Research ArticleTransformation Groups for a Schwarzschild-Type Geometry in119891(119877) Gravity

Emre Dil1 and Talha Zafer2

1Department of Physics Sinop University 57000 Sinop Turkey2Department of Physics Sakarya University 54187 Sakarya Turkey

Correspondence should be addressed to Emre Dil emredilsakaryaedutr

Received 28 June 2016 Revised 17 October 2016 Accepted 23 October 2016

Academic Editor Sergei D Odintsov

Copyright copy 2016 E Dil and T Zafer This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Weknow that the Lorentz transformations are special relativistic coordinate transformations between inertial framesWhat happensif we would like to find the coordinate transformations between noninertial reference frames Noninertial frames are known to beaccelerated frames with respect to an inertial frameTherefore these should be considered in the framework of general relativity orits modified versions We assume that the inertial frames are flat space-times and noninertial frames are curved space-times thenwe investigate the deformation and coordinate transformation groups between a flat space-time and a curved space-time whichis curved by a Schwarzschild-type black hole in the framework of 119891(119877) gravity We firstly study the deformation transformationgroups by relating the metrics of the flat and curved space-times in spherical coordinates after the deformation transformationswe concentrate on the coordinate transformations Later on we investigate the same deformation and coordinate transformationsin Cartesian coordinates Finally we obtain two different sets of transformation groups for the spherical and Cartesian coordinates

1 Introduction

The coordinate transformations are of great importancein physics such as nonrelativistic Galilean transforma-tions between flat Euclidean spaces and special relativis-tic Lorentz transformations between flat Minkowski space-times Another coordinate transformation the generalizedLorentz transformations of Nelson in noninertial frames inflat space-time can be viewed as in nongeneral relativisticpicture [1] The related effects and applications of Nelsonrsquosgeneralized Lorentz transformations in locally useful fieldscan be viewed as Nikolicrsquos works on length contractionsin noninertial frames [2 3] and Gronrsquos works [4 5] Afterthese special relativistic flat space-time coordinate trans-formations we can consider the general relativistic curvedRiemann space-time coordinate transformations betweennoninertial frames While the flat space-time is referred to asthe reference frame of an inertial observer the curved space-time is referred to as the noninertial observer [6]

In general relativity and similarly in the modified 119891(119877)gravity theory the noninertial reference frame may beviewed as a gravitationally accelerated object due to the

energy-momentum tensor of a massive object The space-time curvature is related to the gravitational force of themassive object Therefore as a source of curvature a blackhole can be taken into account In this study we take thespherically symmetric Schwarzschild-type black hole for thesource of the curved space-time in which we investigate thedeformation and coordinate transformation groups in theframework of 119891(119877) gravity [7ndash9]

The modified gravitation theories have become urgentin recent researches because of the missing matter incon-sistency and accelerated expansion of universe Instead ofsearching new matter forms of energy to modify the actionintegral of gravity 119891(119877) gravity comes into play as a geomet-rically modified theory by rearranging the Ricci scalar 119877 toa function of the scalar as 119891(119877) in the action integral of thegravity [10]

The extended gravity theories were studied for the prob-lems having complete solutions in general relativity (GR)such as the initial value problem [11] spherically and axiallysymmetric solutions of black hole and active galactic nucleiThe solutions of the extended gravity theories should beconsistent with the GR They can be analogous and modified

Hindawi Publishing CorporationJournal of GravityVolume 2016 Article ID 7636493 8 pageshttpdxdoiorg10115520167636493

2 Journal of Gravity

solutions of Schwarzschild and Kerr solutions of GR forexample So the methods for the extended theories should bechecked with the solutions of GR [12]

For the spherically symmetric Schwarzschild black holesolutions of GR we will refer to the spherically symmetriccorresponding Schwarzschild-type black hole solutions ofCapozziello et al [13] Then in this curved and expandedspace of Schwarzschild-type black hole we will study thecoordinate transformations from a noninertial frame to aninertial frame being the tangent vector space-time of theSchwarzschild-type space-time and this flat space-time isrepresented by the local Lorentz coordinates (LLC) Fromthese transformations we will obtain the related effects suchas length contraction and time dilation for the 119891(119877) gravityof Schwarzschild-type black hole

The coordinate transformation between flat tangent vec-tor space-time and the curved space-time of Schwarzschild-type black hole of 119891(119877) gravity will be constructed by thedeformation transformations of the flat space-time metric[14ndash16] In spherical coordinates this transformation is aconformal transformation between the flat and curved space-time metrics Similarly by using the transformations fromspherical to Cartesian coordinates we obtain the metricsof both space-times in Cartesian coordinates and thenwe construct the deformation transformation matrices ofthese two metrics in Cartesian coordinates and obtain thecoordinates transformations in Cartesian coordinates for aSchwarzschild-type black hole in 119891(119877) gravity

Consequently we firstly find the deformation transfor-mations of metrics in spherical coordinates and the coor-dinate transformations by using the deformation matrixThen we find the deformation transformation groups and thecoordinate transformation groups in spherical coordinatesSecondly we convert the metrics in Cartesian coordinatesand follow the same procedure within the spherical coor-dinates Then we have the deformation transformation andthe coordinate transformation groups in Cartesian coordi-nates for the same flat space-times and curved 119891(119877) gravitySchwarzschild-type black hole space-time

2 Transformation Groups inSpherical Coordinates

Metric tensors of GR and 119891(119877) gravity can be transformedinto each other by the conformal rescaling transformation119892ℎ119903 = Φℎ119903 = 1198911015840119877 (119877) ℎ119903 (1)

where119892ℎ119903 is themetric tensor in Einstein frame inGR and ℎ119903is themetric tensor in Jordan frame in119891(119877) gravity HereΦ =1198911015840119877(119877) is the potential related to the expansion of the universe1198911015840119877(119877) = 120597119891(119877)120597119877 and 119891(119877) is the modified function of theRicci scalar in119891(119877) gravity theory and119891(119877) is used in place ofthe Ricci scalar in Einstein-Hilbert actionThe solution of theaction integral with 119891(119877) modified Einstein-Hilbert actionand the matter Lagrangian action gives the 119891(119877) gravity fieldequations [17ndash19]

In 119891(119877) gravity we can solve the analogous problemsfor the GR by using modified metric and modified field

equations of 119891(119877) gravity The interested problem here is thespherical symmetric Schwarzschild-type black hole solutionsanalogous to GR We will use Capozziellorsquos approach for thesolution [16 19 20] The general solution of this problemis for the constant Ricci Scalar 119877 = 1198770 the metric of thecurved space by the spherical symmetric Schwarzschild-typeblack hole in the perfect fluid expanding universe with stress-energy tensor 119879119894ℎ = (120588 + 119901)119906119894119906ℎ minus 119901119892119894ℎ1198891199042 = minus(1 + 1198961119903 + 13 (119902120594120588 minus 120582) 1199032)1198891199052+ (1 + 1198961119903 + 13 (119902120594120588 minus 120582) 1199032)minus1 1198891199032 + 11990321198891205792+ 1199032sin21205791198891206012

(2)

when 119901 = minus120588 120582 = minus119891(1198770)21198911015840(1198770) 120594 = 8120587119866119888minus4 and119902minus1 = 1198911015840(1198770) Also when we look at (1) and compare itwith the usual spherical symmetric Schwarzschild black holemetric in GR we confirm the 11989200 and 11989211 terms are inverseof one another and there is a conformal transformation onGR and119891(119877)metrics in these termsThe 11989222 and 11989233 terms in119891(119877) are equal to the ones in GR and the spherical symmetryis seen

The coordinate transformations between flat tangentvector space-time and curved spherically symmetricSchwarzschild-type black hole space-time in the frameworkof 119891(119877) gravity can now be investigated Firstly thedeformation matrix between these two diagonal metrictensors of the flat and curved space-time can be found andwith the use of this matrix the coordinate transformations arefound between these two space-times Then the deformationtransformation is

119892119886119887 = 1205971199091198861015840120597119909119886 1205971199091198871015840120597119909119887 11989211988610158401198871015840 (3)

where the indices run as 119886 = (119905 119903 120579 120601) and 1198861015840 = (1199051015840 1199031015840 1205791015840 1206011015840)for the flat and curved space-times respectively Since themetric tensors are diagonal four by four matrices here (3)gives the deformation matrix between two metric tensorssuch as

(119892119905119905 0 0 00 119892119903119903 0 00 0 119892120579120579 00 0 0 119892120601120601)=119872(11989211990510158401199051015840 0 0 00 11989211990310158401199031015840 0 00 0 11989212057910158401205791015840 00 0 0 11989212060110158401206011015840)

(4)

Journal of Gravity 3

119872 =(((((((((

(1205971199051015840120597119905 )2 (1205971199031015840120597119905 )2 (1205971205791015840120597119905 )2 (1205971206011015840120597119905 )2(1205971199051015840120597119903 )2 (1205971199031015840120597119903 )2 (1205971205791015840120597119903 )2 (1205971206011015840120597119903 )2(1205971199051015840120597120579 )2 (1205971199031015840120597120579 )2 (1205971205791015840120597120579 )2 (1205971206011015840120597120579 )2(1205971199051015840120597120601)2 (1205971199031015840120597120601 )2 (1205971205791015840120597120601 )2 (1205971206011015840120597120601 )2))))))))) (5)

The deformations transformations in (3) are called the con-formal transformations of the metrics [16 21] We infer thatthe coordinate transformation matrix is a diagonal four byfour matrix from (4) The metrics for flat space-time andspherically symmetric Schwarzschild-type black hole space-time in the framework of 119891(119877) gravity are1198891199042 = minus1198891199052 + 1198891199032 + 11990321198891205792 + 1199032sin21205791198891206012 (6)1198891199042 = minus1198801198891199052 + 119880minus11198891199032 + 11990321198891205792 + 1199032sin21205791198891206012 (7)

where119880 = 1+(1198961119903)+(119902120594120588minus120582)11990323 respectively [7ndash9 13 20]For this work in spherical coordinates the nonzero coordinatetransformation matrix elements are1205971199051015840120597119905 = radic(1 + 1198961119903 + 13 (119902120594120588 minus 120582) 1199032)minus1 = radic119880minus11205971199031015840120597119903 = radic(1 + 1198961119903 + 13 (119902120594120588 minus 120582) 1199032) = radic1198801205971205791015840120597120579 = 11205971206011015840120597120601 = 1

(8)

Summing all of these by using the coordinate transforma-tions rule 1199091198861015840 = 1205971199091198861015840120597119909119886 119909119886 (9)

we obtain the coordinate transformations from flat tangentvector space-time to curved space-time of 119891(119877) gravity suchas 1199051015840 = 119905radic(1 + 1198961119903 + 13 (119902120594120588 minus 120582) 1199032)minus1 (10)

1199031015840 = 119903radic(1 + 1198961119903 + 13 (119902120594120588 minus 120582) 1199032) (11)1205791015840 = 120579 (12)1206011015840 = 120601 (13)

It is the convenient part for the study of related effectsof length contraction and time dilation between inertial

and noninertial frames in 119891(119877) Schwarzschild-type blackhole curved space-time So the time dilation and lengthcontraction can be inferred from the solutions of (10) and(11) for point effects The point effect means that when a timetransformation is made between frames it is assumed to bethe space coordinate of the event that is the same or if aspace transformation is made then the time is assumed tobe simultaneous between two events in the framesThe resultfollows as we get the time dilation equation by using (10)Δ1199051015840 = radic(1 + 1198961119903 + 13 (119902120594120588 minus 120582) 1199032)minus1Δ119905 (14)

and for length contraction from (11)Δ1199031015840 = radic(1 + 1198961119903 + 13 (119902120594120588 minus 120582) 1199032)Δ119903 (15)

If we call the four by four coordinate transformation matrixfor these space-times having these metrics as Λ it has theform

Λ =((((((

1205971199051015840120597119905 0 0 00 1205971199031015840120597119903 0 00 0 1205971205791015840120597120579 00 0 0 1205971206011015840120597120601))))))

=(radic119880minus1 0 0 00 radic119880 0 00 0 1 00 0 0 1)(16)

In this study we investigate the symmetry group of thecoordinate transformation matrix Λ The coordinate trans-formations leave the space-time interval 1198891199042 invariant so thesymmetry occurs for these transformations It is obvious thatthe determinant of the matrix Λ is

detΛ = 1 (17)

Therefore the symmetry group of the matrix Λ is found to beSL(4) for the coordinate transformations in spherical coor-dinates In addition the symmetry group of the conformaldeformation matrix119872 is again obtained by the determinantof119872 such that

119872 =(119880minus1 0 0 00 119880 0 00 0 1 00 0 0 1) (18)

Computing the determinant of119872 obviously gives

det119872 = 1 (19)

4 Journal of Gravity

We obtain a special linear symmetry group for the four byfour conformal deformation matrix119872 So it also belongs toSL(4) as the matrix Λ does The deformation matrices andconsequently the coordinate transformation matrices whichcan directly be obtained from the deformation matricesbelong to general linear groupGL(4) that is a general fact [16]Here these transformation matrices between flat and curvedspherically symmetric Schwarzschild-type black hole space-time in the framework of119891(119877) gravity are found to be in SL(4)symmetry group

Now we will investigate the deformation transformationand coordinate transformations in the Cartesian coordinatesfor the flat space-time and the Schwarzschild-type black holespace-time in the framework of 119891(119877) gravity3 Transformation Groups in

Cartesian Coordinates

It is a common example that in 2-dimensional Euclideanspace a vector can be transformed as a rotation by anorthogonal matrix belonging to SO(2) groups Here we infact imply that the transformations for the components ofthe vector are considered to be in Cartesian coordinates Ifwe consider the components of the transformed vector inthe spherical coordinates we obtain a GL(2) transformationmatrix for the same event Therefore for the same events inSection 2 we now investigate the symmetry groups of thedeformation transformations and the coordinate transforma-tions in Cartesian coordinates

For this we should first transform the metrics fromthe spherical coordinates to the Cartesian coordinates forthe Schwarzschild-type black hole curved space-time Sincethe space-time interval is invariant for both spherical andCartesian coordinates we use the fact that1198921205831015840]1015840 = 12059711990911988610158401205971199091205831015840 1205971199091198871015840120597119909]1015840 11989211988610158401198871015840 (20)

where 1198861015840 = (1199051015840 1199031015840 1205791015840 1206011015840) and 1205831015840 = (1199051015840 1199091015840 1199101015840 1199111015840) We needto find the elements of the metric tensor in the Cartesiancoordinates from (20) For this we will use the elements of themetric tensor in the spherical coordinates from (7) We alsoneed the transformation elements 12059711990911988610158401205971199091205831015840 from sphericalto Cartesian coordinate To find these elements we use 1199091015840 =1199031015840 sin 1205791015840 cos1206011015840 1199101015840 = 1199031015840 sin 1205791015840 sin1206011015840 1199111015840 = 1199031015840 cos 1205791015840 and 119903 =radic11990910158402 + 11991010158402 + 11991110158402 After a little calculus we obtain the elementsof the metric tensor in Cartesian coordinates but in terms ofthe spherical coordinates11989211990510158401199051015840 = minus(1 + 1198961119903 + 13 (119902120594120588 minus 120582) 1199032) = minus11988011989211990910158401199091015840 = (sin 1205791015840 cos1206011015840)2119880minus1 + (cos 1205791015840 cos1206011015840)2+ (sin1206011015840)2 11989211991010158401199101015840 = (sin 1205791015840 sin1206011015840)2119880minus1 + (cos 1205791015840 sin1206011015840)2

+ (cos1206011015840)2 11989211991110158401199111015840 = (cos 1205791015840)2119880minus1 + (sin 1205791015840)2 11989211990910158401199101015840 = 11989211991010158401199091015840= [(sin 1205791015840)2119880minus1 + (cos 1205791015840)2 minus 1] cos1206011015840 sin120601101584011989211991110158401199091015840 = 11989211990910158401199111015840= (sin 1205791015840 cos1206011015840) cos 1205791015840119880minus1+ (cos 1205791015840 cos1206011015840) (minus sin 1205791015840) 11989211991110158401199101015840 = 11989211991010158401199111015840= (sin 1205791015840 sin1206011015840) cos 1205791015840119880minus1+ (cos 1205791015840 sin1206011015840) (minus sin 1205791015840) (21)

Nowwe should use the Cartesian coordinates for all sphericalcoordinates in the elements of the metric such that11989211990510158401199051015840 = minus119880 (22)11989211990910158401199091015840 = 11990910158402(11990910158402 + 11991010158402 + 11991110158402)119880minus1+ 1199091015840211991110158402(11990910158402 + 11991010158402 + 11991110158402) (11990910158402 + 11991010158402) + 11991010158402(11990910158402 + 11991010158402) (23)

11989211991010158401199101015840 = 11991010158402(11990910158402 + 11991010158402 + 11991110158402)119880minus1+ 1199101015840211991110158402(11990910158402 + 11991010158402 + 11991110158402) (11990910158402 + 11991010158402) + 11990910158402(11990910158402 + 11991010158402) (24)

11989211991110158401199111015840 = 11991110158402(11990910158402 + 11991010158402 + 11991110158402)119880minus1 + 11990910158402 + 11991010158402(11990910158402 + 11991010158402 + 11991110158402) (25)11989211990910158401199101015840 = 11989211991010158401199091015840= 11990910158401199101015840(11990910158402 + 11991010158402 + 11991110158402)119880minus1+ 1199091015840119910101584011991110158402(11990910158402 + 11991010158402 + 11991110158402) (11990910158402 + 11991010158402) minus 11990910158401199101015840(11990910158402 + 11991010158402) (26)

11989211990910158401199111015840 = 11989211991110158401199091015840= 11990910158401199111015840(11990910158402 + 11991010158402 + 11991110158402)119880minus1 minus 11990910158401199111015840(11990910158402 + 11991010158402 + 11991110158402) (27)

11989211991010158401199111015840 = 11989211991110158401199101015840= 11991010158401199111015840(11990910158402 + 11991010158402 + 11991110158402)119880minus1 minus 11991010158401199111015840(11990910158402 + 11991010158402 + 11991110158402) (28)

Journal of Gravity 5

Finally we obtain the elements of the metric tensor for thecurved space-time of Schwarzschild-type black hole in termsof the Cartesian coordinates The indices for this metricwere shown by the primed Greek letters Since we investigatethe transformations between flat and curved space-times inCartesian coordinates we use the flat Cartesian metric 119892120583]whose indices are shown by the unprimed Greek letters suchthat

119892120583] =(minus1 0 0 00 1 0 00 0 1 00 0 0 1) (29)

Then the deformation transformations between the metrics119892120583] and 1198921205831015840]1015840 are 119892120583] = 1205971199091205831015840120597119909120583 120597119909]1015840120597119909] 1198921205831015840]1015840 (30)

We cannot call the deformation transformations in (30)conformal transformations of the metrics because both ofthe metrics are not in diagonal form As seen in (26)ndash(28)the curved space-time metric has nondiagonal terms Nowthe deformation matrix can be constructed by arranging (30)as below

(119892119905119905 0 0 00 119892119909119909 0 00 0 119892119910119910 00 0 0 119892119911119911)=119872(11989211990510158401199051015840 0 0 00 11989211990910158401199091015840 11989211990910158401199101015840 119892119909101584011991110158400 11989211991010158401199091015840 11989211991010158401199101015840 119892119910101584011991110158400 11989211991110158401199091015840 11989211991110158401199101015840 11989211991110158401199111015840)

(31)

and only nonzero elements of the matrix119872 forms as

119872 =(119860 0 0 00 119861 119864 1198650 119864 119862 1198660 119865 119866 119863) (32)

To find the deformation matrix M from (31) is not so easyInstead of that we write the inverse transformations in (31)and obtain the inverse deformation matrix119872minus1 in an easiermannerThen we find the matrix119872 by evaluating the inverseof the matrix119872minus1 Consequently the matrix119872 is found tohave the elements below

