We investigate asymptotic symmetries regularly defined on spherically symmetric Killing horizons in the Einstein theory with or without the cosmological constant. Those asymptotic symmetries are described by asymptotic Killing vectors, along which the Lie derivatives of perturbed metrics vanish on a Killing horizon. We derive the general form of asymptotic Killing vectors and find that the group of the asymptotic symmetries consists of rigid O(3) rotations of a horizon two-sphere and supertranslations along the null direction on the horizon, which depend arbitrarily on the null coordinate as well as the angular coordinates. By introducing the notion of asymptotic Killing horizons, we also show that local properties of Killing horizons are preserved under not only diffeomorphisms but also non-trivial transformations generated by the asymptotic symmetry group. Although the asymptotic symmetry group contains the Diff (S 1 ) subgroup, which results from the supertranslations dependent only on the null coordinate, it is shown that the Poisson bracket algebra of the conserved charges conjugate to asymptotic Killing vectors does not acquire non-trivial central charges. Finally, by considering extended symmetries, we discuss that unnatural reduction of the symmetry group is necessary in order to obtain the Virasoro algebra with non-trivial central charges, which will not be justified when we respect the spherical symmetry of Killing horizons.
Recently, Carlip proposed a derivation of the entropy of the two-dimensional dilatonic black hole by investigating the Virasoro algebra associated with a newly introduced near-horizon conformal symmetry. We point out not only that the algebra of these conformal transformations is not well defined on the horizon, but also that the correct use of the eigenvalue of the operator L 0 yields vanishing entropy. It has been shown that these problems can be resolved by choosing a different basis of the conformal transformations which is regular even at the horizon. We also show the generalization of Carlip's derivation to any higher dimensional case in pure Einstein gravity. The entropy obtained is proportional to the area of the event horizon, but it also depends linearly on the product of the surface gravity and the parameter length of a horizon segment in consideration. We finally point out that this derivation of black hole entropy is quite different from the ones proposed so far, and several features of this method and some open issues are also discussed.
We consider n-dimensional asymptotically anti-de Sitter spacetimes in higher curvature gravitational theories with n ≥ 4, by employing the conformal completion technique. We first argue that a condition on the Ricci tensor should be supplemented to define an asymptotically anti-de Sitter spacetime in higher curvature gravitational theories and propose an alternative definition of an asymptotically anti-de Sitter spacetime. Based on that definition, we then derive a conservation law of the gravitational field and construct conserved quantities in two classes of higher curvature gravitational theories. We also show that these conserved quantities satisfy a balance equation in the same sense as in Einstein gravity and that they reproduce the results derived elsewhere. These conserved quantities are shown to be expressed as an integral of the electric part of the Weyl tensor alone and hence they vanish identically in the pure anti-de Sitter spacetime as in the case of Einstein gravity.
We consider universal properties that arise from a local geometric structure of a Killing horizon. We first introduce a non-perturbative definition of such a local geometric structure, which we call an asymptotic Killing horizon. It is shown that infinitely many asymptotic Killing horizons reside on a common null hypersurface, once there exists one asymptotic Killing horizon. The acceleration of the orbits of the vector that generates an asymptotic Killing horizon is then considered. We show that there exists the diff(S 1 ) or diff(R 1 ) sub-algebra on an asymptotic Killing horizon universally, which is picked out naturally based on the behavior of the acceleration. We also argue that the discrepancy between string theory and the Euclidean approach in the entropy of an extreme black hole may be resolved, if the microscopic states responsible for black hole thermodynamics are connected with asymptotic Killing horizons. a (non-extreme) black hole in Einstein gravity is shown to be given by one quarter of the area of the black hole horizon in Planck units, which is known as the Bekenstein-Hawking formula [2,5]. Even in generalized theories of gravity, one can show that black hole entropy is expressed by geometric quantities on the black hole horizon [6]. Furthermore, it has been shown [7,8] that the thermal feature is possessed not only by a black hole horizon, but also by any causal horizons, including a cosmological horizon and an acceleration horizon, while the geometric structures far from the horizon differ between them. These facts may possibly suggest that the thermal feature of a black hole and the microscopic physics responsible for it are intimately connected with a universal and local geometric structure particular to presence of a horizon. However, it is quite difficult to understand, in terms of a universal and local geometric structure of a horizon, the statistical physics of the microscopic states of black hole thermodynamics. Although the Euclidean approach to black hole thermodynamics [9] gives the relation between the classical gravitational field of a black hole and the partition function of black hole thermodynamics, it does not clarify the nature of these microscopic states. The Euclidean approach shows only that there will exist the microscopic states associated with a single classical (on-shell) configuration in the phase space.One of the attempts to embark on this issue is to consider asymptotic symmetries on the horizon of an arbitrary black hole [10] (see [3,4] for recent reviews and the references therein), following the success in the case of the B.T.Z. black hole whose entropy has been reproduced based on the asymptotic symmetries at infinity [11]. This attempt would be very interesting, if it worked in the same way as the case of the B.T.Z. black hole, since it would suggest that there is a sort of correspondence also between a black hole horizon and the conformal field theory, much like the AdS/CFT correspondence [12]. However, it seems that the fatal Universal properties from local ge...
We consider the first law of black hole thermodynamics in an asymptotically anti-de Sitter spacetime in the class of gravitational theories whose gravitational Lagrangian is an arbitrary function of the Ricci scalar. We first show that the conserved quantities in this class of gravitational theories constructed through conformal completion remain unchanged under the conformal transformation into the Einstein frame. We then prove that the mass and the angular momenta defined by these conserved quantities, along with the entropy defined by the Noether charge, satisfy the first law of black hole thermodynamics, not only in Einstein gravity but also in the higher curvature gravity within the class under consideration. We also point out that it is naturally understood in the symplectic formalism that the mass satisfying the first law should be necessarily defined associated with the timelike Killing vector nonrotating at infinity. Finally, a possible generalization into a wider class of gravitational theories is discussed.
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