired space-times withLorentzian distributionrdquo The European Physical Journal C vol74 no 12 article 3213 2014

[16] F Rahaman S Karmakar I Karar and S Ray ldquoWormholeinspired by non-commutative geometryrdquo Physics Letters B vol746 pp 73ndash78 2015

[17] M Jamil F Rahaman R Myrzakulov P Kuhfittig N Ahmedand U Mondal ldquoNonommutative wormholes in f(R) gravityrdquoJournal of the Korean Physical Society vol 65 no 6 pp 917ndash9252014

[18] A Jawad and S Rani ldquoLorentz distributed noncommutativewormhole solutions in extended teleparallel gravityrdquoThe Euro-pean Physical Journal C vol 75 article 173 2015

[19] B Bhawal and S Kar ldquoLorentzian wormholes in Einstein-Gauss-Bonnet theoryrdquo Physical Review D vol 46 no 6 pp2464ndash2468 1992

[20] M R Mehdizadeh M K Zangeneh and F S N LoboldquoEinstein-Gauss-Bonnet traversable wormholes satisfying theweak energy conditionrdquo Physical ReviewD vol 91 no 8 ArticleID 084004 2015

[21] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[22] S H Mehdipour ldquoEntropic force approach to noncommutativeSchwarzschild black holes signals a failure of current physicalideasrdquo The European Physical Journal Plus vol 127 article 802012

[23] S Sushkov ldquoWormholes supported by a phantom energyrdquoPhysical Review D vol 71 no 4 Article ID 043520 2005

[24] P Nicolini and E Spallucci ldquoNoncommutative geometry-inspired dirty black holesrdquo Classical and Quantum Gravity vol27 no 1 Article ID 015010 2010

[25] F Rahaman P K Kuhfittig K Chakraborty A A Usmani andS Ray ldquoGalactic rotation curves inspired by a noncommutative-geometry backgroundrdquo General Relativity and Gravitation vol44 no 4 pp 905ndash916 2012

[26] P K F Kuhfittig ldquoOn the stability of thin-shell wormholes innoncommutative geometryrdquo Advances in High Energy Physicsvol 2012 Article ID 462493 12 pages 2012

[27] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquo Journal of High Energy Physics vol 1999 no 9 article032 1999

[28] J P de Leon ldquoMass and charge in brane-world and non-compact Kaluza-KLEin theories in 5 dimrdquo General Relativityand Gravitation vol 35 no 8 pp 1365ndash1384 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 5

025 030 035 040020r

000

002

120588

004

006

008

010

(a) Evolution of 120588+versus 119903

000

002

004

006

008

010

120588

025 030 035 040020r

(b) Evolution of 120588minusversus 119903

Figure 2 Plots of (a) 120588+and (b) 120588

minusversus 119903 in Gaussian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed) 119899 = 6 120598 = 3 (purple)

and 119899 = 6 120598 = minus3 (purple dashed)

possibility of wormhole scenario with respect to 119903 for positiveroot solution as compared to negative root solution of shapefunction

Figure 2 shows the behavior of energy density versus119903 as positively decreasing behavior for both dimensionscorresponding to 119899 = 5 and 6 dimensions For 119904

+and positive

120598 the plot of 120588+119901119903represents positive behavior in decreasing

manner for 119899 = 5 and 6 while representing negative behaviorfor negative 120598 as shown in Figure 3(a) In Figure 3(b) weexamine same behavior for negative root solution Figure 4(a)depicts the opposite behavior to Figure 3(a) that is for 119899 =

5 6 dimensions and positive 120598 120588 + 119901119905demonstrates negative

behavior Thus these plots express the violation of WECincorporating the case of 119904

+ For 119904

minus we obtain the same

behavior as in Figure 3(b) which indicates positive behaviorfor positive 120598 and 119899 = 5 6 and negative behavior for negative120598 and 119899 = 5 6 This implies that WEC is satisfied fornegative root solutionwith positiveGauss-Bonnet coefficientThus we obtain physically acceptable wormhole solutionssatisfying WEC for both dimensions

32 Lorentzian Distributed Non-Commutative FrameworkNow we consider the case of noncommutative geometrywith Lorentzian distributionThe energy density of point-likesource under this distribution becomes [8 15]

120588Lnc =119872radic120601

1205872(1199032+ 120601)1198992

(14)

where 120601 is the noncommutative parameter in Lorentziandistribution Inserting 120588Lnc in (7) the differential equationtakes the following form

1199041015840

=

1

1 + 21199041205981199033[

21199032

119872radic120601

1205872(119899 minus 2) (119903

2+ 120601)1198992

+

119887

119903

4 minus 119899 + (7 minus 119899)

119904120598

1199033]

(15)

The solution of this equation is given by

119904 (119903) =

1199033

2120598120587

[minus120587 plusmn

1

(119899 minus 2) (119899 minus 1)

1205872

(1198992

minus 3119899 + 2) (119903119899

+ 41199031205982

1198882)

+ 8119872120598119903119899

120601(1minus119899)2Hypergeometric 2F1 [1

2

(119899 minus 1)

119899

2

119899 + 1

2

minus

1199032

120601

]

12

]

(16)

where 1198882is an integration constant We again assign both

solutions as 119904+and 119904minusand explore the wormhole solutions

In this case we choose constants as 120601 = 2 119872 = 2 and1198882= 05 for same dimensions and Gauss-Bonnet coefficient

as for noncommutative backgroundIn Figure 5(a) we plot 1 minus 119904

+119903 versus 119903 which represents

that the condition for wormhole geometry that is 1minus119904119903 gt 0holds for 5 and 6 dimensions with negative 120598 For positive 120598the positivity of this expression depends on some ranges sothat it remains positive for 119903 lt 19 for 119899 = 5 while 119899 = 6

observes the range 119903 lt 21 Figure 5(b) corresponds to theplot of 1 minus 119904

minus119903 with respect to 119903 showing positive behavior

for positive 120598 For 119899 = 5 it expresses a very short range119903 lt 05 for positivity whereas it remains negative for 6-dimensional solutionThat is we have no wormhole solutionfor 119899 = 6 120598 = minus3 taking negative root solution So we skipthis case in further discussion The behavior of 120588

+and 120588

minus

remains positive for all cases except 119899 = 6 120598 = minus3 for whichit describes negative behavior as shown in Figures 6(a) and6(b) This implies that we have no wormhole solutions for 6-dimensional case with negative Gauss-Bonnet coefficient forboth solutions

Figure 7(a) expresses the behavior of 120588 + 119901119903for positive

root solution versus 119903 which remains positive for the range119903 lt 1 for 119899 = 5 and 6 with positive 120598 and then turns towardsnegative behavior It demonstrates negative behavior for 119903 lt 1and then moves to positive region Incorporating 119904

minussolution

120588 + 119901119903shows positive behavior only for the case 119899 = 5

6 Advances in High Energy Physics

pr+120588

minus06

minus04

minus02

00

02

04

06

0402 06 08 10 12 1400r

(a) Evolution of (120588 + 119901119903)+ versus 119903

pr+120588

minus3

minus2

minus1

0

1

2

3

04 06 08 10 12 1402r

(b) Evolution of (120588 + 119901119903)minus versus 119903

Figure 3 Plots of (a) 120588 + 119901119903for 119904+and (b) 120588 + 119901

119903for 119904minusversus 119903 in Gaussian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

119899 = 6 120598 = 3 (purple) and 119899 = 6 120598 = minus3 (purple dashed)

pt+120588

pt+120588

n = 5 120598 = minus1

minus04

minus02

00

02

04

05 10 15 20 25 3000r

000001500000200000250000030000035

17 18 19 2016r

(a) Evolution of (120588 + 119901119905)+ versus 119903

pt+120588

minus10

minus5

0

5

10

05 10 15 20 25 3000r

(b) Evolution of (120588 + 119901119905)minus versus 119903

Figure 4 Plots of (a) 120588 + 119901119905for 119904+and (b) 120588 + 119901

119905for 119904minusversus 119903 for in Gaussian distribution 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

119899 = 6 120598 = 3 (purple) and 119899 = 6 120598 = minus3 (purple dashed)

n = 5 120598 = minus1

minus2

minus1

0

1

2

05 10 15 20 25 3000r

0985

0990

0995

1000

005 010 015 020000

r

1minuss +r

1minuss +r

(a)

05 10 15 20 25 3000r

minus4

minus2

0

2

4

1minuss minusr

(b)

Figure 5 Plots of (a) 1minus 119904+119903 and (b) 1minus 119904

minus119903 versus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed) and 119899 = 6

120598 = 3 (purple)

Advances in High Energy Physics 7

04 05 06 07 08 09 10050100150200

r

05 10 15 20

0

05 10 15 20r

n = 6 120598 = minus3

n = 6 120598 = 3

minus1000minus2000minus3000minus4000minus5000minus6000

001

002

120588

120588

120588

003

004

005

(a) Evolution of 120588+versus 119903

05 10 15 20000

001

002

003

004

005

r

120588

(b) Evolution of 120588minusversus 119903

Figure 6 Plots of (a) 120588+and (b) 120588

minusversus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed) and 119899 = 6 120598 = 3

(purple) For (a) 120588+ 119899 = 6 and 120598 = minus3 (purple dashed)

05 10 15 20 25 30

0

5

10

15

r

minus10

minus5

pr+120588

(a) Evolution of (120588 + 119901119903)+versus 119903

02 04 06 08 101020304050

02 04 06 08 10

0

n = 5 120598 = 1

r

rminus8000

minus6000

minus4000

minus2000

pr+120588

pr+120588

(b) Evolution of (120588 + 119901119903)minusversus 119903

Figure 7 Plots of (a) 120588 + 119901119903for 119904+and (b) 120588 + 119901

119903for 119904minusversus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

and 119899 = 6 120598 = 3 (purple)

120598 gt 0 and negative behavior for the remaining two casesas shown in Figure 7(b) In Figures 8(a) and 8(b) we draw120588 + 119901119905for both solutions versus 119903 This expression represents

the negative behavior for (119899 = 5 120598 = 1) (119899 = 5 120598 = minus1) with119904+and (119899 = 5 120598 = minus1) with 119904

minusand preserves positivity for

(119899 = 6 120598 = 3) with 119904+ (119899 = 5 120598 = 1) (119899 = 6 120598 = 3) with 119904

minus

solution

4 Equilibrium Condition

In order to find the equilibrium configuration of thewormhole solutions in Gaussian as well as Lorentzian dis-tributed noncommutative backgrounds we use the general-ized Tolman-Oppenheimer-Volkoff equation This equationis derived by solving the Einstein equations for a general time-invariant spherically symmetric metric having metric tensor119892120572120573

= (119890120591(119903)

minus119890120596(119903)

minus1199032

minus1199032sin2120579) where 120572120573 represents

only diagonal entries and 120591 120596 are general metric functionsdependent on 119903 The generalized Tolman-Oppenheimer-Volkoff equation is

119889119901119903

119889119903

+

1205911015840

2

(120588 + 119901119903) +

2

119903

(119901119903minus 119901119905) = 0 (17)

Keeping in mind the above equation de Leon [28] proposedan equation for anisotropicmass distributionwhich naturallygives the equilibrium for the wormhole subject It is given by

2

119903

(119901119905minus 119901119903) minus

119890(120596minus120591)2

119898eff1199032

(120588 + 119901119903) minus

119889119901119903

119889119903

= 0 (18)

where effective gravitational mass 119898eff = (12)1199032

119890(120591minus120596)2

1205911015840 is

measured from throat to some arbitrary radius 119903 Accord-ingly the gravitational hydrostatic and anisotropic force dueto anisotropic matter distribution are defied as follows

119891119892= minus

1205911015840

(120588 + 119901119903)

2

119891ℎ= minus

119889119901119903

119889119903

119891119886=

2 (119901119905minus 119901119903)

119903

(19)