119860 = minus 111989211990510158401199051015840 119861 = minus1198921199101015840119910101584011989211991110158401199111015840 + 119892119910101584011991110158402minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402 119862 = minus1198921199091015840119909101584011989211991110158401199111015840 + 119892119909101584011991110158402minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402 119863 = minus1198921199091015840119909101584011989211991010158401199101015840 + 119892119909101584011991010158402minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402 119864 = 1198921199111015840119911101584011989211990910158401199101015840 minus 1198921199101015840119911101584011989211990910158401199111015840minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402 119865 = 1198921199101015840119910101584011989211990910158401199111015840 minus 1198921199091015840119910101584011989211991010158401199111015840minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402 119866 = 1198921199091015840119909101584011989211991010158401199111015840 minus 1198921199091015840119910101584011989211990910158401199111015840minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402

(33)

By using (22)ndash(28) in (33) and making the simplifications forthese matrix elements we finally obtain these elements asfollows 119860 = 1119880 (34)

119861 = 1198801199092 + 1199102 + 11991121199092 + 1199102 + 1199112 (35)

119862 = 1198801199102 + 1199092 + 11991121199092 + 1199102 + 1199112 (36)

119863 = 1198801199112 + 1199092 + 11991021199092 + 1199102 + 1199112 (37)

119864 = 119880119909119910 minus 1199091199101199092 + 1199102 + 1199112 (38)

6 Journal of Gravity

119865 = 119880119909119911 minus 1199091199111199092 + 1199102 + 1199112 (39)119866 = 119880119910119911 minus 1199101199111199092 + 1199102 + 1199112 (40)

These are the elements of the deformation matrix119872 To findthe determinant of matrix119872 we use (34)ndash(40) in (32) andthen we evaluate the determinant such as

det119872 = 1 (41)

We understand that our matrix belongs to SL(4) because ofthe determinant This is in accordance with the general factthat the symmetry group of the deformation matrices belongto GL(4)

Now we are to find the coordinate transformationsbetween these two space-times having the deformationtransformations given by (30)ndash(32) We need to find theinvertible matrix elements 1205971199091205831015840120597119909] between flat and curvedspace-times in Cartesian coordinates Therefore we write thematrix multiplication in (31) explicitly Then we also writethe deformation equations with summation convention in(30) explicitly We equate the metric elements 119892120583] in bothexplicit forms then we obtain the invertible matrix elements1205971199091205831015840120597119909] From these elements we construct the coordinatetransformation matrix Λ directly as given below

Λ =(((((((

1205971199051015840120597119905 0 0 00 1205971199091015840120597119909 1205971199091015840120597119910 12059711990910158401205971199110 1205971199101015840120597119909 1205971199101015840120597119910 12059711991010158401205971199110 1205971199111015840120597119909 1205971199111015840120597119910 1205971199111015840120597119911)))))))

=((((radic119860 0 0 00 radic119861 119864radic119862 119865radic1198630 119864radic119861 radic119862 119866radic1198630 119865radic119861 119866radic119862 radic119863

))))

(42)

We can write these elements more explicitly by using (34)ndash(40) and obtain 1205971199051015840120597119905 = radic 1119880 (43)1205971199091015840120597119909 = radic1198801199092 + 1199102 + 11991121199092 + 1199102 + 1199112 (44)

1205971199101015840120597119910 = radic1199092 + 1198801199102 + 11991121199092 + 1199102 + 1199112 (45)

1205971199111015840120597119911 = radic1199092 + 1199102 + 11988011991121199092 + 1199102 + 1199112 (46)

1205971199091015840120597119910 = (119880119909119910 minus 119909119910) (1199092 + 1199102 + 1199112)radic(1199092 + 1198801199102 + 1199112) (1199092 + 1199102 + 1199112) (47)

1205971199091015840120597119911 = (119880119909119911 minus 119909119911) (1199092 + 1199102 + 1199112)radic(1199092 + 1199102 + 1198801199112) (1199092 + 1199102 + 1199112) (48)

1205971199101015840120597119909 = (119880119909119910 minus 119909119910) (1199092 + 1199102 + 1199112)radic(1198801199092 + 1199102 + 1199112) (1199092 + 1199102 + 1199112) (49)

1205971199101015840120597119911 = (119880119910119911 minus 119910119911) (1199092 + 1199102 + 1199112)radic(1199092 + 1199102 + 1198801199112) (1199092 + 1199102 + 1199112) (50)

1205971199111015840120597119909 = (119880119909119911 minus 119909119911) (1199092 + 1199102 + 1199112)radic(1198801199092 + 1199102 + 1199112) (1199092 + 1199102 + 1199112) (51)

1205971199111015840120597119910 = (119880119910119911 minus 119910119911) (1199092 + 1199102 + 1199112)radic(1199092 + 1198801199102 + 1199112) (1199092 + 1199102 + 1199112) (52)

When we insert these elements into the coordinate transfor-mation matrix Λ we evaluate its determinant as follows

det119872= radic119880(1199092 + 1199102 + 1199112) (1199092 + 1199102 + 1199112) (1199092 + 1199102 + 1199112)radic(1198801199092 + 1199102 + 1199112) (1199092 + 1198801199102 + 1199112) (1199092 + 1199102 + 1198801199112) (53)

or

det119872= radic119880(1199092 + 1199102 + 1199112)32radic(1198801199092 + 1199102 + 1199112) (1199092 + 1198801199102 + 1199112) (1199092 + 1199102 + 1198801199112) (54)

We infer from the nonzero determinant of the coordinatetransformation matrix Λ that it is invertible and it belongsto the GL(4) symmetry group since it is a four by fourmatrixThis is again in accordance with the general fact aboutthe deformation matrices and their resulting coordinatetransformation matrices

The deformation matrix 119872 and the coordinate trans-formation matrix Λ are shown to belong to SL(4) andGL(4) respectively In Section 2 for the spherical coordinatescase we found that both matrices belong to SL(4) forthe transformations from flat tangent vector space-time tocurved Schwarzschild-type black hole space-time in theframework of 119891(119877) gravity These symmetry groups resultfrom the invariance of the space-time interval1198891199042 under thesetransformations

Similarly one can also find the deformation and coor-dinate transformation groups between the inertial frames

Journal of Gravity 7

and the noninertial frames of 119891(119877) gravity axially symmet-ric Kerr-Type and charged Reissner-Nordstrom-Type blackholes because their metric tensors have nondiagonal termsas in the case of Cartesian coordinates representations (22)ndash(28) for the metric of Schwarzschild-type black hole curvedspace-time (31) The metric tensor of a charged and rotating119891(119877) gravity black hole is given as [22]1198891199042 = minusΔ 1199031205882 [119889119905 minus 119886 sin2120579Ξ 119889120601]2 + 1205882Δ 119903 1198891199032 + 1205882Δ 120579 1198891205792+ Δ 120579sin21205791205882 [119886119889119905 minus 1199032 + 1198862Ξ 119889120601]2 (55)

whereΔ 119903 = (1199032 + 1198862) (1 + 119877012 1199032) minus 2119872119903 + 11987621 + 1198911015840 (1198770) Ξ = 1 minus 1198770121198862Δ 120579 = 1 minus 1198770121198862cos2120579(56)

Also 119886 and 119876 are the angular momentum and the charge ofthe 119891(119877) gravity black hole respectively If 119886 = 0 (55) turnsto be a Reissner-Nordstrom-Type black hole in 119891(119877) gravitywhereas if 119876 = 0 it gives a Kerr-Type 119891(119877) gravity blackholeWe can usemetric (55) for bothKerr-Type andReissner-Nordstrom-Type 119891(119877) gravity black holes and rewrite it as

1198891199042 = 1198862Δ 120579sin2120579 minus Δ 1199031205882 1198891199052+ 2119886 sin21205791205882Ξ [Δ 119903 minus Δ 120579 (1199032 + 1198862)] 119889119905119889120601+ 1205882Δ 119903 1198891199032 + 1205882Δ 120579 1198891205792+ Δ 120579sin2120579 (1199032 + 1198862)2 minus Δ 1199031198862sin41205791205882 1198891206012

(57)

and in matrix form

119892120583] =((((((((

1198862Δ 120579sin2120579 minus Δ 1199031205882 0 0 2119886 sin21205791205882Ξ [Δ 119903 minus Δ 120579 (1199032 + 1198862)]0 1205882Δ 119903 0 00 0 1205882Δ 120579 02119886 sin21205791205882Ξ [Δ 119903 minus Δ 120579 (1199032 + 1198862)] 0 0 Δ 120579 sin2120579 (1199032 + 1198862)2 minus Δ 1199031198862sin41205791205882)))))))) (58)

By using the similar steps in (31) and (32) we can obtainthe deformation matrix 119872 for the inertial observer andthe noninertial observer in the space-time curved by Kerr-Type or Reissner-Nordstrom-Type 119891(119877) gravity black holesAccordingly from the same step in (42) we then obtainthe coordinate transformation matrix Λ for these observersrsquoframe

4 Conclusions

In this paper we consider the transformations between acurved and flat space-time being the tangent vector space-time of the curved space-time and also the coordinates ofthis tangent space are called the local Lorentz coordinatesThe curvature of the space-time used here results fromthe Schwarzschild-type black hole whose solution for themetric tensor is obtained in the frameworks of modified119891(119877) gravity [7ndash9] We investigated the symmetry groups ofthe deformation matrix for the transformations of metricsand then the coordinate transformation matrix obtained by

this deformation matrix The extended and straightforwardphysical meaning and interpretations of the deformationsand conformal transformations can be found in the literaturesee [16 23 24]

These deformation and coordinate transformationmatri-ces are the identity matrices for the case 119880 = 1 This meansthat the both space-times are same flat space-timesThereforethe transformation matrices for deformation and coordinatetransformations must be identity matrix in both coordinatesystems

When we use 119880 = 1 in spherical coordinates forthe transformation matrices 119872 and Λ in (18) and (16)119908respectively we can easily realize that they are the identitymatrices Similarly if we again use 119880 = 1 in Cartesiancoordinates we find thematrices119872 andΛ in (32) and (42) arethe identity matrices Namely for 119880 = 1 all the nondiagonalelements of 119872 in (38) and (40) vanish In addition all thenondiagonal elements of Λ in (47) and (52) again vanish for119880 = 1 Taking 119880 = 1 can be viewed as a limiting case forchecking the calculations for transformation matrices whosesymmetry groups are investigated

8 Journal of Gravity

We infer that the deformation matrices the coordinatetransformation matrices and their symmetry groups can beinvestigated for the other type of black hole solutions in theframework ofmodified119891(119877) gravity Alsowe can consider thepresent transformation groups for the other type of modifiedgravitation theories for these various black hole solutions asthe candidate studies related to this topic

Competing Interests

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

References

[1] R A Nelson ldquoGeneralized Lorentz transformation for an accel-erated rotating frame of referencerdquo Journal of MathematicalPhysics vol 28 no 10 pp 2379ndash2383 1987

[2] H Nikolic ldquoRelativistic contraction of an accelerated rodrdquoAmerican Journal of Physics vol 67 no 11 pp 1007ndash1012 1999

[3] H Nikolic ldquoRelativistic contraction and related effects innoninertial framesrdquo Physical Review A vol 61 no 3 Article ID032109 8 pages 2000

[4] O Gron ldquoAcceleration and weight of extended bodies in thetheory of relativityrdquo American Journal of Physics vol 45 no 1pp 65ndash70 1977

[5] O Gron ldquoRelativictic description of a rotating diskrdquo AmericanJournal of Physics vol 43 pp 869ndash876 1975

[6] R DrsquoInverno Introtucing Einsteinrsquos Relativity OxfordUniversityPress Oxford UK 1992

[7] S Capozziello and M D Laurentis ldquoBlack holes and stellarstructures in f(R)-gravityrdquo httpsarxivorgabs12020394

[8] S Nojiri and S D Odintsov ldquoUnified cosmic history inmodified gravity from F(R) theory to Lorentz non-invariantmodelsrdquo Physics Reports vol 505 no 2ndash4 pp 59ndash144 2011

[9] S Nojiri and S D Odintsov ldquoIntroduction to modified gravityand gravitational alternative for dark energyrdquo InternationalJournal of Geometric Methods in Modern Physics vol 4 no 1pp 115ndash145 2007

[10] S Capozziello ldquoCurvature quintessencerdquo International Journalof Modern Physics D vol 11 no 4 pp 483ndash491 2002

[11] S Capozziello and S Vignolo ldquoA comment on lsquoThe Cauchyproblem of 119891(119877) gravityrsquordquo Classical and Quantum Gravity vol26 no 16 Article ID 168001 2009

[12] S Capozziello A Stabile and A Troisi ldquoSpherically symmetricsolutions in f (R) gravity via the Noether symmetry approachrdquoClassical and Quantum Gravity vol 24 no 8 pp 2153ndash21662007

[13] S Capozziello A Stabile and A Troisi ldquoSpherical symmetryin 119891(119877)-gravityrdquo Classical and Quantum Gravity vol 25 no 8Article ID 085004 2008

[14] B Coll ldquoA universal law of gravitational deformation forGeneral Relativityrdquo in Proceedings of the Spanish RelativisticMeeting EREs Salamanca Spain 1998

[15] B Coll J Llosa and D Soler ldquoThree-dimensional metrics asdeformations of a constant curvaturemetricrdquoGeneral Relativityand Gravitation vol 34 no 2 pp 269ndash282 2002

[16] S Capozziello and M D Laurentis ldquoExtended theories ofgravityrdquo Physics Reports vol 509 no 4-5 pp 167ndash321 2011

[17] H A Buchdahl ldquoMonthly notices of the royal astronomicalsocietyrdquoMonthly Notices of the Royal Astronomical Society vol150 no 1 pp 1ndash8 1970

[18] A A Starobinsky ldquoA new type of isotropic cosmological modelswithout singularityrdquo Physics Letters B vol 91 no 1 pp 99ndash1021980

[19] H Kleinert and H-J Schmidt ldquoCosmology with curvature-saturated gravitational lagrangian 119877radic1 + 11989741198772rdquo General Rela-tivity and Gravitation vol 34 no 8 pp 1295ndash1318 2002

[20] J D Barrow and A C Ottewill ldquoThe stability of general rela-tivistic cosmological theoryrdquo Journal of Physics AMathematicaland General vol 16 no 12 pp 2757ndash2776 1983

[21] M CarmeliGroupTheory and General Relativity McGraw-HillInternational Book London UK 1977

[22] A Larranaga ldquoA rotating charged black hole solution in 119891(119877)gravityrdquo Pramana vol 78 no 5 pp 697ndash703 2012

[23] N Lanahan-Tremblay and V Faraoni ldquoThe Cauchy problem off (R) gravityrdquo Classical and QuantumGravity vol 24 no 22 pp5667ndash5679 2007

[24] G Allemandi M Capone S Capozziello and M FrancaviglialdquoConformal aspects of the Palatini approach in extended theo-ries of gravityrdquo General Relativity and Gravitation vol 38 no 1pp 33ndash60 2006

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 Journal of Gravity

solutions of Schwarzschild and Kerr solutions of GR forexample So the methods for the extended theories should bechecked with the solutions of GR [12]

For the spherically symmetric Schwarzschild black holesolutions of GR we will refer to the spherically symmetriccorresponding Schwarzschild-type black hole solutions ofCapozziello et al [13] Then in this curved and expandedspace of Schwarzschild-type black hole we will study thecoordinate transformations from a noninertial frame to aninertial frame being the tangent vector space-time of theSchwarzschild-type space-time and this flat space-time isrepresented by the local Lorentz coordinates (LLC) Fromthese transformations we will obtain the related effects suchas length contraction and time dilation for the 119891(119877) gravityof Schwarzschild-type black hole

The coordinate transformation between flat tangent vec-tor space-time and the curved space-time of Schwarzschild-type black hole of 119891(119877) gravity will be constructed by thedeformation transformations of the flat space-time metric[14ndash16] In spherical coordinates this transformation is aconformal transformation between the flat and curved space-time metrics Similarly by using the transformations fromspherical to Cartesian coordinates we obtain the metricsof both space-times in Cartesian coordinates and thenwe construct the deformation transformation matrices ofthese two metrics in Cartesian coordinates and obtain thecoordinates transformations in Cartesian coordinates for aSchwarzschild-type black hole in 119891(119877) gravity

Consequently we firstly find the deformation transfor-mations of metrics in spherical coordinates and the coor-dinate transformations by using the deformation matrixThen we find the deformation transformation groups and thecoordinate transformation groups in spherical coordinatesSecondly we convert the metrics in Cartesian coordinatesand follow the same procedure within the spherical coor-dinates Then we have the deformation transformation andthe coordinate transformation groups in Cartesian coordi-nates for the same flat space-times and curved 119891(119877) gravitySchwarzschild-type black hole space-time

2 Transformation Groups inSpherical Coordinates

Metric tensors of GR and 119891(119877) gravity can be transformedinto each other by the conformal rescaling transformation119892ℎ119903 = Φℎ119903 = 1198911015840119877 (119877) ℎ119903 (1)

where119892ℎ119903 is themetric tensor in Einstein frame inGR and ℎ119903is themetric tensor in Jordan frame in119891(119877) gravity HereΦ =1198911015840119877(119877) is the potential related to the expansion of the universe1198911015840119877(119877) = 120597119891(119877)120597119877 and 119891(119877) is the modified function of theRicci scalar in119891(119877) gravity theory and119891(119877) is used in place ofthe Ricci scalar in Einstein-Hilbert actionThe solution of theaction integral with 119891(119877) modified Einstein-Hilbert actionand the matter Lagrangian action gives the 119891(119877) gravity fieldequations [17ndash19]

In 119891(119877) gravity we can solve the analogous problemsfor the GR by using modified metric and modified field

equations of 119891(119877) gravity The interested problem here is thespherical symmetric Schwarzschild-type black hole solutionsanalogous to GR We will use Capozziellorsquos approach for thesolution [16 19 20] The general solution of this problemis for the constant Ricci Scalar 119877 = 1198770 the metric of thecurved space by the spherical symmetric Schwarzschild-typeblack hole in the perfect fluid expanding universe with stress-energy tensor 119879119894ℎ = (120588 + 119901)119906119894119906ℎ minus 119901119892119894ℎ1198891199042 = minus(1 + 1198961119903 + 13 (119902120594120588 minus 120582) 1199032)1198891199052+ (1 + 1198961119903 + 13 (119902120594120588 minus 120582) 1199032)minus1 1198891199032 + 11990321198891205792+ 1199032sin21205791198891206012

(2)

when 119901 = minus120588 120582 = minus119891(1198770)21198911015840(1198770) 120594 = 8120587119866119888minus4 and119902minus1 = 1198911015840(1198770) Also when we look at (1) and compare itwith the usual spherical symmetric Schwarzschild black holemetric in GR we confirm the 11989200 and 11989211 terms are inverseof one another and there is a conformal transformation onGR and119891(119877)metrics in these termsThe 11989222 and 11989233 terms in119891(119877) are equal to the ones in GR and the spherical symmetryis seen

The coordinate transformations between flat tangentvector space-time and curved spherically symmetricSchwarzschild-type black hole space-time in the frameworkof 119891(119877) gravity can now be investigated Firstly thedeformation matrix between these two diagonal metrictensors of the flat and curved space-time can be found andwith the use of this matrix the coordinate transformations arefound between these two space-times Then the deformationtransformation is

119892119886119887 = 1205971199091198861015840120597119909119886 1205971199091198871015840120597119909119887 11989211988610158401198871015840 (3)

where the indices run as 119886 = (119905 119903 120579 120601) and 1198861015840 = (1199051015840 1199031015840 1205791015840 1206011015840)for the flat and curved space-times respectively Since themetric tensors are diagonal four by four matrices here (3)gives the deformation matrix between two metric tensorssuch as

