• Sonuç bulunamadı

Higher dimensional Bell-Szekeres metric

N/A
N/A
Protected

Academic year: 2021

Share "Higher dimensional Bell-Szekeres metric"

Copied!
5
0
0

Yükleniyor.... (view fulltext now)

Tam metin

(1)

Higher dimensional Bell-Szekeres metric

M. Gu¨rses*

Department of Mathematics, Faculty of Sciences, Bilkent University, 06533 Ankara, Turkey Y. I˙pekog˘lu,†A. Karasu,‡and C¸ . S¸entu¨rk§

Department of Physics, Faculty of Arts and Sciences, Middle East Technical University, 06531 Ankara, Turkey 共Received 4 June 2003; published 22 October 2003兲

The collision of pure electromagnetic plane waves with collinear polarization in N-dimensional (N⫽2 ⫹n) Einstein-Maxwell theory is considered. A class of exact solutions for the higher dimensional Bell-Szekeres metric is obtained and its singularity structure is examined.

DOI: 10.1103/PhysRevD.68.084007 PACS number共s兲: 04.20.Jb, 04.30.⫺w

I. INTRODUCTION

One of the main fields of interest in general relativity is the collision of gravitational plane waves. Colliding plane wave space-times have been investigated in detail in general relativity 关1兴. The first exact solution of the Einstein-Maxwell equations representing colliding plane shock elec-tromagnetic waves with collinear polarizations was obtained by Bell and Szekeres 共BS兲 关2兴. This solution is conformally flat in the interaction region and its singularity structure has been considered by Matzner and Tipler关3兴, Clarke and Hay-ward关4兴, and Helliwell and Konkowski 关5兴. Later Halil 关6兴, Gu¨rses and Halilsoy 关7兴, Griffiths 关8兴, and Chandrasekhar and Xanthopoulos关9兴 studied exact solutions of the Einstein-Maxwell equations describing the collision of gravitational and electromagnetic waves. Furthermore, Gu¨rses and Ser-mutlu关10兴, and more recently Halilsoy and Sakallı 关11兴, have obtained the extensions of the BS solution in the Einstein-Maxwell-dilaton and Einstein-Maxwell-axion theories, re-spectively.

One of the purposes of this work is to observe which of the relevant physical properties of BS metric are conveyed to higher dimensions. Another motivation is that the BS metric has attracted many researchers working in the context of string theory. Plane wave metrics in various dimensions pro-vide exact solutions in string theory关12兴. It is interesting to study the collision of plane waves at least in the low energy limit of string theory. There have been some attempts in this direction 关10,13–19兴. In addition, the collision of plane waves may shed some light on string cosmology 共see 关16兴 and references therein兲.

In this work we give a higher dimensional generalization of the BS metric. We present an exact solution generalizing the BS solution and examine the singularity structure of the corresponding space-times in the context of curvature and Maxwell invariants. We show that this space-time, unlike the BS metric, is not conformally flat.

In Sec. II we give a brief review of the BS solution and in

Sec. III we formulate the N-dimensional Einstein-Maxwell equations. In Sec. IV we present the N-dimensional colliding exact plane wave solutions describing the collision of shock electromagnetic waves. We also examine the singularity structure of the corresponding space-times and show that in-teraction region of our solution admits curvature singulari-ties.

II. THE BELL-SZEKERES METRIC

The BS metric is given by

ds2⫽2dudv⫹e⫺U共eVdx2⫹e⫺Vdy2兲 共1兲

where the metric functions U and V depend on the null co-ordinates u andv. The electromagnetic vector potential has a single nonzero component A⫽(0,0,0,A), where A is func-tions of u and v. The complete solution of the Einstein-Maxwell equations is