It is required that119891119892+119891ℎ+119891119886= 0must hold for the wormhole

solutions to be in equilibrium In the underlying cases we

8 Advances in High Energy Physics

05 10 15 20 25 30

0

r

r

150 152 154 156 158 160

r

05 10 15 200

500

1000

1500

2000

pt+120588

pt+120588

pt+120588

minus0058

minus15

minus10

minus5

minus0056

minus0054

minus0052

minus0050n = 5 120598 = minus1

n = 5 120598 = 1

(a) Evolution of (120588 + 119901119905)+versus 119903

r

05 10 15 20

r

00 05 10 15 200

5

10

15

20

pt+120588 p

t+120588

minus70

minus60

minus50

minus40

minus30

minus20

minus10

n = 5 120598 = minus1

(b) Evolution of (120588 + 119901119905)minusversus 119903

Figure 8 Plots of (a) 120588 + 119901119905for 119904+and (b) 120588 + 119901

119905for 119904minusversus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

and 119899 = 6 120598 = 3 (purple)

proceedwith constant redshift function 120582 = 0which vanishesthe gravitational contribution119891

119892in the equilibrium equation

that is 120591 = 2120582 leads to 1205911015840 = 0 for constant 120582 Thus we are leftwith hydrostatic and anisotropic forces with correspondingequilibrium condition as

119891119886+ 119891ℎ= 0 (20)

Using (8) and (9) we obtain the following expressions for thehydrostatic and anisotropic forces

119891119886=

1

1199033(119899 minus 1)

119904

119903

minus 1199041015840

(119899 minus 3 + (119899 minus 5)

119904120598

1199033)

minus

119904

1199034[(119899 minus 3) (119899 minus 4) + (119899 minus 5) (119899 minus 6)

119904120598

1199033]

119891ℎ=

(119899 minus 2)

21199033

[119899 minus 3 + (119899 minus 5)

119904120598

1199033+

119904120598 (119899 minus 5)

1199033

]

sdot (1199041015840

minus

3119904

119903

)

(21)

Figure 9 represents the plots of anisotropic as well ashydrostatic forces for the wormhole solutions in Gaussiandistributed framework Figures 9(a) and 9(c) describe theequilibrium for positive root solution for 119899 = 5 120598 =

1 minus1 and 119899 = 6 120598 = 3 minus3 through opposite behaviorand hence they cancel each other in order to satisfy (20)For negative root solution 119904

minus Figures 9(b) and 9(d) show

the stable configuration of wormhole solutions for fifthdimensional case only The behaviors of both forces (whichare not in opposite manner) do not cancel each other sothere is no equilibrium configuration examined for sixthdimensional wormhole solutions In the case of Lorentziannoncommutative background Figure 10 expresses that allwormhole solutions are in equilibrium by satisfying theequilibrium condition for both dimensions

Table 1 Wormhole solutions with Gaussian distributed noncom-mutative framework for 119904

+

Expressions 119899 = 5 120598 =

1

119899 = 5 120598 =

minus1

119899 = 6 120598 =

3

119899 = 6 120598 =

minus3

1 minus 119904+119903 Positive Positive

for 119903 lt 1 Positive Positivefor 119903 lt 18

120588 Positive Positive Positive Positive120588 + 119901

119903 Positive Negative Positive Negative120588 + 119901

119905 Negative Positive Negative PositiveWEC Violates Violates Violates ViolatesEC Holds Holds Holds Holds

5 Conclusion

It is a well-known fact that the existence of wormhole solu-tions is based on violation of NEC Since the normal mattersatisfies the energy conditions this violation is associatedwith an energy-momentum tensor which provides exoticmatter a hypothetical form of matter To explore realisticmodel or physically acceptable wormhole solutions it isnecessary to find such a source which gives the violationof NEC while normal matter meets the energy conditionsHere in this paper we have explored wormhole solutionsin 119899-dimensional Einstein Gauss-Bonnet gravity with Gaus-sian and Lorentzian noncommutative backgrounds We haverestricted ourselves to fifth and sixth dimensional cases withpositive as well as negativeGauss-Bonnet coefficient Also wehave checked the condition of equilibrium for the wormholesolutions

The results are summarized in Tables 1 and 2 It is notedthat the 119904

+(Tables 1 and 3) and 119904

minus(Tables 2 and 4) represent

the two roots of the solutions in both backgrounds and ECdenotes the equilibrium condition

In the Lorentzan distributed noncommutative frame-work we have found negative energy density for 119904

+while

Advances in High Energy Physics 9

02 04 06 1008

0

2

4

r

fa

minus2

minus4

(a) Evolution of 119891119886versus 119903

r

02 04 06 08 10

0

2

4

fa

minus2

minus4

(b) Evolution of 119891119886versus 119903

02 04 06 08 10

0

2

4

r

fh

minus2

minus4

(c) Evolution of 119891ℎversus 119903

02 04 06 08 10

0

2

4

r

fh

minus2

minus4

(d) Evolution of 119891ℎversus 119903

Figure 9 Plots of (a) 119891119886for 119904+ (b) 119891

119886for 119904minus (c) 119891

ℎfor 119904+ (d) 119891

ℎfor 119904minusversus 119903 with Gaussian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1

(red dashed) and 119899 = 6 120598 = 3 (purple)

Table 2 Wormhole solutions with Gaussian distributed noncom-mutative framework for 119904

minus

Expressions 119899 = 5 120598 =

1

119899 = 5 120598 =

minus1

119899 = 6 120598 =

3

119899 = 6 120598 =

minus3

1 minus 119904minus119903 Positive Positive

for 119903 lt 4 Positive Positivefor 119903 lt 8

120588 Positive Positive Positive Positive120588 + 119901

119903 Positive Negative Positive Negative120588 + 119901

119905 Positive Negative Positive NegativeWEC Holds Violates Holds Violates

EC Holds Holds Does nothold

Does nothold

violation of 1 minus 119904119903 gt 0 incorporating 119904minusfor the case 119899 = 6

120598 = minus3 so we skipped this case The remaining results aresummarized in Tables 3 and 4

In the paper [20] higher dimensional asymptotically flatwormhole solutions have been explored in the frameworkof Gauss-Bonnet gravity by considering a specific choicefor a radial dependent redshift function and by imposinga particular equation of state The WEC is satisfied at thethroat by considering a negative Gauss-Bonnet coupling

Table 3 Wormhole solutions with Lorentzian noncommutativebackground for 119904

+

Expressions 119899 = 5 120598 = 1 119899 = 5 120598 = minus1 119899 = 6 120598 = 3

1 minus 119904+119903

Positive for119903 lt 19

Positive Positive for119903 lt 21

120588 Positive Positive Positive

120588 + 119901119903

Positive for119903 lt 1

Positive for119903 gt 1

Positive for119903 lt 1

120588 + 119901119905 Negative Negative Positive

WEC Does nothold

Does nothold

Holds for119903 lt 1

EC Holds Holds Holds

constant Furthermore they have considered a constantredshift function and shown specifically that for negativeGauss-Bonnet coupling constant one may have normalmatter in a determined radial region and that the increaseof coupling constant enlarges the normal matter regionIn the present paper we have taken energy density undernoncommutative geometry distributions instead of particularequation of stateWe have obtained results for positiveGauss-Bonnet coefficient satisfying energy conditions It contains

10 Advances in High Energy Physics

02 04 06 08 10

0

200

400

r

fa

minus400

minus200

(a) Evolution of 119891119886versus 119903

02 04 06 08 10

0

200

400

r

fa

minus400

minus200

(b) Evolution of 119891119886versus 119903

02 04 06 08 10r

minus400

minus200

0

200

400

fh

(c) Evolution of 119891ℎversus 119903

02 04 06 08 10

0

200

400

r

fh

minus400

minus200

(d) Evolution of 119891ℎversus 119903

Figure 10 Plots of (a) 119891119886for 119904+ (b) 119891

119886for 119904minus (c) 119891

ℎfor 119904+ (d) 119891

ℎfor 119904minusversus 119903 with Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5

120598 = minus1 (red dashed) and 119899 = 6 120598 = 3 (purple)

Table 4 Wormhole solutions with Lorentzian noncommutativebackground for 119904

minus

Expressions 119899 = 5 120598 = 1 119899 = 5 120598 = minus1 119899 = 6 120598 = 3

1 minus 119904minus119903 Positive Positive for

119903 lt 05

Positive

120588 Positive Positive Positive120588 + 119901

119903 Positive Negative Negative120588 + 119901

119905 Positive Negative Positive

WEC Holds Does nothold

Does nothold

EC Holds Holds Holds

fifth dimensional wormhole solutions in both backgroundssatisfying equilibrium condition and sixth dimensional withdisequilibrium in noncommutative background with 119904

minussolu-

tion Also there is possibility for the existence of wormholein equilibrium satisfying WEC for 119899 = 6 120598 = 3 taking intoaccount 119904

+solution for the range 119903 lt 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M S Morris and K S Thorne ldquoWormholes in spacetimeand their use for interstellar travel a tool for teaching generalrelativityrdquo American Journal of Physics vol 56 no 5 pp 395ndash412 1988

[2] M Sharif and A Jawad ldquoPhantom-like generalized cosmicchaplygin gas and traversable wormhole solutionsrdquo The Euro-pean Physical Journal Plus vol 129 article 15 2014

[3] F S N Lobo F Parsaei and N Riazi ldquoNew asymptotically flatphantom wormhole solutionsrdquo Physical Review D vol 87 no 8Article ID 084030 2013

[4] S-W Kim andH Lee ldquoExact solutions of a charged wormholerdquoPhysical Review D vol 63 no 6 Article ID 064014 5 pages2001

[5] M Jamil and M U Farooq ldquoPhantom wormholes in (2 + 1)-dimensionsrdquo International Journal ofTheoretical Physics vol 49no 4 pp 835ndash841 2010

[6] M U Farooq M Akbar and M Jamil ldquoDynamics and ther-modynamics of (2+1)minusdimensional evolving lorentzian worm-holerdquo in Proceedings of the 3rd AlgerianWorkshop on Astronomyand Astrophysics vol 1295 of AIP Conference Proceedings pp176ndash190 Constantine Algeria June 2010

[7] P K Kuhfittig ldquoGravitational lensing of wormholes in thegalactic halo regionrdquo The European Physical Journal C vol 74no 3 article 2818 2014

Advances in High Energy Physics 11

[8] F Rahaman A Banerjee M Jamil A K Yadav and H IdrisldquoNoncommutative wormholes in f(R) gravity with lorentziandistributionrdquo International Journal ofTheoretical Physics vol 53no 6 pp 1910ndash1919 2014

[9] M Sharif and S Rani ldquoDynamical wormhole solutions in f(T)gravityrdquo General Relativity and Gravitation vol 45 no 11 pp2389ndash2402 2013

[10] M Sharif and S Rani ldquoCharged noncommutative wormholesolutions in f (T) gravityrdquo The European Physical Journal Plusvol 129 no 10 article 237 2014

[11] M Sharif and S Rani ldquoGalactic halo wormhole solutions in119891(119879) gravityrdquoAdvances in High Energy Physics vol 2014 ArticleID 691497 9 pages 2014

[12] F S N Lobo andM A Oliveira ldquoWormhole geometries in f (R)modified theories of gravityrdquo Physical Review D vol 80 ArticleID 104012 2009

[13] M Sharif and S Rani ldquoWormhole solutions in f(T) gravity withnoncommutativerdquo Physical Review D vol 88 no 12 Article ID123501 2013

[14] F Rahaman S Islam P K F Kuhfittig and S Ray ldquoSearch-ing for higher-dimensional wormholes with noncommutativegeometryrdquo Physical Review D vol 86 no 10 Article ID 1060102012

[15] P Bhar and F Rahaman ldquoSearch of wormholes in differ-ent dimensional non-commutative inspired space-times withLorentzian distributionrdquo The European Physical Journal C vol74 no 12 article 3213 2014

[16] F Rahaman S Karmakar I Karar and S Ray ldquoWormholeinspired by non-commutative geometryrdquo Physics Letters B vol746 pp 73ndash78 2015

[17] M Jamil F Rahaman R Myrzakulov P Kuhfittig N Ahmedand U Mondal ldquoNonommutative wormholes in f(R) gravityrdquoJournal of the Korean Physical Society vol 65 no 6 pp 917ndash9252014

[18] A Jawad and S Rani ldquoLorentz distributed noncommutativewormhole solutions in extended teleparallel gravityrdquoThe Euro-pean Physical Journal C vol 75 article 173 2015