(119892119905119905 0 0 00 119892119903119903 0 00 0 119892120579120579 00 0 0 119892120601120601)=119872(11989211990510158401199051015840 0 0 00 11989211990310158401199031015840 0 00 0 11989212057910158401205791015840 00 0 0 11989212060110158401206011015840)

(4)

Journal of Gravity 3

119872 =(((((((((

(1205971199051015840120597119905 )2 (1205971199031015840120597119905 )2 (1205971205791015840120597119905 )2 (1205971206011015840120597119905 )2(1205971199051015840120597119903 )2 (1205971199031015840120597119903 )2 (1205971205791015840120597119903 )2 (1205971206011015840120597119903 )2(1205971199051015840120597120579 )2 (1205971199031015840120597120579 )2 (1205971205791015840120597120579 )2 (1205971206011015840120597120579 )2(1205971199051015840120597120601)2 (1205971199031015840120597120601 )2 (1205971205791015840120597120601 )2 (1205971206011015840120597120601 )2))))))))) (5)

The deformations transformations in (3) are called the con-formal transformations of the metrics [16 21] We infer thatthe coordinate transformation matrix is a diagonal four byfour matrix from (4) The metrics for flat space-time andspherically symmetric Schwarzschild-type black hole space-time in the framework of 119891(119877) gravity are1198891199042 = minus1198891199052 + 1198891199032 + 11990321198891205792 + 1199032sin21205791198891206012 (6)1198891199042 = minus1198801198891199052 + 119880minus11198891199032 + 11990321198891205792 + 1199032sin21205791198891206012 (7)

where119880 = 1+(1198961119903)+(119902120594120588minus120582)11990323 respectively [7ndash9 13 20]For this work in spherical coordinates the nonzero coordinatetransformation matrix elements are1205971199051015840120597119905 = radic(1 + 1198961119903 + 13 (119902120594120588 minus 120582) 1199032)minus1 = radic119880minus11205971199031015840120597119903 = radic(1 + 1198961119903 + 13 (119902120594120588 minus 120582) 1199032) = radic1198801205971205791015840120597120579 = 11205971206011015840120597120601 = 1

(8)

Summing all of these by using the coordinate transforma-tions rule 1199091198861015840 = 1205971199091198861015840120597119909119886 119909119886 (9)

we obtain the coordinate transformations from flat tangentvector space-time to curved space-time of 119891(119877) gravity suchas 1199051015840 = 119905radic(1 + 1198961119903 + 13 (119902120594120588 minus 120582) 1199032)minus1 (10)

1199031015840 = 119903radic(1 + 1198961119903 + 13 (119902120594120588 minus 120582) 1199032) (11)1205791015840 = 120579 (12)1206011015840 = 120601 (13)

It is the convenient part for the study of related effectsof length contraction and time dilation between inertial

and noninertial frames in 119891(119877) Schwarzschild-type blackhole curved space-time So the time dilation and lengthcontraction can be inferred from the solutions of (10) and(11) for point effects The point effect means that when a timetransformation is made between frames it is assumed to bethe space coordinate of the event that is the same or if aspace transformation is made then the time is assumed tobe simultaneous between two events in the framesThe resultfollows as we get the time dilation equation by using (10)Δ1199051015840 = radic(1 + 1198961119903 + 13 (119902120594120588 minus 120582) 1199032)minus1Δ119905 (14)

and for length contraction from (11)Δ1199031015840 = radic(1 + 1198961119903 + 13 (119902120594120588 minus 120582) 1199032)Δ119903 (15)

If we call the four by four coordinate transformation matrixfor these space-times having these metrics as Λ it has theform

Λ =((((((

1205971199051015840120597119905 0 0 00 1205971199031015840120597119903 0 00 0 1205971205791015840120597120579 00 0 0 1205971206011015840120597120601))))))

=(radic119880minus1 0 0 00 radic119880 0 00 0 1 00 0 0 1)(16)

In this study we investigate the symmetry group of thecoordinate transformation matrix Λ The coordinate trans-formations leave the space-time interval 1198891199042 invariant so thesymmetry occurs for these transformations It is obvious thatthe determinant of the matrix Λ is

detΛ = 1 (17)

Therefore the symmetry group of the matrix Λ is found to beSL(4) for the coordinate transformations in spherical coor-dinates In addition the symmetry group of the conformaldeformation matrix119872 is again obtained by the determinantof119872 such that

119872 =(119880minus1 0 0 00 119880 0 00 0 1 00 0 0 1) (18)

Computing the determinant of119872 obviously gives

det119872 = 1 (19)

4 Journal of Gravity

We obtain a special linear symmetry group for the four byfour conformal deformation matrix119872 So it also belongs toSL(4) as the matrix Λ does The deformation matrices andconsequently the coordinate transformation matrices whichcan directly be obtained from the deformation matricesbelong to general linear groupGL(4) that is a general fact [16]Here these transformation matrices between flat and curvedspherically symmetric Schwarzschild-type black hole space-time in the framework of119891(119877) gravity are found to be in SL(4)symmetry group

Now we will investigate the deformation transformationand coordinate transformations in the Cartesian coordinatesfor the flat space-time and the Schwarzschild-type black holespace-time in the framework of 119891(119877) gravity3 Transformation Groups in

Cartesian Coordinates

It is a common example that in 2-dimensional Euclideanspace a vector can be transformed as a rotation by anorthogonal matrix belonging to SO(2) groups Here we infact imply that the transformations for the components ofthe vector are considered to be in Cartesian coordinates Ifwe consider the components of the transformed vector inthe spherical coordinates we obtain a GL(2) transformationmatrix for the same event Therefore for the same events inSection 2 we now investigate the symmetry groups of thedeformation transformations and the coordinate transforma-tions in Cartesian coordinates

For this we should first transform the metrics fromthe spherical coordinates to the Cartesian coordinates forthe Schwarzschild-type black hole curved space-time Sincethe space-time interval is invariant for both spherical andCartesian coordinates we use the fact that1198921205831015840]1015840 = 12059711990911988610158401205971199091205831015840 1205971199091198871015840120597119909]1015840 11989211988610158401198871015840 (20)

where 1198861015840 = (1199051015840 1199031015840 1205791015840 1206011015840) and 1205831015840 = (1199051015840 1199091015840 1199101015840 1199111015840) We needto find the elements of the metric tensor in the Cartesiancoordinates from (20) For this we will use the elements of themetric tensor in the spherical coordinates from (7) We alsoneed the transformation elements 12059711990911988610158401205971199091205831015840 from sphericalto Cartesian coordinate To find these elements we use 1199091015840 =1199031015840 sin 1205791015840 cos1206011015840 1199101015840 = 1199031015840 sin 1205791015840 sin1206011015840 1199111015840 = 1199031015840 cos 1205791015840 and 119903 =radic11990910158402 + 11991010158402 + 11991110158402 After a little calculus we obtain the elementsof the metric tensor in Cartesian coordinates but in terms ofthe spherical coordinates11989211990510158401199051015840 = minus(1 + 1198961119903 + 13 (119902120594120588 minus 120582) 1199032) = minus11988011989211990910158401199091015840 = (sin 1205791015840 cos1206011015840)2119880minus1 + (cos 1205791015840 cos1206011015840)2+ (sin1206011015840)2 11989211991010158401199101015840 = (sin 1205791015840 sin1206011015840)2119880minus1 + (cos 1205791015840 sin1206011015840)2

+ (cos1206011015840)2 11989211991110158401199111015840 = (cos 1205791015840)2119880minus1 + (sin 1205791015840)2 11989211990910158401199101015840 = 11989211991010158401199091015840= [(sin 1205791015840)2119880minus1 + (cos 1205791015840)2 minus 1] cos1206011015840 sin120601101584011989211991110158401199091015840 = 11989211990910158401199111015840= (sin 1205791015840 cos1206011015840) cos 1205791015840119880minus1+ (cos 1205791015840 cos1206011015840) (minus sin 1205791015840) 11989211991110158401199101015840 = 11989211991010158401199111015840= (sin 1205791015840 sin1206011015840) cos 1205791015840119880minus1+ (cos 1205791015840 sin1206011015840) (minus sin 1205791015840) (21)

Nowwe should use the Cartesian coordinates for all sphericalcoordinates in the elements of the metric such that11989211990510158401199051015840 = minus119880 (22)11989211990910158401199091015840 = 11990910158402(11990910158402 + 11991010158402 + 11991110158402)119880minus1+ 1199091015840211991110158402(11990910158402 + 11991010158402 + 11991110158402) (11990910158402 + 11991010158402) + 11991010158402(11990910158402 + 11991010158402) (23)

11989211991010158401199101015840 = 11991010158402(11990910158402 + 11991010158402 + 11991110158402)119880minus1+ 1199101015840211991110158402(11990910158402 + 11991010158402 + 11991110158402) (11990910158402 + 11991010158402) + 11990910158402(11990910158402 + 11991010158402) (24)

11989211991110158401199111015840 = 11991110158402(11990910158402 + 11991010158402 + 11991110158402)119880minus1 + 11990910158402 + 11991010158402(11990910158402 + 11991010158402 + 11991110158402) (25)11989211990910158401199101015840 = 11989211991010158401199091015840= 11990910158401199101015840(11990910158402 + 11991010158402 + 11991110158402)119880minus1+ 1199091015840119910101584011991110158402(11990910158402 + 11991010158402 + 11991110158402) (11990910158402 + 11991010158402) minus 11990910158401199101015840(11990910158402 + 11991010158402) (26)

11989211990910158401199111015840 = 11989211991110158401199091015840= 11990910158401199111015840(11990910158402 + 11991010158402 + 11991110158402)119880minus1 minus 11990910158401199111015840(11990910158402 + 11991010158402 + 11991110158402) (27)

11989211991010158401199111015840 = 11989211991110158401199101015840= 11991010158401199111015840(11990910158402 + 11991010158402 + 11991110158402)119880minus1 minus 11991010158401199111015840(11990910158402 + 11991010158402 + 11991110158402) (28)

Journal of Gravity 5

Finally we obtain the elements of the metric tensor for thecurved space-time of Schwarzschild-type black hole in termsof the Cartesian coordinates The indices for this metricwere shown by the primed Greek letters Since we investigatethe transformations between flat and curved space-times inCartesian coordinates we use the flat Cartesian metric 119892120583]whose indices are shown by the unprimed Greek letters suchthat

119892120583] =(minus1 0 0 00 1 0 00 0 1 00 0 0 1) (29)

Then the deformation transformations between the metrics119892120583] and 1198921205831015840]1015840 are 119892120583] = 1205971199091205831015840120597119909120583 120597119909]1015840120597119909] 1198921205831015840]1015840 (30)

We cannot call the deformation transformations in (30)conformal transformations of the metrics because both ofthe metrics are not in diagonal form As seen in (26)ndash(28)the curved space-time metric has nondiagonal terms Nowthe deformation matrix can be constructed by arranging (30)as below

(119892119905119905 0 0 00 119892119909119909 0 00 0 119892119910119910 00 0 0 119892119911119911)=119872(11989211990510158401199051015840 0 0 00 11989211990910158401199091015840 11989211990910158401199101015840 119892119909101584011991110158400 11989211991010158401199091015840 11989211991010158401199101015840 119892119910101584011991110158400 11989211991110158401199091015840 11989211991110158401199101015840 11989211991110158401199111015840)

(31)

and only nonzero elements of the matrix119872 forms as

119872 =(119860 0 0 00 119861 119864 1198650 119864 119862 1198660 119865 119866 119863) (32)

To find the deformation matrix M from (31) is not so easyInstead of that we write the inverse transformations in (31)and obtain the inverse deformation matrix119872minus1 in an easiermannerThen we find the matrix119872 by evaluating the inverseof the matrix119872minus1 Consequently the matrix119872 is found tohave the elements below

119860 = minus 111989211990510158401199051015840 119861 = minus1198921199101015840119910101584011989211991110158401199111015840 + 119892119910101584011991110158402minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402 119862 = minus1198921199091015840119909101584011989211991110158401199111015840 + 119892119909101584011991110158402minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402 119863 = minus1198921199091015840119909101584011989211991010158401199101015840 + 119892119909101584011991010158402minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402 119864 = 1198921199111015840119911101584011989211990910158401199101015840 minus 1198921199101015840119911101584011989211990910158401199111015840minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402 119865 = 1198921199101015840119910101584011989211990910158401199111015840 minus 1198921199091015840119910101584011989211991010158401199111015840minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402 119866 = 1198921199091015840119909101584011989211991010158401199111015840 minus 1198921199091015840119910101584011989211990910158401199111015840minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402

(33)

By using (22)ndash(28) in (33) and making the simplifications forthese matrix elements we finally obtain these elements asfollows 119860 = 1119880 (34)

119861 = 1198801199092 + 1199102 + 11991121199092 + 1199102 + 1199112 (35)

119862 = 1198801199102 + 1199092 + 11991121199092 + 1199102 + 1199112 (36)

119863 = 1198801199112 + 1199092 + 11991021199092 + 1199102 + 1199112 (37)

119864 = 119880119909119910 minus 1199091199101199092 + 1199102 + 1199112 (38)

6 Journal of Gravity

119865 = 119880119909119911 minus 1199091199111199092 + 1199102 + 1199112 (39)119866 = 119880119910119911 minus 1199101199111199092 + 1199102 + 1199112 (40)

These are the elements of the deformation matrix119872 To findthe determinant of matrix119872 we use (34)ndash(40) in (32) andthen we evaluate the determinant such as

det119872 = 1 (41)

We understand that our matrix belongs to SL(4) because ofthe determinant This is in accordance with the general factthat the symmetry group of the deformation matrices belongto GL(4)

Now we are to find the coordinate transformationsbetween these two space-times having the deformationtransformations given by (30)ndash(32) We need to find theinvertible matrix elements 1205971199091205831015840120597119909] between flat and curvedspace-times in Cartesian coordinates Therefore we write thematrix multiplication in (31) explicitly Then we also writethe deformation equations with summation convention in(30) explicitly We equate the metric elements 119892120583] in bothexplicit forms then we obtain the invertible matrix elements1205971199091205831015840120597119909] From these elements we construct the coordinatetransformation matrix Λ directly as given below

Λ =(((((((

1205971199051015840120597119905 0 0 00 1205971199091015840120597119909 1205971199091015840120597119910 12059711990910158401205971199110 1205971199101015840120597119909 1205971199101015840120597119910 12059711991010158401205971199110 1205971199111015840120597119909 1205971199111015840120597119910 1205971199111015840120597119911)))))))

=((((radic119860 0 0 00 radic119861 119864radic119862 119865radic1198630 119864radic119861 radic119862 119866radic1198630 119865radic119861 119866radic119862 radic119863

))))

(42)

We can write these elements more explicitly by using (34)ndash(40) and obtain 1205971199051015840120597119905 = radic 1119880 (43)1205971199091015840120597119909 = radic1198801199092 + 1199102 + 11991121199092 + 1199102 + 1199112 (44)

1205971199101015840120597119910 = radic1199092 + 1198801199102 + 11991121199092 + 1199102 + 1199112 (45)

1205971199111015840120597119911 = radic1199092 + 1199102 + 11988011991121199092 + 1199102 + 1199112 (46)

1205971199091015840120597119910 = (119880119909119910 minus 119909119910) (1199092 + 1199102 + 1199112)radic(1199092 + 1198801199102 + 1199112) (1199092 + 1199102 + 1199112) (47)

1205971199091015840120597119911 = (119880119909119911 minus 119909119911) (1199092 + 1199102 + 1199112)radic(1199092 + 1199102 + 1198801199112) (1199092 + 1199102 + 1199112) (48)

1205971199101015840120597119909 = (119880119909119910 minus 119909119910) (1199092 + 1199102 + 1199112)radic(1198801199092 + 1199102 + 1199112) (1199092 + 1199102 + 1199112) (49)

1205971199101015840120597119911 = (119880119910119911 minus 119910119911) (1199092 + 1199102 + 1199112)radic(1199092 + 1199102 + 1198801199112) (1199092 + 1199102 + 1199112) (50)

1205971199111015840120597119909 = (119880119909119911 minus 119909119911) (1199092 + 1199102 + 1199112)radic(1198801199092 + 1199102 + 1199112) (1199092 + 1199102 + 1199112) (51)

1205971199111015840120597119910 = (119880119910119911 minus 119910119911) (1199092 + 1199102 + 1199112)radic(1199092 + 1198801199102 + 1199112) (1199092 + 1199102 + 1199112) (52)

When we insert these elements into the coordinate transfor-mation matrix Λ we evaluate its determinant as follows

det119872= radic119880(1199092 + 1199102 + 1199112) (1199092 + 1199102 + 1199112) (1199092 + 1199102 + 1199112)radic(1198801199092 + 1199102 + 1199112) (1199092 + 1198801199102 + 1199112) (1199092 + 1199102 + 1198801199112) (53)

or

det119872= radic119880(1199092 + 1199102 + 1199112)32radic(1198801199092 + 1199102 + 1199112) (1199092 + 1198801199102 + 1199112) (1199092 + 1199102 + 1198801199112) (54)

We infer from the nonzero determinant of the coordinatetransformation matrix Λ that it is invertible and it belongsto the GL(4) symmetry group since it is a four by fourmatrixThis is again in accordance with the general fact aboutthe deformation matrices and their resulting coordinatetransformation matrices

The deformation matrix 119872 and the coordinate trans-formation matrix Λ are shown to belong to SL(4) andGL(4) respectively In Section 2 for the spherical coordinatescase we found that both matrices belong to SL(4) forthe transformations from flat tangent vector space-time tocurved Schwarzschild-type black hole space-time in theframework of 119891(119877) gravity These symmetry groups resultfrom the invariance of the space-time interval1198891199042 under thesetransformations

Similarly one can also find the deformation and coor-dinate transformation groups between the inertial frames

Journal of Gravity 7

and the noninertial frames of 119891(119877) gravity axially symmet-ric Kerr-Type and charged Reissner-Nordstrom-Type blackholes because their metric tensors have nondiagonal termsas in the case of Cartesian coordinates representations (22)ndash(28) for the metric of Schwarzschild-type black hole curvedspace-time (31) The metric tensor of a charged and rotating119891(119877) gravity black hole is given as [22]1198891199042 = minusΔ 1199031205882 [119889119905 minus 119886 sin2120579Ξ 119889120601]2 + 1205882Δ 119903 1198891199032 + 1205882Δ 120579 1198891205792+ Δ 120579sin21205791205882 [119886119889119905 minus 1199032 + 1198862Ξ 119889120601]2 (55)

whereΔ 119903 = (1199032 + 1198862) (1 + 119877012 1199032) minus 2119872119903 + 11987621 + 1198911015840 (1198770) Ξ = 1 minus 1198770121198862Δ 120579 = 1 minus 1198770121198862cos2120579(56)

Also 119886 and 119876 are the angular momentum and the charge ofthe 119891(119877) gravity black hole respectively If 119886 = 0 (55) turnsto be a Reissner-Nordstrom-Type black hole in 119891(119877) gravitywhereas if 119876 = 0 it gives a Kerr-Type 119891(119877) gravity blackholeWe can usemetric (55) for bothKerr-Type andReissner-Nordstrom-Type 119891(119877) gravity black holes and rewrite it as

1198891199042 = 1198862Δ 120579sin2120579 minus Δ 1199031205882 1198891199052+ 2119886 sin21205791205882Ξ [Δ 119903 minus Δ 120579 (1199032 + 1198862)] 119889119905119889120601+ 1205882Δ 119903 1198891199032 + 1205882Δ 120579 1198891205792+ Δ 120579sin2120579 (1199032 + 1198862)2 minus Δ 1199031198862sin41205791205882 1198891206012

(57)

and in matrix form

119892120583] =((((((((

1198862Δ 120579sin2120579 minus Δ 1199031205882 0 0 2119886 sin21205791205882Ξ [Δ 119903 minus Δ 120579 (1199032 + 1198862)]0 1205882Δ 119903 0 00 0 1205882Δ 120579 02119886 sin21205791205882Ξ [Δ 119903 minus Δ 120579 (1199032 + 1198862)] 0 0 Δ 120579 sin2120579 (1199032 + 1198862)2 minus Δ 1199031198862sin41205791205882)))))))) (58)

