research article complex structure of the four...

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2013 Article ID 509749 6 pageshttpdxdoiorg1011552013509749

Research ArticleComplex Structure of the Four-Dimensional Kerr GeometryStringy System Kerr Theorem and Calabi-Yau Twofold

Alexander Burinskii

Theoretical Physics Laboratory NSI Russian Academy of Sciences B Tulskaya 52 Moscow 115191 Russia

Correspondence should be addressed to Alexander Burinskii buribraeacru

Received 12 November 2012 Accepted 9 February 2013

Academic Editor P Bussey

Copyright copy 2013 Alexander BurinskiiThis 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

The 4D Kerr geometry displays many wonderful relations with quantum world and in particular with superstring theory Thelightlike structure of fields near the Kerr singular ring is similar to the structure of Sen solution for a closed heterotic string Anotherstring open and complex appears in the complex representation of the Kerr geometry initiated by Newman Combination of thesestrings forms a membrane source of the Kerr geometry which is parallel to the structure of M-theory In this paper we give onemore evidence of this relationship emergence of the Calabi-Yau twofold (K3 surface) in twistorial structure of the Kerr geometryas a consequence of the Kerr theorem Finally we indicate that the Kerr stringy system may correspond to a complex embeddingof the critical119873 = 2 superstring

1 Introduction

Theblack hole solutions of diverse dimensions represent nowone of the basic objects for study in superstring theory Recentideas and methods in the black hole physics are based oncomplex analyticity and conformal field theory which unifiesthe black hole physics with superstring theory and physics ofelementary particles The Kerr solution plays in this respectespecial role Being obtained as a metric of a ldquospinning massrdquo[1] with angular momentum 119869 = 119898|119886| the Kerr solutionfound basic application as a metric of rotating black holeIn the four-dimensional Kerr solution parameter 119886 = 119869119898

is radius of the Kerr singular ring For |119886| lt 119898 the ringis covered by horizon but for parameters of the elementaryparticles |119886| ≫ 119898 the black hole horizons disappear andthe Kerr singular ring turns out to be naked Following thecensorship principle it should be covered by a source Duringfour decades of investigations structure of Kerrrsquos source wasspecified step by stepOne of the earliermodelswas themodelof theKerr ring as a closed string [2 3] It has been obtained in[4] that structure of the fields around the Kerr ring is similarto the structure of the heterotic string in the solutions to lowenergy string theory obtained by Sen [5] However the Kerrstring is branch line of the Kerr spacetime into two sheets[6] and this bizarre peculiarity created an alternative line of

investigations of the problem of Kerrrsquos source [7ndash12] whichled to conclusion that the source of the Kerr-Newman (KN)solution should form a rigidly rotating membrane or to bemore precise a highly oblate ellipsoidal bubble with a flatvacuum interior [9 12]

The charged KN solution [13] has found application as aconsistency with gravity classical model of spinning particle[2 8 9 14ndash16] which has gyromagnetic ratio 119892 = 2 as thatof the Dirac electron [14 15] (In fact the four observableparameters of the electron mass spin charge and magneticmoment indicate unambiguously that theKN solution is to bethe electron background geometry [17 18]) The KN solutiondisplays also other relationships with the Dirac electron [17ndash21] as well as the relationships with twistor theory [6 22ndash24]models of the soliton [10ndash12 25] and with basic structures ofsuperstring theory [3 4 17 18 24 26ndash29]

In this paper we consider complex structure of the Kerrgeometry [26ndash29] and reveal one new evidence of its inher-ent parallelism with twistor theory and superstring theoryNamely we show the presence of the Calabi-Yau twofold (K3surface) in complex structure of the Kerr geometry whichappears as a consequence of the Kerr theorem in the formof a quartic equation in the projective twistor space CP3 InSection 2 we describe briefly the real structure of the Kerr

2 Advances in High Energy Physics

geometry and the Kerr theorem which determines Kerrrsquosprincipal null congruence (PNC) in twistor terms

On the way to our principal result there appear afew important intermediate structures First of all it is thecomplexKerr geometry itself which is generated by theAppelcomplex shift method [30] and by the Newman complexretarded-time construction [31 32] We describe them inSection 3

In Section 4 we show that the source of the complex Kerrgeometry is an open complex string It is based on the oldremarks by Ooguri and Vafa that the complex world lines(CWL) parametrized by complex time parameter 120591 = 119905 + 119894120590

turns into a worldsheet of a complex string [33] and thecomplex Kerr string generating the complex Kerr geometryis to be an open string with orientifold worldsheet [26ndash29]

Finally we observe that the structure of the membranesource of the real Kerr geometry is parallel to formation ofthe membrane by the transfer from superstring theory to M-theory the closed Kerr string of the real Kerr geometry growsby extra worldsheet parameter from the open complex Kerrstring in analogue with the known superstringM-theorycorrespondence [34 35]