[19] B Bhawal and S Kar ldquoLorentzian wormholes in Einstein-Gauss-Bonnet theoryrdquo Physical Review D vol 46 no 6 pp2464ndash2468 1992

[20] M R Mehdizadeh M K Zangeneh and F S N LoboldquoEinstein-Gauss-Bonnet traversable wormholes satisfying theweak energy conditionrdquo Physical ReviewD vol 91 no 8 ArticleID 084004 2015

[21] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[22] S H Mehdipour ldquoEntropic force approach to noncommutativeSchwarzschild black holes signals a failure of current physicalideasrdquo The European Physical Journal Plus vol 127 article 802012

[23] S Sushkov ldquoWormholes supported by a phantom energyrdquoPhysical Review D vol 71 no 4 Article ID 043520 2005

[24] P Nicolini and E Spallucci ldquoNoncommutative geometry-inspired dirty black holesrdquo Classical and Quantum Gravity vol27 no 1 Article ID 015010 2010

[25] F Rahaman P K Kuhfittig K Chakraborty A A Usmani andS Ray ldquoGalactic rotation curves inspired by a noncommutative-geometry backgroundrdquo General Relativity and Gravitation vol44 no 4 pp 905ndash916 2012

[26] P K F Kuhfittig ldquoOn the stability of thin-shell wormholes innoncommutative geometryrdquo Advances in High Energy Physicsvol 2012 Article ID 462493 12 pages 2012

[27] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquo Journal of High Energy Physics vol 1999 no 9 article032 1999

[28] J P de Leon ldquoMass and charge in brane-world and non-compact Kaluza-KLEin theories in 5 dimrdquo General Relativityand Gravitation vol 35 no 8 pp 1365ndash1384 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

6 Advances in High Energy Physics

pr+120588

minus06

minus04

minus02

00

02

04

06

0402 06 08 10 12 1400r

(a) Evolution of (120588 + 119901119903)+ versus 119903

pr+120588

minus3

minus2

minus1

0

1

2

3

04 06 08 10 12 1402r

(b) Evolution of (120588 + 119901119903)minus versus 119903

Figure 3 Plots of (a) 120588 + 119901119903for 119904+and (b) 120588 + 119901

119903for 119904minusversus 119903 in Gaussian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

119899 = 6 120598 = 3 (purple) and 119899 = 6 120598 = minus3 (purple dashed)

pt+120588

pt+120588

n = 5 120598 = minus1

minus04

minus02

00

02

04

05 10 15 20 25 3000r

000001500000200000250000030000035

17 18 19 2016r

(a) Evolution of (120588 + 119901119905)+ versus 119903

pt+120588

minus10

minus5

0

5

10

05 10 15 20 25 3000r

(b) Evolution of (120588 + 119901119905)minus versus 119903

Figure 4 Plots of (a) 120588 + 119901119905for 119904+and (b) 120588 + 119901

119905for 119904minusversus 119903 for in Gaussian distribution 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

119899 = 6 120598 = 3 (purple) and 119899 = 6 120598 = minus3 (purple dashed)

n = 5 120598 = minus1

minus2

minus1

0

1

2

05 10 15 20 25 3000r

0985

0990

0995

1000

005 010 015 020000

r

1minuss +r

1minuss +r

(a)

05 10 15 20 25 3000r

minus4

minus2

0

2

4

1minuss minusr

(b)

Figure 5 Plots of (a) 1minus 119904+119903 and (b) 1minus 119904

minus119903 versus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed) and 119899 = 6

120598 = 3 (purple)

Advances in High Energy Physics 7

04 05 06 07 08 09 10050100150200

r

05 10 15 20

0

05 10 15 20r

n = 6 120598 = minus3

n = 6 120598 = 3

minus1000minus2000minus3000minus4000minus5000minus6000

001

002

120588

120588

120588

003

004

005

(a) Evolution of 120588+versus 119903

05 10 15 20000

001

002

003

004

005

r

120588

(b) Evolution of 120588minusversus 119903

Figure 6 Plots of (a) 120588+and (b) 120588

minusversus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed) and 119899 = 6 120598 = 3

(purple) For (a) 120588+ 119899 = 6 and 120598 = minus3 (purple dashed)

05 10 15 20 25 30

0

5

10

15

r

minus10

minus5

pr+120588

(a) Evolution of (120588 + 119901119903)+versus 119903

02 04 06 08 101020304050

02 04 06 08 10

0

n = 5 120598 = 1

r

rminus8000

minus6000

minus4000

minus2000

pr+120588

pr+120588

(b) Evolution of (120588 + 119901119903)minusversus 119903

Figure 7 Plots of (a) 120588 + 119901119903for 119904+and (b) 120588 + 119901

119903for 119904minusversus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

and 119899 = 6 120598 = 3 (purple)

120598 gt 0 and negative behavior for the remaining two casesas shown in Figure 7(b) In Figures 8(a) and 8(b) we draw120588 + 119901119905for both solutions versus 119903 This expression represents

the negative behavior for (119899 = 5 120598 = 1) (119899 = 5 120598 = minus1) with119904+and (119899 = 5 120598 = minus1) with 119904

minusand preserves positivity for

(119899 = 6 120598 = 3) with 119904+ (119899 = 5 120598 = 1) (119899 = 6 120598 = 3) with 119904

minus

solution

4 Equilibrium Condition

In order to find the equilibrium configuration of thewormhole solutions in Gaussian as well as Lorentzian dis-tributed noncommutative backgrounds we use the general-ized Tolman-Oppenheimer-Volkoff equation This equationis derived by solving the Einstein equations for a general time-invariant spherically symmetric metric having metric tensor119892120572120573

= (119890120591(119903)

minus119890120596(119903)

minus1199032

minus1199032sin2120579) where 120572120573 represents

only diagonal entries and 120591 120596 are general metric functionsdependent on 119903 The generalized Tolman-Oppenheimer-Volkoff equation is

119889119901119903

119889119903

+

1205911015840

2

(120588 + 119901119903) +

2

119903

(119901119903minus 119901119905) = 0 (17)

Keeping in mind the above equation de Leon [28] proposedan equation for anisotropicmass distributionwhich naturallygives the equilibrium for the wormhole subject It is given by

2

119903

(119901119905minus 119901119903) minus

119890(120596minus120591)2

119898eff1199032

(120588 + 119901119903) minus

119889119901119903

119889119903

= 0 (18)

where effective gravitational mass 119898eff = (12)1199032

119890(120591minus120596)2

1205911015840 is

measured from throat to some arbitrary radius 119903 Accord-ingly the gravitational hydrostatic and anisotropic force dueto anisotropic matter distribution are defied as follows

119891119892= minus

1205911015840

(120588 + 119901119903)

2

119891ℎ= minus

119889119901119903

119889119903

119891119886=

2 (119901119905minus 119901119903)

119903

(19)

It is required that119891119892+119891ℎ+119891119886= 0must hold for the wormhole

solutions to be in equilibrium In the underlying cases we

8 Advances in High Energy Physics

05 10 15 20 25 30

0

r

r

150 152 154 156 158 160

r

05 10 15 200

500

1000

1500

2000

pt+120588

pt+120588

pt+120588

minus0058

minus15

minus10

minus5

minus0056

minus0054

minus0052

minus0050n = 5 120598 = minus1

n = 5 120598 = 1

(a) Evolution of (120588 + 119901119905)+versus 119903

r

05 10 15 20

r

00 05 10 15 200

5

10

15

20

pt+120588 p

t+120588

minus70

minus60

minus50

minus40

minus30

minus20

minus10

n = 5 120598 = minus1

(b) Evolution of (120588 + 119901119905)minusversus 119903

Figure 8 Plots of (a) 120588 + 119901119905for 119904+and (b) 120588 + 119901

119905for 119904minusversus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

and 119899 = 6 120598 = 3 (purple)

proceedwith constant redshift function 120582 = 0which vanishesthe gravitational contribution119891

119892in the equilibrium equation

that is 120591 = 2120582 leads to 1205911015840 = 0 for constant 120582 Thus we are leftwith hydrostatic and anisotropic forces with correspondingequilibrium condition as

119891119886+ 119891ℎ= 0 (20)

Using (8) and (9) we obtain the following expressions for thehydrostatic and anisotropic forces

119891119886=

1

1199033(119899 minus 1)

119904

119903

minus 1199041015840

(119899 minus 3 + (119899 minus 5)

119904120598

1199033)

minus

119904

1199034[(119899 minus 3) (119899 minus 4) + (119899 minus 5) (119899 minus 6)

119904120598

1199033]

119891ℎ=

(119899 minus 2)

21199033

[119899 minus 3 + (119899 minus 5)

119904120598

1199033+

119904120598 (119899 minus 5)

1199033

]

sdot (1199041015840

minus

3119904

119903

)

(21)

Figure 9 represents the plots of anisotropic as well ashydrostatic forces for the wormhole solutions in Gaussiandistributed framework Figures 9(a) and 9(c) describe theequilibrium for positive root solution for 119899 = 5 120598 =

1 minus1 and 119899 = 6 120598 = 3 minus3 through opposite behaviorand hence they cancel each other in order to satisfy (20)For negative root solution 119904

minus Figures 9(b) and 9(d) show

the stable configuration of wormhole solutions for fifthdimensional case only The behaviors of both forces (whichare not in opposite manner) do not cancel each other sothere is no equilibrium configuration examined for sixthdimensional wormhole solutions In the case of Lorentziannoncommutative background Figure 10 expresses that allwormhole solutions are in equilibrium by satisfying theequilibrium condition for both dimensions

Table 1 Wormhole solutions with Gaussian distributed noncom-mutative framework for 119904

+

Expressions 119899 = 5 120598 =

1

119899 = 5 120598 =

minus1

119899 = 6 120598 =

3

119899 = 6 120598 =

minus3

1 minus 119904+119903 Positive Positive

for 119903 lt 1 Positive Positivefor 119903 lt 18

120588 Positive Positive Positive Positive120588 + 119901

119903 Positive Negative Positive Negative120588 + 119901

119905 Negative Positive Negative PositiveWEC Violates Violates Violates ViolatesEC Holds Holds Holds Holds

5 Conclusion

It is a well-known fact that the existence of wormhole solu-tions is based on violation of NEC Since the normal mattersatisfies the energy conditions this violation is associatedwith an energy-momentum tensor which provides exoticmatter a hypothetical form of matter To explore realisticmodel or physically acceptable wormhole solutions it isnecessary to find such a source which gives the violationof NEC while normal matter meets the energy conditionsHere in this paper we have explored wormhole solutionsin 119899-dimensional Einstein Gauss-Bonnet gravity with Gaus-sian and Lorentzian noncommutative backgrounds We haverestricted ourselves to fifth and sixth dimensional cases withpositive as well as negativeGauss-Bonnet coefficient Also wehave checked the condition of equilibrium for the wormholesolutions

The results are summarized in Tables 1 and 2 It is notedthat the 119904

+(Tables 1 and 3) and 119904

minus(Tables 2 and 4) represent

the two roots of the solutions in both backgrounds and ECdenotes the equilibrium condition

In the Lorentzan distributed noncommutative frame-work we have found negative energy density for 119904

+while

Advances in High Energy Physics 9

02 04 06 1008

0

2

4

r

fa

minus2

minus4

(a) Evolution of 119891119886versus 119903

r

02 04 06 08 10

0

2

4

fa

minus2

minus4

(b) Evolution of 119891119886versus 119903

02 04 06 08 10

0

2

4

r

fh

minus2

minus4

(c) Evolution of 119891ℎversus 119903

02 04 06 08 10

0

2

4

r

fh

minus2

minus4

(d) Evolution of 119891ℎversus 119903

Figure 9 Plots of (a) 119891119886for 119904+ (b) 119891

119886for 119904minus (c) 119891

ℎfor 119904+ (d) 119891

ℎfor 119904minusversus 119903 with Gaussian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1

(red dashed) and 119899 = 6 120598 = 3 (purple)