By using the similar steps in (31) and (32) we can obtainthe deformation matrix 119872 for the inertial observer andthe noninertial observer in the space-time curved by Kerr-Type or Reissner-Nordstrom-Type 119891(119877) gravity black holesAccordingly from the same step in (42) we then obtainthe coordinate transformation matrix Λ for these observersrsquoframe

4 Conclusions

In this paper we consider the transformations between acurved and flat space-time being the tangent vector space-time of the curved space-time and also the coordinates ofthis tangent space are called the local Lorentz coordinatesThe curvature of the space-time used here results fromthe Schwarzschild-type black hole whose solution for themetric tensor is obtained in the frameworks of modified119891(119877) gravity [7ndash9] We investigated the symmetry groups ofthe deformation matrix for the transformations of metricsand then the coordinate transformation matrix obtained by

this deformation matrix The extended and straightforwardphysical meaning and interpretations of the deformationsand conformal transformations can be found in the literaturesee [16 23 24]

These deformation and coordinate transformationmatri-ces are the identity matrices for the case 119880 = 1 This meansthat the both space-times are same flat space-timesThereforethe transformation matrices for deformation and coordinatetransformations must be identity matrix in both coordinatesystems

When we use 119880 = 1 in spherical coordinates forthe transformation matrices 119872 and Λ in (18) and (16)119908respectively we can easily realize that they are the identitymatrices Similarly if we again use 119880 = 1 in Cartesiancoordinates we find thematrices119872 andΛ in (32) and (42) arethe identity matrices Namely for 119880 = 1 all the nondiagonalelements of 119872 in (38) and (40) vanish In addition all thenondiagonal elements of Λ in (47) and (52) again vanish for119880 = 1 Taking 119880 = 1 can be viewed as a limiting case forchecking the calculations for transformation matrices whosesymmetry groups are investigated

8 Journal of Gravity

We infer that the deformation matrices the coordinatetransformation matrices and their symmetry groups can beinvestigated for the other type of black hole solutions in theframework ofmodified119891(119877) gravity Alsowe can consider thepresent transformation groups for the other type of modifiedgravitation theories for these various black hole solutions asthe candidate studies related to this topic

Competing Interests

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

References

[1] R A Nelson ldquoGeneralized Lorentz transformation for an accel-erated rotating frame of referencerdquo Journal of MathematicalPhysics vol 28 no 10 pp 2379ndash2383 1987

[2] H Nikolic ldquoRelativistic contraction of an accelerated rodrdquoAmerican Journal of Physics vol 67 no 11 pp 1007ndash1012 1999

[3] H Nikolic ldquoRelativistic contraction and related effects innoninertial framesrdquo Physical Review A vol 61 no 3 Article ID032109 8 pages 2000

[4] O Gron ldquoAcceleration and weight of extended bodies in thetheory of relativityrdquo American Journal of Physics vol 45 no 1pp 65ndash70 1977

[5] O Gron ldquoRelativictic description of a rotating diskrdquo AmericanJournal of Physics vol 43 pp 869ndash876 1975

[6] R DrsquoInverno Introtucing Einsteinrsquos Relativity OxfordUniversityPress Oxford UK 1992

[7] S Capozziello and M D Laurentis ldquoBlack holes and stellarstructures in f(R)-gravityrdquo httpsarxivorgabs12020394

[8] S Nojiri and S D Odintsov ldquoUnified cosmic history inmodified gravity from F(R) theory to Lorentz non-invariantmodelsrdquo Physics Reports vol 505 no 2ndash4 pp 59ndash144 2011

[9] S Nojiri and S D Odintsov ldquoIntroduction to modified gravityand gravitational alternative for dark energyrdquo InternationalJournal of Geometric Methods in Modern Physics vol 4 no 1pp 115ndash145 2007

[10] S Capozziello ldquoCurvature quintessencerdquo International Journalof Modern Physics D vol 11 no 4 pp 483ndash491 2002

[11] S Capozziello and S Vignolo ldquoA comment on lsquoThe Cauchyproblem of 119891(119877) gravityrsquordquo Classical and Quantum Gravity vol26 no 16 Article ID 168001 2009

[12] S Capozziello A Stabile and A Troisi ldquoSpherically symmetricsolutions in f (R) gravity via the Noether symmetry approachrdquoClassical and Quantum Gravity vol 24 no 8 pp 2153ndash21662007

[13] S Capozziello A Stabile and A Troisi ldquoSpherical symmetryin 119891(119877)-gravityrdquo Classical and Quantum Gravity vol 25 no 8Article ID 085004 2008

[14] B Coll ldquoA universal law of gravitational deformation forGeneral Relativityrdquo in Proceedings of the Spanish RelativisticMeeting EREs Salamanca Spain 1998

[15] B Coll J Llosa and D Soler ldquoThree-dimensional metrics asdeformations of a constant curvaturemetricrdquoGeneral Relativityand Gravitation vol 34 no 2 pp 269ndash282 2002

[16] S Capozziello and M D Laurentis ldquoExtended theories ofgravityrdquo Physics Reports vol 509 no 4-5 pp 167ndash321 2011

[17] H A Buchdahl ldquoMonthly notices of the royal astronomicalsocietyrdquoMonthly Notices of the Royal Astronomical Society vol150 no 1 pp 1ndash8 1970

[18] A A Starobinsky ldquoA new type of isotropic cosmological modelswithout singularityrdquo Physics Letters B vol 91 no 1 pp 99ndash1021980

[19] H Kleinert and H-J Schmidt ldquoCosmology with curvature-saturated gravitational lagrangian 119877radic1 + 11989741198772rdquo General Rela-tivity and Gravitation vol 34 no 8 pp 1295ndash1318 2002

[20] J D Barrow and A C Ottewill ldquoThe stability of general rela-tivistic cosmological theoryrdquo Journal of Physics AMathematicaland General vol 16 no 12 pp 2757ndash2776 1983

[21] M CarmeliGroupTheory and General Relativity McGraw-HillInternational Book London UK 1977

[22] A Larranaga ldquoA rotating charged black hole solution in 119891(119877)gravityrdquo Pramana vol 78 no 5 pp 697ndash703 2012

[23] N Lanahan-Tremblay and V Faraoni ldquoThe Cauchy problem off (R) gravityrdquo Classical and QuantumGravity vol 24 no 22 pp5667ndash5679 2007

[24] G Allemandi M Capone S Capozziello and M FrancaviglialdquoConformal aspects of the Palatini approach in extended theo-ries of gravityrdquo General Relativity and Gravitation vol 38 no 1pp 33ndash60 2006

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

Journal of Gravity 3

119872 =(((((((((

(1205971199051015840120597119905 )2 (1205971199031015840120597119905 )2 (1205971205791015840120597119905 )2 (1205971206011015840120597119905 )2(1205971199051015840120597119903 )2 (1205971199031015840120597119903 )2 (1205971205791015840120597119903 )2 (1205971206011015840120597119903 )2(1205971199051015840120597120579 )2 (1205971199031015840120597120579 )2 (1205971205791015840120597120579 )2 (1205971206011015840120597120579 )2(1205971199051015840120597120601)2 (1205971199031015840120597120601 )2 (1205971205791015840120597120601 )2 (1205971206011015840120597120601 )2))))))))) (5)

The deformations transformations in (3) are called the con-formal transformations of the metrics [16 21] We infer thatthe coordinate transformation matrix is a diagonal four byfour matrix from (4) The metrics for flat space-time andspherically symmetric Schwarzschild-type black hole space-time in the framework of 119891(119877) gravity are1198891199042 = minus1198891199052 + 1198891199032 + 11990321198891205792 + 1199032sin21205791198891206012 (6)1198891199042 = minus1198801198891199052 + 119880minus11198891199032 + 11990321198891205792 + 1199032sin21205791198891206012 (7)

where119880 = 1+(1198961119903)+(119902120594120588minus120582)11990323 respectively [7ndash9 13 20]For this work in spherical coordinates the nonzero coordinatetransformation matrix elements are1205971199051015840120597119905 = radic(1 + 1198961119903 + 13 (119902120594120588 minus 120582) 1199032)minus1 = radic119880minus11205971199031015840120597119903 = radic(1 + 1198961119903 + 13 (119902120594120588 minus 120582) 1199032) = radic1198801205971205791015840120597120579 = 11205971206011015840120597120601 = 1

(8)

Summing all of these by using the coordinate transforma-tions rule 1199091198861015840 = 1205971199091198861015840120597119909119886 119909119886 (9)

we obtain the coordinate transformations from flat tangentvector space-time to curved space-time of 119891(119877) gravity suchas 1199051015840 = 119905radic(1 + 1198961119903 + 13 (119902120594120588 minus 120582) 1199032)minus1 (10)

1199031015840 = 119903radic(1 + 1198961119903 + 13 (119902120594120588 minus 120582) 1199032) (11)1205791015840 = 120579 (12)1206011015840 = 120601 (13)

It is the convenient part for the study of related effectsof length contraction and time dilation between inertial

and noninertial frames in 119891(119877) Schwarzschild-type blackhole curved space-time So the time dilation and lengthcontraction can be inferred from the solutions of (10) and(11) for point effects The point effect means that when a timetransformation is made between frames it is assumed to bethe space coordinate of the event that is the same or if aspace transformation is made then the time is assumed tobe simultaneous between two events in the framesThe resultfollows as we get the time dilation equation by using (10)Δ1199051015840 = radic(1 + 1198961119903 + 13 (119902120594120588 minus 120582) 1199032)minus1Δ119905 (14)

and for length contraction from (11)Δ1199031015840 = radic(1 + 1198961119903 + 13 (119902120594120588 minus 120582) 1199032)Δ119903 (15)

If we call the four by four coordinate transformation matrixfor these space-times having these metrics as Λ it has theform

Λ =((((((

1205971199051015840120597119905 0 0 00 1205971199031015840120597119903 0 00 0 1205971205791015840120597120579 00 0 0 1205971206011015840120597120601))))))

=(radic119880minus1 0 0 00 radic119880 0 00 0 1 00 0 0 1)(16)

In this study we investigate the symmetry group of thecoordinate transformation matrix Λ The coordinate trans-formations leave the space-time interval 1198891199042 invariant so thesymmetry occurs for these transformations It is obvious thatthe determinant of the matrix Λ is

detΛ = 1 (17)

Therefore the symmetry group of the matrix Λ is found to beSL(4) for the coordinate transformations in spherical coor-dinates In addition the symmetry group of the conformaldeformation matrix119872 is again obtained by the determinantof119872 such that

119872 =(119880minus1 0 0 00 119880 0 00 0 1 00 0 0 1) (18)

Computing the determinant of119872 obviously gives

det119872 = 1 (19)

4 Journal of Gravity

We obtain a special linear symmetry group for the four byfour conformal deformation matrix119872 So it also belongs toSL(4) as the matrix Λ does The deformation matrices andconsequently the coordinate transformation matrices whichcan directly be obtained from the deformation matricesbelong to general linear groupGL(4) that is a general fact [16]Here these transformation matrices between flat and curvedspherically symmetric Schwarzschild-type black hole space-time in the framework of119891(119877) gravity are found to be in SL(4)symmetry group

Now we will investigate the deformation transformationand coordinate transformations in the Cartesian coordinatesfor the flat space-time and the Schwarzschild-type black holespace-time in the framework of 119891(119877) gravity3 Transformation Groups in

Cartesian Coordinates

It is a common example that in 2-dimensional Euclideanspace a vector can be transformed as a rotation by anorthogonal matrix belonging to SO(2) groups Here we infact imply that the transformations for the components ofthe vector are considered to be in Cartesian coordinates Ifwe consider the components of the transformed vector inthe spherical coordinates we obtain a GL(2) transformationmatrix for the same event Therefore for the same events inSection 2 we now investigate the symmetry groups of thedeformation transformations and the coordinate transforma-tions in Cartesian coordinates

For this we should first transform the metrics fromthe spherical coordinates to the Cartesian coordinates forthe Schwarzschild-type black hole curved space-time Sincethe space-time interval is invariant for both spherical andCartesian coordinates we use the fact that1198921205831015840]1015840 = 12059711990911988610158401205971199091205831015840 1205971199091198871015840120597119909]1015840 11989211988610158401198871015840 (20)

where 1198861015840 = (1199051015840 1199031015840 1205791015840 1206011015840) and 1205831015840 = (1199051015840 1199091015840 1199101015840 1199111015840) We needto find the elements of the metric tensor in the Cartesiancoordinates from (20) For this we will use the elements of themetric tensor in the spherical coordinates from (7) We alsoneed the transformation elements 12059711990911988610158401205971199091205831015840 from sphericalto Cartesian coordinate To find these elements we use 1199091015840 =1199031015840 sin 1205791015840 cos1206011015840 1199101015840 = 1199031015840 sin 1205791015840 sin1206011015840 1199111015840 = 1199031015840 cos 1205791015840 and 119903 =radic11990910158402 + 11991010158402 + 11991110158402 After a little calculus we obtain the elementsof the metric tensor in Cartesian coordinates but in terms ofthe spherical coordinates11989211990510158401199051015840 = minus(1 + 1198961119903 + 13 (119902120594120588 minus 120582) 1199032) = minus11988011989211990910158401199091015840 = (sin 1205791015840 cos1206011015840)2119880minus1 + (cos 1205791015840 cos1206011015840)2+ (sin1206011015840)2 11989211991010158401199101015840 = (sin 1205791015840 sin1206011015840)2119880minus1 + (cos 1205791015840 sin1206011015840)2

+ (cos1206011015840)2 11989211991110158401199111015840 = (cos 1205791015840)2119880minus1 + (sin 1205791015840)2 11989211990910158401199101015840 = 11989211991010158401199091015840= [(sin 1205791015840)2119880minus1 + (cos 1205791015840)2 minus 1] cos1206011015840 sin120601101584011989211991110158401199091015840 = 11989211990910158401199111015840= (sin 1205791015840 cos1206011015840) cos 1205791015840119880minus1+ (cos 1205791015840 cos1206011015840) (minus sin 1205791015840) 11989211991110158401199101015840 = 11989211991010158401199111015840= (sin 1205791015840 sin1206011015840) cos 1205791015840119880minus1+ (cos 1205791015840 sin1206011015840) (minus sin 1205791015840) (21)

Nowwe should use the Cartesian coordinates for all sphericalcoordinates in the elements of the metric such that11989211990510158401199051015840 = minus119880 (22)11989211990910158401199091015840 = 11990910158402(11990910158402 + 11991010158402 + 11991110158402)119880minus1+ 1199091015840211991110158402(11990910158402 + 11991010158402 + 11991110158402) (11990910158402 + 11991010158402) + 11991010158402(11990910158402 + 11991010158402) (23)

11989211991010158401199101015840 = 11991010158402(11990910158402 + 11991010158402 + 11991110158402)119880minus1+ 1199101015840211991110158402(11990910158402 + 11991010158402 + 11991110158402) (11990910158402 + 11991010158402) + 11990910158402(11990910158402 + 11991010158402) (24)

11989211991110158401199111015840 = 11991110158402(11990910158402 + 11991010158402 + 11991110158402)119880minus1 + 11990910158402 + 11991010158402(11990910158402 + 11991010158402 + 11991110158402) (25)11989211990910158401199101015840 = 11989211991010158401199091015840= 11990910158401199101015840(11990910158402 + 11991010158402 + 11991110158402)119880minus1+ 1199091015840119910101584011991110158402(11990910158402 + 11991010158402 + 11991110158402) (11990910158402 + 11991010158402) minus 11990910158401199101015840(11990910158402 + 11991010158402) (26)

11989211990910158401199111015840 = 11989211991110158401199091015840= 11990910158401199111015840(11990910158402 + 11991010158402 + 11991110158402)119880minus1 minus 11990910158401199111015840(11990910158402 + 11991010158402 + 11991110158402) (27)

11989211991010158401199111015840 = 11989211991110158401199101015840= 11991010158401199111015840(11990910158402 + 11991010158402 + 11991110158402)119880minus1 minus 11991010158401199111015840(11990910158402 + 11991010158402 + 11991110158402) (28)

Journal of Gravity 5

Finally we obtain the elements of the metric tensor for thecurved space-time of Schwarzschild-type black hole in termsof the Cartesian coordinates The indices for this metricwere shown by the primed Greek letters Since we investigatethe transformations between flat and curved space-times inCartesian coordinates we use the flat Cartesian metric 119892120583]whose indices are shown by the unprimed Greek letters suchthat

119892120583] =(minus1 0 0 00 1 0 00 0 1 00 0 0 1) (29)

Then the deformation transformations between the metrics119892120583] and 1198921205831015840]1015840 are 119892120583] = 1205971199091205831015840120597119909120583 120597119909]1015840120597119909] 1198921205831015840]1015840 (30)

We cannot call the deformation transformations in (30)conformal transformations of the metrics because both ofthe metrics are not in diagonal form As seen in (26)ndash(28)the curved space-time metric has nondiagonal terms Nowthe deformation matrix can be constructed by arranging (30)as below

(119892119905119905 0 0 00 119892119909119909 0 00 0 119892119910119910 00 0 0 119892119911119911)=119872(11989211990510158401199051015840 0 0 00 11989211990910158401199091015840 11989211990910158401199101015840 119892119909101584011991110158400 11989211991010158401199091015840 11989211991010158401199101015840 119892119910101584011991110158400 11989211991110158401199091015840 11989211991110158401199101015840 11989211991110158401199111015840)

(31)

and only nonzero elements of the matrix119872 forms as

119872 =(119860 0 0 00 119861 119864 1198650 119864 119862 1198660 119865 119866 119863) (32)

To find the deformation matrix M from (31) is not so easyInstead of that we write the inverse transformations in (31)and obtain the inverse deformation matrix119872minus1 in an easiermannerThen we find the matrix119872 by evaluating the inverseof the matrix119872minus1 Consequently the matrix119872 is found tohave the elements below

119860 = minus 111989211990510158401199051015840 119861 = minus1198921199101015840119910101584011989211991110158401199111015840 + 119892119910101584011991110158402minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402 119862 = minus1198921199091015840119909101584011989211991110158401199111015840 + 119892119909101584011991110158402minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402 119863 = minus1198921199091015840119909101584011989211991010158401199101015840 + 119892119909101584011991010158402minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402 119864 = 1198921199111015840119911101584011989211990910158401199101015840 minus 1198921199101015840119911101584011989211990910158401199111015840minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402 119865 = 1198921199101015840119910101584011989211990910158401199111015840 minus 1198921199091015840119910101584011989211991010158401199111015840minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402 119866 = 1198921199091015840119909101584011989211991010158401199111015840 minus 1198921199091015840119910101584011989211990910158401199111015840minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402

(33)

By using (22)ndash(28) in (33) and making the simplifications forthese matrix elements we finally obtain these elements asfollows 119860 = 1119880 (34)

119861 = 1198801199092 + 1199102 + 11991121199092 + 1199102 + 1199112 (35)

119862 = 1198801199102 + 1199092 + 11991121199092 + 1199102 + 1199112 (36)

119863 = 1198801199112 + 1199092 + 11991021199092 + 1199102 + 1199112 (37)

119864 = 119880119909119910 minus 1199091199101199092 + 1199102 + 1199112 (38)

6 Journal of Gravity

119865 = 119880119909119911 minus 1199091199111199092 + 1199102 + 1199112 (39)119866 = 119880119910119911 minus 1199101199111199092 + 1199102 + 1199112 (40)

These are the elements of the deformation matrix119872 To findthe determinant of matrix119872 we use (34)ndash(40) in (32) andthen we evaluate the determinant such as

det119872 = 1 (41)

We understand that our matrix belongs to SL(4) because ofthe determinant This is in accordance with the general factthat the symmetry group of the deformation matrices belongto GL(4)

