Note on cosmological Levi-Civita spacetimes in higher dimensions

Ozgur Sar�oglu* and Bayram Tekin†

Department of Physics, Faculty of Arts and Sciences, Middle East Technical University, 06531, Ankara, Turkey(Received 18 February 2009; published 8 April 2009)

We find a class of solutions to cosmological Einstein equations that generalizes the four dimensional

cylindrically symmetric spacetimes to higher dimensions. The AdS soliton is a special member of this

class with a unique singularity structure.

DOI: 10.1103/PhysRevD.79.087502 PACS numbers: 04.20.�q, 04.20.Jb, 04.50.Gh

In this note, we generalize the four-dimensionalcylindrically-symmetric static spacetimes that are solu-tions to the Einstein equations, with a negative cosmologi-cal constant �< 0, to higher dimensions. The flat space(� ¼ 0) analogs of these solutions are known as the Levi-Civita spacetimes [1].

The original four dimensional solutions with a nonvan-ishing�were found in [2,3]. Nonsingular sheet sources forthese spacetimes were recently constructed in [4]. Thereare subtle issues in the interpretation of the parameters thatappear in the four dimensional solution in terms of thephysical properties, such as the mass, of material sources.According to [4], ‘‘for the�< 0 case the asymptotic formsof the metrics due to material cylinders are more closelyrelated to the asymptotics of bounded sources than for the� ¼ 0 case.’’

Here we provide a new class of solutions to the cosmo-logical Einstein equations (with a redefined cosmologicalconstant)

R�� ¼ � n

‘2g��

in D ¼ nþ 1 � 4 dimensions. It reads

ds2 ¼ r2

‘2

���1� rn0

rn

�p0

dt2 þ Xn�1

k¼1

�1� rn0

rn

�pkðdxkÞ2

�

þ�1� rn0

rn

��1 ‘2

r2dr2; (1)

where r0 is a free parameter, and the constants pk, with 0 �k � n� 1, satisfy two Kasner-type conditions:

Xn�1

k¼0

pk ¼Xn�1

k¼0

ðpkÞ2 ¼ 1: (2)

This is a higher dimensional generalization of the metricsin [2,3]. Specifically, the metric studied by [4] follows from(1) after solving the constraints (2) for n ¼ 3.1

It immediately follows that as r ! 1, (1) asymptoti-cally approaches the usual maximally symmetric AdS

spacetime in horospheric coordinates. Another importantobservation is that the AdS soliton of [5] is just a specialmember in this class: It is obtained simply by taking one ofthe pk ¼ 1, where 1 � k � n� 1, and setting all theothers, including p0, to zero. In lower dimensions, whenn ¼ 1 and n ¼ 2, (1) becomes the usual two and threedimensional AdS spacetimes, respectively. Here we shouldalso note that special forms of (1) with specific choices forthe constants pk have already appeared in the literature forD ¼ 5 and D ¼ 7 [6–8].The singularity structure of the solution (1) is apparent

from its Kretschmann scalar, which we give here only forthe n ¼ 3 and n ¼ 4 cases.2 For n ¼ 3, it reads

R����R���� ¼ 24

‘4þ 12

‘4r60r6

� 81

‘4r90r9

ð1þ hðrÞÞhðrÞ2

Y2k¼0

pk;

where hðrÞ � 1� r30=r3. As for the n ¼ 4 case, it will be

convenient to first define fðrÞ � 1� r40=r4 and � �

p0p1p2 þ p0p1p3 þ p0p2p3 þ p1p2p3. Then

R����R���� ¼ 40

‘4þ 72

‘4r80r8

� 64

‘4r120r12

1

fðrÞ2

���ð4þ 5fðrÞÞ þ 2

r40r4

Y3k¼0

pk

�:

For the generic choice of the constants pk, one clearly findsnaked singularities at r ¼ r0 and r ¼ 0. Remarkably, theAdS soliton is the unique solution with no naked singular-ities provided that r � r0 and the conical singularity at r0is avoided by a proper compactification of the correspond-ing coordinate [5].It may be of interest to write (1) in different coordinate

systems. A somewhat obvious possibility is to consider the

coordinate transformation r ¼ r0cosh2=nðn�=ð2‘ÞÞ which

takes (1) to*sarioglu@metu.edu.tr†btekin@metu.edu.tr1Their parameter � can be thought of as the remaining uncon-

strained p.