The parallelism of the complex Kerr geometry with thebasic structures of the superstring theory and in particularthe inherent existence of the K3 surface in twistorial CP3space of the principal null congruences allows us to supposethat the complex Kerr string represents a complex realizationof the critical119873 = 2 superstring theory embedding of whichinto complex Kerr geometry looks admissible leading to thepoint of view that complexification may be considered as analternative to compactification of higher dimensions

2 Real Structure of the KN Geometry

KNmetric is represented in the Kerr-Schild (KS) form [15] asfollows

119892120583120584= 120578120583120584+ 2ℎ1198903

1205831198903

120584 (1)

where 120578120583120584

is auxiliary Minkowski background in Cartesiancoordinates x = 119909120583 = (119905 119909 119910 119911)

ℎ = 1198752 119898119903 minus 119890

22

1199032 + 1198862cos2120579 119875 =

(1 + 119884119884)

radic2 (2)

and 1198903(x) is a tangent direction to a principal null congruence(PNC) which is determined by the form

1198903

120583119889119909120583= 119889119906 + 119884119889120577 + 119884119889120577 minus 119884119884119889119907 (3)

via function 119884(x) which is obtained from the Kerr theorem[15 36ndash42] (Here 120577 = (119909 + 119894119910)radic2 120577 = (119909 minus 119894119910)radic2 119906 = (119911 +119905)radic2 and 119907 = (119911 minus 119905)radic2 are the null Cartesian coordinates119903 120579 and 120601 are the Kerr oblate spheroidal coordinates and119884(x) = 119890

119894120601 tan(1205792) is a projective angular coordinate Theused signature is (minus + ++))

The PNC forms a caustic at the Kerr singular ring 119903 =

cos 120579 = 0 As a result the KN metric (1) and electromagneticpotential

119860120583= minus119875minus2 Re 119890

(119903 + 119894119886 cos 120579)1198903

120583 (4)

are aligned with Kerr PNC and concentrate near the Kerrring forming a closed stringmdashwaveguide for traveling elec-tromagnetic waves [2 3 17 18] Analysis of the Kerr-Sensolution to low energy string theory [5] showed that similarityof the Kerr ring with a closed strings is not only analoguebut it has really the structure of a fundamental heteroticstring [4] Along with this closed string the KN geometrycontains also a complex open string [26ndash29] which appearsin the complex representation of Kerr geometry initiated byNewman [31 32] This string gives an extradimension 120579 tothe stringy source (120579 isin [0 120587]) resulting in its extension to amembrane (bubble source [9 12]) A superstring counterpartof this extension is a transfer from superstring theory to 11-dimensional M-theory and M2-brane [35]

Kerr theorem determines the shear free null congruenceswith tangent direction (3) by means of the solution 119884(x) ofthe equation

119865 (119879119860) = 0 (5)

where 119865(119879119860) is an arbitrary holomorphic function in theprojective twistor space with CP3 coordinates

119879119860= 119884 120582

1= 120577 minus 119884119907 120582

2= 119906 + 119884120577 (6)

Using the Cartesian coordinates 119909120583 one can rearrangevariables and reduce function 119865(119879

119860) to the form 119865(119884 119909

120583)

which allows one to get solution of (5) in the form 119884(119909120583)

For the Kerr and KN solutions the function 119865(119884 119909120583)

turns out to be quadratic in 119884 as follows

119865 = 119860 (119909120583) 1198842+ 119861 (119909

120583) 119884 + 119862 (119909

120583) (7)

and (5) represents a quadric in the projective twistor spaceCP3 with a nondegenerate determinant Δ = (119861

2minus 4119860119862)

12

which determines the complex radial distance [39 43] asfollows

119903 = minusΔ = minus(1198612minus 4119860119862)

12

(8)

This case is explicitly resolved and yields two solutions

119884plusmn(119909120583) =

(minus119861 ∓ 119903)

2119860 (9)

which allows one to restore two PNC by means of (3)One can easily obtain from (7) and (9) that complex radial

distance 119903may also be determined from the relation

119903 = minus119889119865

119889119884 (10)

Advances in High Energy Physics 3

and therefore the Kerr singular ring 119903 = 0 corresponds tocaustics of the Kerr congruence as follows

119889119865

119889119884= 0 (11)

As a consequence of the Vietarsquos formulas the quadratic in119884 function (7) may be expressed via the solutions 119884plusmn(119909120583) inthe form

119865 (119884 119909120583) = 119860 (119884 minus 119884

+(119909120583)) (119884 minus 119884

minus(119909120583)) (12)

3 Complex Kerr Geometry and the ComplexRetarded-Time Construction

KN solution was initially obtained by a ldquocomplex trickrdquo [13]and Newman and Lind [31 32] showed that linearized KNsolution may be generated by a complex world line Thiscomplex trick was first described by Appel in 1887 [30] as acomplex shift Appel noticed that Coulomb solution

120601 () =119890

119903=

119890

radic1199092 + 1199102 + 1199112(13)

is invariant under the shift rarr + 119886 and consideredcomplex shift of the origin (119909