Table 2 Wormhole solutions with Gaussian distributed noncom-mutative framework for 119904

minus

Expressions 119899 = 5 120598 =

1

119899 = 5 120598 =

minus1

119899 = 6 120598 =

3

119899 = 6 120598 =

minus3

1 minus 119904minus119903 Positive Positive

for 119903 lt 4 Positive Positivefor 119903 lt 8

120588 Positive Positive Positive Positive120588 + 119901

119903 Positive Negative Positive Negative120588 + 119901

119905 Positive Negative Positive NegativeWEC Holds Violates Holds Violates

EC Holds Holds Does nothold

Does nothold

violation of 1 minus 119904119903 gt 0 incorporating 119904minusfor the case 119899 = 6

120598 = minus3 so we skipped this case The remaining results aresummarized in Tables 3 and 4

In the paper [20] higher dimensional asymptotically flatwormhole solutions have been explored in the frameworkof Gauss-Bonnet gravity by considering a specific choicefor a radial dependent redshift function and by imposinga particular equation of state The WEC is satisfied at thethroat by considering a negative Gauss-Bonnet coupling

Table 3 Wormhole solutions with Lorentzian noncommutativebackground for 119904

+

Expressions 119899 = 5 120598 = 1 119899 = 5 120598 = minus1 119899 = 6 120598 = 3

1 minus 119904+119903

Positive for119903 lt 19

Positive Positive for119903 lt 21

120588 Positive Positive Positive

120588 + 119901119903

Positive for119903 lt 1

Positive for119903 gt 1

Positive for119903 lt 1

120588 + 119901119905 Negative Negative Positive

WEC Does nothold

Does nothold

Holds for119903 lt 1

EC Holds Holds Holds

constant Furthermore they have considered a constantredshift function and shown specifically that for negativeGauss-Bonnet coupling constant one may have normalmatter in a determined radial region and that the increaseof coupling constant enlarges the normal matter regionIn the present paper we have taken energy density undernoncommutative geometry distributions instead of particularequation of stateWe have obtained results for positiveGauss-Bonnet coefficient satisfying energy conditions It contains

10 Advances in High Energy Physics

02 04 06 08 10

0

200

400

r

fa

minus400

minus200

(a) Evolution of 119891119886versus 119903

02 04 06 08 10

0

200

400

r

fa

minus400

minus200

(b) Evolution of 119891119886versus 119903

02 04 06 08 10r

minus400

minus200

0

200

400

fh

(c) Evolution of 119891ℎversus 119903

02 04 06 08 10

0

200

400

r

fh

minus400

minus200

(d) Evolution of 119891ℎversus 119903

Figure 10 Plots of (a) 119891119886for 119904+ (b) 119891

119886for 119904minus (c) 119891

ℎfor 119904+ (d) 119891

ℎfor 119904minusversus 119903 with Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5

120598 = minus1 (red dashed) and 119899 = 6 120598 = 3 (purple)

Table 4 Wormhole solutions with Lorentzian noncommutativebackground for 119904

minus

Expressions 119899 = 5 120598 = 1 119899 = 5 120598 = minus1 119899 = 6 120598 = 3

1 minus 119904minus119903 Positive Positive for

119903 lt 05

Positive

120588 Positive Positive Positive120588 + 119901

119903 Positive Negative Negative120588 + 119901

119905 Positive Negative Positive

WEC Holds Does nothold

Does nothold

EC Holds Holds Holds

fifth dimensional wormhole solutions in both backgroundssatisfying equilibrium condition and sixth dimensional withdisequilibrium in noncommutative background with 119904

minussolu-

tion Also there is possibility for the existence of wormholein equilibrium satisfying WEC for 119899 = 6 120598 = 3 taking intoaccount 119904

+solution for the range 119903 lt 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M S Morris and K S Thorne ldquoWormholes in spacetimeand their use for interstellar travel a tool for teaching generalrelativityrdquo American Journal of Physics vol 56 no 5 pp 395ndash412 1988

[2] M Sharif and A Jawad ldquoPhantom-like generalized cosmicchaplygin gas and traversable wormhole solutionsrdquo The Euro-pean Physical Journal Plus vol 129 article 15 2014

[3] F S N Lobo F Parsaei and N Riazi ldquoNew asymptotically flatphantom wormhole solutionsrdquo Physical Review D vol 87 no 8Article ID 084030 2013

[4] S-W Kim andH Lee ldquoExact solutions of a charged wormholerdquoPhysical Review D vol 63 no 6 Article ID 064014 5 pages2001

[5] M Jamil and M U Farooq ldquoPhantom wormholes in (2 + 1)-dimensionsrdquo International Journal ofTheoretical Physics vol 49no 4 pp 835ndash841 2010

[6] M U Farooq M Akbar and M Jamil ldquoDynamics and ther-modynamics of (2+1)minusdimensional evolving lorentzian worm-holerdquo in Proceedings of the 3rd AlgerianWorkshop on Astronomyand Astrophysics vol 1295 of AIP Conference Proceedings pp176ndash190 Constantine Algeria June 2010

[7] P K Kuhfittig ldquoGravitational lensing of wormholes in thegalactic halo regionrdquo The European Physical Journal C vol 74no 3 article 2818 2014

Advances in High Energy Physics 11

[8] F Rahaman A Banerjee M Jamil A K Yadav and H IdrisldquoNoncommutative wormholes in f(R) gravity with lorentziandistributionrdquo International Journal ofTheoretical Physics vol 53no 6 pp 1910ndash1919 2014

[9] M Sharif and S Rani ldquoDynamical wormhole solutions in f(T)gravityrdquo General Relativity and Gravitation vol 45 no 11 pp2389ndash2402 2013

[10] M Sharif and S Rani ldquoCharged noncommutative wormholesolutions in f (T) gravityrdquo The European Physical Journal Plusvol 129 no 10 article 237 2014

[11] M Sharif and S Rani ldquoGalactic halo wormhole solutions in119891(119879) gravityrdquoAdvances in High Energy Physics vol 2014 ArticleID 691497 9 pages 2014

[12] F S N Lobo andM A Oliveira ldquoWormhole geometries in f (R)modified theories of gravityrdquo Physical Review D vol 80 ArticleID 104012 2009

[13] M Sharif and S Rani ldquoWormhole solutions in f(T) gravity withnoncommutativerdquo Physical Review D vol 88 no 12 Article ID123501 2013

[14] F Rahaman S Islam P K F Kuhfittig and S Ray ldquoSearch-ing for higher-dimensional wormholes with noncommutativegeometryrdquo Physical Review D vol 86 no 10 Article ID 1060102012

[15] P Bhar and F Rahaman ldquoSearch of wormholes in differ-ent dimensional non-commutative inspired space-times withLorentzian distributionrdquo The European Physical Journal C vol74 no 12 article 3213 2014

[16] F Rahaman S Karmakar I Karar and S Ray ldquoWormholeinspired by non-commutative geometryrdquo Physics Letters B vol746 pp 73ndash78 2015

[17] M Jamil F Rahaman R Myrzakulov P Kuhfittig N Ahmedand U Mondal ldquoNonommutative wormholes in f(R) gravityrdquoJournal of the Korean Physical Society vol 65 no 6 pp 917ndash9252014

[18] A Jawad and S Rani ldquoLorentz distributed noncommutativewormhole solutions in extended teleparallel gravityrdquoThe Euro-pean Physical Journal C vol 75 article 173 2015

[19] B Bhawal and S Kar ldquoLorentzian wormholes in Einstein-Gauss-Bonnet theoryrdquo Physical Review D vol 46 no 6 pp2464ndash2468 1992

[20] M R Mehdizadeh M K Zangeneh and F S N LoboldquoEinstein-Gauss-Bonnet traversable wormholes satisfying theweak energy conditionrdquo Physical ReviewD vol 91 no 8 ArticleID 084004 2015

[21] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[22] S H Mehdipour ldquoEntropic force approach to noncommutativeSchwarzschild black holes signals a failure of current physicalideasrdquo The European Physical Journal Plus vol 127 article 802012

[23] S Sushkov ldquoWormholes supported by a phantom energyrdquoPhysical Review D vol 71 no 4 Article ID 043520 2005

[24] P Nicolini and E Spallucci ldquoNoncommutative geometry-inspired dirty black holesrdquo Classical and Quantum Gravity vol27 no 1 Article ID 015010 2010

[25] F Rahaman P K Kuhfittig K Chakraborty A A Usmani andS Ray ldquoGalactic rotation curves inspired by a noncommutative-geometry backgroundrdquo General Relativity and Gravitation vol44 no 4 pp 905ndash916 2012

[26] P K F Kuhfittig ldquoOn the stability of thin-shell wormholes innoncommutative geometryrdquo Advances in High Energy Physicsvol 2012 Article ID 462493 12 pages 2012

[27] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquo Journal of High Energy Physics vol 1999 no 9 article032 1999

[28] J P de Leon ldquoMass and charge in brane-world and non-compact Kaluza-KLEin theories in 5 dimrdquo General Relativityand Gravitation vol 35 no 8 pp 1365ndash1384 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 7

04 05 06 07 08 09 10050100150200

r

05 10 15 20

0

05 10 15 20r

n = 6 120598 = minus3

n = 6 120598 = 3

minus1000minus2000minus3000minus4000minus5000minus6000

001

002

120588

120588

120588

003

004

005

(a) Evolution of 120588+versus 119903

05 10 15 20000

001

002

003

004

005

r

120588

(b) Evolution of 120588minusversus 119903

Figure 6 Plots of (a) 120588+and (b) 120588

minusversus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed) and 119899 = 6 120598 = 3

(purple) For (a) 120588+ 119899 = 6 and 120598 = minus3 (purple dashed)

05 10 15 20 25 30

0

5

10

15

r

minus10

minus5

pr+120588

(a) Evolution of (120588 + 119901119903)+versus 119903

02 04 06 08 101020304050

02 04 06 08 10

0

n = 5 120598 = 1

r

rminus8000

minus6000

minus4000

minus2000

pr+120588

pr+120588

(b) Evolution of (120588 + 119901119903)minusversus 119903

Figure 7 Plots of (a) 120588 + 119901119903for 119904+and (b) 120588 + 119901

119903for 119904minusversus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

and 119899 = 6 120598 = 3 (purple)

120598 gt 0 and negative behavior for the remaining two casesas shown in Figure 7(b) In Figures 8(a) and 8(b) we draw120588 + 119901119905for both solutions versus 119903 This expression represents

the negative behavior for (119899 = 5 120598 = 1) (119899 = 5 120598 = minus1) with119904+and (119899 = 5 120598 = minus1) with 119904

minusand preserves positivity for

(119899 = 6 120598 = 3) with 119904+ (119899 = 5 120598 = 1) (119899 = 6 120598 = 3) with 119904

minus

solution

4 Equilibrium Condition

In order to find the equilibrium configuration of thewormhole solutions in Gaussian as well as Lorentzian dis-tributed noncommutative backgrounds we use the general-ized Tolman-Oppenheimer-Volkoff equation This equationis derived by solving the Einstein equations for a general time-invariant spherically symmetric metric having metric tensor119892120572120573

= (119890120591(119903)

minus119890120596(119903)

minus1199032

minus1199032sin2120579) where 120572120573 represents

only diagonal entries and 120591 120596 are general metric functionsdependent on 119903 The generalized Tolman-Oppenheimer-Volkoff equation is

119889119901119903

119889119903

+

1205911015840

2

(120588 + 119901119903) +

2

119903

(119901119903minus 119901119905) = 0 (17)

Keeping in mind the above equation de Leon [28] proposedan equation for anisotropicmass distributionwhich naturallygives the equilibrium for the wormhole subject It is given by

2

119903

(119901119905minus 119901119903) minus

119890(120596minus120591)2

119898eff1199032

(120588 + 119901119903) minus

119889119901119903

119889119903

= 0 (18)

where effective gravitational mass 119898eff = (12)1199032

119890(120591minus120596)2

1205911015840 is

measured from throat to some arbitrary radius 119903 Accord-ingly the gravitational hydrostatic and anisotropic force dueto anisotropic matter distribution are defied as follows

119891119892= minus

1205911015840

(120588 + 119901119903)

2

119891ℎ= minus

119889119901119903

119889119903

119891119886=

2 (119901119905minus 119901119903)

119903

(19)

