Slowly rotating black holes modeled by Solv geometries
aa r X i v : . [ g r- q c ] F e b Slowly rotating black holes modeled by Solv geometries
Jos´e Figueroa and Marcelo Oyarzo
Departamento de F´ısica, Universidad de Concepci´on, Casilla, 160-C, Concepci´on, Chile.
Abstract
We present a slowly rotating generalization of a black hole modeled by a Solv horizon geometry,in five dimensional General Relativity. A separable ansatz compatible with Einstein equations isproposed, which is integrated in terms of hypergeometric and exponential functions. The solutionhighly simplifies after requiring to maintain the same asymptotic behaviour as the non-rotatingblack hole and to have finite, non-vanishing charges. We compute the charges of the solution andalso comment on its importance for studying the phase space of this sector of the theory. . INTRODUCTION Black hole solutions in General Relativity with negative cosmological constant Λ canhave a horizon geometry given either by flat, spherical or hyperbolic ( d − E , spherical S or hyperbolic H spaces. These are a subset of the eight Thurston geometries, which arethe building blocks of 3-dimensional Euclidean spaces, as stated by the Geometrizationconjecture [3]. The remaining five geometries are denoted by S × E , H × E , N il , Sol and SL (2 , R ). Interestingly, General Relativity in five dimension with negative Λ also admitsblack holes with horizon geometry given by some of the previously mentioned Thurstongeometries [4]. One of them is the Sol black hole, whose metric is ds = − (cid:18) − r − µr (cid:19) dt + (cid:18) − r − µr (cid:19) − dr + 3 r − Λ (cid:0) e z dx + e − z dy (cid:1) + 3 − Λ dz , (1)where µ is an integration constant related to the mass. When µ = 0 one finds a kind ofvacuum for this sector of the theory. Spacetimes approaching to that vacuum at infinitybelong to a different phase space than those which are asymptotically AdS. Then, arises thenatural question of whether the phase space of the Sol black holes can be enlarged, including,for example, rotating Sol black holes. In this work we constructed a new deformation of theblack hole (1) at the linear level, representing a slowly rotating solution, and we compute itscharges using the covariant phase space formalism. We also find a linear hair of gravitationalorigin. II. SOL BLACK HOLE AND VACUUM
Einstein equations in five dimensions admit the metric (1) as a solution. As stated before,this solution represents a black hole, with event horizon located at r = r + = ( − µ/ Λ) / .The geometry of the horizon is modeled by the Sol geometry, namely, an homogeneous2uclidean space, with a solvable isometry algebra. This spacetime can also be interpretedas an inhomogeneous black string, since at each z = cte slice the geometry of the horizongets modified.Asymptotically the black hole goes as the geometry with µ = 0, namely ds = − (cid:18) − r (cid:19) dt + (cid:18) − r (cid:19) − dr + 3 r − Λ (cid:0) e z dx + e − z dy (cid:1) + 3 − Λ dz , (2)which is an Einstein manifold that is neither of constant curvature, nor conformally flat.This vacuum has a five dimensional isometry algebra generated by ξ (1) = − x∂ x + y∂ y + ∂ z ,ξ (2) = − t∂ t + r∂ r − x∂ x − y∂ y ,ξ (3) = ∂ t , ξ (4) = ∂ y , ξ (5) = ∂ x with the following commutation relations[ ξ , ξ ] = − ξ , [ ξ , ξ ] = ξ [ ξ , ξ ] = 23 ξ , [ ξ , ξ ] = 23 ξ , [ ξ , ξ ] = 23 ξ . The only isometry broken by the mass of the black hole is the one generated by ξ (2) . Notealso that this algebra is not semisimple, it has an Abelian ideal, so it may acquire a non-trivial central extension. Now, in order to understand better the family of metrics thatapproach the background (2), we turn to the construction of the slowly rotating black hole. III. SLOWLY ROTATING SOL BLACK HOLE
For planar Schwarzschild-AdS, the rotating solution can be obtained by a boost, sinceit has ∂ x and ∂ y as Killing vectors. Now we have the same symmetries, but even thoughthe horizon of the Sol black hole is homogeneous, it has a non-vanishing curvature, and theslowly “rotating” solution is not locally related to the static one.Therefore, we propose the following ansatz for the slowly rotating metric ds = − f ( r ) dt + dr f ( r ) + d ˜ s + aF − ( r ) G − ( z ) dtdx + bF + ( r ) G + ( z ) dtdy . (3)with d ˜ s = r (cid:0) e z dx + e − z dy (cid:1) + dz −