0 1199100 1199110) = (0 0 0) along 119911-

axis (1199090 1199100 1199110) = (0 0 minus119894119886) On the real slice he obtained

the complex potential

120601119886() = Re 119890

119903 (14)

with complex radial coordinate 119903 = 119903 + 119894119886 cos 120579 It wasshown in [2 39 43] that potential (14) corresponds exactlyto KN electromagnetic field and the exact KN solutionmay be described as a field generated by a complex sourcepropagating along a complex worldline parametrized bycomplex time 120591

119871as follows

119909120583

119871(120591119871) = 119909120583

0(0) + 119906

120583120591119871+119894119886

2119896120583

119871minus 119896120583

119877 (15)

where 119906120583 = (1 0 0 1) 119896119877= (1 0 0 minus1) and 119896

119871= (1 0 0 1)

Index 119871 labels it as a left structure and we should add acomplex conjugate right structure

119909120583

119877(120591119877) = 119909120583

0(0) + 119906

120583120591119877minus119894119886

2119896120583

119871minus 119896120583

119877 (16)

Therefore theKerr geometrymay be obtained by the complexshift from the Schwarzschild one indicating that from thecomplex point of view the Kerr and Schwarzschild geometriesare equivalent and differ only by the relative positions of thereal slice which for theKerr solution is to be aside of its centerComplex shift turns the Schwarzschild radial directions 119899 =

119903|119903| into complex radial directions 119903 of the twisted Kerrcongruence (Figure 1)

4 Complex Kerrrsquos String

It was obtained [26ndash29 33] that the complex world line119909120583

0(120591) parametrized by complex time 120591 = 119905 + 119894120590 represents

119911

10

5

0

minus5

minus1010

10550

0minus5minus5

minus10minus10

Figure 1 Twistor null lines of the Kerr congruence are focused onthe Kerr singular ring forming a two-sheeted spacetime branchedby closed string

really a two-dimensional surface which takes an intermediateposition between particle and string The correspondingldquohyperbolic stringrdquo equation [33] 120597

1205911205971205911199090(119905 120590) = 0 yields the

general solution

1199090(119905 120590) = 119909

119871(120591) + 119909

119877(120591) (17)

as sum of the analytic and antianalytic modes 119909119871(120591) 119909

119877(120591)

which are not necessarily complex conjugate For each realpoint 119909120583 the parameters 120591 and 120591 should be determined by acomplex retarded-time construction Complex source of theKN solution corresponds to two straight complex conjugateworldlines (15) (16) Contrary to the real case the complexretarded and advanced times 120591∓ = 119905 ∓ 119903 may be determinedby two different (left or right) complex null planes (Figure 2)which are generators of the complex light cone It yields fourdifferent roots for the left and right complex structures [3943] as follows

120591∓

119871= 119905 ∓ (119903

119871+ 119894119886 cos 120579

119871)

120591∓

119877= 119905 ∓ (119903

119877+ 119894119886 cos 120579

119877)

(18)

The real slice condition determines correlation betweenparameter 120590 = 119886 cos 120579 and angular directions 120579 of the nullrays of the Kerr congruence in the diapason 120579 isin [0 120587] Thisputs restriction 120590 isin [minus119886 119886] indicating that the complex stringis open and its endpoints 120590 = plusmn119886 may be associated withthe Chan-Paton charges of a quark-antiquark pair In the realslice the complex endpoints of the string are mapped to thenorth and south twistor null lines 120579 = 0 120587 see Figure 3Orientifold The complex open string boundary conditions[26ndash29] require the worldsheet orientifold structure [35 44ndash47] which turns the open string in a closed but folded oneThe worldsheet parity transformation Ω 120590 rarr minus120590 reversesorientation of the worldsheet and covers it for the secondtime in mirror direction Simultaneously the left and rightmodes are exchanged (Two oriented copies of the intervalΣ = [minus119886 119886] Σ+ = [minus119886 119886] and Σ

minus= [minus119886 119886] are joined


Real slice

Leftcomplex

world line 119909119871

Complexconjuate

world line 119909119877

Leftnull plane

Rightnull plane

Figure 2 The complex conjugate left and right null planes generate the left and right retarded and advanced roots

Σ119871

Σ119877

minus119886

119911minus

119911+

119911

210

minus1

minus2

minus3

3

3

3

2

2

1

1

0

0

minus1

minus1

minus2

minus2minus3

minus3

+119886

Figure 3 Ends of the open complex string associatedwith quantumnumbers of quark-antiquark pair are mapped onto the real half-infinite 119911+ 119911minus axial strings Dotted lines indicate orientifold projec-tion

forming a circle 1198781 = Σ+cup Σminus parametrized by 120579 and map

120579 rarr 120590 = 119886 cos 120579 covers theworldsheet twice)TheprojectionΩ is combined with space reflection 119877 119903 rarr minus119903 resulting in119877Ω 119903 rarr minus119903 which relates the retarded and advanced foldsof the complex light cones as follows