Now we are to find the coordinate transformationsbetween these two space-times having the deformationtransformations given by (30)ndash(32) We need to find theinvertible matrix elements 1205971199091205831015840120597119909] between flat and curvedspace-times in Cartesian coordinates Therefore we write thematrix multiplication in (31) explicitly Then we also writethe deformation equations with summation convention in(30) explicitly We equate the metric elements 119892120583] in bothexplicit forms then we obtain the invertible matrix elements1205971199091205831015840120597119909] From these elements we construct the coordinatetransformation matrix Λ directly as given below

Λ =(((((((

1205971199051015840120597119905 0 0 00 1205971199091015840120597119909 1205971199091015840120597119910 12059711990910158401205971199110 1205971199101015840120597119909 1205971199101015840120597119910 12059711991010158401205971199110 1205971199111015840120597119909 1205971199111015840120597119910 1205971199111015840120597119911)))))))

=((((radic119860 0 0 00 radic119861 119864radic119862 119865radic1198630 119864radic119861 radic119862 119866radic1198630 119865radic119861 119866radic119862 radic119863

))))

(42)

We can write these elements more explicitly by using (34)ndash(40) and obtain 1205971199051015840120597119905 = radic 1119880 (43)1205971199091015840120597119909 = radic1198801199092 + 1199102 + 11991121199092 + 1199102 + 1199112 (44)

1205971199101015840120597119910 = radic1199092 + 1198801199102 + 11991121199092 + 1199102 + 1199112 (45)

1205971199111015840120597119911 = radic1199092 + 1199102 + 11988011991121199092 + 1199102 + 1199112 (46)

1205971199091015840120597119910 = (119880119909119910 minus 119909119910) (1199092 + 1199102 + 1199112)radic(1199092 + 1198801199102 + 1199112) (1199092 + 1199102 + 1199112) (47)

1205971199091015840120597119911 = (119880119909119911 minus 119909119911) (1199092 + 1199102 + 1199112)radic(1199092 + 1199102 + 1198801199112) (1199092 + 1199102 + 1199112) (48)

1205971199101015840120597119909 = (119880119909119910 minus 119909119910) (1199092 + 1199102 + 1199112)radic(1198801199092 + 1199102 + 1199112) (1199092 + 1199102 + 1199112) (49)

1205971199101015840120597119911 = (119880119910119911 minus 119910119911) (1199092 + 1199102 + 1199112)radic(1199092 + 1199102 + 1198801199112) (1199092 + 1199102 + 1199112) (50)

1205971199111015840120597119909 = (119880119909119911 minus 119909119911) (1199092 + 1199102 + 1199112)radic(1198801199092 + 1199102 + 1199112) (1199092 + 1199102 + 1199112) (51)

1205971199111015840120597119910 = (119880119910119911 minus 119910119911) (1199092 + 1199102 + 1199112)radic(1199092 + 1198801199102 + 1199112) (1199092 + 1199102 + 1199112) (52)

When we insert these elements into the coordinate transfor-mation matrix Λ we evaluate its determinant as follows

det119872= radic119880(1199092 + 1199102 + 1199112) (1199092 + 1199102 + 1199112) (1199092 + 1199102 + 1199112)radic(1198801199092 + 1199102 + 1199112) (1199092 + 1198801199102 + 1199112) (1199092 + 1199102 + 1198801199112) (53)

or

det119872= radic119880(1199092 + 1199102 + 1199112)32radic(1198801199092 + 1199102 + 1199112) (1199092 + 1198801199102 + 1199112) (1199092 + 1199102 + 1198801199112) (54)

We infer from the nonzero determinant of the coordinatetransformation matrix Λ that it is invertible and it belongsto the GL(4) symmetry group since it is a four by fourmatrixThis is again in accordance with the general fact aboutthe deformation matrices and their resulting coordinatetransformation matrices

The deformation matrix 119872 and the coordinate trans-formation matrix Λ are shown to belong to SL(4) andGL(4) respectively In Section 2 for the spherical coordinatescase we found that both matrices belong to SL(4) forthe transformations from flat tangent vector space-time tocurved Schwarzschild-type black hole space-time in theframework of 119891(119877) gravity These symmetry groups resultfrom the invariance of the space-time interval1198891199042 under thesetransformations

Similarly one can also find the deformation and coor-dinate transformation groups between the inertial frames

Journal of Gravity 7

and the noninertial frames of 119891(119877) gravity axially symmet-ric Kerr-Type and charged Reissner-Nordstrom-Type blackholes because their metric tensors have nondiagonal termsas in the case of Cartesian coordinates representations (22)ndash(28) for the metric of Schwarzschild-type black hole curvedspace-time (31) The metric tensor of a charged and rotating119891(119877) gravity black hole is given as [22]1198891199042 = minusΔ 1199031205882 [119889119905 minus 119886 sin2120579Ξ 119889120601]2 + 1205882Δ 119903 1198891199032 + 1205882Δ 120579 1198891205792+ Δ 120579sin21205791205882 [119886119889119905 minus 1199032 + 1198862Ξ 119889120601]2 (55)

whereΔ 119903 = (1199032 + 1198862) (1 + 119877012 1199032) minus 2119872119903 + 11987621 + 1198911015840 (1198770) Ξ = 1 minus 1198770121198862Δ 120579 = 1 minus 1198770121198862cos2120579(56)

Also 119886 and 119876 are the angular momentum and the charge ofthe 119891(119877) gravity black hole respectively If 119886 = 0 (55) turnsto be a Reissner-Nordstrom-Type black hole in 119891(119877) gravitywhereas if 119876 = 0 it gives a Kerr-Type 119891(119877) gravity blackholeWe can usemetric (55) for bothKerr-Type andReissner-Nordstrom-Type 119891(119877) gravity black holes and rewrite it as

1198891199042 = 1198862Δ 120579sin2120579 minus Δ 1199031205882 1198891199052+ 2119886 sin21205791205882Ξ [Δ 119903 minus Δ 120579 (1199032 + 1198862)] 119889119905119889120601+ 1205882Δ 119903 1198891199032 + 1205882Δ 120579 1198891205792+ Δ 120579sin2120579 (1199032 + 1198862)2 minus Δ 1199031198862sin41205791205882 1198891206012

(57)

and in matrix form

119892120583] =((((((((

1198862Δ 120579sin2120579 minus Δ 1199031205882 0 0 2119886 sin21205791205882Ξ [Δ 119903 minus Δ 120579 (1199032 + 1198862)]0 1205882Δ 119903 0 00 0 1205882Δ 120579 02119886 sin21205791205882Ξ [Δ 119903 minus Δ 120579 (1199032 + 1198862)] 0 0 Δ 120579 sin2120579 (1199032 + 1198862)2 minus Δ 1199031198862sin41205791205882)))))))) (58)

By using the similar steps in (31) and (32) we can obtainthe deformation matrix 119872 for the inertial observer andthe noninertial observer in the space-time curved by Kerr-Type or Reissner-Nordstrom-Type 119891(119877) gravity black holesAccordingly from the same step in (42) we then obtainthe coordinate transformation matrix Λ for these observersrsquoframe

4 Conclusions

In this paper we consider the transformations between acurved and flat space-time being the tangent vector space-time of the curved space-time and also the coordinates ofthis tangent space are called the local Lorentz coordinatesThe curvature of the space-time used here results fromthe Schwarzschild-type black hole whose solution for themetric tensor is obtained in the frameworks of modified119891(119877) gravity [7ndash9] We investigated the symmetry groups ofthe deformation matrix for the transformations of metricsand then the coordinate transformation matrix obtained by

this deformation matrix The extended and straightforwardphysical meaning and interpretations of the deformationsand conformal transformations can be found in the literaturesee [16 23 24]

These deformation and coordinate transformationmatri-ces are the identity matrices for the case 119880 = 1 This meansthat the both space-times are same flat space-timesThereforethe transformation matrices for deformation and coordinatetransformations must be identity matrix in both coordinatesystems

When we use 119880 = 1 in spherical coordinates forthe transformation matrices 119872 and Λ in (18) and (16)119908respectively we can easily realize that they are the identitymatrices Similarly if we again use 119880 = 1 in Cartesiancoordinates we find thematrices119872 andΛ in (32) and (42) arethe identity matrices Namely for 119880 = 1 all the nondiagonalelements of 119872 in (38) and (40) vanish In addition all thenondiagonal elements of Λ in (47) and (52) again vanish for119880 = 1 Taking 119880 = 1 can be viewed as a limiting case forchecking the calculations for transformation matrices whosesymmetry groups are investigated

8 Journal of Gravity

We infer that the deformation matrices the coordinatetransformation matrices and their symmetry groups can beinvestigated for the other type of black hole solutions in theframework ofmodified119891(119877) gravity Alsowe can consider thepresent transformation groups for the other type of modifiedgravitation theories for these various black hole solutions asthe candidate studies related to this topic

Competing Interests

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

References

[1] R A Nelson ldquoGeneralized Lorentz transformation for an accel-erated rotating frame of referencerdquo Journal of MathematicalPhysics vol 28 no 10 pp 2379ndash2383 1987

[2] H Nikolic ldquoRelativistic contraction of an accelerated rodrdquoAmerican Journal of Physics vol 67 no 11 pp 1007ndash1012 1999

[3] H Nikolic ldquoRelativistic contraction and related effects innoninertial framesrdquo Physical Review A vol 61 no 3 Article ID032109 8 pages 2000

[4] O Gron ldquoAcceleration and weight of extended bodies in thetheory of relativityrdquo American Journal of Physics vol 45 no 1pp 65ndash70 1977

[5] O Gron ldquoRelativictic description of a rotating diskrdquo AmericanJournal of Physics vol 43 pp 869ndash876 1975

[6] R DrsquoInverno Introtucing Einsteinrsquos Relativity OxfordUniversityPress Oxford UK 1992

[7] S Capozziello and M D Laurentis ldquoBlack holes and stellarstructures in f(R)-gravityrdquo httpsarxivorgabs12020394

[8] S Nojiri and S D Odintsov ldquoUnified cosmic history inmodified gravity from F(R) theory to Lorentz non-invariantmodelsrdquo Physics Reports vol 505 no 2ndash4 pp 59ndash144 2011

[9] S Nojiri and S D Odintsov ldquoIntroduction to modified gravityand gravitational alternative for dark energyrdquo InternationalJournal of Geometric Methods in Modern Physics vol 4 no 1pp 115ndash145 2007

[10] S Capozziello ldquoCurvature quintessencerdquo International Journalof Modern Physics D vol 11 no 4 pp 483ndash491 2002

[11] S Capozziello and S Vignolo ldquoA comment on lsquoThe Cauchyproblem of 119891(119877) gravityrsquordquo Classical and Quantum Gravity vol26 no 16 Article ID 168001 2009

[12] S Capozziello A Stabile and A Troisi ldquoSpherically symmetricsolutions in f (R) gravity via the Noether symmetry approachrdquoClassical and Quantum Gravity vol 24 no 8 pp 2153ndash21662007

[13] S Capozziello A Stabile and A Troisi ldquoSpherical symmetryin 119891(119877)-gravityrdquo Classical and Quantum Gravity vol 25 no 8Article ID 085004 2008

[14] B Coll ldquoA universal law of gravitational deformation forGeneral Relativityrdquo in Proceedings of the Spanish RelativisticMeeting EREs Salamanca Spain 1998

[15] B Coll J Llosa and D Soler ldquoThree-dimensional metrics asdeformations of a constant curvaturemetricrdquoGeneral Relativityand Gravitation vol 34 no 2 pp 269ndash282 2002

[16] S Capozziello and M D Laurentis ldquoExtended theories ofgravityrdquo Physics Reports vol 509 no 4-5 pp 167ndash321 2011

[17] H A Buchdahl ldquoMonthly notices of the royal astronomicalsocietyrdquoMonthly Notices of the Royal Astronomical Society vol150 no 1 pp 1ndash8 1970

[18] A A Starobinsky ldquoA new type of isotropic cosmological modelswithout singularityrdquo Physics Letters B vol 91 no 1 pp 99ndash1021980

[19] H Kleinert and H-J Schmidt ldquoCosmology with curvature-saturated gravitational lagrangian 119877radic1 + 11989741198772rdquo General Rela-tivity and Gravitation vol 34 no 8 pp 1295ndash1318 2002

[20] J D Barrow and A C Ottewill ldquoThe stability of general rela-tivistic cosmological theoryrdquo Journal of Physics AMathematicaland General vol 16 no 12 pp 2757ndash2776 1983

[21] M CarmeliGroupTheory and General Relativity McGraw-HillInternational Book London UK 1977

[22] A Larranaga ldquoA rotating charged black hole solution in 119891(119877)gravityrdquo Pramana vol 78 no 5 pp 697ndash703 2012

[23] N Lanahan-Tremblay and V Faraoni ldquoThe Cauchy problem off (R) gravityrdquo Classical and QuantumGravity vol 24 no 22 pp5667ndash5679 2007

[24] G Allemandi M Capone S Capozziello and M FrancaviglialdquoConformal aspects of the Palatini approach in extended theo-ries of gravityrdquo General Relativity and Gravitation vol 38 no 1pp 33ndash60 2006

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 Journal of Gravity

We obtain a special linear symmetry group for the four byfour conformal deformation matrix119872 So it also belongs toSL(4) as the matrix Λ does The deformation matrices andconsequently the coordinate transformation matrices whichcan directly be obtained from the deformation matricesbelong to general linear groupGL(4) that is a general fact [16]Here these transformation matrices between flat and curvedspherically symmetric Schwarzschild-type black hole space-time in the framework of119891(119877) gravity are found to be in SL(4)symmetry group

Now we will investigate the deformation transformationand coordinate transformations in the Cartesian coordinatesfor the flat space-time and the Schwarzschild-type black holespace-time in the framework of 119891(119877) gravity3 Transformation Groups in

Cartesian Coordinates

It is a common example that in 2-dimensional Euclideanspace a vector can be transformed as a rotation by anorthogonal matrix belonging to SO(2) groups Here we infact imply that the transformations for the components ofthe vector are considered to be in Cartesian coordinates Ifwe consider the components of the transformed vector inthe spherical coordinates we obtain a GL(2) transformationmatrix for the same event Therefore for the same events inSection 2 we now investigate the symmetry groups of thedeformation transformations and the coordinate transforma-tions in Cartesian coordinates

For this we should first transform the metrics fromthe spherical coordinates to the Cartesian coordinates forthe Schwarzschild-type black hole curved space-time Sincethe space-time interval is invariant for both spherical andCartesian coordinates we use the fact that1198921205831015840]1015840 = 12059711990911988610158401205971199091205831015840 1205971199091198871015840120597119909]1015840 11989211988610158401198871015840 (20)

where 1198861015840 = (1199051015840 1199031015840 1205791015840 1206011015840) and 1205831015840 = (1199051015840 1199091015840 1199101015840 1199111015840) We needto find the elements of the metric tensor in the Cartesiancoordinates from (20) For this we will use the elements of themetric tensor in the spherical coordinates from (7) We alsoneed the transformation elements 12059711990911988610158401205971199091205831015840 from sphericalto Cartesian coordinate To find these elements we use 1199091015840 =1199031015840 sin 1205791015840 cos1206011015840 1199101015840 = 1199031015840 sin 1205791015840 sin1206011015840 1199111015840 = 1199031015840 cos 1205791015840 and 119903 =radic11990910158402 + 11991010158402 + 11991110158402 After a little calculus we obtain the elementsof the metric tensor in Cartesian coordinates but in terms ofthe spherical coordinates11989211990510158401199051015840 = minus(1 + 1198961119903 + 13 (119902120594120588 minus 120582) 1199032) = minus11988011989211990910158401199091015840 = (sin 1205791015840 cos1206011015840)2119880minus1 + (cos 1205791015840 cos1206011015840)2+ (sin1206011015840)2 11989211991010158401199101015840 = (sin 1205791015840 sin1206011015840)2119880minus1 + (cos 1205791015840 sin1206011015840)2

+ (cos1206011015840)2 11989211991110158401199111015840 = (cos 1205791015840)2119880minus1 + (sin 1205791015840)2 11989211990910158401199101015840 = 11989211991010158401199091015840= [(sin 1205791015840)2119880minus1 + (cos 1205791015840)2 minus 1] cos1206011015840 sin120601101584011989211991110158401199091015840 = 11989211990910158401199111015840= (sin 1205791015840 cos1206011015840) cos 1205791015840119880minus1+ (cos 1205791015840 cos1206011015840) (minus sin 1205791015840) 11989211991110158401199101015840 = 11989211991010158401199111015840= (sin 1205791015840 sin1206011015840) cos 1205791015840119880minus1+ (cos 1205791015840 sin1206011015840) (minus sin 1205791015840) (21)

Nowwe should use the Cartesian coordinates for all sphericalcoordinates in the elements of the metric such that11989211990510158401199051015840 = minus119880 (22)11989211990910158401199091015840 = 11990910158402(11990910158402 + 11991010158402 + 11991110158402)119880minus1+ 1199091015840211991110158402(11990910158402 + 11991010158402 + 11991110158402) (11990910158402 + 11991010158402) + 11991010158402(11990910158402 + 11991010158402) (23)

11989211991010158401199101015840 = 11991010158402(11990910158402 + 11991010158402 + 11991110158402)119880minus1+ 1199101015840211991110158402(11990910158402 + 11991010158402 + 11991110158402) (11990910158402 + 11991010158402) + 11990910158402(11990910158402 + 11991010158402) (24)

11989211991110158401199111015840 = 11991110158402(11990910158402 + 11991010158402 + 11991110158402)119880minus1 + 11990910158402 + 11991010158402(11990910158402 + 11991010158402 + 11991110158402) (25)11989211990910158401199101015840 = 11989211991010158401199091015840= 11990910158401199101015840(11990910158402 + 11991010158402 + 11991110158402)119880minus1+ 1199091015840119910101584011991110158402(11990910158402 + 11991010158402 + 11991110158402) (11990910158402 + 11991010158402) minus 11990910158401199101015840(11990910158402 + 11991010158402) (26)

11989211990910158401199111015840 = 11989211991110158401199091015840= 11990910158401199111015840(11990910158402 + 11991010158402 + 11991110158402)119880minus1 minus 11990910158401199111015840(11990910158402 + 11991010158402 + 11991110158402) (27)

11989211991010158401199111015840 = 11989211991110158401199101015840= 11991010158401199111015840(11990910158402 + 11991010158402 + 11991110158402)119880minus1 minus 11991010158401199111015840(11990910158402 + 11991010158402 + 11991110158402) (28)

Journal of Gravity 5

Finally we obtain the elements of the metric tensor for thecurved space-time of Schwarzschild-type black hole in termsof the Cartesian coordinates The indices for this metricwere shown by the primed Greek letters Since we investigatethe transformations between flat and curved space-times inCartesian coordinates we use the flat Cartesian metric 119892120583]whose indices are shown by the unprimed Greek letters suchthat

119892120583] =(minus1 0 0 00 1 0 00 0 1 00 0 0 1) (29)

Then the deformation transformations between the metrics119892120583] and 1198921205831015840]1015840 are 119892120583] = 1205971199091205831015840120597119909120583 120597119909]1015840120597119909] 1198921205831015840]1015840 (30)

We cannot call the deformation transformations in (30)conformal transformations of the metrics because both ofthe metrics are not in diagonal form As seen in (26)ndash(28)the curved space-time metric has nondiagonal terms Nowthe deformation matrix can be constructed by arranging (30)as below

(119892119905119905 0 0 00 119892119909119909 0 00 0 119892119910119910 00 0 0 119892119911119911)=119872(11989211990510158401199051015840 0 0 00 11989211990910158401199091015840 11989211990910158401199101015840 119892119909101584011991110158400 11989211991010158401199091015840 11989211991010158401199101015840 119892119910101584011991110158400 11989211991110158401199091015840 11989211991110158401199101015840 11989211991110158401199111015840)