3. Interpreting this as a perturbation of the static black holeone can obtain, in the presence of a Maxwell field, the response properties of the dual system[5].Interestingly, Einstein equations are compatible with the separation we have assumed,and of course at leading order the metric function f ( r ) is the same as in the static Blackhole.At linear order in the rotation parameters a and b , we obtain an equation involving bothpair of functions F ± ( r ) and G ± ( z ), that can be separated finding d F ± ( r ) dr − (cid:0) f ( r ) − (cid:0) c ± − (cid:1) r (cid:1) f ( r ) r F ± ( r ) = 0 (4)and d G ± ( z ) dz ± dG ± ( z ) dz − (cid:0) c ± − (cid:1) G ± ( z ) = 0 , (5)where c + and c − are separation constants. The slowly rotating Kerr metric can be obtainedin the same way, but leads to slightly different equations. In this case, the equation for G ± ( z ) can be solved as G ± ( z ) = C , ± e ∓ (1+ c ± ) z + C , ± e ∓ (1 − c ± ) z (6)with C , ± and C , ± integration constants. The equations for F ± ( r ) can be integrated interms of hypergeometric functions, leading to the following asymptotic behaviors F ± = D , ± r + √ − c ± (cid:0) O (cid:0) r − (cid:1)(cid:1) + D , ± r − √ − c ± (cid:0) O (cid:0) r − (cid:1)(cid:1) . (7)The branches with D , ± have to be discarded because they modify the asymptotic behavior,and can even move us out from the perturbative approach, while the branches with D , ± goto zero at infinity provided − √ < c ± < √ . (8)Then, the solution that decays at infinity written for arbitrary values of the radial coor-dinate reads F ± ( r ) = D , ± r − − √ − c ± f ( r ) F (cid:18) s ± , s ± + 1 , s ± , µ r (cid:19) , (9)where F stands for a hypergeometric function F and s ± := + p − c ± .In the next section we will see how analyzing the charges we can impose further constraintson the slowly rotating solution, leading to a considerable simplification of the metric.4 V. CHARGES AND FURTHER CONSTRAINTS
In order to obtain the charges we used the Covariant Phase Space Formalism. Theexpression for the variation of the charges is known [6] and it is given by δQ = I S k ξ [ δ Φ; Φ] , (10)where k µνξ = √− g πG (cid:18) ξ µ ∇ σ h νσ − ξ µ ∇ v h + ξ σ ∇ v h µσ + 12 h ∇ ν ξ µ − h ρv ∇ ρ ξ µ + 12 h νσ ∇ µ ξ σ (cid:19) . (11)For our ansatz (3), we obtain the following charges δQ ( ∂ t ) = 3 √ π δµ Z dxdydz , (12) δQ ( ∂ x ) = lim r →∞ − √ π δa Z (cid:18) r ∂H − ( r, z ) ∂r − H − ( r, z ) (cid:19) rdxdydz ! , (13) δQ ( ∂ y ) = lim r →∞ − √ π δb Z (cid:18) r ∂H + ( r, z ) ∂r − H + ( r, z ) (cid:19) rdxdydz ! . (14)Where H ± ( r, z ) = F ± ( r ) G ± ( z ). Analyzing the integrand, we found that c ± ≤ δQ ( ∂ x ) and δQ ( ∂ y ) as r → ∞ .For c ± <
1, both charges δQ ( ∂ x,y ) will vanish, and the off-diagonal term must be in-terpreted as a linear, regular hair on the black hole background. Remarkably, only for c ± = 1, the charges δQ ( ∂ x,y ) receive a non-vanishing contribution from the surface integralas r → + ∞ . Even more, in this case, in order to avoid divergences in the z direction onecan turn off the corresponding integration constant in the G ± ( z ) solution, giving rise to thefollowing the spacetime metric ds = − f ( r ) dt + dr f ( r ) + r (cid:0) e z dx + e − z dy (cid:1) + dz + ar dtdx + br dtdy , (15)which, defining V T = R dxdydz , has the following charges Q ( ∂ t ) = M = 3 √ π µV T (16) Q ( ∂ x ) = P x = 9 √ π aV T (17) Q ( ∂ y ) = P y = 9 √ π bV T . (18)5 . CONCLUDING REMARKS We have constructed new, slowly rotating solutions of General Relativity in five dimen-sions, with a negative cosmological constant. These represent the stationary extension ofthe static black hole found in [4] with a horizon modeled by the Sol Thurston geometry.By computing the charges we have also shown that these solution contain a real parameterthat is bounded from above. Below the bound, the off-diagonal terms lead to a hair of agravitational origin since the only non-vanishing global charge is the mass of the spacetime.When the bound is saturated the spacetime acquires two extra non-vanishing global charges,namely Q ( ∂ x ) and Q ( ∂ y ).It has been recently shown in [7] that the static Sol solution can be embedded in N = 2gauged supergravity in five dimensions, in the presence of vector multiplets. Neverthelesssuch construction does not lead to supersymmetric black holes [7]. It would be interestingto explore whether the presence of a non-trivial rotation allows to by pass such result.It is also interesting to notice that in the presence of matter fields without a supersym-metric origin, one can still construct black holes with Sol horizons, with dyonic sources [8].The construction of the rotating versions of such geometries is an open problem. Acknowledgment
The authors thank the organizers of XXII Simposio Chileno de F´ısica for the opportunityto present our work. Special thanks to Julio Oliva for his contribution and guidance. J.Fand M.O also want to thank the financial support of Agencia Nacional de Investigaci´on yDesarrollo (ANID) through the Fellowships No. 22191705. and No. 22201618 , respectively.This work is also partialy soported by FONDECYT grant 1181047. [1] R. B. Mann, Class. and Quantum Grav. 14, L109 (1997)[2] D. Klemm, V. Moretti and L. Vanzo, Phys. Rev. D , 6127-6137 (1998) [erratum: Phys. Rev.D , 109902 (1999)] doi:10.1103/PhysRevD.60.109902 [arXiv:gr-qc/9710123 [gr-qc]].[3] W. P. Thurston, Three-Dimensional Geometry and Topology, Ed. by S. Levy, Princeton Uni-versity Press (1997)
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