2Its calculation gets rather complicated beyond these dimen-sions, but is not different in general features.

PHYSICAL REVIEW D 79, 087502 (2009)

1550-7998=2009=79(8)=087502(2) 087502-1 � 2009 The American Physical Society

ds2 ¼ d�2 þ r20‘2

cosh4=n�n�

2‘

���tanh2p0

�n�

2‘

�dt2

þ Xn�1

k¼1

tanh2pk

�n�

2‘

�ðdxkÞ2

�; (3)

where again the constants pk are subject to (2), of course.The interpretation of the constants pk and r0 in terms of

physical parameters would be of interest. As a step in thisdirection, we calculate the mass of the solutions (1). Forthis purpose, we use the procedure described in [9,10] thatrequires a choice of a background and a perturbation aboutit. The correct background in this case is the usual AdSmetric obtained from (1) by simply setting r0 to zero. Thebackground Killing vector that leads to mass/energy is��� ¼ �ð@=@tÞ�, which in our case yields

M ¼ Vn�1

4GD�n�2

rn0‘nþ1

ðnp0 � 1Þ; (4)

where Vn�1 is the volume of the transverse dimensions.The mass of the AdS soliton, for which there is only onenonzero pk where k � 0, has already been considered in

[5,11]. It was conjectured in [5] that the AdS soliton, withits negative mass, has the lowest possible energy, (hencethe new ‘‘positive’’ mass conjecture) in its asymptoticclass. At first sight, the result (4) seems to contradict thissince one can have a solution with p0 < 0 leading to M<Msoliton. However, one then has naked singularities inwhich case any kind of ‘‘positive’’ mass theorem fails.Observing that the AdS soliton is the unique solutionwith no naked singularities, our result lends further supportto the new positive mass conjecture of [5] and the ‘‘unique-ness’’ result of [12]. Amusingly, if one chooses p0 ¼ 1=n,one ends up with massless nonflat (n� 1)-branes.To conclude, we have found new Einstein spaces in

higher dimensions, generalizing the four dimensional cos-mological cylindrically symmetric Levi-Civita spacetimes.We have calculated the mass of these metrics and identifiedthe AdS soliton as a rather unique member.

This work is partially supported by the Scientific andTechnological Research Council of Turkey (TUBITAK).B. T. is also partially supported by the TUBITAK KariyerGrant No. 104T177.

[1] H. Stephani, D. Kramer, M.A.H. MacCallum, C.Hoenselaers, and E. Herlt, Exact Solutions of Einstein’sField Equations (Cambridge University Press, Cambridge,2003), 2nd ed..

[2] B. Linet, J. Math. Phys. (N.Y.) 27, 1817 (1986).[3] Q. Tian, Phys. Rev. D 33, 3549 (1986).[4] M. Zofka and J. Bicak, Classical Quantum Gravity 25,

015011 (2008).[5] G. T. Horowitz and R. C. Myers, Phys. Rev. D 59, 026005

(1998).

[6] R. C. Myers, Phys. Rev. D 60, 046002 (1999).[7] J. G. Russo, Phys. Lett. B 435, 284 (1998).[8] Y. Kiem and D. Park, Phys. Rev. D 59, 044010 (1999).[9] S. Deser and B. Tekin, Phys. Rev. Lett. 89, 101101 (2002).[10] S. Deser and B. Tekin, Phys. Rev. D 67, 084009 (2003).[11] H. Cebeci, O. Sar�oglu, and B. Tekin, Phys. Rev. D 73,

064020 (2006).[12] G. J. Galloway, S. Surya, and E. Woolgar, Phys. Rev. Lett.

88, 101102 (2002).

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