(31)

and only nonzero elements of the matrix119872 forms as

119872 =(119860 0 0 00 119861 119864 1198650 119864 119862 1198660 119865 119866 119863) (32)

To find the deformation matrix M from (31) is not so easyInstead of that we write the inverse transformations in (31)and obtain the inverse deformation matrix119872minus1 in an easiermannerThen we find the matrix119872 by evaluating the inverseof the matrix119872minus1 Consequently the matrix119872 is found tohave the elements below

119860 = minus 111989211990510158401199051015840 119861 = minus1198921199101015840119910101584011989211991110158401199111015840 + 119892119910101584011991110158402minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402 119862 = minus1198921199091015840119909101584011989211991110158401199111015840 + 119892119909101584011991110158402minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402 119863 = minus1198921199091015840119909101584011989211991010158401199101015840 + 119892119909101584011991010158402minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402 119864 = 1198921199111015840119911101584011989211990910158401199101015840 minus 1198921199101015840119911101584011989211990910158401199111015840minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402 119865 = 1198921199101015840119910101584011989211990910158401199111015840 minus 1198921199091015840119910101584011989211991010158401199111015840minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402 119866 = 1198921199091015840119909101584011989211991010158401199111015840 minus 1198921199091015840119910101584011989211990910158401199111015840minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402

(33)

By using (22)ndash(28) in (33) and making the simplifications forthese matrix elements we finally obtain these elements asfollows 119860 = 1119880 (34)

119861 = 1198801199092 + 1199102 + 11991121199092 + 1199102 + 1199112 (35)

119862 = 1198801199102 + 1199092 + 11991121199092 + 1199102 + 1199112 (36)

119863 = 1198801199112 + 1199092 + 11991021199092 + 1199102 + 1199112 (37)

119864 = 119880119909119910 minus 1199091199101199092 + 1199102 + 1199112 (38)

6 Journal of Gravity

119865 = 119880119909119911 minus 1199091199111199092 + 1199102 + 1199112 (39)119866 = 119880119910119911 minus 1199101199111199092 + 1199102 + 1199112 (40)

These are the elements of the deformation matrix119872 To findthe determinant of matrix119872 we use (34)ndash(40) in (32) andthen we evaluate the determinant such as

det119872 = 1 (41)

We understand that our matrix belongs to SL(4) because ofthe determinant This is in accordance with the general factthat the symmetry group of the deformation matrices belongto GL(4)

Now we are to find the coordinate transformationsbetween these two space-times having the deformationtransformations given by (30)ndash(32) We need to find theinvertible matrix elements 1205971199091205831015840120597119909] between flat and curvedspace-times in Cartesian coordinates Therefore we write thematrix multiplication in (31) explicitly Then we also writethe deformation equations with summation convention in(30) explicitly We equate the metric elements 119892120583] in bothexplicit forms then we obtain the invertible matrix elements1205971199091205831015840120597119909] From these elements we construct the coordinatetransformation matrix Λ directly as given below

Λ =(((((((

1205971199051015840120597119905 0 0 00 1205971199091015840120597119909 1205971199091015840120597119910 12059711990910158401205971199110 1205971199101015840120597119909 1205971199101015840120597119910 12059711991010158401205971199110 1205971199111015840120597119909 1205971199111015840120597119910 1205971199111015840120597119911)))))))

=((((radic119860 0 0 00 radic119861 119864radic119862 119865radic1198630 119864radic119861 radic119862 119866radic1198630 119865radic119861 119866radic119862 radic119863

))))

(42)

We can write these elements more explicitly by using (34)ndash(40) and obtain 1205971199051015840120597119905 = radic 1119880 (43)1205971199091015840120597119909 = radic1198801199092 + 1199102 + 11991121199092 + 1199102 + 1199112 (44)

1205971199101015840120597119910 = radic1199092 + 1198801199102 + 11991121199092 + 1199102 + 1199112 (45)

1205971199111015840120597119911 = radic1199092 + 1199102 + 11988011991121199092 + 1199102 + 1199112 (46)

1205971199091015840120597119910 = (119880119909119910 minus 119909119910) (1199092 + 1199102 + 1199112)radic(1199092 + 1198801199102 + 1199112) (1199092 + 1199102 + 1199112) (47)

1205971199091015840120597119911 = (119880119909119911 minus 119909119911) (1199092 + 1199102 + 1199112)radic(1199092 + 1199102 + 1198801199112) (1199092 + 1199102 + 1199112) (48)

1205971199101015840120597119909 = (119880119909119910 minus 119909119910) (1199092 + 1199102 + 1199112)radic(1198801199092 + 1199102 + 1199112) (1199092 + 1199102 + 1199112) (49)

1205971199101015840120597119911 = (119880119910119911 minus 119910119911) (1199092 + 1199102 + 1199112)radic(1199092 + 1199102 + 1198801199112) (1199092 + 1199102 + 1199112) (50)

1205971199111015840120597119909 = (119880119909119911 minus 119909119911) (1199092 + 1199102 + 1199112)radic(1198801199092 + 1199102 + 1199112) (1199092 + 1199102 + 1199112) (51)

1205971199111015840120597119910 = (119880119910119911 minus 119910119911) (1199092 + 1199102 + 1199112)radic(1199092 + 1198801199102 + 1199112) (1199092 + 1199102 + 1199112) (52)

When we insert these elements into the coordinate transfor-mation matrix Λ we evaluate its determinant as follows

det119872= radic119880(1199092 + 1199102 + 1199112) (1199092 + 1199102 + 1199112) (1199092 + 1199102 + 1199112)radic(1198801199092 + 1199102 + 1199112) (1199092 + 1198801199102 + 1199112) (1199092 + 1199102 + 1198801199112) (53)

or

det119872= radic119880(1199092 + 1199102 + 1199112)32radic(1198801199092 + 1199102 + 1199112) (1199092 + 1198801199102 + 1199112) (1199092 + 1199102 + 1198801199112) (54)

We infer from the nonzero determinant of the coordinatetransformation matrix Λ that it is invertible and it belongsto the GL(4) symmetry group since it is a four by fourmatrixThis is again in accordance with the general fact aboutthe deformation matrices and their resulting coordinatetransformation matrices

The deformation matrix 119872 and the coordinate trans-formation matrix Λ are shown to belong to SL(4) andGL(4) respectively In Section 2 for the spherical coordinatescase we found that both matrices belong to SL(4) forthe transformations from flat tangent vector space-time tocurved Schwarzschild-type black hole space-time in theframework of 119891(119877) gravity These symmetry groups resultfrom the invariance of the space-time interval1198891199042 under thesetransformations

Similarly one can also find the deformation and coor-dinate transformation groups between the inertial frames

Journal of Gravity 7

and the noninertial frames of 119891(119877) gravity axially symmet-ric Kerr-Type and charged Reissner-Nordstrom-Type blackholes because their metric tensors have nondiagonal termsas in the case of Cartesian coordinates representations (22)ndash(28) for the metric of Schwarzschild-type black hole curvedspace-time (31) The metric tensor of a charged and rotating119891(119877) gravity black hole is given as [22]1198891199042 = minusΔ 1199031205882 [119889119905 minus 119886 sin2120579Ξ 119889120601]2 + 1205882Δ 119903 1198891199032 + 1205882Δ 120579 1198891205792+ Δ 120579sin21205791205882 [119886119889119905 minus 1199032 + 1198862Ξ 119889120601]2 (55)

whereΔ 119903 = (1199032 + 1198862) (1 + 119877012 1199032) minus 2119872119903 + 11987621 + 1198911015840 (1198770) Ξ = 1 minus 1198770121198862Δ 120579 = 1 minus 1198770121198862cos2120579(56)

Also 119886 and 119876 are the angular momentum and the charge ofthe 119891(119877) gravity black hole respectively If 119886 = 0 (55) turnsto be a Reissner-Nordstrom-Type black hole in 119891(119877) gravitywhereas if 119876 = 0 it gives a Kerr-Type 119891(119877) gravity blackholeWe can usemetric (55) for bothKerr-Type andReissner-Nordstrom-Type 119891(119877) gravity black holes and rewrite it as

1198891199042 = 1198862Δ 120579sin2120579 minus Δ 1199031205882 1198891199052+ 2119886 sin21205791205882Ξ [Δ 119903 minus Δ 120579 (1199032 + 1198862)] 119889119905119889120601+ 1205882Δ 119903 1198891199032 + 1205882Δ 120579 1198891205792+ Δ 120579sin2120579 (1199032 + 1198862)2 minus Δ 1199031198862sin41205791205882 1198891206012

(57)

and in matrix form

119892120583] =((((((((

1198862Δ 120579sin2120579 minus Δ 1199031205882 0 0 2119886 sin21205791205882Ξ [Δ 119903 minus Δ 120579 (1199032 + 1198862)]0 1205882Δ 119903 0 00 0 1205882Δ 120579 02119886 sin21205791205882Ξ [Δ 119903 minus Δ 120579 (1199032 + 1198862)] 0 0 Δ 120579 sin2120579 (1199032 + 1198862)2 minus Δ 1199031198862sin41205791205882)))))))) (58)

By using the similar steps in (31) and (32) we can obtainthe deformation matrix 119872 for the inertial observer andthe noninertial observer in the space-time curved by Kerr-Type or Reissner-Nordstrom-Type 119891(119877) gravity black holesAccordingly from the same step in (42) we then obtainthe coordinate transformation matrix Λ for these observersrsquoframe

4 Conclusions

In this paper we consider the transformations between acurved and flat space-time being the tangent vector space-time of the curved space-time and also the coordinates ofthis tangent space are called the local Lorentz coordinatesThe curvature of the space-time used here results fromthe Schwarzschild-type black hole whose solution for themetric tensor is obtained in the frameworks of modified119891(119877) gravity [7ndash9] We investigated the symmetry groups ofthe deformation matrix for the transformations of metricsand then the coordinate transformation matrix obtained by

this deformation matrix The extended and straightforwardphysical meaning and interpretations of the deformationsand conformal transformations can be found in the literaturesee [16 23 24]

These deformation and coordinate transformationmatri-ces are the identity matrices for the case 119880 = 1 This meansthat the both space-times are same flat space-timesThereforethe transformation matrices for deformation and coordinatetransformations must be identity matrix in both coordinatesystems

When we use 119880 = 1 in spherical coordinates forthe transformation matrices 119872 and Λ in (18) and (16)119908respectively we can easily realize that they are the identitymatrices Similarly if we again use 119880 = 1 in Cartesiancoordinates we find thematrices119872 andΛ in (32) and (42) arethe identity matrices Namely for 119880 = 1 all the nondiagonalelements of 119872 in (38) and (40) vanish In addition all thenondiagonal elements of Λ in (47) and (52) again vanish for119880 = 1 Taking 119880 = 1 can be viewed as a limiting case forchecking the calculations for transformation matrices whosesymmetry groups are investigated

8 Journal of Gravity

We infer that the deformation matrices the coordinatetransformation matrices and their symmetry groups can beinvestigated for the other type of black hole solutions in theframework ofmodified119891(119877) gravity Alsowe can consider thepresent transformation groups for the other type of modifiedgravitation theories for these various black hole solutions asthe candidate studies related to this topic

Competing Interests

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

References

[1] R A Nelson ldquoGeneralized Lorentz transformation for an accel-erated rotating frame of referencerdquo Journal of MathematicalPhysics vol 28 no 10 pp 2379ndash2383 1987

[2] H Nikolic ldquoRelativistic contraction of an accelerated rodrdquoAmerican Journal of Physics vol 67 no 11 pp 1007ndash1012 1999

[3] H Nikolic ldquoRelativistic contraction and related effects innoninertial framesrdquo Physical Review A vol 61 no 3 Article ID032109 8 pages 2000

[4] O Gron ldquoAcceleration and weight of extended bodies in thetheory of relativityrdquo American Journal of Physics vol 45 no 1pp 65ndash70 1977

[5] O Gron ldquoRelativictic description of a rotating diskrdquo AmericanJournal of Physics vol 43 pp 869ndash876 1975

[6] R DrsquoInverno Introtucing Einsteinrsquos Relativity OxfordUniversityPress Oxford UK 1992

[7] S Capozziello and M D Laurentis ldquoBlack holes and stellarstructures in f(R)-gravityrdquo httpsarxivorgabs12020394

[8] S Nojiri and S D Odintsov ldquoUnified cosmic history inmodified gravity from F(R) theory to Lorentz non-invariantmodelsrdquo Physics Reports vol 505 no 2ndash4 pp 59ndash144 2011

[9] S Nojiri and S D Odintsov ldquoIntroduction to modified gravityand gravitational alternative for dark energyrdquo InternationalJournal of Geometric Methods in Modern Physics vol 4 no 1pp 115ndash145 2007

[10] S Capozziello ldquoCurvature quintessencerdquo International Journalof Modern Physics D vol 11 no 4 pp 483ndash491 2002

[11] S Capozziello and S Vignolo ldquoA comment on lsquoThe Cauchyproblem of 119891(119877) gravityrsquordquo Classical and Quantum Gravity vol26 no 16 Article ID 168001 2009

[12] S Capozziello A Stabile and A Troisi ldquoSpherically symmetricsolutions in f (R) gravity via the Noether symmetry approachrdquoClassical and Quantum Gravity vol 24 no 8 pp 2153ndash21662007

[13] S Capozziello A Stabile and A Troisi ldquoSpherical symmetryin 119891(119877)-gravityrdquo Classical and Quantum Gravity vol 25 no 8Article ID 085004 2008

[14] B Coll ldquoA universal law of gravitational deformation forGeneral Relativityrdquo in Proceedings of the Spanish RelativisticMeeting EREs Salamanca Spain 1998

[15] B Coll J Llosa and D Soler ldquoThree-dimensional metrics asdeformations of a constant curvaturemetricrdquoGeneral Relativityand Gravitation vol 34 no 2 pp 269ndash282 2002

[16] S Capozziello and M D Laurentis ldquoExtended theories ofgravityrdquo Physics Reports vol 509 no 4-5 pp 167ndash321 2011

[17] H A Buchdahl ldquoMonthly notices of the royal astronomicalsocietyrdquoMonthly Notices of the Royal Astronomical Society vol150 no 1 pp 1ndash8 1970

[18] A A Starobinsky ldquoA new type of isotropic cosmological modelswithout singularityrdquo Physics Letters B vol 91 no 1 pp 99ndash1021980

[19] H Kleinert and H-J Schmidt ldquoCosmology with curvature-saturated gravitational lagrangian 119877radic1 + 11989741198772rdquo General Rela-tivity and Gravitation vol 34 no 8 pp 1295ndash1318 2002

[20] J D Barrow and A C Ottewill ldquoThe stability of general rela-tivistic cosmological theoryrdquo Journal of Physics AMathematicaland General vol 16 no 12 pp 2757ndash2776 1983

[21] M CarmeliGroupTheory and General Relativity McGraw-HillInternational Book London UK 1977

[22] A Larranaga ldquoA rotating charged black hole solution in 119891(119877)gravityrdquo Pramana vol 78 no 5 pp 697ndash703 2012

[23] N Lanahan-Tremblay and V Faraoni ldquoThe Cauchy problem off (R) gravityrdquo Classical and QuantumGravity vol 24 no 22 pp5667ndash5679 2007

[24] G Allemandi M Capone S Capozziello and M FrancaviglialdquoConformal aspects of the Palatini approach in extended theo-ries of gravityrdquo General Relativity and Gravitation vol 38 no 1pp 33ndash60 2006

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

Journal of Gravity 5

Finally we obtain the elements of the metric tensor for thecurved space-time of Schwarzschild-type black hole in termsof the Cartesian coordinates The indices for this metricwere shown by the primed Greek letters Since we investigatethe transformations between flat and curved space-times inCartesian coordinates we use the flat Cartesian metric 119892120583]whose indices are shown by the unprimed Greek letters suchthat

119892120583] =(minus1 0 0 00 1 0 00 0 1 00 0 0 1) (29)

Then the deformation transformations between the metrics119892120583] and 1198921205831015840]1015840 are 119892120583] = 1205971199091205831015840120597119909120583 120597119909]1015840120597119909] 1198921205831015840]1015840 (30)

We cannot call the deformation transformations in (30)conformal transformations of the metrics because both ofthe metrics are not in diagonal form As seen in (26)ndash(28)the curved space-time metric has nondiagonal terms Nowthe deformation matrix can be constructed by arranging (30)as below

(119892119905119905 0 0 00 119892119909119909 0 00 0 119892119910119910 00 0 0 119892119911119911)=119872(11989211990510158401199051015840 0 0 00 11989211990910158401199091015840 11989211990910158401199101015840 119892119909101584011991110158400 11989211991010158401199091015840 11989211991010158401199101015840 119892119910101584011991110158400 11989211991110158401199091015840 11989211991110158401199101015840 11989211991110158401199111015840)

(31)

and only nonzero elements of the matrix119872 forms as

119872 =(119860 0 0 00 119861 119864 1198650 119864 119862 1198660 119865 119866 119863) (32)

To find the deformation matrix M from (31) is not so easyInstead of that we write the inverse transformations in (31)and obtain the inverse deformation matrix119872minus1 in an easiermannerThen we find the matrix119872 by evaluating the inverseof the matrix119872minus1 Consequently the matrix119872 is found tohave the elements below

119860 = minus 111989211990510158401199051015840 119861 = minus1198921199101015840119910101584011989211991110158401199111015840 + 119892119910101584011991110158402minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402 119862 = minus1198921199091015840119909101584011989211991110158401199111015840 + 119892119909101584011991110158402minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402 119863 = minus1198921199091015840119909101584011989211991010158401199101015840 + 119892119909101584011991010158402minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402 119864 = 1198921199111015840119911101584011989211990910158401199101015840 minus 1198921199101015840119911101584011989211990910158401199111015840minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402 119865 = 1198921199101015840119910101584011989211990910158401199111015840 minus 1198921199091015840119910101584011989211991010158401199111015840minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402 119866 = 1198921199091015840119909101584011989211991010158401199111015840 minus 1198921199091015840119910101584011989211990910158401199111015840minus119892119909101584011990910158401198921199101015840119910101584011989211991110158401199111015840 minus 2119892119909101584011991010158401198921199101015840119911101584011989211990910158401199111015840 + 11989211990910158401199091015840119892119910101584011991110158402 + 11989211991110158401199111015840119892119909101584011991010158402 + 11989211991010158401199101015840119892119909101584011991110158402

(33)

By using (22)ndash(28) in (33) and making the simplifications forthese matrix elements we finally obtain these elements asfollows 119860 = 1119880 (34)

119861 = 1198801199092 + 1199102 + 11991121199092 + 1199102 + 1199112 (35)

119862 = 1198801199102 + 1199092 + 11991121199092 + 1199102 + 1199112 (36)

119863 = 1198801199112 + 1199092 + 11991021199092 + 1199102 + 1199112 (37)

119864 = 119880119909119910 minus 1199091199101199092 + 1199102 + 1199112 (38)

6 Journal of Gravity

119865 = 119880119909119911 minus 1199091199111199092 + 1199102 + 1199112 (39)119866 = 119880119910119911 minus 1199101199111199092 + 1199102 + 1199112 (40)

These are the elements of the deformation matrix119872 To findthe determinant of matrix119872 we use (34)ndash(40) in (32) andthen we evaluate the determinant such as

det119872 = 1 (41)

We understand that our matrix belongs to SL(4) because ofthe determinant This is in accordance with the general factthat the symmetry group of the deformation matrices belongto GL(4)

Now we are to find the coordinate transformationsbetween these two space-times having the deformationtransformations given by (30)ndash(32) We need to find theinvertible matrix elements 1205971199091205831015840120597119909] between flat and curvedspace-times in Cartesian coordinates Therefore we write thematrix multiplication in (31) explicitly Then we also writethe deformation equations with summation convention in(30) explicitly We equate the metric elements 119892120583] in bothexplicit forms then we obtain the invertible matrix elements1205971199091205831015840120597119909] From these elements we construct the coordinatetransformation matrix Λ directly as given below