It is required that119891119892+119891ℎ+119891119886= 0must hold for the wormhole

solutions to be in equilibrium In the underlying cases we

8 Advances in High Energy Physics

05 10 15 20 25 30

0

r

r

150 152 154 156 158 160

r

05 10 15 200

500

1000

1500

2000

pt+120588

pt+120588

pt+120588

minus0058

minus15

minus10

minus5

minus0056

minus0054

minus0052

minus0050n = 5 120598 = minus1

n = 5 120598 = 1

(a) Evolution of (120588 + 119901119905)+versus 119903

r

05 10 15 20

r

00 05 10 15 200

5

10

15

20

pt+120588 p

t+120588

minus70

minus60

minus50

minus40

minus30

minus20

minus10

n = 5 120598 = minus1

(b) Evolution of (120588 + 119901119905)minusversus 119903

Figure 8 Plots of (a) 120588 + 119901119905for 119904+and (b) 120588 + 119901

119905for 119904minusversus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

and 119899 = 6 120598 = 3 (purple)

proceedwith constant redshift function 120582 = 0which vanishesthe gravitational contribution119891

119892in the equilibrium equation

that is 120591 = 2120582 leads to 1205911015840 = 0 for constant 120582 Thus we are leftwith hydrostatic and anisotropic forces with correspondingequilibrium condition as

119891119886+ 119891ℎ= 0 (20)

Using (8) and (9) we obtain the following expressions for thehydrostatic and anisotropic forces

119891119886=

1

1199033(119899 minus 1)

119904

119903

minus 1199041015840

(119899 minus 3 + (119899 minus 5)

119904120598

1199033)

minus

119904

1199034[(119899 minus 3) (119899 minus 4) + (119899 minus 5) (119899 minus 6)

119904120598

1199033]

119891ℎ=

(119899 minus 2)

21199033

[119899 minus 3 + (119899 minus 5)

119904120598

1199033+

119904120598 (119899 minus 5)

1199033

]

sdot (1199041015840

minus

3119904

119903

)

(21)

Figure 9 represents the plots of anisotropic as well ashydrostatic forces for the wormhole solutions in Gaussiandistributed framework Figures 9(a) and 9(c) describe theequilibrium for positive root solution for 119899 = 5 120598 =

1 minus1 and 119899 = 6 120598 = 3 minus3 through opposite behaviorand hence they cancel each other in order to satisfy (20)For negative root solution 119904

minus Figures 9(b) and 9(d) show

the stable configuration of wormhole solutions for fifthdimensional case only The behaviors of both forces (whichare not in opposite manner) do not cancel each other sothere is no equilibrium configuration examined for sixthdimensional wormhole solutions In the case of Lorentziannoncommutative background Figure 10 expresses that allwormhole solutions are in equilibrium by satisfying theequilibrium condition for both dimensions

Table 1 Wormhole solutions with Gaussian distributed noncom-mutative framework for 119904

+

Expressions 119899 = 5 120598 =

1

119899 = 5 120598 =

minus1

119899 = 6 120598 =

3

119899 = 6 120598 =

minus3

1 minus 119904+119903 Positive Positive

for 119903 lt 1 Positive Positivefor 119903 lt 18

120588 Positive Positive Positive Positive120588 + 119901

119903 Positive Negative Positive Negative120588 + 119901

119905 Negative Positive Negative PositiveWEC Violates Violates Violates ViolatesEC Holds Holds Holds Holds

5 Conclusion

It is a well-known fact that the existence of wormhole solu-tions is based on violation of NEC Since the normal mattersatisfies the energy conditions this violation is associatedwith an energy-momentum tensor which provides exoticmatter a hypothetical form of matter To explore realisticmodel or physically acceptable wormhole solutions it isnecessary to find such a source which gives the violationof NEC while normal matter meets the energy conditionsHere in this paper we have explored wormhole solutionsin 119899-dimensional Einstein Gauss-Bonnet gravity with Gaus-sian and Lorentzian noncommutative backgrounds We haverestricted ourselves to fifth and sixth dimensional cases withpositive as well as negativeGauss-Bonnet coefficient Also wehave checked the condition of equilibrium for the wormholesolutions

The results are summarized in Tables 1 and 2 It is notedthat the 119904

+(Tables 1 and 3) and 119904

minus(Tables 2 and 4) represent

the two roots of the solutions in both backgrounds and ECdenotes the equilibrium condition

In the Lorentzan distributed noncommutative frame-work we have found negative energy density for 119904

+while

Advances in High Energy Physics 9

02 04 06 1008

0

2

4

r

fa

minus2

minus4

(a) Evolution of 119891119886versus 119903

r

02 04 06 08 10

0

2

4

fa

minus2

minus4

(b) Evolution of 119891119886versus 119903

02 04 06 08 10

0

2

4

r

fh

minus2

minus4

(c) Evolution of 119891ℎversus 119903

02 04 06 08 10

0

2

4

r

fh

minus2

minus4

(d) Evolution of 119891ℎversus 119903

Figure 9 Plots of (a) 119891119886for 119904+ (b) 119891

119886for 119904minus (c) 119891

ℎfor 119904+ (d) 119891

ℎfor 119904minusversus 119903 with Gaussian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1

(red dashed) and 119899 = 6 120598 = 3 (purple)

Table 2 Wormhole solutions with Gaussian distributed noncom-mutative framework for 119904

minus

Expressions 119899 = 5 120598 =

1

119899 = 5 120598 =

minus1

119899 = 6 120598 =

3

119899 = 6 120598 =

minus3

1 minus 119904minus119903 Positive Positive

for 119903 lt 4 Positive Positivefor 119903 lt 8

120588 Positive Positive Positive Positive120588 + 119901

119903 Positive Negative Positive Negative120588 + 119901

119905 Positive Negative Positive NegativeWEC Holds Violates Holds Violates

EC Holds Holds Does nothold

Does nothold

violation of 1 minus 119904119903 gt 0 incorporating 119904minusfor the case 119899 = 6

120598 = minus3 so we skipped this case The remaining results aresummarized in Tables 3 and 4

In the paper [20] higher dimensional asymptotically flatwormhole solutions have been explored in the frameworkof Gauss-Bonnet gravity by considering a specific choicefor a radial dependent redshift function and by imposinga particular equation of state The WEC is satisfied at thethroat by considering a negative Gauss-Bonnet coupling

Table 3 Wormhole solutions with Lorentzian noncommutativebackground for 119904

+

Expressions 119899 = 5 120598 = 1 119899 = 5 120598 = minus1 119899 = 6 120598 = 3

1 minus 119904+119903

Positive for119903 lt 19

Positive Positive for119903 lt 21

120588 Positive Positive Positive

120588 + 119901119903

Positive for119903 lt 1

Positive for119903 gt 1

Positive for119903 lt 1

120588 + 119901119905 Negative Negative Positive

WEC Does nothold

Does nothold

Holds for119903 lt 1

EC Holds Holds Holds

constant Furthermore they have considered a constantredshift function and shown specifically that for negativeGauss-Bonnet coupling constant one may have normalmatter in a determined radial region and that the increaseof coupling constant enlarges the normal matter regionIn the present paper we have taken energy density undernoncommutative geometry distributions instead of particularequation of stateWe have obtained results for positiveGauss-Bonnet coefficient satisfying energy conditions It contains

10 Advances in High Energy Physics

02 04 06 08 10

0

200

400

r

fa

minus400

minus200

(a) Evolution of 119891119886versus 119903

02 04 06 08 10

0

200

400

r

fa

minus400

minus200

(b) Evolution of 119891119886versus 119903

02 04 06 08 10r

minus400

minus200

0

200

400

fh

(c) Evolution of 119891ℎversus 119903

02 04 06 08 10

0

200

400

r

fh

minus400

minus200

(d) Evolution of 119891ℎversus 119903

Figure 10 Plots of (a) 119891119886for 119904+ (b) 119891

119886for 119904minus (c) 119891

ℎfor 119904+ (d) 119891

ℎfor 119904minusversus 119903 with Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5

120598 = minus1 (red dashed) and 119899 = 6 120598 = 3 (purple)

Table 4 Wormhole solutions with Lorentzian noncommutativebackground for 119904

minus

Expressions 119899 = 5 120598 = 1 119899 = 5 120598 = minus1 119899 = 6 120598 = 3

1 minus 119904minus119903 Positive Positive for

119903 lt 05

Positive

120588 Positive Positive Positive120588 + 119901

119903 Positive Negative Negative120588 + 119901

119905 Positive Negative Positive

WEC Holds Does nothold

Does nothold

EC Holds Holds Holds

fifth dimensional wormhole solutions in both backgroundssatisfying equilibrium condition and sixth dimensional withdisequilibrium in noncommutative background with 119904

minussolu-

tion Also there is possibility for the existence of wormholein equilibrium satisfying WEC for 119899 = 6 120598 = 3 taking intoaccount 119904

+solution for the range 119903 lt 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M S Morris and K S Thorne ldquoWormholes in spacetimeand their use for interstellar travel a tool for teaching generalrelativityrdquo American Journal of Physics vol 56 no 5 pp 395ndash412 1988

[2] M Sharif and A Jawad ldquoPhantom-like generalized cosmicchaplygin gas and traversable wormhole solutionsrdquo The Euro-pean Physical Journal Plus vol 129 article 15 2014

[3] F S N Lobo F Parsaei and N Riazi ldquoNew asymptotically flatphantom wormhole solutionsrdquo Physical Review D vol 87 no 8Article ID 084030 2013

[4] S-W Kim andH Lee ldquoExact solutions of a charged wormholerdquoPhysical Review D vol 63 no 6 Article ID 064014 5 pages2001

[5] M Jamil and M U Farooq ldquoPhantom wormholes in (2 + 1)-dimensionsrdquo International Journal ofTheoretical Physics vol 49no 4 pp 835ndash841 2010

[6] M U Farooq M Akbar and M Jamil ldquoDynamics and ther-modynamics of (2+1)minusdimensional evolving lorentzian worm-holerdquo in Proceedings of the 3rd AlgerianWorkshop on Astronomyand Astrophysics vol 1295 of AIP Conference Proceedings pp176ndash190 Constantine Algeria June 2010

[7] P K Kuhfittig ldquoGravitational lensing of wormholes in thegalactic halo regionrdquo The European Physical Journal C vol 74no 3 article 2818 2014

Advances in High Energy Physics 11

[8] F Rahaman A Banerjee M Jamil A K Yadav and H IdrisldquoNoncommutative wormholes in f(R) gravity with lorentziandistributionrdquo International Journal ofTheoretical Physics vol 53no 6 pp 1910ndash1919 2014

[9] M Sharif and S Rani ldquoDynamical wormhole solutions in f(T)gravityrdquo General Relativity and Gravitation vol 45 no 11 pp2389ndash2402 2013

[10] M Sharif and S Rani ldquoCharged noncommutative wormholesolutions in f (T) gravityrdquo The European Physical Journal Plusvol 129 no 10 article 237 2014

[11] M Sharif and S Rani ldquoGalactic halo wormhole solutions in119891(119879) gravityrdquoAdvances in High Energy Physics vol 2014 ArticleID 691497 9 pages 2014

[12] F S N Lobo andM A Oliveira ldquoWormhole geometries in f (R)modified theories of gravityrdquo Physical Review D vol 80 ArticleID 104012 2009

[13] M Sharif and S Rani ldquoWormhole solutions in f(T) gravity withnoncommutativerdquo Physical Review D vol 88 no 12 Article ID123501 2013

[14] F Rahaman S Islam P K F Kuhfittig and S Ray ldquoSearch-ing for higher-dimensional wormholes with noncommutativegeometryrdquo Physical Review D vol 86 no 10 Article ID 1060102012

[15] P Bhar and F Rahaman ldquoSearch of wormholes in differ-ent dimensional non-commutative inspired space-times withLorentzian distributionrdquo The European Physical Journal C vol74 no 12 article 3213 2014

[16] F Rahaman S Karmakar I Karar and S Ray ldquoWormholeinspired by non-commutative geometryrdquo Physics Letters B vol746 pp 73ndash78 2015

[17] M Jamil F Rahaman R Myrzakulov P Kuhfittig N Ahmedand U Mondal ldquoNonommutative wormholes in f(R) gravityrdquoJournal of the Korean Physical Society vol 65 no 6 pp 917ndash9252014

