Bekenstein-Hawking area law for black objects with conical singularities

Abstract

We argue that, when working with the appropriate set of thermodynamical variables, the BekensteinHawking law still holds for asymptotically flat black objects with conical singularities. The mass-energy which enters the first law of thermodynamics does not, however, coincide with the ADM mass; it differs from the latter by the energy associated with the conical singularity, as seen by an asymptotic, static observer. These statements are supported by a number of examples: the Bach-Weyl (double-Schwarzschild) solution, its dihole generalisation in Einstein-Maxwell theory and the five dimensional static black ring. IntroductionConical singularities have a topological nature. They arise from identifications along the orbits of a U (1) isometry, in the neighbourhood of a fixed point of this symmetry. It is therefore somewhat surprising that there are circumstances under which the local equations of general relativity demand the existence of conical singularities. One such circumstance is realised in non-extremal multi-black hole solutions. The long range interactions in these solutions are unable to provide equilibrium between the various black holes. The role of the conical singularities is to provide the force balance that allows the existence of such solutions in a non-linear theory. The basic example is the Bach-Weyl (BW) or double-Schwarzschild solution [1], which describes two static black holes in four dimensional vacuum general relativity. This configuration belongs to the general class of static Ricci flat vacua with a U (1) isometry found by Weyl [2], and it was generalised to N collinear Schwarzschild black holes by Israel and Khan [3]. The occurrence of conical singularities in the BW solutions was first discussed by Einstein and Rosen [4]; the generic solution must have these singularities, albeit their precise location is a matter of choice. Depending on the choice of periodicity for the coordinate along the U (1) orbits, the conical singularity can be put either in between the two black holes -in which case it is interpreted as a strut -or connecting either black hole to infinity -in which case it is interpreted as a string. In order for the spacetime to be asymptotically flat (without any conical singularities at spatial infinity), we shall take the former viewpoint. The strut energy is then interpreted as the interaction energy between the black holes, while its pressure prevents the gravitational collapse of the system.Another such circumstance is realised in unbalanced single black hole solutions. In more than four spacetime dimensions, there are black objects with a topologically non-spherical horizon that, in vacuum, can only be balanced in a regular (on and outside the event horizon) geometry by introducing rotation. Their static version possesses conical singularities. The basic example is the static black ring [5,6] in five spacetime dimensions. As before, the precise location of the conical singularity is a matter of choice. Demanding regularity at spatial infinity one finds a (spatia...

Charged dilatonic black rings and black saturns and their thermodynamics

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Thermodynamics of nonextremal Kaluza-Klein multiblack holes in five dimensions

“…As generally expected on physical grounds, since these multiblack hole configurations are static, the presence of the conical singularities in between the black holes is unavoidable since they are needed to provide the necessary forces to balance the gravitational attraction among the black holes and keep the black hole system in equilibrium. For asymptotically flat spaces it turns out that even in the presence of conical singularities such singular geometries still admit a reasonable thermodynamic description as recently shown in [40,41,42]. In this work we generalize that thermodynamic description to spaces with KK asymptotics in presence of conical singularities.…”

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Generalized Weyl solutions in d = 5 Einstein-Gauss-Bonnet theory: the static black ring

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