119877Ω 120591+997888rarr 120591minus (19)

preserving analyticity of the worldsheet The string modes119909119871(120591) 119909

119877(120591) are extended on the second half cycle by the

well-known extrapolation [35 47] as follows

119909119871(120591+) = 119909119877(120591minus) 119909

119877(120591+) = 119909119871(120591minus) (20)

which forms the folded string in with the retarded andadvanced modes are exchanged every half cycle

The real KN solution is generated by the straight complexworld line (CWL) (15) and by its conjugate right counterpart(16) When the complex string is excited the orientifoldcondition (20) becomes inconsistent with the complex con-jugation of the string ends and the world lines 119909

119871(120591) and

119909119877(120591) should be considered as independent complex sources

The projectionT = 119877Ω sets parity between the positive Kerrsheet determined by the right retarded time and the negativesheet of the left advanced time It allows one to escape theantianalytical right complex structure replacing it by the leftadvanced one and the problem is reduced to self-interactionof the retarded and advanced sources determined by thetime parameters 120591plusmn For any nontrivial (not straight) CWLthe Kerr theorem will generate different congruences for 120591+and 120591minus Each of these sources produces a two-sheeted Kerr-Schild geometry and the formal description of the resultingfourfolded congruence should be based on the multiparticleKerr-Schild solutions [40ndash42] (Physical motivation of such asplitting of the sources is discussed in seminal paper by DeWitt and Breme [48] where authors introduce the similarldquobitensorrdquo fields which are predecessors of the two-pointGreenrsquos functions and Feynman propagator The problemof physical interpretation goes beyond frame of this paperand will be considered elsewhere) The corresponding two-particle generating function of the Kerr theorem will be

1198652(119879119860) = 119865119871(119879119860) 119865119877(119879119860) (21)

where 119865119871and 119865

119877are determined by 119909

119871(120591+) and 119909

119871(120591minus) The

both factors are quadratic in119879119860The corresponding equation

1198652(119879119860) = 0 (22)

describes a quartic in CP3 which is the well-known Calabi-Yau twofold [35 47] We arrive at the result that excitationsof the Kerr complex string generate a Calabi-Yau twofold orK3 surface on the projective twistor space CP3

5 Outlook

One sees that the Kerr-Schild geometry displays strikingparallelism with basic structures of superstring theory Inthe recent paper [17 18] we argued that it is not accidentalbecause gravity is a fundamental part of the superstringtheory However the Kerr-Schild gravity being based ontwistor theory displays also some inherent relationships with


superstring theory which is confirmed by the principal resultof this papermdashpresence of the Calabi-Yau twofold in thecomplex twistorial structure of the Kerr geometry

In many respects the Kerr-Schild gravity resemblesthe twistor-string theory [24 49 50] which is also four-dimensional based on twistors and related with experi-mental particle physics On the other hand the complexKerr string has much in common with the complex critical119873 = 2 superstring [33 47 51] which is also related withtwistors and has the complex critical dimension two (fourreal dimensions) indicating that 119873 = 2 superstring maybe consistent with four-dimensional spacetime Howeversignature of the 119873 = 2 string may only be (2 2) or (4 0)which caused the obstacles for embedding of this stringin the spacetimes with minkowskian signature Up to ourknowledge this trouble was not resolved so far and theinitially enormous interest to the 119873 = 2 string seems to bedampened Meanwhile embedding of the119873 = 2 string in thecomplexified Kerr geometry is almost trivial task It hints thatstring-like structures of the real and complex Kerr geometryare not simply analogues but reflect the underlying dynamicsof the119873 = 2 superstring theory

In the same time along with wonderful parallelism thestringy systemof the four-dimensional KNgeometry displaysvery essential peculiarities

(i) The supplementary Kaluza-Klein space is absent andthe role of compactification circle is played by theKerr singular ring which realizes a ldquocompactificationwithout compactificationrdquo [17 18]

(ii) The lightlike twistorial rays are tangent to the Kerrsingular ring indicating that the Kerr ring is thelightlike string playing the role parallel to DLCQcircle in M-theory [52]

(iii) Consistency of the KN solution with gravitationalbackground of the electron [8 9 14 17 18] showsthat the characteristic length of the Kerr ring 119886 =

119869119898 should correspond to the Compton scale ofelementary particlesThe smallmasses and great spinsof the elementary particles indicate unambiguouslythat the black hole horizons disappear and the Kerrring is to be naked This offers new applications nextto the traditional attention of superstring theory toquantum black holes

The considered stringy structures of the real and complexKerr geometry set a parallelism between the 4D Kerr geom-etry and superstring theory indicating the potential role ofthis alliance in the particle physics On the other hand theserelationships show that complexification of the Kerr geome-trymay serve an alternative to traditional compactification ofhigher dimensions in superstring theory

