The integrability of Virasoro charges for axisymmetric Killing horizons
Lin-Qing Chen
备注：
19 Pages. v.1: The notes is submitted to match the same releasing day with a separate independent investigation on the same topic. v.2: Add the analysis of boundary term and the result on the past horizon. v.3: minor typos are corrected, add references. v.4: Improved discussion section, published version
Through the analysis of null symplectic structure, we derive the condition for integrable Virasoro generators on the covariant phase space of axisymmetric Killing horizons. A weak boundary condition selects a special relationship between the two temperatures for the putative CFT. When the integrability is satisfied for both future and past horizons, the two central charges are equal. At the end we discuss the physical implications.
We treat D-dimensional black holes with Killing horizon for extended Gauss-Bonnet gravity. We use Carlip method and impose boundary conditions on horizon what enables us to identify Virasoro algebra and evaluate its central charge and Hamiltonian eigenvalue. The Cardy formula allows then to calculate the number of states and thus provides for microscopic interpretation of entropy.
We show that the gravitational phase space for the near-horizon region of a bifurcate, axisymmetric Killing horizon in any dimension admits a 2D conformal symmetry algebra with central charges proportional to the area. This extends the construction of [Haco et. al., JHEP 12, 098 (2018)] to generic Killing horizons appearing in solutions of Einstein's equations, and motivates a holographic description in terms of a 2D conformal field theory. The Cardy entropy in such a field theory agrees with the Bekenstein-Hawking entropy of the horizon, suggesting a microscopic interpretation.
We study some geometric properties of Killing horizons in 4-dimensional stationary and axisymmetric space-times with electromagnetic field and cosmological constant. Using a $(1+1+2)$ space-time split, we construct relations between the space-time Riemann tensor components and components of the Riemann tensor corresponding to the horizon surface. The Einstein equations allow to derive the space-time scalar curvature invariants, Kretschmann, Chern-Pontryagin, and Euler, on the 2-dimensional spacelike horizon surface. The derived relations generalize the relations known for Killing horizons of s…
We impose a certain class of boundary conditions on Killing horizon and show for Lagrangians with arbitrary curvature dependence that one can identify a Virasoro algebra with nontrivial central charge and calculable Hamiltonian eigenvalue. Entropy can then be calculated from Cardy formula.
The set of solutions to the AdS$_3$ Einstein gravity with Brown-Henneaux boundary conditions is known to be a family of metrics labeled by two arbitrary periodic functions, respectively left and right-moving. It turns out that there exists an appropriate presymplectic form which vanishes on-shell. This promotes this set of metrics to a phase space in which the Brown-Henneaux asymptotic symmetries become symplectic symmetries in the bulk of spacetime. Moreover, any element in the phase space admits two global Killing vectors. We show that the conserved charges associated with these Killing vect…
The covariant phase technique is used to compute the constraint algebra of the stationary axisymmetric charged black hole. A standard Virasoro subalgebra with corresponding central charge is constructed at a Killing horizon with Carlip's boundary conditions. For the Kerr-Newman black hole and the Kerr-Newman-AdS black hole, the density of states determined by conformal fields theory methods yields the statistical entropy which agrees with the Bekenstein-Hawking entropy.
A key result in the proof of black hole uniqueness in 4-dimensions is that a stationary black hole that is ``rotating''--i.e., is such that the stationary Killing field is not everywhere normal to the horizon--must be axisymmetric. The proof of this result in 4-dimensions relies on the fact that the orbits of the stationary Killing field on the horizon have the property that they must return to the same null geodesic generator of the horizon after a certain period, $P$. This latter property follows, in turn, from the fact that the cross-sections of the horizon are two-dimensional spheres. Howe…
Noncommutative QFT and Relative Entropy on Axisymmetric Bifurcate Killing Horizons
Institut für Theoretische Physik, Universität Leipzig（莱比锡大学理论物理研究所）；Department of Applied Physics and Astronomy, College of Sciences, University of Sharjah（沙迦大学理学院应用物理与天文学系）
In the Carlip-Majhi-Padmanabhan approach, we calculate microscopic entropy of trapping (apparent) horizon of the FRW metric. We solve Killing equations for $t,r$ part of the metric without fixing {\it a piori} the form of the scaling factor $a(t)$ which is determined from the requirement of consistency of Killing equations. Further restrictions on the form of the Killing vector follow from the requirement that Killing vector is null at the trapping horizon at all $t$. Applying the technique used to calculate microscopic entropy of Killing horizons, we calculate microscopic entropy of trapping …
We provide a simple and straightforward procedure for defining a Virasoro algebra based on the diffeomorphisms near a null surface in a spacetime and obtain the entropy density of the null surface from its central charge. We use the off-shell Noether current corresponding to the diffeomorphism invariance of a gravitational Lagrangian $L(g_{ab},R_{abcd})$ and define the Virasoro algebra from its variation. This allows us to identify the central charge and the zero mode eigenvalue using which we obtain the entropy density of the Killing horizon. Our approach works for all Lanczos-Lovelock models…
While the argument by Zamolodchikov and Polchinski suggests global conformal invariance implies Virasoro invariance in two-dimensional unitary conformal field theories with discrete dilatation spectrum, it is not the case in more general situations without these assumptions. We indeed show that almost all the globally conformal invariant differential equations in two dimensions are neither Virasoro invariant nor Weyl invariant. The only exceptions are the higher spin conservation laws, conformal Killing tensor equations and the Laplace equation of a conformal scalar.
