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⫽
冉
1 2⫹ f冊
1/2 , p⫽冉
1 2⫺ f冊
1/2 , w⫽冉
1 2⫹g冊
1/2 , q⫽冉
1 2⫺g冊
1/2 共3兲 with f⫽1 2⫺ sin 2P, g⫽1 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]
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
再
f g fu 2 gv2⫹w2p2q2r2fuugvv ⫹14关共3g⫺ f 兲r2p2f uugv 2 ⫹共3 f ⫺g兲w2q2g vvfu 2兴冎
⫽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
冋
f r2p2fu 2⫹ f uu册
. 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␣ dx␣ dx
⫽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a䉮blog
冑
det H, 共11兲 RAB⫽⫺ 1 2共g abH AB,b兲,a⫺ 1 2g abH AB,b冋
共冑
det g兲,a冑
det g ⫹共冑
det H兲,a冑
det H册
⫹ 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
冋
g ␣F F␣⫺ 1 4gF␣F ␣册
共17兲 are Tab⫽ 1 4冋
H ABF aAFbB⫺ 1 2gabF 2册
, TAB⫽ 1 4冋
g abF aAFbB⫺ 1 2HABF 2册
, TaA⫽0, 共18兲where F2⫽FaDFaD. Then the Einstein field equations are
R⫽
冋
T⫹ 12⫺NgT
册
共19兲 where the trace of the energy momentum tensor T isT⫽ 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a䉮blog
冑
det H ⫽ 4H ABF aAFbB⫺ 4ngabF 2, 共21兲a关
冑
det Hg g abHAS bHAB兴 ⫽⫺2冑
det Hg冋
HASgabFAaFBb⫺ ␦B S n F 2册
, 共22兲 and a关冑
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⫺M ab, 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,dcd 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⫽
冉
eV1 䊊 䊊 eVn冊
with det h⫽1, i.e., 兺kn⫽1⫺1Vk⫹Vn⫽0.
Now taking the signature of the flat-space metric with null coordinates ⫽
冉
0 1 1 0冊
, x 1⫽u, x2⫽v, and AA⫽共0, . . . ,A兲the Einstein-Maxwell equations become
2Uuv⫺nUuUv⫽0, 共27兲 ⫺n 2UuVkv⫺ n 2UvVku⫹2Vkuv⫽ ne U⫺VnA uAv, 共28兲 ⫺n 2UuVnv⫺ n 2UvVnu⫹2Vnuv⫽ 共1⫺n兲 n e U⫺VnA uAv, 共29兲
冉
n⫺22
冊
共UuAv⫹UvAu兲⫹VnuAv⫹VnvAu⫽2Auv, 共30兲 ⫺n2Uu2⫺1 2k兺
⫽1 n⫺1 共Vku兲2⫺ 1 2共Vnu兲 2⫹nU uu⫹nMuUu ⫽ 2e U⫺Vn共A u兲2, 共31兲 ⫺n2Uv2⫺1 2k兺
⫽1 n⫺1 共Vkv兲2⫺ 1 2共Vnv兲 2⫹nU vv⫹nMvUv ⫽ 2e U⫺Vn共A v兲2, 共32兲 2 Muv⫺ n 2UuUv⫺ 1 2k兺
⫽1 n⫺1 VkvVku⫺ 1 2VnvVnu⫹nUuv ⫽ 2n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 ⫺n2UfVkg⫺ n 2UgVk f⫹2Vk f g⫽ ne U⫺VnA fAg, 共35兲 ⫺n 2UfVng⫺ n 2UgVn f⫹2Vn f g⫽ 共1⫺n兲 n e U⫺VnA fAg, 共36兲
冉
n⫺22
冊
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
冋
k兺
⫽1 n⫺1 共Vku兲2⫹共Vnu兲2 ⫹ eU⫺Vn共Au兲 2册
, 共38兲 Mv⫽⫺gvv gv ⫹ 共n⫺1兲 n gv f⫹g ⫺ 共 f ⫹g兲 4gv冋
k兺
⫽1 n⫺1 共Vkv兲2⫹共Vnv兲2 ⫹eU⫺Vn共A v兲2册
. 共39兲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 2n 共43兲
for all k⫽1, . . . ,n⫺1, and
兺
k⫽1 n⫺1 ␣k⫽␣⫽ 2 n, k兺
⫽1 n⫺1 k⫽⫽ 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⫺M⫽ fugv 共 f ⫹g兲(n⫺1)/ne ⫺S 共45兲 where S satisfies Sf⫽⫺ 共 f ⫹g兲 4
冋
k兺
⫽1 n⫺1 共Vk f兲2⫹共Vn f兲2⫹ eU⫺Vn共Af兲2册
, 共46兲 Sg⫽⫺ 共 f ⫹g兲 4冋
k兺
⫽1 n⫺1 共Vkg兲2⫹共Vng兲2⫹ eU⫺Vn共Ag兲2册
. 共47兲 Therefore we may write the metric function M asM⫽⫺ log 共c fugv兲⫹
冋
n⫺1 n ⫺ 4⫹n2m1 4n2册
log共 f ⫹g兲 ⫹冉
n 4共n⫺1兲冊
log冉
1 2⫺ f冊
⫹冉
n 4共n⫺1兲冊
log冉
1 2⫹ f冊
⫹冉
4共n⫺1兲n冊
log冉
1 2⫺g冊
⫹冉
n 4共n⫺1兲冊
log冉
1 2⫹g冊
⫹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兺
k⫽1 n⫺1 ␣k 2⫽m 1,兺
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
兺
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, g⫽1
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
冋
1⫺冉
1 2⫺ f冊
2/n1册
1/2冋
1⫺冉
1 2⫺g冊
2/n1册
1/2冉
1 2⫹ f冊
1⫺(1/n1)冉
1 2⫹g冊
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
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
C0n0n⫽ fu 2 8
冉
1 2⫹ f冊
冋
⫺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 ⫻冉
1 2⫺ f冊
⫺1⫹2/n1冉
1 2⫹ f冊
1⫺冉
1 2⫺ f冊
2/n1册
⫹ 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.