Acknowledgments

The author thanks Theo M Nieuwenhuizen for permanentinterest to this work and useful conversations and alsovery thankful to Dirk Bouwmeester for invitation to LeidenUniversity where this work was finally crystallized in the

process of the numerous discussions with him and membersof his group J W Dalhuisen V A L Thompson and J M SSwearngin Author is also very thankful to D Galrsquotsov and LJarv for discussion at the Tallinn conference ldquo3Quatumrdquo andfor the given reference to the related paper [34]

References

[1] R P Kerr ldquoGravitational field of a spinning mass as an exampleof algebraically special metricsrdquo Physical Review Letters vol 11pp 237ndash238 1963

[2] A Burinskii ldquoMicrogeons with spinrdquo Journal of Experimentaland Theoretical Physics vol 39 no 2 p 193 1974

[3] D Ivanenko and A Burinskii ldquoGravitational strings in themodels of spinning elementary particlesrdquo Izv VUZ Fiz vol 5p 135 1975

[4] A Burinskii ldquoSome properties of the Kerr solution to lowenergy string theoryrdquo Physical Review D vol 52 no 10 pp5826ndash5831 1995

[5] A Sen ldquoRotating charged black hole solution in heterotic stringtheoryrdquo Physical Review Letters vol 69 no 7 pp 1006ndash10091992

[6] A Burinskii ldquoInstability of black hole horizon with respect toelectromagnetic excitationsrdquoGeneral Relativity andGravitationvol 41 no 10 pp 2281ndash2286 2009

[7] H Keres ldquoPhysical Interpretation of Solutions of the EinsteinEquationsrdquo Journal of Experimental andTheoretical Physics vol25 no 3 p 504 1967 (Russian)

[8] W Israel ldquoSource of the Kerr metricrdquo Physical Review D vol 2pp 641ndash646 1970

[9] C A L Lopez ldquoExtended model of the electron in generalrelativityrdquo Physical Review D vol 30 no 3 pp 313ndash316 1984

[10] A Burinskii ldquoSupersymmetric superconducting bag as the coreof a Kerr spinning particlerdquoGravitationampCosmology vol 8 no4 pp 261ndash271 2002

[11] A Burinskii ldquoRenormalization by gravity and theKerr spinningparticlerdquo Journal of Physics A vol 39 no 21 p 6209 2006

[12] A Burinskii ldquoRegularized Kerr-Newman solution as a gravi-tating solitonrdquo Journal of Physics A vol 43 no 39 Article ID392001 2010

[13] E TNewman E Couch K Chinnapared A Exton A Prakashand R Torrence ldquoMetric of a rotating chargedmassrdquo Journal ofMathematical Physics vol 6 pp 918ndash9920 1965

[14] B Carter ldquoGlobal structure of the Kerr family ofgravitationalfieldsrdquo Physical Review Letters vol 174 p 1559 1968

[15] G C Debney R P Kerr and A Schild ldquoSolutions of the Ein-stein and Einstein-Maxwell equationsrdquo Journal of MathematicalPhysics vol 10 pp 1842ndash1854 1969

[16] M Th Nieuwenhuizen ldquoThe electron and neutrino as solitonsin classical electromagnetismrdquo in Beyond the Quantum MTh Nieuwenhuizen et al Ed pp 332ndash342 World ScientificSingapore 2007

[17] A Burinskii ldquoGravitational strings beyond quantum theoryelectron as a closed heterotic stringrdquo Journal of Physics vol 361

[18] A Burinskii ldquoGravity vs Quantum theory is electron reallypointlikerdquo Journal of Physics vol 343 2012

[19] A Burinskii ldquoThe Dirac-Kerr-Newman electronrdquo Gravitationand Cosmology vol 14 no 2 pp 109ndash122 2008

[20] A Burinskii ldquoAxial stringy system of the Kerr spinning parti-clerdquo Gravitation amp Cosmology vol 10 no 1-2 pp 50ndash60 2004


[21] A Burinskii ldquoKerr geometry beyond the quantum theoryrdquoin Proceedings of the Conference lsquoBeyond the Quantumrsquo MTh Nieuwenhuizen et al Ed pp 319ndash321 World ScientificSingapore 2007

[22] A Burinskii ldquoTwistor-beam excitations of Black-Holes and pre-quantum Kerr-Schild geometryrdquo Theoretical and MathematicalPhysics vol 163 no 3 pp 782ndash787 2010

[23] A Burinskii ldquoFluctuating twistor-beam solutions and holo-graphic pre-quantum Kerr-Schild geometryrdquo Journal of Physicsvol 222 no 1 Article ID 012044 2010

[24] A Burinskii ldquoTwistorial analyticity and three stringy systemsof the Kerr spinning particlerdquo Physical Review D vol 70 no 8Article ID 086006 2004

[25] I Dymnikova ldquoSpinning superconducting electrovacuum soli-tonrdquo Physics Letters B vol 639 no 3-4 pp 368ndash372 2006