On a manifold with boundary, the constraint algebra of general relativity may acquire a central extension, which can be computed using covariant phase space techniques. When the boundary is a (local) Killing horizon, a natural set of boundary conditions leads to a Virasoro subalgebra with a calculable central charge. Conformal field theory methods may then be used to determine the density of states at the boundary. I consider a number of cases---black holes, Rindler space, de Sitter space, Taub-NUT and Taub-Bolt spaces, and dilaton gravity---and show that the resulting density of states yields…
Several investigations show that in a very small length scale there exists corrections to the entropy of black hole horizon. Due to fluctuations of the background metric and the external fields the action incorporates corrections. In the low energy regime, the one loop effective action in four dimensions leads to trace anomaly. We start from the Noether current corresponding to the Einstein-Hilbert plus the one loop effective action to calculate the charge for the diffeomorphisms which preserve the Killing horizon structure. Then a bracket among the charges is calculated. We show that the Four…
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 w…
At Most One Inner Killing Horizon in Stationary Black Holes
Tianjin University（天津大学）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Hangzhou Institute for Advanced Study, UCAS（杭州高等研究院，中国科学院大学）
We show that microscopic black hole entropy formula based on Virasoro algebra can be derived from usual properties of stationary Killing horizons alone and absence of singularities of curvature invariants on them. In such a way some usual additional assumptions are shown to be fulfilled. In addition, for all quantities power expansion near horizon and thus explicit insight of the limiting procedure is given. More important the near horizon conformal symmetry proposed by Carlip together with its consequences on microscopic entropy is given a clear geometric origin.
There are certain black hole solutions in general relativity (GR) which are conformally related to the stationary solutions in GR. It is not obvious that the horizon entropy of these spacetimes is also one quarter of the area of horizon, like the stationary ones. Here I study this topic in the context of Virasoro algebra and Cardy formula. Using the fact that the conformal spacetime admits conformal Killing vector and the horizon is determined by the vanishing of the norm of it, the diffemorphisms are obtained which keep the near horizon structure invariant. The Noether charge and a bracket am…
We discuss various properties of rotating Killing horizons in generic $F(R)$ theories of gravity in dimension four for spacetimes endowed with two commuting Killing vector fields. Assuming there is no curvature singularity anywhere on or outside the horizon, we construct a suitable $(3+1)$-foliation. We show that similar to Einstein's gravity, we must have $T_{ab}k^ak^b=0$ on the Killing horizon, where $k^a$ is a null geodesic tangent to the horizon. For axisymmetric spacetimes, the effective gravitational coupling $\sim\,F'^{-1}(R)$ should usually depend upon the polar coordinate and hence ne…
We study conformal symmetries on the horizon of a general stationary and axisymmetric black hole. We find that there exist physically reasonable boundary conditions that uniquely determine a set of symmetry generators, which form one copy of the Virasoro algebra. For extremal black holes, Cardy's formula reproduces exactly the Bekenstein-Hawking entropy.
We determine the most general three-dimensional vacuum spacetime with a negative cosmological constant containing a non-singular Killing horizon. We show that the general solution with a spatially compact horizon possesses a second commuting Killing field and deduce that it must be related to the BTZ black hole (or its near-horizon geometry) by a diffeomorphism. We show there is a general class of asymptotically AdS$_3$ extreme black holes with arbitrary charges with respect to one of the asymptotic-symmetry Virasoro algebras and vanishing charges with respect to the other. We interpret these …
We show that the asymptotic symmetries close to nonextremal black hole horizons are generated by an extension of supertranslations. This group is generated by a semidirect sum of Virasoro and Abelian currents. The charges associated with the asymptotic Killing symmetries satisfy the same algebra. When considering the special case of a stationary black hole, the zero mode charges correspond to the angular momentum and the entropy at the horizon.