U⫽⫺ log 关 f共u兲⫹g共v兲兴, A⫽共pw⫺rq兲,

V⫽ log 共rw⫺pq兲⫺ log 共rw⫹pq兲, 共2兲 where r

1 2⫹ f

1/2 , p

1 2⫺ f

1/2 , w

1 2⫹g

1/2 , q

1 2⫺g

1/2 共3兲 with f⫽1 2⫺ sin 2P, g1 2⫺ sin 2Q. 共4兲

Here P⫽au⌰(u), Q⫽bv⌰(v), where ⌰ is the Heaviside unit step function, a and b are arbitrary constants, and ␥2 ⫽8␲/␬ with␬ being the gravitational constant. The nature of the space-time singularity of the BS solution has been considered by Matzner and Tipler 关3兴, Clarke and Hayward 关4兴, and more recently by Helliwell and Konkowski 关5兴. To investigate the singularity structure of a space-time one needs all curvature invariants. Due to the simplicity of BS *Email address: [email protected]

Email address: [email protected]Email address: [email protected] §Email address: [email protected]

(2)

metric in the collision region the invariant I⫽R␮␯␣␤R␮␯␣␤is constant everywhere in the interaction region. It is given by

IBS⫽ 8 w2p2q2r2共 f ⫹g兲2

f g fu 2 gv2⫹w2p2q2r2fuugvv ⫹14关共3g⫺ f 兲r2p2f uugv 2 ⫹共3 f ⫺g兲w2q2g vvfu 2

⫽32a2b2. 共5兲

In the BS solution, the singularity that occurs when f ⫹g⫽0 corresponds to a Cauchy horizon. This solution is conformally flat in the interaction region; as an example one of the components of the Weyl tensor is given by

C0202⫽⫺ wq 2r p共rw⫹pq兲2共 f ⫹g兲

f r2p2fu 2⫹ f uu

. 共6兲

The global structure of the BS solution has been analyzed in detail by Clarke and Hayward关4兴. They have shown that this solution possesses quasiregular singularities at the null boundaries. Finally, the invariant

F␣␤F␣␤⫽⫺2␥2ab 共7兲

which is also a constant quantity in the interaction region. The BS solution in the interaction region is isometric to the Bertotti-Robinson space-time关1,4兴.

III. N-DIMENSIONAL EINSTEIN-MAXWELL EQUATIONS

Let M be an N⫽2⫹n dimensional manifold with a metric

ds2⫽g␣␤ dxdx

⫽gab共xc兲dxadxb⫹HAB共xc兲dyAd yB 共8兲

where x⫽(xa, yA), xa denote the local coordinates on a two-dimensional manifold, and yA denote the local coordi-nates on a n-dimensional manifold, thus a,b⫽1,2, A,B ⫽1,2, . . . ,n. The Christoffel symbols of the metric g␣␤can be calculated to give ⌫Ba A 1 2H AD H DB,a, ⌫AB a ⫽⫺1 2g ab H AB,b, 共9兲 ⌫BD A ⫽⌫ ab A ⫽⌫ Ab a ⫽0,⌫¯ bc a ⫽⌫ bc a 共10兲

where the ⌫bca are the Christoffel symbols of the two-dimensional metric gab.