[26] A Burinskii ldquoString-like structures in complex Kerr geometryrdquoin Relativity Today R P Kerr and Z Perjes Eds p 149Akademiai Kiado Budapest Hungary 1994 httparxivorgabsgr-qc9303003

[27] A Burinskii Complex String as Source of Kerr GeometryProblems of Modern Physics Especial Space Explorations vol9 1995 httparxivorgabshep-th9503094

[28] A Burinskii ldquoThe Kerr geometry complex world lines andhyperbolic stringsrdquo Physics Letters A vol 185 no 5-6 pp 441ndash445 1994

[29] A Burinskii ldquoKerr spinning particle strings and superparticlemodelsrdquo Physical Review D vol 57 no 4 pp 2392ndash2396 1998

[30] E TWhittaker andG NWatsonACourse ofModern AnalysisCambridge University Press New York USA 1996

[31] E T Newman ldquoMaxwellrsquos equations and complex Minkowskispacerdquo Journal of Mathematical Physics vol 14 pp 102ndash1031973

[32] R W Lind and E T Newman ldquoComplexification of the alge-braically special gravitational fieldsrdquo Journal of MathematicalPhysics vol 15 pp 1103ndash1112 1974

[33] H Ooguri and C Vafa ldquo119873 = 2 heterotic stringsrdquo NuclearPhysics B vol 361 no 1 p 469 1991 Erratum Nuclear PhysicsB vol 83 pp 83ndash104 1991

[34] L M Dyson L Jarv and C V Johnson ldquoOblate toroidal andother shapes for the enhanconrdquo Journal of High Energy Physicsvol 2002 article 19 2002

[35] K Becker M Becker and J H Schwarz String Theory and M-TheorymdashA Modern Introduction Cambridge University Press2007

[36] D Kramer H Stephani E Herlt and M MacCallum ExactSolutions of Einsteinrsquos Field Equations Cambridge UniversityPress 1980

[37] R Penrose ldquoTwistor algebrardquo Journal of Mathematical Physicsvol 8 pp 345ndash366 1967

[38] R Penrose and W Rindler Spinors and Space-Time Vol 2Cambridge University Press Cambridge UK 1986

[39] A Burinskii and R P Kerr ldquoNonstationary Kerr congruencesrdquohttparxivorgabsgr-qc9501012

[40] A Burinskii ldquoWonderful consequences of the Kerr theoremmultiparticle Kerr-Schild solutionsrdquo Gravitation amp Cosmologyvol 11 no 4 pp 301ndash304 2005

[41] A Burinskii ldquoThe Kerr theorem Kerr-Schild formalism andmulti-particle Kerr-Schild solutionsrdquo Gravitation and Cosmol-ogy vol 12 no 2 pp 119ndash125 2006

[42] A Burinskii ldquoThe Kerr theorem and multi-particle Kerr-schild solutionsrdquo International Journal of Geometric Methods inModern Physics vol 4 no 3 Article ID p 437 2007

[43] A Burinskii ldquoComplex Kerr geometry and nonstationary Kerrsolutionsrdquo Physical Review D vol 67 no 12 Article ID 124024p 11 2003

[44] L Dixon J A Harvey C Vafa and E Witten ldquoStrings onorbifoldsrdquo Nuclear Physics B vol 261 pp 678ndash686 1985Erratum Nuclear Physics B vol 274 p 258 1986

[45] S Hamidi and C Vafa ldquoInteractions on orbifoldsrdquo NuclearPhysics B vol 279 no 3-4 pp 465ndash513 1987

[46] P Horava ldquoStrings on world-sheet orbifoldsrdquoNuclear Physics Bvol 327 no 2 pp 461ndash484 1989

[47] M B Green J Schwarz and E Witten Superstring Theory volI Cambridge University Press 1987

[48] B S De Witt and R W Breme ldquoRadiation damping in agravitational fieldrdquo Annals of Physics vol 9 no 2 pp 220ndash2591960

[49] V P Nair ldquoA current algebra for some gauge theory amplitudesrdquoPhysics Letters B vol 214 no 2 pp 215ndash218 1988

[50] E Witten ldquoPerturbative gauge theory as a string theory intwistor spacerdquo Communications in Mathematical Physics vol252 no 1ndash3 pp 189ndash258 2004

[51] A DrsquoAdda and F Lizzi ldquoSpace dimensions from supersymmetryfor the 119873 = 2 spinning string a four-dimensional modelrdquoPhysics Letters B vol 191 no 1-2 pp 85ndash90 1987

[52] L Susskind ldquoAnother Conjecture about M(atrix) Theoryrdquohttparxivorgabshep-th9704080

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geometry and the Kerr theorem which determines Kerrrsquosprincipal null congruence (PNC) in twistor terms

On the way to our principal result there appear afew important intermediate structures First of all it is thecomplexKerr geometry itself which is generated by theAppelcomplex shift method [30] and by the Newman complexretarded-time construction [31 32] We describe them inSection 3