[18] A Jawad and S Rani ldquoLorentz distributed noncommutativewormhole solutions in extended teleparallel gravityrdquoThe Euro-pean Physical Journal C vol 75 article 173 2015

[19] B Bhawal and S Kar ldquoLorentzian wormholes in Einstein-Gauss-Bonnet theoryrdquo Physical Review D vol 46 no 6 pp2464ndash2468 1992

[20] M R Mehdizadeh M K Zangeneh and F S N LoboldquoEinstein-Gauss-Bonnet traversable wormholes satisfying theweak energy conditionrdquo Physical ReviewD vol 91 no 8 ArticleID 084004 2015

[21] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[22] S H Mehdipour ldquoEntropic force approach to noncommutativeSchwarzschild black holes signals a failure of current physicalideasrdquo The European Physical Journal Plus vol 127 article 802012

[23] S Sushkov ldquoWormholes supported by a phantom energyrdquoPhysical Review D vol 71 no 4 Article ID 043520 2005

[24] P Nicolini and E Spallucci ldquoNoncommutative geometry-inspired dirty black holesrdquo Classical and Quantum Gravity vol27 no 1 Article ID 015010 2010

[25] F Rahaman P K Kuhfittig K Chakraborty A A Usmani andS Ray ldquoGalactic rotation curves inspired by a noncommutative-geometry backgroundrdquo General Relativity and Gravitation vol44 no 4 pp 905ndash916 2012

[26] P K F Kuhfittig ldquoOn the stability of thin-shell wormholes innoncommutative geometryrdquo Advances in High Energy Physicsvol 2012 Article ID 462493 12 pages 2012

[27] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquo Journal of High Energy Physics vol 1999 no 9 article032 1999

[28] J P de Leon ldquoMass and charge in brane-world and non-compact Kaluza-KLEin theories in 5 dimrdquo General Relativityand Gravitation vol 35 no 8 pp 1365ndash1384 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

8 Advances in High Energy Physics

05 10 15 20 25 30

0

r

r

150 152 154 156 158 160

r

05 10 15 200

500

1000

1500

2000

pt+120588

pt+120588

pt+120588

minus0058

minus15

minus10

minus5

minus0056

minus0054

minus0052

minus0050n = 5 120598 = minus1

n = 5 120598 = 1

(a) Evolution of (120588 + 119901119905)+versus 119903

r

05 10 15 20

r

00 05 10 15 200

5

10

15

20

pt+120588 p

t+120588

minus70

minus60

minus50

minus40

minus30

minus20

minus10

n = 5 120598 = minus1

(b) Evolution of (120588 + 119901119905)minusversus 119903

Figure 8 Plots of (a) 120588 + 119901119905for 119904+and (b) 120588 + 119901

119905for 119904minusversus 119903 in Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1 (red dashed)

and 119899 = 6 120598 = 3 (purple)

proceedwith constant redshift function 120582 = 0which vanishesthe gravitational contribution119891

119892in the equilibrium equation

that is 120591 = 2120582 leads to 1205911015840 = 0 for constant 120582 Thus we are leftwith hydrostatic and anisotropic forces with correspondingequilibrium condition as

119891119886+ 119891ℎ= 0 (20)

Using (8) and (9) we obtain the following expressions for thehydrostatic and anisotropic forces

119891119886=

1

1199033(119899 minus 1)

119904

119903

minus 1199041015840

(119899 minus 3 + (119899 minus 5)

119904120598

1199033)

minus

119904

1199034[(119899 minus 3) (119899 minus 4) + (119899 minus 5) (119899 minus 6)

119904120598

1199033]

119891ℎ=

(119899 minus 2)

21199033

[119899 minus 3 + (119899 minus 5)

119904120598

1199033+

119904120598 (119899 minus 5)

1199033

]

sdot (1199041015840

minus

3119904

119903

)

(21)

Figure 9 represents the plots of anisotropic as well ashydrostatic forces for the wormhole solutions in Gaussiandistributed framework Figures 9(a) and 9(c) describe theequilibrium for positive root solution for 119899 = 5 120598 =

1 minus1 and 119899 = 6 120598 = 3 minus3 through opposite behaviorand hence they cancel each other in order to satisfy (20)For negative root solution 119904

minus Figures 9(b) and 9(d) show

the stable configuration of wormhole solutions for fifthdimensional case only The behaviors of both forces (whichare not in opposite manner) do not cancel each other sothere is no equilibrium configuration examined for sixthdimensional wormhole solutions In the case of Lorentziannoncommutative background Figure 10 expresses that allwormhole solutions are in equilibrium by satisfying theequilibrium condition for both dimensions

Table 1 Wormhole solutions with Gaussian distributed noncom-mutative framework for 119904

+

Expressions 119899 = 5 120598 =

1

119899 = 5 120598 =

minus1

119899 = 6 120598 =

3

119899 = 6 120598 =

minus3

1 minus 119904+119903 Positive Positive

for 119903 lt 1 Positive Positivefor 119903 lt 18

120588 Positive Positive Positive Positive120588 + 119901

119903 Positive Negative Positive Negative120588 + 119901

119905 Negative Positive Negative PositiveWEC Violates Violates Violates ViolatesEC Holds Holds Holds Holds

5 Conclusion

It is a well-known fact that the existence of wormhole solu-tions is based on violation of NEC Since the normal mattersatisfies the energy conditions this violation is associatedwith an energy-momentum tensor which provides exoticmatter a hypothetical form of matter To explore realisticmodel or physically acceptable wormhole solutions it isnecessary to find such a source which gives the violationof NEC while normal matter meets the energy conditionsHere in this paper we have explored wormhole solutionsin 119899-dimensional Einstein Gauss-Bonnet gravity with Gaus-sian and Lorentzian noncommutative backgrounds We haverestricted ourselves to fifth and sixth dimensional cases withpositive as well as negativeGauss-Bonnet coefficient Also wehave checked the condition of equilibrium for the wormholesolutions

The results are summarized in Tables 1 and 2 It is notedthat the 119904

+(Tables 1 and 3) and 119904

minus(Tables 2 and 4) represent

the two roots of the solutions in both backgrounds and ECdenotes the equilibrium condition

In the Lorentzan distributed noncommutative frame-work we have found negative energy density for 119904

+while

Advances in High Energy Physics 9

02 04 06 1008

0

2

4

r

fa

minus2

minus4

(a) Evolution of 119891119886versus 119903

r

02 04 06 08 10

0

2

4

fa

minus2

minus4

(b) Evolution of 119891119886versus 119903

02 04 06 08 10

0

2

4

r

fh

minus2

minus4

(c) Evolution of 119891ℎversus 119903

02 04 06 08 10

0

2

4

r

fh

minus2

minus4

(d) Evolution of 119891ℎversus 119903

Figure 9 Plots of (a) 119891119886for 119904+ (b) 119891

119886for 119904minus (c) 119891

ℎfor 119904+ (d) 119891

ℎfor 119904minusversus 119903 with Gaussian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1

(red dashed) and 119899 = 6 120598 = 3 (purple)

Table 2 Wormhole solutions with Gaussian distributed noncom-mutative framework for 119904

minus

Expressions 119899 = 5 120598 =

1

119899 = 5 120598 =

minus1

119899 = 6 120598 =

3

119899 = 6 120598 =

minus3

1 minus 119904minus119903 Positive Positive

for 119903 lt 4 Positive Positivefor 119903 lt 8

120588 Positive Positive Positive Positive120588 + 119901

119903 Positive Negative Positive Negative120588 + 119901

119905 Positive Negative Positive NegativeWEC Holds Violates Holds Violates

EC Holds Holds Does nothold

Does nothold

violation of 1 minus 119904119903 gt 0 incorporating 119904minusfor the case 119899 = 6

120598 = minus3 so we skipped this case The remaining results aresummarized in Tables 3 and 4

In the paper [20] higher dimensional asymptotically flatwormhole solutions have been explored in the frameworkof Gauss-Bonnet gravity by considering a specific choicefor a radial dependent redshift function and by imposinga particular equation of state The WEC is satisfied at thethroat by considering a negative Gauss-Bonnet coupling

Table 3 Wormhole solutions with Lorentzian noncommutativebackground for 119904

+

Expressions 119899 = 5 120598 = 1 119899 = 5 120598 = minus1 119899 = 6 120598 = 3

1 minus 119904+119903

Positive for119903 lt 19

Positive Positive for119903 lt 21

120588 Positive Positive Positive

120588 + 119901119903

Positive for119903 lt 1

Positive for119903 gt 1

Positive for119903 lt 1

120588 + 119901119905 Negative Negative Positive

WEC Does nothold

Does nothold

Holds for119903 lt 1

EC Holds Holds Holds

constant Furthermore they have considered a constantredshift function and shown specifically that for negativeGauss-Bonnet coupling constant one may have normalmatter in a determined radial region and that the increaseof coupling constant enlarges the normal matter regionIn the present paper we have taken energy density undernoncommutative geometry distributions instead of particularequation of stateWe have obtained results for positiveGauss-Bonnet coefficient satisfying energy conditions It contains

10 Advances in High Energy Physics

02 04 06 08 10

0

200

400

r

fa

minus400

minus200

(a) Evolution of 119891119886versus 119903

02 04 06 08 10

0

200

400

r

fa

minus400

minus200

(b) Evolution of 119891119886versus 119903

02 04 06 08 10r

minus400

minus200

0

200

400

fh

(c) Evolution of 119891ℎversus 119903

02 04 06 08 10

0

200

400

r

fh

minus400

minus200

(d) Evolution of 119891ℎversus 119903

Figure 10 Plots of (a) 119891119886for 119904+ (b) 119891

119886for 119904minus (c) 119891

ℎfor 119904+ (d) 119891

ℎfor 119904minusversus 119903 with Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5

120598 = minus1 (red dashed) and 119899 = 6 120598 = 3 (purple)

Table 4 Wormhole solutions with Lorentzian noncommutativebackground for 119904

minus

Expressions 119899 = 5 120598 = 1 119899 = 5 120598 = minus1 119899 = 6 120598 = 3

1 minus 119904minus119903 Positive Positive for

119903 lt 05

Positive

120588 Positive Positive Positive120588 + 119901

119903 Positive Negative Negative120588 + 119901

119905 Positive Negative Positive

WEC Holds Does nothold

Does nothold

EC Holds Holds Holds

fifth dimensional wormhole solutions in both backgroundssatisfying equilibrium condition and sixth dimensional withdisequilibrium in noncommutative background with 119904

minussolu-

tion Also there is possibility for the existence of wormholein equilibrium satisfying WEC for 119899 = 6 120598 = 3 taking intoaccount 119904

+solution for the range 119903 lt 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M S Morris and K S Thorne ldquoWormholes in spacetimeand their use for interstellar travel a tool for teaching generalrelativityrdquo American Journal of Physics vol 56 no 5 pp 395ndash412 1988

[2] M Sharif and A Jawad ldquoPhantom-like generalized cosmicchaplygin gas and traversable wormhole solutionsrdquo The Euro-pean Physical Journal Plus vol 129 article 15 2014

[3] F S N Lobo F Parsaei and N Riazi ldquoNew asymptotically flatphantom wormhole solutionsrdquo Physical Review D vol 87 no 8Article ID 084030 2013

[4] S-W Kim andH Lee ldquoExact solutions of a charged wormholerdquoPhysical Review D vol 63 no 6 Article ID 064014 5 pages2001

[5] M Jamil and M U Farooq ldquoPhantom wormholes in (2 + 1)-dimensionsrdquo International Journal ofTheoretical Physics vol 49no 4 pp 835ndash841 2010

[6] M U Farooq M Akbar and M Jamil ldquoDynamics and ther-modynamics of (2+1)minusdimensional evolving lorentzian worm-holerdquo in Proceedings of the 3rd AlgerianWorkshop on Astronomyand Astrophysics vol 1295 of AIP Conference Proceedings pp176ndash190 Constantine Algeria June 2010

[7] P K Kuhfittig ldquoGravitational lensing of wormholes in thegalactic halo regionrdquo The European Physical Journal C vol 74no 3 article 2818 2014

Advances in High Energy Physics 11

[8] F Rahaman A Banerjee M Jamil A K Yadav and H IdrisldquoNoncommutative wormholes in f(R) gravity with lorentziandistributionrdquo International Journal ofTheoretical Physics vol 53no 6 pp 1910ndash1919 2014