The components of the Ricci tensor are given by

Rab⫽Ra␣b⫽Rab⫹ 1 4tr共⳵aH ⫺1 bH兲⫺䉮ablog

det H, 共11兲 RAB⫽⫺ 1 2共g abH AB,b,a⫺ 1 2g abH AB,b

det g,a

det g ⫹共

det H,a

det H

⫹ 1 2g abH EA,bH EDH DB,a, 共12兲 RaA⫽0, 共13兲

where Rab is the Ricci tensor of the two-dimensional metric

gab. The Maxwell potential 1-formA is

A⫽AAdy

A. 共14兲

The components of the electromagnetic field

F⫽12F␣␤dx⵩dx␤ 共15兲

are

FaA⫽AA,a, Fab⫽0, FAB⫽0. 共16兲

The components of the energy-momentum tensor

T␮␯⫽ 1 4␲

g ␣␤F ␮␤F␯␣⫺ 1 4g␮␯F␣␤F ␣␤

共17兲 are Tab⫽ 1 4␲

H ABF aAFbB⫺ 1 2gabF 2

, TAB⫽ 1 4␲

g abF aAFbB⫺ 1 2HABF 2

, TaA⫽0, 共18兲

where F2⫽FaDFaD. Then the Einstein field equations are

R␮␯⫽␬

T␮␯⫹ 1

2⫺Ng␮␯T

共19兲 where the trace of the energy momentum tensor T is

T⫽ 1

8␲共2⫺n兲F

2. 共20兲

The Einstein-Maxwell equations are

Rab⫹ 1 4tr共⳵aH ⫺1 bH兲⫺䉮ablog

det H ⫽ ␬ 4␲H ABF aAFbB⫺ ␬ 4␲ngabF 2, 共21兲

(3)

a

det Hg g abHAS bHAB兴 ⫽⫺2

det Hg

HASgabFAaFBb⫺ ␦B S n F 2

, 共22兲 and ⳵a

det H g FAa兴⫽0, 共23兲

where䉮 is the covariant differentiation with respect to the connection ⌫bca 共or with respect to metric gab). We may

rewrite the two-dimensional metric as

gab⫽e⫺Mab, 共24兲

where␩ is the metric of flat 2-geometry with arbitrary sig-nature (0 or ⫾2)the function M depends on the local coor-dinates xa. The corresponding Ricci tensor and the Christof-fel symbols are

Rab⫽ 1 2共䉮␩ 2 M兲 ␩ab, ⌫ab c 1 2关⫺M,b␦a c⫺M,a␦ b c⫹M,d␩cd ab兴. 共25兲

IV. HIGHER DIMENSIONAL BELL-SZEKERES METRIC

In this section we give the higher dimensional colliding exact plane wave metric generalizing the BS metric. For this purpose let H be a diagonal matrix

H⫽e⫺Uh 共26兲 where h

eV1  䊊 䊊  eVn

with det h⫽1, i.e., 兺kn⫽1⫺1Vk⫹Vn⫽0.

Now taking the signature of the flat-space metric with null coordinates ␩⫽

0 1 1 0

, x 1⫽u, x2⫽v, and AA⫽共0, . . . ,A兲

the Einstein-Maxwell equations become

2Uuv⫺nUuUv⫽0, 共27兲 ⫺n 2UuVkvn 2UvVku⫹2Vkuv⫽ ␬ ␲ne U⫺VnA uAv, 共28兲 ⫺n 2UuVnvn 2UvVnu⫹2Vnuv⫽ ␬共1⫺n兲n e U⫺VnA uAv, 共29兲

n⫺2

2

共UuAv⫹UvAu兲⫹VnuAv⫹VnvAu⫽2Auv, 共30兲 ⫺n2Uu2⫺1 2k

⫽1 n⫺1 共Vku兲2⫺ 1 2共Vnu兲 2⫹nU uu⫹nMuUu ⫽ ␬ 2␲e U⫺Vn共A u兲2, 共31兲 ⫺n2Uv2⫺1 2k

⫽1 n⫺1 共Vkv兲2⫺ 1 2共Vnv兲 2⫹nU vv⫹nMvUv ⫽ ␬ 2␲e U⫺Vn共A v兲2, 共32兲 2 Muvn 2UuUv⫺ 1 2k

⫽1 n⫺1 VkvVku⫺ 1 2VnvVnu⫹nUuv ⫽ ␬ 2␲n共2⫺n兲e U⫺VnA uAv, 共33兲

where k⫽1, . . . ,n⫺1. Note that the last equation is not in-dependent. It can be obtained from the other equations. The most general solution to Eq. 共27兲 is given by