In Section 4 we show that the source of the complex Kerrgeometry is an open complex string It is based on the oldremarks by Ooguri and Vafa that the complex world lines(CWL) parametrized by complex time parameter 120591 = 119905 + 119894120590

turns into a worldsheet of a complex string [33] and thecomplex Kerr string generating the complex Kerr geometryis to be an open string with orientifold worldsheet [26ndash29]

Finally we observe that the structure of the membranesource of the real Kerr geometry is parallel to formation ofthe membrane by the transfer from superstring theory to M-theory the closed Kerr string of the real Kerr geometry growsby extra worldsheet parameter from the open complex Kerrstring in analogue with the known superstringM-theorycorrespondence [34 35]

The parallelism of the complex Kerr geometry with thebasic structures of the superstring theory and in particularthe inherent existence of the K3 surface in twistorial CP3space of the principal null congruences allows us to supposethat the complex Kerr string represents a complex realizationof the critical119873 = 2 superstring theory embedding of whichinto complex Kerr geometry looks admissible leading to thepoint of view that complexification may be considered as analternative to compactification of higher dimensions

2 Real Structure of the KN Geometry

KNmetric is represented in the Kerr-Schild (KS) form [15] asfollows

119892120583120584= 120578120583120584+ 2ℎ1198903

1205831198903

120584 (1)

where 120578120583120584

is auxiliary Minkowski background in Cartesiancoordinates x = 119909120583 = (119905 119909 119910 119911)

ℎ = 1198752 119898119903 minus 119890

22

1199032 + 1198862cos2120579 119875 =

(1 + 119884119884)

radic2 (2)

and 1198903(x) is a tangent direction to a principal null congruence(PNC) which is determined by the form

1198903

120583119889119909120583= 119889119906 + 119884119889120577 + 119884119889120577 minus 119884119884119889119907 (3)

via function 119884(x) which is obtained from the Kerr theorem[15 36ndash42] (Here 120577 = (119909 + 119894119910)radic2 120577 = (119909 minus 119894119910)radic2 119906 = (119911 +119905)radic2 and 119907 = (119911 minus 119905)radic2 are the null Cartesian coordinates119903 120579 and 120601 are the Kerr oblate spheroidal coordinates and119884(x) = 119890

119894120601 tan(1205792) is a projective angular coordinate Theused signature is (minus + ++))

The PNC forms a caustic at the Kerr singular ring 119903 =

cos 120579 = 0 As a result the KN metric (1) and electromagneticpotential

119860120583= minus119875minus2 Re 119890

(119903 + 119894119886 cos 120579)1198903

120583 (4)

are aligned with Kerr PNC and concentrate near the Kerrring forming a closed stringmdashwaveguide for traveling elec-tromagnetic waves [2 3 17 18] Analysis of the Kerr-Sensolution to low energy string theory [5] showed that similarityof the Kerr ring with a closed strings is not only analoguebut it has really the structure of a fundamental heteroticstring [4] Along with this closed string the KN geometrycontains also a complex open string [26ndash29] which appearsin the complex representation of Kerr geometry initiated byNewman [31 32] This string gives an extradimension 120579 tothe stringy source (120579 isin [0 120587]) resulting in its extension to amembrane (bubble source [9 12]) A superstring counterpartof this extension is a transfer from superstring theory to 11-dimensional M-theory and M2-brane [35]

Kerr theorem determines the shear free null congruenceswith tangent direction (3) by means of the solution 119884(x) ofthe equation

119865 (119879119860) = 0 (5)

where 119865(119879119860) is an arbitrary holomorphic function in theprojective twistor space with CP3 coordinates

119879119860= 119884 120582

1= 120577 minus 119884119907 120582

2= 119906 + 119884120577 (6)

Using the Cartesian coordinates 119909120583 one can rearrangevariables and reduce function 119865(119879

119860) to the form 119865(119884 119909

120583)

which allows one to get solution of (5) in the form 119884(119909120583)

For the Kerr and KN solutions the function 119865(119884 119909120583)

turns out to be quadratic in 119884 as follows

119865 = 119860 (119909120583) 1198842+ 119861 (119909

120583) 119884 + 119862 (119909

120583) (7)

and (5) represents a quadric in the projective twistor spaceCP3 with a nondegenerate determinant Δ = (119861

2minus 4119860119862)

12

which determines the complex radial distance [39 43] asfollows

119903 = minusΔ = minus(1198612minus 4119860119862)

12

(8)

This case is explicitly resolved and yields two solutions

119884plusmn(119909120583) =

(minus119861 ∓ 119903)

2119860 (9)

which allows one to restore two PNC by means of (3)One can easily obtain from (7) and (9) that complex radial

distance 119903may also be determined from the relation

119903 = minus119889119865

119889119884 (10)



119889119865

119889119884= 0 (11)


119865 (119884 119909120583) = 119860 (119884 minus 119884

+(119909120583)) (119884 minus 119884

minus(119909120583)) (12)



120601 () =119890

119903=

119890

radic1199092 + 1199102 + 1199112(13)