We discuss the structure of horizons in spacetimes with two metrics, with applications to the Vainshtein mechanism and other examples. We show, without using the field equations, that if the two metrics are static, spherically symmetric, nonsingular, and diagonal in a common coordinate system, then a Killing horizon for one must also be a Killing horizon for the other. We then generalize this result to the axisymmetric case. We also show that the surface gravities must agree if the bifurcation surface in one spacetime lies smoothly in the interior of the spacetime of the other metric. These re…
The Chern-Simons-like theories of gravity (CSLTG) are formulated at first order formalism. In this formalism, the derivation of the entropy of a black hole on bifurcation surface, as a quasi-local conserved charge is problematic. In this paper we overcome to these problems by considering the concept of total variation and the Lorentz-Lie derivative. We firstly find an expression for the ADT conserved current in the context of the CSLTG which is based on the concept of the Killing vector fields. Then, we generalize it to be conserved for all diffeomorphism generators. Thus, we can extract an of…
Static axisymmetric Einstein-Maxwell-Dilaton and stationary axisymmetric Einstein-Maxwell-Dilaton-Axion (EMDA) theories in four space-time dimensions are shown to be integrable by means of the inverse scattering transform method. The proof is based on the coset-space representation of the 4-dim theory in a space-time admitting a Killing vector field. Hidden symmetry group of the four-dimensional EMDA theory, unifying T and S string dualities, is shown to be Sp(2, R) acting transitively on the coset Sp(2, R)/U(2). In the case of two-parameter Abelian space-time isometry group, the hidden symmet…
The asymptotic structure of AdS spacetimes in the context of General Relativity coupled to the Maxwell field in three spacetime dimensions is analyzed. Although the fall-off of the fields is relaxed with respect to that of Brown and Henneaux, the variation of the canonical generators associated to the asymptotic Killing vectors can be shown to be finite once required to span the Lie derivative of the fields. The corresponding surface integrals then acquire explicit contributions from the electromagnetic field, and become well-defined provided they fulfill suitable integrability conditions, imp…
We consider a metric which describes Ba$\tilde{\text{n}}$ados geometries and show that the considered metric is a solution of generalized minimal massive gravity (GMMG) model. We consider the Killing vector field which preserves the form of considered metric. Using the off-shell quasi-local approach we obtain the asymptotic conserved charges of given solution. Similar to the Einstein gravity in the presence of negative cosmological constant, for the GMMG model also, we show that the algebra among the asymptotic conserved charges is isomorphic to two copies of the Virasoro algebra. Eventually, …
We prove that non-extremal black holes in four-dimensional general relativity exhibit an infinite-dimensional symmetry in their near horizon region. By prescribing a physically sensible set of boundary conditions at the horizon, we derive the algebra of asymptotic Killing vectors, which is shown to be infinite-dimensional and includes, in particular, two sets of supertranslations and two mutually commuting copies of the Virasoro algebra. We define the surface charges associated to the asymptotic diffeomorphisms that preserve the boundary conditions and discuss the subtleties of this definition…
We initiate a comprehensive study of a set of solutions of topologically massive gravity known as null warped anti-de Sitter spacetimes. These are pp-wave extensions of three-dimensional anti-de Sitter space. We first perform a careful analysis of the linearized stability of black holes in these spacetimes. We find two qualitatively different types of solutions to the linearized equations of motion: the first set has an exponential time dependence, the second - a polynomial time dependence. The solutions polynomial in time induce severe pathologies and moreover survive at the non-linear level.…
The asymptotically flat structure of $\mathcal{N}=(2,0)$ supergravity in three spacetime dimensions is explored. The asymptotic symmetries are spanned by an extension of the super-BMS$_3$ algebra, with two independent $\hat{u}(1)$ currents of electric and magnetic type. These currents are associated to $U(1)$ fields being even and odd under parity, respectively. Remarkably, although the $U(1)$ fields do not generate a backreaction on the metric, they provide nontrivial Sugawara-like contributions to the BMS$_3$ generators, and hence to the energy and the angular momentum. The entropy of flat c…
(Abridged) A solution is obtained for the interior structure of an uncharged rotating black hole that accretes a collisionless fluid. The solution is conformally stationary, axisymmetric, and conformally separable, possessing a conformal Killing tensor. Hyper-relativistic counter-streaming between collisionless ingoing and outgoing streams drives inflation at (just above) the inner horizon, followed by collapse. As ingoing and outgoing streams approach the inner horizon, they focus into twin narrow beams directed along the ingoing and outgoing principal null directions, regardless of the initi…