Λ =(((((((

1205971199051015840120597119905 0 0 00 1205971199091015840120597119909 1205971199091015840120597119910 12059711990910158401205971199110 1205971199101015840120597119909 1205971199101015840120597119910 12059711991010158401205971199110 1205971199111015840120597119909 1205971199111015840120597119910 1205971199111015840120597119911)))))))

=((((radic119860 0 0 00 radic119861 119864radic119862 119865radic1198630 119864radic119861 radic119862 119866radic1198630 119865radic119861 119866radic119862 radic119863

))))

(42)

We can write these elements more explicitly by using (34)ndash(40) and obtain 1205971199051015840120597119905 = radic 1119880 (43)1205971199091015840120597119909 = radic1198801199092 + 1199102 + 11991121199092 + 1199102 + 1199112 (44)

1205971199101015840120597119910 = radic1199092 + 1198801199102 + 11991121199092 + 1199102 + 1199112 (45)

1205971199111015840120597119911 = radic1199092 + 1199102 + 11988011991121199092 + 1199102 + 1199112 (46)

1205971199091015840120597119910 = (119880119909119910 minus 119909119910) (1199092 + 1199102 + 1199112)radic(1199092 + 1198801199102 + 1199112) (1199092 + 1199102 + 1199112) (47)

1205971199091015840120597119911 = (119880119909119911 minus 119909119911) (1199092 + 1199102 + 1199112)radic(1199092 + 1199102 + 1198801199112) (1199092 + 1199102 + 1199112) (48)

1205971199101015840120597119909 = (119880119909119910 minus 119909119910) (1199092 + 1199102 + 1199112)radic(1198801199092 + 1199102 + 1199112) (1199092 + 1199102 + 1199112) (49)

1205971199101015840120597119911 = (119880119910119911 minus 119910119911) (1199092 + 1199102 + 1199112)radic(1199092 + 1199102 + 1198801199112) (1199092 + 1199102 + 1199112) (50)

1205971199111015840120597119909 = (119880119909119911 minus 119909119911) (1199092 + 1199102 + 1199112)radic(1198801199092 + 1199102 + 1199112) (1199092 + 1199102 + 1199112) (51)

1205971199111015840120597119910 = (119880119910119911 minus 119910119911) (1199092 + 1199102 + 1199112)radic(1199092 + 1198801199102 + 1199112) (1199092 + 1199102 + 1199112) (52)

When we insert these elements into the coordinate transfor-mation matrix Λ we evaluate its determinant as follows

det119872= radic119880(1199092 + 1199102 + 1199112) (1199092 + 1199102 + 1199112) (1199092 + 1199102 + 1199112)radic(1198801199092 + 1199102 + 1199112) (1199092 + 1198801199102 + 1199112) (1199092 + 1199102 + 1198801199112) (53)

or

det119872= radic119880(1199092 + 1199102 + 1199112)32radic(1198801199092 + 1199102 + 1199112) (1199092 + 1198801199102 + 1199112) (1199092 + 1199102 + 1198801199112) (54)

We infer from the nonzero determinant of the coordinatetransformation matrix Λ that it is invertible and it belongsto the GL(4) symmetry group since it is a four by fourmatrixThis is again in accordance with the general fact aboutthe deformation matrices and their resulting coordinatetransformation matrices

The deformation matrix 119872 and the coordinate trans-formation matrix Λ are shown to belong to SL(4) andGL(4) respectively In Section 2 for the spherical coordinatescase we found that both matrices belong to SL(4) forthe transformations from flat tangent vector space-time tocurved Schwarzschild-type black hole space-time in theframework of 119891(119877) gravity These symmetry groups resultfrom the invariance of the space-time interval1198891199042 under thesetransformations

Similarly one can also find the deformation and coor-dinate transformation groups between the inertial frames

Journal of Gravity 7

and the noninertial frames of 119891(119877) gravity axially symmet-ric Kerr-Type and charged Reissner-Nordstrom-Type blackholes because their metric tensors have nondiagonal termsas in the case of Cartesian coordinates representations (22)ndash(28) for the metric of Schwarzschild-type black hole curvedspace-time (31) The metric tensor of a charged and rotating119891(119877) gravity black hole is given as [22]1198891199042 = minusΔ 1199031205882 [119889119905 minus 119886 sin2120579Ξ 119889120601]2 + 1205882Δ 119903 1198891199032 + 1205882Δ 120579 1198891205792+ Δ 120579sin21205791205882 [119886119889119905 minus 1199032 + 1198862Ξ 119889120601]2 (55)

whereΔ 119903 = (1199032 + 1198862) (1 + 119877012 1199032) minus 2119872119903 + 11987621 + 1198911015840 (1198770) Ξ = 1 minus 1198770121198862Δ 120579 = 1 minus 1198770121198862cos2120579(56)

Also 119886 and 119876 are the angular momentum and the charge ofthe 119891(119877) gravity black hole respectively If 119886 = 0 (55) turnsto be a Reissner-Nordstrom-Type black hole in 119891(119877) gravitywhereas if 119876 = 0 it gives a Kerr-Type 119891(119877) gravity blackholeWe can usemetric (55) for bothKerr-Type andReissner-Nordstrom-Type 119891(119877) gravity black holes and rewrite it as

1198891199042 = 1198862Δ 120579sin2120579 minus Δ 1199031205882 1198891199052+ 2119886 sin21205791205882Ξ [Δ 119903 minus Δ 120579 (1199032 + 1198862)] 119889119905119889120601+ 1205882Δ 119903 1198891199032 + 1205882Δ 120579 1198891205792+ Δ 120579sin2120579 (1199032 + 1198862)2 minus Δ 1199031198862sin41205791205882 1198891206012

(57)

and in matrix form

119892120583] =((((((((

1198862Δ 120579sin2120579 minus Δ 1199031205882 0 0 2119886 sin21205791205882Ξ [Δ 119903 minus Δ 120579 (1199032 + 1198862)]0 1205882Δ 119903 0 00 0 1205882Δ 120579 02119886 sin21205791205882Ξ [Δ 119903 minus Δ 120579 (1199032 + 1198862)] 0 0 Δ 120579 sin2120579 (1199032 + 1198862)2 minus Δ 1199031198862sin41205791205882)))))))) (58)

By using the similar steps in (31) and (32) we can obtainthe deformation matrix 119872 for the inertial observer andthe noninertial observer in the space-time curved by Kerr-Type or Reissner-Nordstrom-Type 119891(119877) gravity black holesAccordingly from the same step in (42) we then obtainthe coordinate transformation matrix Λ for these observersrsquoframe

4 Conclusions

In this paper we consider the transformations between acurved and flat space-time being the tangent vector space-time of the curved space-time and also the coordinates ofthis tangent space are called the local Lorentz coordinatesThe curvature of the space-time used here results fromthe Schwarzschild-type black hole whose solution for themetric tensor is obtained in the frameworks of modified119891(119877) gravity [7ndash9] We investigated the symmetry groups ofthe deformation matrix for the transformations of metricsand then the coordinate transformation matrix obtained by

this deformation matrix The extended and straightforwardphysical meaning and interpretations of the deformationsand conformal transformations can be found in the literaturesee [16 23 24]

These deformation and coordinate transformationmatri-ces are the identity matrices for the case 119880 = 1 This meansthat the both space-times are same flat space-timesThereforethe transformation matrices for deformation and coordinatetransformations must be identity matrix in both coordinatesystems

When we use 119880 = 1 in spherical coordinates forthe transformation matrices 119872 and Λ in (18) and (16)119908respectively we can easily realize that they are the identitymatrices Similarly if we again use 119880 = 1 in Cartesiancoordinates we find thematrices119872 andΛ in (32) and (42) arethe identity matrices Namely for 119880 = 1 all the nondiagonalelements of 119872 in (38) and (40) vanish In addition all thenondiagonal elements of Λ in (47) and (52) again vanish for119880 = 1 Taking 119880 = 1 can be viewed as a limiting case forchecking the calculations for transformation matrices whosesymmetry groups are investigated

8 Journal of Gravity

We infer that the deformation matrices the coordinatetransformation matrices and their symmetry groups can beinvestigated for the other type of black hole solutions in theframework ofmodified119891(119877) gravity Alsowe can consider thepresent transformation groups for the other type of modifiedgravitation theories for these various black hole solutions asthe candidate studies related to this topic

Competing Interests

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

References

[1] R A Nelson ldquoGeneralized Lorentz transformation for an accel-erated rotating frame of referencerdquo Journal of MathematicalPhysics vol 28 no 10 pp 2379ndash2383 1987

[2] H Nikolic ldquoRelativistic contraction of an accelerated rodrdquoAmerican Journal of Physics vol 67 no 11 pp 1007ndash1012 1999

[3] H Nikolic ldquoRelativistic contraction and related effects innoninertial framesrdquo Physical Review A vol 61 no 3 Article ID032109 8 pages 2000

[4] O Gron ldquoAcceleration and weight of extended bodies in thetheory of relativityrdquo American Journal of Physics vol 45 no 1pp 65ndash70 1977

[5] O Gron ldquoRelativictic description of a rotating diskrdquo AmericanJournal of Physics vol 43 pp 869ndash876 1975

[6] R DrsquoInverno Introtucing Einsteinrsquos Relativity OxfordUniversityPress Oxford UK 1992

[7] S Capozziello and M D Laurentis ldquoBlack holes and stellarstructures in f(R)-gravityrdquo httpsarxivorgabs12020394

[8] S Nojiri and S D Odintsov ldquoUnified cosmic history inmodified gravity from F(R) theory to Lorentz non-invariantmodelsrdquo Physics Reports vol 505 no 2ndash4 pp 59ndash144 2011

[9] S Nojiri and S D Odintsov ldquoIntroduction to modified gravityand gravitational alternative for dark energyrdquo InternationalJournal of Geometric Methods in Modern Physics vol 4 no 1pp 115ndash145 2007

[10] S Capozziello ldquoCurvature quintessencerdquo International Journalof Modern Physics D vol 11 no 4 pp 483ndash491 2002

[11] S Capozziello and S Vignolo ldquoA comment on lsquoThe Cauchyproblem of 119891(119877) gravityrsquordquo Classical and Quantum Gravity vol26 no 16 Article ID 168001 2009

[12] S Capozziello A Stabile and A Troisi ldquoSpherically symmetricsolutions in f (R) gravity via the Noether symmetry approachrdquoClassical and Quantum Gravity vol 24 no 8 pp 2153ndash21662007

[13] S Capozziello A Stabile and A Troisi ldquoSpherical symmetryin 119891(119877)-gravityrdquo Classical and Quantum Gravity vol 25 no 8Article ID 085004 2008

[14] B Coll ldquoA universal law of gravitational deformation forGeneral Relativityrdquo in Proceedings of the Spanish RelativisticMeeting EREs Salamanca Spain 1998

[15] B Coll J Llosa and D Soler ldquoThree-dimensional metrics asdeformations of a constant curvaturemetricrdquoGeneral Relativityand Gravitation vol 34 no 2 pp 269ndash282 2002

[16] S Capozziello and M D Laurentis ldquoExtended theories ofgravityrdquo Physics Reports vol 509 no 4-5 pp 167ndash321 2011

[17] H A Buchdahl ldquoMonthly notices of the royal astronomicalsocietyrdquoMonthly Notices of the Royal Astronomical Society vol150 no 1 pp 1ndash8 1970

[18] A A Starobinsky ldquoA new type of isotropic cosmological modelswithout singularityrdquo Physics Letters B vol 91 no 1 pp 99ndash1021980

[19] H Kleinert and H-J Schmidt ldquoCosmology with curvature-saturated gravitational lagrangian 119877radic1 + 11989741198772rdquo General Rela-tivity and Gravitation vol 34 no 8 pp 1295ndash1318 2002

[20] J D Barrow and A C Ottewill ldquoThe stability of general rela-tivistic cosmological theoryrdquo Journal of Physics AMathematicaland General vol 16 no 12 pp 2757ndash2776 1983

[21] M CarmeliGroupTheory and General Relativity McGraw-HillInternational Book London UK 1977

[22] A Larranaga ldquoA rotating charged black hole solution in 119891(119877)gravityrdquo Pramana vol 78 no 5 pp 697ndash703 2012

[23] N Lanahan-Tremblay and V Faraoni ldquoThe Cauchy problem off (R) gravityrdquo Classical and QuantumGravity vol 24 no 22 pp5667ndash5679 2007

[24] G Allemandi M Capone S Capozziello and M FrancaviglialdquoConformal aspects of the Palatini approach in extended theo-ries of gravityrdquo General Relativity and Gravitation vol 38 no 1pp 33ndash60 2006

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 Journal of Gravity

119865 = 119880119909119911 minus 1199091199111199092 + 1199102 + 1199112 (39)119866 = 119880119910119911 minus 1199101199111199092 + 1199102 + 1199112 (40)

These are the elements of the deformation matrix119872 To findthe determinant of matrix119872 we use (34)ndash(40) in (32) andthen we evaluate the determinant such as

det119872 = 1 (41)

We understand that our matrix belongs to SL(4) because ofthe determinant This is in accordance with the general factthat the symmetry group of the deformation matrices belongto GL(4)

Now we are to find the coordinate transformationsbetween these two space-times having the deformationtransformations given by (30)ndash(32) We need to find theinvertible matrix elements 1205971199091205831015840120597119909] between flat and curvedspace-times in Cartesian coordinates Therefore we write thematrix multiplication in (31) explicitly Then we also writethe deformation equations with summation convention in(30) explicitly We equate the metric elements 119892120583] in bothexplicit forms then we obtain the invertible matrix elements1205971199091205831015840120597119909] From these elements we construct the coordinatetransformation matrix Λ directly as given below

Λ =(((((((

1205971199051015840120597119905 0 0 00 1205971199091015840120597119909 1205971199091015840120597119910 12059711990910158401205971199110 1205971199101015840120597119909 1205971199101015840120597119910 12059711991010158401205971199110 1205971199111015840120597119909 1205971199111015840120597119910 1205971199111015840120597119911)))))))

=((((radic119860 0 0 00 radic119861 119864radic119862 119865radic1198630 119864radic119861 radic119862 119866radic1198630 119865radic119861 119866radic119862 radic119863

))))

(42)

We can write these elements more explicitly by using (34)ndash(40) and obtain 1205971199051015840120597119905 = radic 1119880 (43)1205971199091015840120597119909 = radic1198801199092 + 1199102 + 11991121199092 + 1199102 + 1199112 (44)

1205971199101015840120597119910 = radic1199092 + 1198801199102 + 11991121199092 + 1199102 + 1199112 (45)

1205971199111015840120597119911 = radic1199092 + 1199102 + 11988011991121199092 + 1199102 + 1199112 (46)

1205971199091015840120597119910 = (119880119909119910 minus 119909119910) (1199092 + 1199102 + 1199112)radic(1199092 + 1198801199102 + 1199112) (1199092 + 1199102 + 1199112) (47)

1205971199091015840120597119911 = (119880119909119911 minus 119909119911) (1199092 + 1199102 + 1199112)radic(1199092 + 1199102 + 1198801199112) (1199092 + 1199102 + 1199112) (48)

1205971199101015840120597119909 = (119880119909119910 minus 119909119910) (1199092 + 1199102 + 1199112)radic(1198801199092 + 1199102 + 1199112) (1199092 + 1199102 + 1199112) (49)

1205971199101015840120597119911 = (119880119910119911 minus 119910119911) (1199092 + 1199102 + 1199112)radic(1199092 + 1199102 + 1198801199112) (1199092 + 1199102 + 1199112) (50)

1205971199111015840120597119909 = (119880119909119911 minus 119909119911) (1199092 + 1199102 + 1199112)radic(1198801199092 + 1199102 + 1199112) (1199092 + 1199102 + 1199112) (51)

1205971199111015840120597119910 = (119880119910119911 minus 119910119911) (1199092 + 1199102 + 1199112)radic(1199092 + 1198801199102 + 1199112) (1199092 + 1199102 + 1199112) (52)

When we insert these elements into the coordinate transfor-mation matrix Λ we evaluate its determinant as follows

det119872= radic119880(1199092 + 1199102 + 1199112) (1199092 + 1199102 + 1199112) (1199092 + 1199102 + 1199112)radic(1198801199092 + 1199102 + 1199112) (1199092 + 1198801199102 + 1199112) (1199092 + 1199102 + 1198801199112) (53)

or

det119872= radic119880(1199092 + 1199102 + 1199112)32radic(1198801199092 + 1199102 + 1199112) (1199092 + 1198801199102 + 1199112) (1199092 + 1199102 + 1198801199112) (54)

We infer from the nonzero determinant of the coordinatetransformation matrix Λ that it is invertible and it belongsto the GL(4) symmetry group since it is a four by fourmatrixThis is again in accordance with the general fact aboutthe deformation matrices and their resulting coordinatetransformation matrices

The deformation matrix 119872 and the coordinate trans-formation matrix Λ are shown to belong to SL(4) andGL(4) respectively In Section 2 for the spherical coordinatescase we found that both matrices belong to SL(4) forthe transformations from flat tangent vector space-time tocurved Schwarzschild-type black hole space-time in theframework of 119891(119877) gravity These symmetry groups resultfrom the invariance of the space-time interval1198891199042 under thesetransformations

Similarly one can also find the deformation and coor-dinate transformation groups between the inertial frames

Journal of Gravity 7

and the noninertial frames of 119891(119877) gravity axially symmet-ric Kerr-Type and charged Reissner-Nordstrom-Type blackholes because their metric tensors have nondiagonal termsas in the case of Cartesian coordinates representations (22)ndash(28) for the metric of Schwarzschild-type black hole curvedspace-time (31) The metric tensor of a charged and rotating119891(119877) gravity black hole is given as [22]1198891199042 = minusΔ 1199031205882 [119889119905 minus 119886 sin2120579Ξ 119889120601]2 + 1205882Δ 119903 1198891199032 + 1205882Δ 120579 1198891205792+ Δ 120579sin21205791205882 [119886119889119905 minus 1199032 + 1198862Ξ 119889120601]2 (55)

whereΔ 119903 = (1199032 + 1198862) (1 + 119877012 1199032) minus 2119872119903 + 11987621 + 1198911015840 (1198770) Ξ = 1 minus 1198770121198862Δ 120579 = 1 minus 1198770121198862cos2120579(56)

Also 119886 and 119876 are the angular momentum and the charge ofthe 119891(119877) gravity black hole respectively If 119886 = 0 (55) turnsto be a Reissner-Nordstrom-Type black hole in 119891(119877) gravitywhereas if 119876 = 0 it gives a Kerr-Type 119891(119877) gravity blackholeWe can usemetric (55) for bothKerr-Type andReissner-Nordstrom-Type 119891(119877) gravity black holes and rewrite it as

1198891199042 = 1198862Δ 120579sin2120579 minus Δ 1199031205882 1198891199052+ 2119886 sin21205791205882Ξ [Δ 119903 minus Δ 120579 (1199032 + 1198862)] 119889119905119889120601+ 1205882Δ 119903 1198891199032 + 1205882Δ 120579 1198891205792+ Δ 120579sin2120579 (1199032 + 1198862)2 minus Δ 1199031198862sin41205791205882 1198891206012

(57)

and in matrix form

119892120583] =((((((((

1198862Δ 120579sin2120579 minus Δ 1199031205882 0 0 2119886 sin21205791205882Ξ [Δ 119903 minus Δ 120579 (1199032 + 1198862)]0 1205882Δ 119903 0 00 0 1205882Δ 120579 02119886 sin21205791205882Ξ [Δ 119903 minus Δ 120579 (1199032 + 1198862)] 0 0 Δ 120579 sin2120579 (1199032 + 1198862)2 minus Δ 1199031198862sin41205791205882)))))))) (58)