0 1199100 1199110) = (0 0 0) along 119911-



120601119886() = Re 119890

119903 (14)


119871as follows

119909120583

119871(120591119871) = 119909120583

0(0) + 119906

120583120591119871+119894119886

2119896120583

119871minus 119896120583

119877 (15)

where 119906120583 = (1 0 0 1) 119896119877= (1 0 0 minus1) and 119896

119871= (1 0 0 1)


119909120583

119877(120591119877) = 119909120583

0(0) + 119906

120583120591119877minus119894119886

2119896120583

119871minus 119896120583

119877 (16)






119911

10

5

0

minus5

minus1010

10550

0minus5minus5

minus10minus10



1205911205971205911199090(119905 120590) = 0 yields the

general solution

1199090(119905 120590) = 119909

119871(120591) + 119909

119877(120591) (17)


119877(120591)


120591∓

119871= 119905 ∓ (119903

119871+ 119894119886 cos 120579

119871)

120591∓

119877= 119905 ∓ (119903

119877+ 119894119886 cos 120579

119877)

(18)




Real slice

Leftcomplex

world line 119909119871

Complexconjuate

world line 119909119877

Leftnull plane

Rightnull plane


Σ119871

Σ119877

minus119886

119911minus

119911+

119911

210

minus1

minus2

minus3

3

3

3

2

2

1

1

0

0

minus1

minus1

minus2

minus2minus3

minus3

+119886




119877Ω 120591+997888rarr 120591minus (19)




119909119871(120591+) = 119909119877(120591minus) 119909

119877(120591+) = 119909119871(120591minus) (20)



119871(120591) and



1198652(119879119860) = 119865119871(119879119860) 119865119877(119879119860) (21)

where 119865119871and 119865


119871(120591+) and 119909

119871(120591minus) The


1198652(119879119860) = 0 (22)


5 Outlook











Acknowledgments



References



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119889119865

119889119884= 0 (11)


119865 (119884 119909120583) = 119860 (119884 minus 119884

+(119909120583)) (119884 minus 119884

minus(119909120583)) (12)



120601 () =119890

119903=

119890

radic1199092 + 1199102 + 1199112(13)


0 1199100 1199110) = (0 0 0) along 119911-



120601119886() = Re 119890

119903 (14)


119871as follows

119909120583

119871(120591119871) = 119909120583

0(0) + 119906

120583120591119871+119894119886

2119896120583

119871minus 119896120583

119877 (15)

where 119906120583 = (1 0 0 1) 119896119877= (1 0 0 minus1) and 119896

119871= (1 0 0 1)


119909120583

119877(120591119877) = 119909120583

0(0) + 119906

120583120591119877minus119894119886

2119896120583

119871minus 119896120583

119877 (16)






119911

10

5

0

minus5

minus1010

10550

0minus5minus5

minus10minus10



1205911205971205911199090(119905 120590) = 0 yields the

general solution

1199090(119905 120590) = 119909

119871(120591) + 119909

119877(120591) (17)


119877(120591)


120591∓

119871= 119905 ∓ (119903

119871+ 119894119886 cos 120579

119871)

120591∓

119877= 119905 ∓ (119903

119877+ 119894119886 cos 120579

119877)

(18)




Real slice

Leftcomplex

world line 119909119871

Complexconjuate

world line 119909119877

Leftnull plane

Rightnull plane


Σ119871

Σ119877

minus119886

119911minus

119911+

119911

210

minus1

minus2

minus3

3

3

3

2

2

1

1

0

0

minus1

minus1

minus2

minus2minus3

minus3

+119886




119877Ω 120591+997888rarr 120591minus (19)




119909119871(120591+) = 119909119877(120591minus) 119909

119877(120591+) = 119909119871(120591minus) (20)



119871(120591) and



1198652(119879119860) = 119865119871(119879119860) 119865119877(119879119860) (21)

where 119865119871and 119865


119871(120591+) and 119909

119871(120591minus) The


1198652(119879119860) = 0 (22)


5 Outlook











Acknowledgments



References



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Real slice

Leftcomplex

world line 119909119871

Complexconjuate

world line 119909119877

Leftnull plane

Rightnull plane


Σ119871

Σ119877

minus119886

119911minus

119911+

119911

210

minus1

minus2

minus3

3

3

3

2

2

1

1

0

0

minus1

minus1

minus2

minus2minus3

minus3

+119886




119877Ω 120591+997888rarr 120591minus (19)




119909119871(120591+) = 119909119877(120591minus) 119909

119877(120591+) = 119909119871(120591minus) (20)



119871(120591) and



1198652(119879119860) = 119865119871(119879119860) 119865119877(119879119860) (21)

where 119865119871and 119865


119871(120591+) and 119909

119871(120591minus) The


1198652(119879119860) = 0 (22)


5 Outlook











Acknowledgments



References



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics












Acknowledgments



References



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics









































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



research article complex structure of the four...

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