U⫽⫺2

nlog关 f 共u兲⫹g共v兲兴 共34兲

in terms of two arbitrary functions f and g. Now changing variables (u,v) to ( f ,g) the remaining field equations be-come ⫺n2UfVkgn 2UgVk f⫹2Vk f g⫽ ␬ ␲ne U⫺VnA fAg, 共35兲 ⫺n 2UfVngn 2UgVn f⫹2Vn f g⫽ ␬共1⫺n兲n e U⫺VnA fAg, 共36兲

n⫺2

2

共UfAg⫹UgAf兲⫹Vn fAg⫹VngAf⫽2Af g, 共37兲

Mu⫽⫺ fuu fu共n⫺1兲 n fu f⫹g ⫺ 共 f ⫹g兲 4 fu

k

⫽1 n⫺1 共Vku兲2⫹共Vnu兲2 ⫹␬ ␲eU⫺Vn共Au兲 2

, 共38兲 Mv⫽⫺gvv gv共n⫺1兲 n gv f⫹g ⫺ 共 f ⫹g兲 4gv

k

⫽1 n⫺1 共Vkv兲2⫹共Vnv兲2 ⫹eU⫺Vn共A v兲2

. 共39兲

(4)

Equations共35兲, 共36兲, and 共37兲 are integrability conditions for Eqs.共38兲 and 共39兲. An exact solution to the above Eqs. 共35兲, 共36兲, and 共37兲 is Vk⫽␣klog共rw⫺pq兲⫹klog共rw⫹pq兲, 共40兲 Vn⫽⫺␣log共rw⫺pq兲⫺␤log共rw⫹pq兲, 共41兲 A⫽␥共pw⫺rq兲 共42兲 with ␣k⫺␤k⫽ ␬␥2 2␲n 共43兲

for all k⫽1, . . . ,n⫺1, and

k⫽1 n⫺1k⫽␣⫽ 2 n, k

⫽1 n⫺1k⫽␤⫽ 2 n⫺2. 共44兲

Then we may obtain the value of ␥ as ␥2⫽4n␲/␬(n⫺1). It is convenient to put Eqs.共38兲 and 共39兲 in the following form关1兴: e⫺Mfugv 共 f ⫹g兲(n⫺1)/ne ⫺S 共45兲 where S satisfies Sf⫽⫺ 共 f ⫹g兲 4

k

⫽1 n⫺1 共Vk f兲2⫹共Vn f兲2⫹ ␬ ␲eU⫺Vn共Af兲2

, 共46兲 Sg⫽⫺ 共 f ⫹g兲 4

k

⫽1 n⫺1 共Vkg兲2⫹共Vng兲2⫹ ␬ ␲eU⫺Vn共Ag兲2

. 共47兲 Therefore we may write the metric function M as

M⫽⫺ log 共c fugv兲⫹

n⫺1 n ⫺ 4⫹n2m1 4n2

log共 f ⫹g兲

n 4共n⫺1兲

log

1 2⫺ f

n 4共n⫺1兲

log

1 2⫹ f

4共n⫺1兲n

log

1 2⫺g

n 4共n⫺1兲

log

1 2⫹g

8n1 关8⫺4n⫹n共m1⫺m2兲兴 log 共1⫹4 f g⫹4prwq兲 共48兲 where c is a constant and

k⫽1 n⫺1 ␣k 2⫽m 1,

k⫽1 n⫺1 ␤k 2⫽m 2. 共49兲

m1 and m2, using Eq.共43兲, satisfy m1⫹m2⫺2m3⫽ 4 n⫺1, m1⫺m2⫽ 4共2⫺n兲 n共n⫺1兲 共50兲 with

k⫽1 n⫺1 ␣kk⫽m3.

The metric function e⫺M must be continuous across the null boundaries. To make it so we assume that the functions

f and g take the form f⫽1

2⫺ sin

n1P, g1

2⫺ sin

n2Q. 共51兲

Then the metric function e⫺M is continuous across the boundaries if

n1⫽n2⫽

4共n⫺1兲

3n⫺4 . 共52兲

Therefore, the metric function e⫺M reads

e⫺M共1⫹4 f g⫹4pqrw兲k1

1

1 2⫺ f

2/n1

1/2

1⫺

1 2⫺g

2/n1

1/2

1 2⫹ f

1⫺(1/n1)

1 2⫹g

1⫺(1/n1) 共 f ⫹g兲k2 , 共53兲 where k1⫽ n⫺2 2共n⫺1兲, k2⫽ n⫺1 n ⫺ 1 4n2共4⫹n 2m 1兲. 共54兲

It may thus be observed that the constant n1 (⫽n2) is

re-stricted to the range satisfying

2⭓n1⫽n2⬎

4

3. 共55兲

It is also appropriate to choose c⫽1/n12ab.