[9] M Sharif and S Rani ldquoDynamical wormhole solutions in f(T)gravityrdquo General Relativity and Gravitation vol 45 no 11 pp2389ndash2402 2013

[10] M Sharif and S Rani ldquoCharged noncommutative wormholesolutions in f (T) gravityrdquo The European Physical Journal Plusvol 129 no 10 article 237 2014

[11] M Sharif and S Rani ldquoGalactic halo wormhole solutions in119891(119879) gravityrdquoAdvances in High Energy Physics vol 2014 ArticleID 691497 9 pages 2014

[12] F S N Lobo andM A Oliveira ldquoWormhole geometries in f (R)modified theories of gravityrdquo Physical Review D vol 80 ArticleID 104012 2009

[13] M Sharif and S Rani ldquoWormhole solutions in f(T) gravity withnoncommutativerdquo Physical Review D vol 88 no 12 Article ID123501 2013

[14] F Rahaman S Islam P K F Kuhfittig and S Ray ldquoSearch-ing for higher-dimensional wormholes with noncommutativegeometryrdquo Physical Review D vol 86 no 10 Article ID 1060102012

[15] P Bhar and F Rahaman ldquoSearch of wormholes in differ-ent dimensional non-commutative inspired space-times withLorentzian distributionrdquo The European Physical Journal C vol74 no 12 article 3213 2014

[16] F Rahaman S Karmakar I Karar and S Ray ldquoWormholeinspired by non-commutative geometryrdquo Physics Letters B vol746 pp 73ndash78 2015

[17] M Jamil F Rahaman R Myrzakulov P Kuhfittig N Ahmedand U Mondal ldquoNonommutative wormholes in f(R) gravityrdquoJournal of the Korean Physical Society vol 65 no 6 pp 917ndash9252014

[18] A Jawad and S Rani ldquoLorentz distributed noncommutativewormhole solutions in extended teleparallel gravityrdquoThe Euro-pean Physical Journal C vol 75 article 173 2015

[19] B Bhawal and S Kar ldquoLorentzian wormholes in Einstein-Gauss-Bonnet theoryrdquo Physical Review D vol 46 no 6 pp2464ndash2468 1992

[20] M R Mehdizadeh M K Zangeneh and F S N LoboldquoEinstein-Gauss-Bonnet traversable wormholes satisfying theweak energy conditionrdquo Physical ReviewD vol 91 no 8 ArticleID 084004 2015

[21] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[22] S H Mehdipour ldquoEntropic force approach to noncommutativeSchwarzschild black holes signals a failure of current physicalideasrdquo The European Physical Journal Plus vol 127 article 802012

[23] S Sushkov ldquoWormholes supported by a phantom energyrdquoPhysical Review D vol 71 no 4 Article ID 043520 2005

[24] P Nicolini and E Spallucci ldquoNoncommutative geometry-inspired dirty black holesrdquo Classical and Quantum Gravity vol27 no 1 Article ID 015010 2010

[25] F Rahaman P K Kuhfittig K Chakraborty A A Usmani andS Ray ldquoGalactic rotation curves inspired by a noncommutative-geometry backgroundrdquo General Relativity and Gravitation vol44 no 4 pp 905ndash916 2012

[26] P K F Kuhfittig ldquoOn the stability of thin-shell wormholes innoncommutative geometryrdquo Advances in High Energy Physicsvol 2012 Article ID 462493 12 pages 2012

[27] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquo Journal of High Energy Physics vol 1999 no 9 article032 1999

[28] J P de Leon ldquoMass and charge in brane-world and non-compact Kaluza-KLEin theories in 5 dimrdquo General Relativityand Gravitation vol 35 no 8 pp 1365ndash1384 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 9

02 04 06 1008

0

2

4

r

fa

minus2

minus4

(a) Evolution of 119891119886versus 119903

r

02 04 06 08 10

0

2

4

fa

minus2

minus4

(b) Evolution of 119891119886versus 119903

02 04 06 08 10

0

2

4

r

fh

minus2

minus4

(c) Evolution of 119891ℎversus 119903

02 04 06 08 10

0

2

4

r

fh

minus2

minus4

(d) Evolution of 119891ℎversus 119903

Figure 9 Plots of (a) 119891119886for 119904+ (b) 119891

119886for 119904minus (c) 119891

ℎfor 119904+ (d) 119891

ℎfor 119904minusversus 119903 with Gaussian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5 120598 = minus1

(red dashed) and 119899 = 6 120598 = 3 (purple)

Table 2 Wormhole solutions with Gaussian distributed noncom-mutative framework for 119904

minus

Expressions 119899 = 5 120598 =

1

119899 = 5 120598 =

minus1

119899 = 6 120598 =

3

119899 = 6 120598 =

minus3

1 minus 119904minus119903 Positive Positive

for 119903 lt 4 Positive Positivefor 119903 lt 8

120588 Positive Positive Positive Positive120588 + 119901

119903 Positive Negative Positive Negative120588 + 119901

119905 Positive Negative Positive NegativeWEC Holds Violates Holds Violates

EC Holds Holds Does nothold

Does nothold

violation of 1 minus 119904119903 gt 0 incorporating 119904minusfor the case 119899 = 6

120598 = minus3 so we skipped this case The remaining results aresummarized in Tables 3 and 4

In the paper [20] higher dimensional asymptotically flatwormhole solutions have been explored in the frameworkof Gauss-Bonnet gravity by considering a specific choicefor a radial dependent redshift function and by imposinga particular equation of state The WEC is satisfied at thethroat by considering a negative Gauss-Bonnet coupling

Table 3 Wormhole solutions with Lorentzian noncommutativebackground for 119904

+

Expressions 119899 = 5 120598 = 1 119899 = 5 120598 = minus1 119899 = 6 120598 = 3

1 minus 119904+119903

Positive for119903 lt 19

Positive Positive for119903 lt 21

120588 Positive Positive Positive

120588 + 119901119903

Positive for119903 lt 1

Positive for119903 gt 1

Positive for119903 lt 1

120588 + 119901119905 Negative Negative Positive

WEC Does nothold

Does nothold

Holds for119903 lt 1

EC Holds Holds Holds

constant Furthermore they have considered a constantredshift function and shown specifically that for negativeGauss-Bonnet coupling constant one may have normalmatter in a determined radial region and that the increaseof coupling constant enlarges the normal matter regionIn the present paper we have taken energy density undernoncommutative geometry distributions instead of particularequation of stateWe have obtained results for positiveGauss-Bonnet coefficient satisfying energy conditions It contains

10 Advances in High Energy Physics

02 04 06 08 10

0

200

400

r

fa

minus400

minus200

(a) Evolution of 119891119886versus 119903

02 04 06 08 10

0

200

400

r

fa

minus400

minus200

(b) Evolution of 119891119886versus 119903

02 04 06 08 10r

minus400

minus200

0

200

400

fh

(c) Evolution of 119891ℎversus 119903

02 04 06 08 10

0

200

400

r

fh

minus400

minus200

(d) Evolution of 119891ℎversus 119903

Figure 10 Plots of (a) 119891119886for 119904+ (b) 119891

119886for 119904minus (c) 119891

ℎfor 119904+ (d) 119891

ℎfor 119904minusversus 119903 with Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5

120598 = minus1 (red dashed) and 119899 = 6 120598 = 3 (purple)

Table 4 Wormhole solutions with Lorentzian noncommutativebackground for 119904

minus

Expressions 119899 = 5 120598 = 1 119899 = 5 120598 = minus1 119899 = 6 120598 = 3

1 minus 119904minus119903 Positive Positive for

119903 lt 05

Positive

120588 Positive Positive Positive120588 + 119901

119903 Positive Negative Negative120588 + 119901

119905 Positive Negative Positive

WEC Holds Does nothold

Does nothold

EC Holds Holds Holds

fifth dimensional wormhole solutions in both backgroundssatisfying equilibrium condition and sixth dimensional withdisequilibrium in noncommutative background with 119904

minussolu-

tion Also there is possibility for the existence of wormholein equilibrium satisfying WEC for 119899 = 6 120598 = 3 taking intoaccount 119904

+solution for the range 119903 lt 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M S Morris and K S Thorne ldquoWormholes in spacetimeand their use for interstellar travel a tool for teaching generalrelativityrdquo American Journal of Physics vol 56 no 5 pp 395ndash412 1988

[2] M Sharif and A Jawad ldquoPhantom-like generalized cosmicchaplygin gas and traversable wormhole solutionsrdquo The Euro-pean Physical Journal Plus vol 129 article 15 2014

[3] F S N Lobo F Parsaei and N Riazi ldquoNew asymptotically flatphantom wormhole solutionsrdquo Physical Review D vol 87 no 8Article ID 084030 2013

[4] S-W Kim andH Lee ldquoExact solutions of a charged wormholerdquoPhysical Review D vol 63 no 6 Article ID 064014 5 pages2001

[5] M Jamil and M U Farooq ldquoPhantom wormholes in (2 + 1)-dimensionsrdquo International Journal ofTheoretical Physics vol 49no 4 pp 835ndash841 2010

[6] M U Farooq M Akbar and M Jamil ldquoDynamics and ther-modynamics of (2+1)minusdimensional evolving lorentzian worm-holerdquo in Proceedings of the 3rd AlgerianWorkshop on Astronomyand Astrophysics vol 1295 of AIP Conference Proceedings pp176ndash190 Constantine Algeria June 2010

[7] P K Kuhfittig ldquoGravitational lensing of wormholes in thegalactic halo regionrdquo The European Physical Journal C vol 74no 3 article 2818 2014

Advances in High Energy Physics 11

[8] F Rahaman A Banerjee M Jamil A K Yadav and H IdrisldquoNoncommutative wormholes in f(R) gravity with lorentziandistributionrdquo International Journal ofTheoretical Physics vol 53no 6 pp 1910ndash1919 2014

[9] M Sharif and S Rani ldquoDynamical wormhole solutions in f(T)gravityrdquo General Relativity and Gravitation vol 45 no 11 pp2389ndash2402 2013

[10] M Sharif and S Rani ldquoCharged noncommutative wormholesolutions in f (T) gravityrdquo The European Physical Journal Plusvol 129 no 10 article 237 2014

[11] M Sharif and S Rani ldquoGalactic halo wormhole solutions in119891(119879) gravityrdquoAdvances in High Energy Physics vol 2014 ArticleID 691497 9 pages 2014

[12] F S N Lobo andM A Oliveira ldquoWormhole geometries in f (R)modified theories of gravityrdquo Physical Review D vol 80 ArticleID 104012 2009

[13] M Sharif and S Rani ldquoWormhole solutions in f(T) gravity withnoncommutativerdquo Physical Review D vol 88 no 12 Article ID123501 2013

[14] F Rahaman S Islam P K F Kuhfittig and S Ray ldquoSearch-ing for higher-dimensional wormholes with noncommutativegeometryrdquo Physical Review D vol 86 no 10 Article ID 1060102012

[15] P Bhar and F Rahaman ldquoSearch of wormholes in differ-ent dimensional non-commutative inspired space-times withLorentzian distributionrdquo The European Physical Journal C vol74 no 12 article 3213 2014

[16] F Rahaman S Karmakar I Karar and S Ray ldquoWormholeinspired by non-commutative geometryrdquo Physics Letters B vol746 pp 73ndash78 2015

[17] M Jamil F Rahaman R Myrzakulov P Kuhfittig N Ahmedand U Mondal ldquoNonommutative wormholes in f(R) gravityrdquoJournal of the Korean Physical Society vol 65 no 6 pp 917ndash9252014

[18] A Jawad and S Rani ldquoLorentz distributed noncommutativewormhole solutions in extended teleparallel gravityrdquoThe Euro-pean Physical Journal C vol 75 article 173 2015

[19] B Bhawal and S Kar ldquoLorentzian wormholes in Einstein-Gauss-Bonnet theoryrdquo Physical Review D vol 46 no 6 pp2464ndash2468 1992

[20] M R Mehdizadeh M K Zangeneh and F S N LoboldquoEinstein-Gauss-Bonnet traversable wormholes satisfying theweak energy conditionrdquo Physical ReviewD vol 91 no 8 ArticleID 084004 2015