By using the similar steps in (31) and (32) we can obtainthe deformation matrix 119872 for the inertial observer andthe noninertial observer in the space-time curved by Kerr-Type or Reissner-Nordstrom-Type 119891(119877) gravity black holesAccordingly from the same step in (42) we then obtainthe coordinate transformation matrix Λ for these observersrsquoframe

4 Conclusions

In this paper we consider the transformations between acurved and flat space-time being the tangent vector space-time of the curved space-time and also the coordinates ofthis tangent space are called the local Lorentz coordinatesThe curvature of the space-time used here results fromthe Schwarzschild-type black hole whose solution for themetric tensor is obtained in the frameworks of modified119891(119877) gravity [7ndash9] We investigated the symmetry groups ofthe deformation matrix for the transformations of metricsand then the coordinate transformation matrix obtained by

this deformation matrix The extended and straightforwardphysical meaning and interpretations of the deformationsand conformal transformations can be found in the literaturesee [16 23 24]

These deformation and coordinate transformationmatri-ces are the identity matrices for the case 119880 = 1 This meansthat the both space-times are same flat space-timesThereforethe transformation matrices for deformation and coordinatetransformations must be identity matrix in both coordinatesystems

When we use 119880 = 1 in spherical coordinates forthe transformation matrices 119872 and Λ in (18) and (16)119908respectively we can easily realize that they are the identitymatrices Similarly if we again use 119880 = 1 in Cartesiancoordinates we find thematrices119872 andΛ in (32) and (42) arethe identity matrices Namely for 119880 = 1 all the nondiagonalelements of 119872 in (38) and (40) vanish In addition all thenondiagonal elements of Λ in (47) and (52) again vanish for119880 = 1 Taking 119880 = 1 can be viewed as a limiting case forchecking the calculations for transformation matrices whosesymmetry groups are investigated

8 Journal of Gravity

We infer that the deformation matrices the coordinatetransformation matrices and their symmetry groups can beinvestigated for the other type of black hole solutions in theframework ofmodified119891(119877) gravity Alsowe can consider thepresent transformation groups for the other type of modifiedgravitation theories for these various black hole solutions asthe candidate studies related to this topic

Competing Interests

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

References

[1] R A Nelson ldquoGeneralized Lorentz transformation for an accel-erated rotating frame of referencerdquo Journal of MathematicalPhysics vol 28 no 10 pp 2379ndash2383 1987

[2] H Nikolic ldquoRelativistic contraction of an accelerated rodrdquoAmerican Journal of Physics vol 67 no 11 pp 1007ndash1012 1999

[3] H Nikolic ldquoRelativistic contraction and related effects innoninertial framesrdquo Physical Review A vol 61 no 3 Article ID032109 8 pages 2000

[4] O Gron ldquoAcceleration and weight of extended bodies in thetheory of relativityrdquo American Journal of Physics vol 45 no 1pp 65ndash70 1977

[5] O Gron ldquoRelativictic description of a rotating diskrdquo AmericanJournal of Physics vol 43 pp 869ndash876 1975

[6] R DrsquoInverno Introtucing Einsteinrsquos Relativity OxfordUniversityPress Oxford UK 1992

[7] S Capozziello and M D Laurentis ldquoBlack holes and stellarstructures in f(R)-gravityrdquo httpsarxivorgabs12020394

[8] S Nojiri and S D Odintsov ldquoUnified cosmic history inmodified gravity from F(R) theory to Lorentz non-invariantmodelsrdquo Physics Reports vol 505 no 2ndash4 pp 59ndash144 2011

[9] S Nojiri and S D Odintsov ldquoIntroduction to modified gravityand gravitational alternative for dark energyrdquo InternationalJournal of Geometric Methods in Modern Physics vol 4 no 1pp 115ndash145 2007

[10] S Capozziello ldquoCurvature quintessencerdquo International Journalof Modern Physics D vol 11 no 4 pp 483ndash491 2002

[11] S Capozziello and S Vignolo ldquoA comment on lsquoThe Cauchyproblem of 119891(119877) gravityrsquordquo Classical and Quantum Gravity vol26 no 16 Article ID 168001 2009

[12] S Capozziello A Stabile and A Troisi ldquoSpherically symmetricsolutions in f (R) gravity via the Noether symmetry approachrdquoClassical and Quantum Gravity vol 24 no 8 pp 2153ndash21662007

[13] S Capozziello A Stabile and A Troisi ldquoSpherical symmetryin 119891(119877)-gravityrdquo Classical and Quantum Gravity vol 25 no 8Article ID 085004 2008

[14] B Coll ldquoA universal law of gravitational deformation forGeneral Relativityrdquo in Proceedings of the Spanish RelativisticMeeting EREs Salamanca Spain 1998

[15] B Coll J Llosa and D Soler ldquoThree-dimensional metrics asdeformations of a constant curvaturemetricrdquoGeneral Relativityand Gravitation vol 34 no 2 pp 269ndash282 2002

[16] S Capozziello and M D Laurentis ldquoExtended theories ofgravityrdquo Physics Reports vol 509 no 4-5 pp 167ndash321 2011

[17] H A Buchdahl ldquoMonthly notices of the royal astronomicalsocietyrdquoMonthly Notices of the Royal Astronomical Society vol150 no 1 pp 1ndash8 1970

[18] A A Starobinsky ldquoA new type of isotropic cosmological modelswithout singularityrdquo Physics Letters B vol 91 no 1 pp 99ndash1021980

[19] H Kleinert and H-J Schmidt ldquoCosmology with curvature-saturated gravitational lagrangian 119877radic1 + 11989741198772rdquo General Rela-tivity and Gravitation vol 34 no 8 pp 1295ndash1318 2002

[20] J D Barrow and A C Ottewill ldquoThe stability of general rela-tivistic cosmological theoryrdquo Journal of Physics AMathematicaland General vol 16 no 12 pp 2757ndash2776 1983

[21] M CarmeliGroupTheory and General Relativity McGraw-HillInternational Book London UK 1977

[22] A Larranaga ldquoA rotating charged black hole solution in 119891(119877)gravityrdquo Pramana vol 78 no 5 pp 697ndash703 2012

[23] N Lanahan-Tremblay and V Faraoni ldquoThe Cauchy problem off (R) gravityrdquo Classical and QuantumGravity vol 24 no 22 pp5667ndash5679 2007

[24] G Allemandi M Capone S Capozziello and M FrancaviglialdquoConformal aspects of the Palatini approach in extended theo-ries of gravityrdquo General Relativity and Gravitation vol 38 no 1pp 33ndash60 2006

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

Journal of Gravity 7

and the noninertial frames of 119891(119877) gravity axially symmet-ric Kerr-Type and charged Reissner-Nordstrom-Type blackholes because their metric tensors have nondiagonal termsas in the case of Cartesian coordinates representations (22)ndash(28) for the metric of Schwarzschild-type black hole curvedspace-time (31) The metric tensor of a charged and rotating119891(119877) gravity black hole is given as [22]1198891199042 = minusΔ 1199031205882 [119889119905 minus 119886 sin2120579Ξ 119889120601]2 + 1205882Δ 119903 1198891199032 + 1205882Δ 120579 1198891205792+ Δ 120579sin21205791205882 [119886119889119905 minus 1199032 + 1198862Ξ 119889120601]2 (55)

whereΔ 119903 = (1199032 + 1198862) (1 + 119877012 1199032) minus 2119872119903 + 11987621 + 1198911015840 (1198770) Ξ = 1 minus 1198770121198862Δ 120579 = 1 minus 1198770121198862cos2120579(56)

Also 119886 and 119876 are the angular momentum and the charge ofthe 119891(119877) gravity black hole respectively If 119886 = 0 (55) turnsto be a Reissner-Nordstrom-Type black hole in 119891(119877) gravitywhereas if 119876 = 0 it gives a Kerr-Type 119891(119877) gravity blackholeWe can usemetric (55) for bothKerr-Type andReissner-Nordstrom-Type 119891(119877) gravity black holes and rewrite it as

1198891199042 = 1198862Δ 120579sin2120579 minus Δ 1199031205882 1198891199052+ 2119886 sin21205791205882Ξ [Δ 119903 minus Δ 120579 (1199032 + 1198862)] 119889119905119889120601+ 1205882Δ 119903 1198891199032 + 1205882Δ 120579 1198891205792+ Δ 120579sin2120579 (1199032 + 1198862)2 minus Δ 1199031198862sin41205791205882 1198891206012

(57)

and in matrix form

119892120583] =((((((((

1198862Δ 120579sin2120579 minus Δ 1199031205882 0 0 2119886 sin21205791205882Ξ [Δ 119903 minus Δ 120579 (1199032 + 1198862)]0 1205882Δ 119903 0 00 0 1205882Δ 120579 02119886 sin21205791205882Ξ [Δ 119903 minus Δ 120579 (1199032 + 1198862)] 0 0 Δ 120579 sin2120579 (1199032 + 1198862)2 minus Δ 1199031198862sin41205791205882)))))))) (58)

By using the similar steps in (31) and (32) we can obtainthe deformation matrix 119872 for the inertial observer andthe noninertial observer in the space-time curved by Kerr-Type or Reissner-Nordstrom-Type 119891(119877) gravity black holesAccordingly from the same step in (42) we then obtainthe coordinate transformation matrix Λ for these observersrsquoframe

4 Conclusions

In this paper we consider the transformations between acurved and flat space-time being the tangent vector space-time of the curved space-time and also the coordinates ofthis tangent space are called the local Lorentz coordinatesThe curvature of the space-time used here results fromthe Schwarzschild-type black hole whose solution for themetric tensor is obtained in the frameworks of modified119891(119877) gravity [7ndash9] We investigated the symmetry groups ofthe deformation matrix for the transformations of metricsand then the coordinate transformation matrix obtained by

this deformation matrix The extended and straightforwardphysical meaning and interpretations of the deformationsand conformal transformations can be found in the literaturesee [16 23 24]

These deformation and coordinate transformationmatri-ces are the identity matrices for the case 119880 = 1 This meansthat the both space-times are same flat space-timesThereforethe transformation matrices for deformation and coordinatetransformations must be identity matrix in both coordinatesystems

When we use 119880 = 1 in spherical coordinates forthe transformation matrices 119872 and Λ in (18) and (16)119908respectively we can easily realize that they are the identitymatrices Similarly if we again use 119880 = 1 in Cartesiancoordinates we find thematrices119872 andΛ in (32) and (42) arethe identity matrices Namely for 119880 = 1 all the nondiagonalelements of 119872 in (38) and (40) vanish In addition all thenondiagonal elements of Λ in (47) and (52) again vanish for119880 = 1 Taking 119880 = 1 can be viewed as a limiting case forchecking the calculations for transformation matrices whosesymmetry groups are investigated

8 Journal of Gravity

We infer that the deformation matrices the coordinatetransformation matrices and their symmetry groups can beinvestigated for the other type of black hole solutions in theframework ofmodified119891(119877) gravity Alsowe can consider thepresent transformation groups for the other type of modifiedgravitation theories for these various black hole solutions asthe candidate studies related to this topic

Competing Interests

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

References

[1] R A Nelson ldquoGeneralized Lorentz transformation for an accel-erated rotating frame of referencerdquo Journal of MathematicalPhysics vol 28 no 10 pp 2379ndash2383 1987

[2] H Nikolic ldquoRelativistic contraction of an accelerated rodrdquoAmerican Journal of Physics vol 67 no 11 pp 1007ndash1012 1999

[3] H Nikolic ldquoRelativistic contraction and related effects innoninertial framesrdquo Physical Review A vol 61 no 3 Article ID032109 8 pages 2000

[4] O Gron ldquoAcceleration and weight of extended bodies in thetheory of relativityrdquo American Journal of Physics vol 45 no 1pp 65ndash70 1977

[5] O Gron ldquoRelativictic description of a rotating diskrdquo AmericanJournal of Physics vol 43 pp 869ndash876 1975

[6] R DrsquoInverno Introtucing Einsteinrsquos Relativity OxfordUniversityPress Oxford UK 1992

[7] S Capozziello and M D Laurentis ldquoBlack holes and stellarstructures in f(R)-gravityrdquo httpsarxivorgabs12020394

[8] S Nojiri and S D Odintsov ldquoUnified cosmic history inmodified gravity from F(R) theory to Lorentz non-invariantmodelsrdquo Physics Reports vol 505 no 2ndash4 pp 59ndash144 2011

[9] S Nojiri and S D Odintsov ldquoIntroduction to modified gravityand gravitational alternative for dark energyrdquo InternationalJournal of Geometric Methods in Modern Physics vol 4 no 1pp 115ndash145 2007

[10] S Capozziello ldquoCurvature quintessencerdquo International Journalof Modern Physics D vol 11 no 4 pp 483ndash491 2002

[11] S Capozziello and S Vignolo ldquoA comment on lsquoThe Cauchyproblem of 119891(119877) gravityrsquordquo Classical and Quantum Gravity vol26 no 16 Article ID 168001 2009

[12] S Capozziello A Stabile and A Troisi ldquoSpherically symmetricsolutions in f (R) gravity via the Noether symmetry approachrdquoClassical and Quantum Gravity vol 24 no 8 pp 2153ndash21662007

[13] S Capozziello A Stabile and A Troisi ldquoSpherical symmetryin 119891(119877)-gravityrdquo Classical and Quantum Gravity vol 25 no 8Article ID 085004 2008

[14] B Coll ldquoA universal law of gravitational deformation forGeneral Relativityrdquo in Proceedings of the Spanish RelativisticMeeting EREs Salamanca Spain 1998

[15] B Coll J Llosa and D Soler ldquoThree-dimensional metrics asdeformations of a constant curvaturemetricrdquoGeneral Relativityand Gravitation vol 34 no 2 pp 269ndash282 2002

[16] S Capozziello and M D Laurentis ldquoExtended theories ofgravityrdquo Physics Reports vol 509 no 4-5 pp 167ndash321 2011

[17] H A Buchdahl ldquoMonthly notices of the royal astronomicalsocietyrdquoMonthly Notices of the Royal Astronomical Society vol150 no 1 pp 1ndash8 1970

[18] A A Starobinsky ldquoA new type of isotropic cosmological modelswithout singularityrdquo Physics Letters B vol 91 no 1 pp 99ndash1021980

[19] H Kleinert and H-J Schmidt ldquoCosmology with curvature-saturated gravitational lagrangian 119877radic1 + 11989741198772rdquo General Rela-tivity and Gravitation vol 34 no 8 pp 1295ndash1318 2002

[20] J D Barrow and A C Ottewill ldquoThe stability of general rela-tivistic cosmological theoryrdquo Journal of Physics AMathematicaland General vol 16 no 12 pp 2757ndash2776 1983

[21] M CarmeliGroupTheory and General Relativity McGraw-HillInternational Book London UK 1977

[22] A Larranaga ldquoA rotating charged black hole solution in 119891(119877)gravityrdquo Pramana vol 78 no 5 pp 697ndash703 2012

[23] N Lanahan-Tremblay and V Faraoni ldquoThe Cauchy problem off (R) gravityrdquo Classical and QuantumGravity vol 24 no 22 pp5667ndash5679 2007

[24] G Allemandi M Capone S Capozziello and M FrancaviglialdquoConformal aspects of the Palatini approach in extended theo-ries of gravityrdquo General Relativity and Gravitation vol 38 no 1pp 33ndash60 2006

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 Journal of Gravity

We infer that the deformation matrices the coordinatetransformation matrices and their symmetry groups can beinvestigated for the other type of black hole solutions in theframework ofmodified119891(119877) gravity Alsowe can consider thepresent transformation groups for the other type of modifiedgravitation theories for these various black hole solutions asthe candidate studies related to this topic

Competing Interests

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

References

[1] R A Nelson ldquoGeneralized Lorentz transformation for an accel-erated rotating frame of referencerdquo Journal of MathematicalPhysics vol 28 no 10 pp 2379ndash2383 1987

[2] H Nikolic ldquoRelativistic contraction of an accelerated rodrdquoAmerican Journal of Physics vol 67 no 11 pp 1007ndash1012 1999

[3] H Nikolic ldquoRelativistic contraction and related effects innoninertial framesrdquo Physical Review A vol 61 no 3 Article ID032109 8 pages 2000

[4] O Gron ldquoAcceleration and weight of extended bodies in thetheory of relativityrdquo American Journal of Physics vol 45 no 1pp 65ndash70 1977

[5] O Gron ldquoRelativictic description of a rotating diskrdquo AmericanJournal of Physics vol 43 pp 869ndash876 1975

[6] R DrsquoInverno Introtucing Einsteinrsquos Relativity OxfordUniversityPress Oxford UK 1992

[7] S Capozziello and M D Laurentis ldquoBlack holes and stellarstructures in f(R)-gravityrdquo httpsarxivorgabs12020394

[8] S Nojiri and S D Odintsov ldquoUnified cosmic history inmodified gravity from F(R) theory to Lorentz non-invariantmodelsrdquo Physics Reports vol 505 no 2ndash4 pp 59ndash144 2011

[9] S Nojiri and S D Odintsov ldquoIntroduction to modified gravityand gravitational alternative for dark energyrdquo InternationalJournal of Geometric Methods in Modern Physics vol 4 no 1pp 115ndash145 2007

[10] S Capozziello ldquoCurvature quintessencerdquo International Journalof Modern Physics D vol 11 no 4 pp 483ndash491 2002

[11] S Capozziello and S Vignolo ldquoA comment on lsquoThe Cauchyproblem of 119891(119877) gravityrsquordquo Classical and Quantum Gravity vol26 no 16 Article ID 168001 2009

[12] S Capozziello A Stabile and A Troisi ldquoSpherically symmetricsolutions in f (R) gravity via the Noether symmetry approachrdquoClassical and Quantum Gravity vol 24 no 8 pp 2153ndash21662007

[13] S Capozziello A Stabile and A Troisi ldquoSpherical symmetryin 119891(119877)-gravityrdquo Classical and Quantum Gravity vol 25 no 8Article ID 085004 2008

[14] B Coll ldquoA universal law of gravitational deformation forGeneral Relativityrdquo in Proceedings of the Spanish RelativisticMeeting EREs Salamanca Spain 1998

[15] B Coll J Llosa and D Soler ldquoThree-dimensional metrics asdeformations of a constant curvaturemetricrdquoGeneral Relativityand Gravitation vol 34 no 2 pp 269ndash282 2002

[16] S Capozziello and M D Laurentis ldquoExtended theories ofgravityrdquo Physics Reports vol 509 no 4-5 pp 167ndash321 2011

[17] H A Buchdahl ldquoMonthly notices of the royal astronomicalsocietyrdquoMonthly Notices of the Royal Astronomical Society vol150 no 1 pp 1ndash8 1970

[18] A A Starobinsky ldquoA new type of isotropic cosmological modelswithout singularityrdquo Physics Letters B vol 91 no 1 pp 99ndash1021980

[19] H Kleinert and H-J Schmidt ldquoCosmology with curvature-saturated gravitational lagrangian 119877radic1 + 11989741198772rdquo General Rela-tivity and Gravitation vol 34 no 8 pp 1295ndash1318 2002

[20] J D Barrow and A C Ottewill ldquoThe stability of general rela-tivistic cosmological theoryrdquo Journal of Physics AMathematicaland General vol 16 no 12 pp 2757ndash2776 1983

[21] M CarmeliGroupTheory and General Relativity McGraw-HillInternational Book London UK 1977

[22] A Larranaga ldquoA rotating charged black hole solution in 119891(119877)gravityrdquo Pramana vol 78 no 5 pp 697ndash703 2012

[23] N Lanahan-Tremblay and V Faraoni ldquoThe Cauchy problem off (R) gravityrdquo Classical and QuantumGravity vol 24 no 22 pp5667ndash5679 2007

[24] G Allemandi M Capone S Capozziello and M FrancaviglialdquoConformal aspects of the Palatini approach in extended theo-ries of gravityrdquo General Relativity and Gravitation vol 38 no 1pp 33ndash60 2006

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