The space-time line element generalizing the BS metric in

N⫽2⫹n dimensions is

ds2⫽2e⫺Mdu dv⫹e⫺U共eV1dx

1

2•••⫹eVndx n

(5)

where the metric functions are given in Eqs.共40兲, 共41兲, and 共53兲. Because of Eq. 共55兲 the metric we have found is C1for n⬎2 across the null boundaries. In spite of this fact, the

Ricci tensor is regular across the null boundaries due to the Einstein field equations. The above solution reduces to the well known BS solution for n⫽2.

We now discuss the nature of the space-time singularities. We study the behavior of the metric functions U, Vk, Vn,

and M as f⫹g tends to zero. In the BS solution the collision of the two shock electromagnetic plane waves generates im-pulsive gravitational waves along the null boundaries. It is shown that, apart from the impulsive waves themselves, the BS solution has no curvature singularities and the only sin-gularites are of the quasiregular type 关4兴. In general the analysis of space-time singularities requires all invariants. In a higher dimensional case it is not feasible to study all in-variants of the space-time geometry. For this reason we sider only the quadratic Riemann invariant I which is con-stant for the BS solution Eq. 共5兲. However, for n⬎2 this invariant can be shown to have the behavior

I⬃e2 M共 fugv兲 2

共 f ⫹g兲4 共57兲

as f⫹g→0. Using M from Eq. 共53兲 we find

I⬃共 fu gv兲2 共 f ⫹g兲2k2⫺4 共58兲

as f⫹g→0. It is obvious that space-times possess curvature singularities when k2⬍2 and their strength depend on n and

m1. It is also of great importance to further analyze the

glo-bal singularity structure of these space-times. At present the global structure of only a few solutions 共colliding plane waves geometries兲 are known in the four-dimensional case 关3,4兴.

We also investigate the singularity structure of space-times in the context of the Maxwell invariants; one of the invariants is F␣␤F␣␤⫽⫺ ␥ 2n 1 2 2k1⫹1共rwpq兲 1⫺(2/n1)共rw⫹pq兲⫺2k1 ⫻共 f ⫹g兲k2P uQv 共59兲

which has singularities for n⬎2 for the negative values of

k2.

We finally examine the Weyl tensor to see whether our space-time is conformally flat. One of the components of the Weyl tensor in region II for our space-times is

C0n0nfu 2 8

1 2⫹ f

⫺m1⫹ 2 n共n⫺1兲2n 共n⫺1兲 共␣⫹␤兲3 ⫹共1⫺n兲n 共␣⫹␤兲⫹2共␣⫹␤兲 ⫻

1 2⫺ f

⫺1⫹2/n1

1 2⫹ f

1⫺

1 2⫺ f

2/n1

⫹ 共␣⫹␤兲 4 fuu. 共60兲 It can be seen that it vanishes only for n⫽2. Therefore, the higher dimensional extensions of the BS metric are not con-formally flat.

V. CONCLUSION

In this paper, we give a higher dimensional generalization of the BS metric which describes the collision of pure elec-tromagnetic plane waves with collinear polarization in all space-time dimensions. The solution has two free param-eters; the space-time dimension N(⫽2⫹n) and an arbitrary real number m1. We show that these space-times, unlike the

BS metric, are not conformally flat. We find that, even though purely electromagnetic plane wave collision in four-dimensional space-time possesses no curvature singularities, in higher dimensions there exist curvature singularities whose nature depend on the real number m1 and the

space-time dimension.