[21] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[22] S H Mehdipour ldquoEntropic force approach to noncommutativeSchwarzschild black holes signals a failure of current physicalideasrdquo The European Physical Journal Plus vol 127 article 802012

[23] S Sushkov ldquoWormholes supported by a phantom energyrdquoPhysical Review D vol 71 no 4 Article ID 043520 2005

[24] P Nicolini and E Spallucci ldquoNoncommutative geometry-inspired dirty black holesrdquo Classical and Quantum Gravity vol27 no 1 Article ID 015010 2010

[25] F Rahaman P K Kuhfittig K Chakraborty A A Usmani andS Ray ldquoGalactic rotation curves inspired by a noncommutative-geometry backgroundrdquo General Relativity and Gravitation vol44 no 4 pp 905ndash916 2012

[26] P K F Kuhfittig ldquoOn the stability of thin-shell wormholes innoncommutative geometryrdquo Advances in High Energy Physicsvol 2012 Article ID 462493 12 pages 2012

[27] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquo Journal of High Energy Physics vol 1999 no 9 article032 1999

[28] J P de Leon ldquoMass and charge in brane-world and non-compact Kaluza-KLEin theories in 5 dimrdquo General Relativityand Gravitation vol 35 no 8 pp 1365ndash1384 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

10 Advances in High Energy Physics

02 04 06 08 10

0

200

400

r

fa

minus400

minus200

(a) Evolution of 119891119886versus 119903

02 04 06 08 10

0

200

400

r

fa

minus400

minus200

(b) Evolution of 119891119886versus 119903

02 04 06 08 10r

minus400

minus200

0

200

400

fh

(c) Evolution of 119891ℎversus 119903

02 04 06 08 10

0

200

400

r

fh

minus400

minus200

(d) Evolution of 119891ℎversus 119903

Figure 10 Plots of (a) 119891119886for 119904+ (b) 119891

119886for 119904minus (c) 119891

ℎfor 119904+ (d) 119891

ℎfor 119904minusversus 119903 with Lorentzian distribution for 119899 = 5 120598 = 1 (red) 119899 = 5

120598 = minus1 (red dashed) and 119899 = 6 120598 = 3 (purple)

Table 4 Wormhole solutions with Lorentzian noncommutativebackground for 119904

minus

Expressions 119899 = 5 120598 = 1 119899 = 5 120598 = minus1 119899 = 6 120598 = 3

1 minus 119904minus119903 Positive Positive for

119903 lt 05

Positive

120588 Positive Positive Positive120588 + 119901

119903 Positive Negative Negative120588 + 119901

119905 Positive Negative Positive

WEC Holds Does nothold

Does nothold

EC Holds Holds Holds

fifth dimensional wormhole solutions in both backgroundssatisfying equilibrium condition and sixth dimensional withdisequilibrium in noncommutative background with 119904

minussolu-

tion Also there is possibility for the existence of wormholein equilibrium satisfying WEC for 119899 = 6 120598 = 3 taking intoaccount 119904

+solution for the range 119903 lt 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M S Morris and K S Thorne ldquoWormholes in spacetimeand their use for interstellar travel a tool for teaching generalrelativityrdquo American Journal of Physics vol 56 no 5 pp 395ndash412 1988

[2] M Sharif and A Jawad ldquoPhantom-like generalized cosmicchaplygin gas and traversable wormhole solutionsrdquo The Euro-pean Physical Journal Plus vol 129 article 15 2014

[3] F S N Lobo F Parsaei and N Riazi ldquoNew asymptotically flatphantom wormhole solutionsrdquo Physical Review D vol 87 no 8Article ID 084030 2013

[4] S-W Kim andH Lee ldquoExact solutions of a charged wormholerdquoPhysical Review D vol 63 no 6 Article ID 064014 5 pages2001

[5] M Jamil and M U Farooq ldquoPhantom wormholes in (2 + 1)-dimensionsrdquo International Journal ofTheoretical Physics vol 49no 4 pp 835ndash841 2010

[6] M U Farooq M Akbar and M Jamil ldquoDynamics and ther-modynamics of (2+1)minusdimensional evolving lorentzian worm-holerdquo in Proceedings of the 3rd AlgerianWorkshop on Astronomyand Astrophysics vol 1295 of AIP Conference Proceedings pp176ndash190 Constantine Algeria June 2010

[7] P K Kuhfittig ldquoGravitational lensing of wormholes in thegalactic halo regionrdquo The European Physical Journal C vol 74no 3 article 2818 2014

Advances in High Energy Physics 11

[8] F Rahaman A Banerjee M Jamil A K Yadav and H IdrisldquoNoncommutative wormholes in f(R) gravity with lorentziandistributionrdquo International Journal ofTheoretical Physics vol 53no 6 pp 1910ndash1919 2014

[9] M Sharif and S Rani ldquoDynamical wormhole solutions in f(T)gravityrdquo General Relativity and Gravitation vol 45 no 11 pp2389ndash2402 2013

[10] M Sharif and S Rani ldquoCharged noncommutative wormholesolutions in f (T) gravityrdquo The European Physical Journal Plusvol 129 no 10 article 237 2014

[11] M Sharif and S Rani ldquoGalactic halo wormhole solutions in119891(119879) gravityrdquoAdvances in High Energy Physics vol 2014 ArticleID 691497 9 pages 2014

[12] F S N Lobo andM A Oliveira ldquoWormhole geometries in f (R)modified theories of gravityrdquo Physical Review D vol 80 ArticleID 104012 2009

[13] M Sharif and S Rani ldquoWormhole solutions in f(T) gravity withnoncommutativerdquo Physical Review D vol 88 no 12 Article ID123501 2013

[14] F Rahaman S Islam P K F Kuhfittig and S Ray ldquoSearch-ing for higher-dimensional wormholes with noncommutativegeometryrdquo Physical Review D vol 86 no 10 Article ID 1060102012

[15] P Bhar and F Rahaman ldquoSearch of wormholes in differ-ent dimensional non-commutative inspired space-times withLorentzian distributionrdquo The European Physical Journal C vol74 no 12 article 3213 2014

[16] F Rahaman S Karmakar I Karar and S Ray ldquoWormholeinspired by non-commutative geometryrdquo Physics Letters B vol746 pp 73ndash78 2015

[17] M Jamil F Rahaman R Myrzakulov P Kuhfittig N Ahmedand U Mondal ldquoNonommutative wormholes in f(R) gravityrdquoJournal of the Korean Physical Society vol 65 no 6 pp 917ndash9252014

[18] A Jawad and S Rani ldquoLorentz distributed noncommutativewormhole solutions in extended teleparallel gravityrdquoThe Euro-pean Physical Journal C vol 75 article 173 2015

[19] B Bhawal and S Kar ldquoLorentzian wormholes in Einstein-Gauss-Bonnet theoryrdquo Physical Review D vol 46 no 6 pp2464ndash2468 1992

[20] M R Mehdizadeh M K Zangeneh and F S N LoboldquoEinstein-Gauss-Bonnet traversable wormholes satisfying theweak energy conditionrdquo Physical ReviewD vol 91 no 8 ArticleID 084004 2015

[21] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[22] S H Mehdipour ldquoEntropic force approach to noncommutativeSchwarzschild black holes signals a failure of current physicalideasrdquo The European Physical Journal Plus vol 127 article 802012

[23] S Sushkov ldquoWormholes supported by a phantom energyrdquoPhysical Review D vol 71 no 4 Article ID 043520 2005

[24] P Nicolini and E Spallucci ldquoNoncommutative geometry-inspired dirty black holesrdquo Classical and Quantum Gravity vol27 no 1 Article ID 015010 2010

[25] F Rahaman P K Kuhfittig K Chakraborty A A Usmani andS Ray ldquoGalactic rotation curves inspired by a noncommutative-geometry backgroundrdquo General Relativity and Gravitation vol44 no 4 pp 905ndash916 2012

[26] P K F Kuhfittig ldquoOn the stability of thin-shell wormholes innoncommutative geometryrdquo Advances in High Energy Physicsvol 2012 Article ID 462493 12 pages 2012

[27] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquo Journal of High Energy Physics vol 1999 no 9 article032 1999

[28] J P de Leon ldquoMass and charge in brane-world and non-compact Kaluza-KLEin theories in 5 dimrdquo General Relativityand Gravitation vol 35 no 8 pp 1365ndash1384 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 11

[8] F Rahaman A Banerjee M Jamil A K Yadav and H IdrisldquoNoncommutative wormholes in f(R) gravity with lorentziandistributionrdquo International Journal ofTheoretical Physics vol 53no 6 pp 1910ndash1919 2014

[9] M Sharif and S Rani ldquoDynamical wormhole solutions in f(T)gravityrdquo General Relativity and Gravitation vol 45 no 11 pp2389ndash2402 2013

[10] M Sharif and S Rani ldquoCharged noncommutative wormholesolutions in f (T) gravityrdquo The European Physical Journal Plusvol 129 no 10 article 237 2014

[11] M Sharif and S Rani ldquoGalactic halo wormhole solutions in119891(119879) gravityrdquoAdvances in High Energy Physics vol 2014 ArticleID 691497 9 pages 2014

[12] F S N Lobo andM A Oliveira ldquoWormhole geometries in f (R)modified theories of gravityrdquo Physical Review D vol 80 ArticleID 104012 2009

[13] M Sharif and S Rani ldquoWormhole solutions in f(T) gravity withnoncommutativerdquo Physical Review D vol 88 no 12 Article ID123501 2013

[14] F Rahaman S Islam P K F Kuhfittig and S Ray ldquoSearch-ing for higher-dimensional wormholes with noncommutativegeometryrdquo Physical Review D vol 86 no 10 Article ID 1060102012

[15] P Bhar and F Rahaman ldquoSearch of wormholes in differ-ent dimensional non-commutative inspired space-times withLorentzian distributionrdquo The European Physical Journal C vol74 no 12 article 3213 2014

[16] F Rahaman S Karmakar I Karar and S Ray ldquoWormholeinspired by non-commutative geometryrdquo Physics Letters B vol746 pp 73ndash78 2015

[17] M Jamil F Rahaman R Myrzakulov P Kuhfittig N Ahmedand U Mondal ldquoNonommutative wormholes in f(R) gravityrdquoJournal of the Korean Physical Society vol 65 no 6 pp 917ndash9252014

[18] A Jawad and S Rani ldquoLorentz distributed noncommutativewormhole solutions in extended teleparallel gravityrdquoThe Euro-pean Physical Journal C vol 75 article 173 2015

[19] B Bhawal and S Kar ldquoLorentzian wormholes in Einstein-Gauss-Bonnet theoryrdquo Physical Review D vol 46 no 6 pp2464ndash2468 1992

[20] M R Mehdizadeh M K Zangeneh and F S N LoboldquoEinstein-Gauss-Bonnet traversable wormholes satisfying theweak energy conditionrdquo Physical ReviewD vol 91 no 8 ArticleID 084004 2015

[21] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[22] S H Mehdipour ldquoEntropic force approach to noncommutativeSchwarzschild black holes signals a failure of current physicalideasrdquo The European Physical Journal Plus vol 127 article 802012

[23] S Sushkov ldquoWormholes supported by a phantom energyrdquoPhysical Review D vol 71 no 4 Article ID 043520 2005

[24] P Nicolini and E Spallucci ldquoNoncommutative geometry-inspired dirty black holesrdquo Classical and Quantum Gravity vol27 no 1 Article ID 015010 2010

[25] F Rahaman P K Kuhfittig K Chakraborty A A Usmani andS Ray ldquoGalactic rotation curves inspired by a noncommutative-geometry backgroundrdquo General Relativity and Gravitation vol44 no 4 pp 905ndash916 2012

[26] P K F Kuhfittig ldquoOn the stability of thin-shell wormholes innoncommutative geometryrdquo Advances in High Energy Physicsvol 2012 Article ID 462493 12 pages 2012

[27] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquo Journal of High Energy Physics vol 1999 no 9 article032 1999

[28] J P de Leon ldquoMass and charge in brane-world and non-compact Kaluza-KLEin theories in 5 dimrdquo General Relativityand Gravitation vol 35 no 8 pp 1365ndash1384 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of