ACKNOWLEDGMENTS

This work is partially supported by the Scientific and Technical Research Council of Turkey 共TUBITAK兲 and by Turkish Academy of Sciences 共TUBA兲.

关1兴 J.B. Griffiths, Colliding Plane Waves in General Relativity 共Clarendon Press, Oxford, 1991兲.

关2兴 P. Bell and P. Szekeres, Gen. Relativ. Gravit. 5, 275 共1974兲. 关3兴 R.A. Matzner and F.J. Tipler, Phys. Rev. D 29, 1575 共1984兲. 关4兴 C.J.S. Clarke and S.A. Hayward, Class. Quantum Grav. 6, 615

共1989兲.

关5兴 T.M. Helliwell and D.A. Konkowski, Class. Quantum Grav.

16, 2709共1999兲.

关6兴 M. Halilsoy, J. Math. Phys. 20, 120 共1979兲.

关7兴 M. Gu¨rses and M. Halilsoy, Lett. Nuovo Cimento 34, 588 共1982兲.

关8兴 J.B. Griffiths, Phys. Lett. A 54, 269 共1975兲.

关9兴 S. Chandrasekhar and B.C. Xanthopoulos, Proc. R. Soc. Lon-don A410, 311共1987兲.

关10兴 M. Gu¨rses and E. Sermutlu, Phys. Rev. D 52, 809 共1995兲.

关11兴 M. Halilsoy and I. Sakallı, Class. Quantum Grav. 20, 1417 共2003兲.

关12兴 R. Gu¨ven, Phys. Lett. B 191, 275 共1987兲.

关13兴 A. Garcı´a and N. Breto´n, Phys. Rev. D 53, 4351 共1996兲. 关14兴 A. Feinstein, K.E. Kunze, and M.A. Va´zquez-Mozo, Class.

Quantum Grav. 17, 3599共2000兲.

关15兴 V. Bozza and G. Veneziano, J. High Energy Phys. 10, 035 共2000兲.

关16兴 A. Das, J. Maharana, and A. Melikyan, Phys. Lett. B 518, 306 共2001兲.

关17兴 A. Feinstein, Class. Quantum Grav. 19, 5353 共2002兲. 关18兴 M. Gu¨rses and A. Karasu, Class. Quantum Grav. 18, 509

共2001兲.

关19兴 M. Gu¨rses, E.O. Kahya, and A. Karasu Phys. Rev. D 66, 024029共2002兲.

Referanslar

Benzer Belgeler

By presenting an alternative writing portfolio assessment scale and the results of an inter-rater reliability study on instructors’ evaluations using the new writing

Yapının yatay yük taşıma kapasitesini gösteren kapasite eğrisinin elde edilebilmesi için, yapı sabit düşey yükler ve aralarındaki oran sabit kalarak artan

(38) The coefficient of reflection by the black hole is calculated by virtue of the fact that outgoing mode must be absent at the spatial infinity. This is because the Hawking radiation

We have investigated the possibility of thin-shell wormholes in EYMGB theory in higher (d ≥ 5) dimensions with particular emphasis on stability against spherical, linear

By employing the higher (N > 5)-dimensional version of the Wu–Yang ansatz we obtain magnetically charged new black hole solutions in the Einstein–Yang–Mills–Lovelock (EYML)

Ama öyle farklı imgeler kullan­ mıştı ki, hiçbir şiiri birbirine ben­ zemiyordu.. Cansever’i okurken tekrar duygusuna düştüğünüz hemen hemen

4 - Mahlas yerlerinde Yunus Emre’nin hiç kullanmadığı “Âşık Yunus, Derviş Yunus, Yunus Dede, Kul Yunus’lara dikkat edilmek gereklidir.. 5- Yunus

“ Leasing Đşlerinin Muhasebeleştirilmesi (ABD uygulaması), Muhasebe ve Denetime Bakış, Yıl.1, sayı.. 7- Đflas durumunda , finansal kiralamaya konu olan malın mülkiyeti