Kerr-AdS Spacetime Research Articles

General mass formulas for charged Kerr-AdS black holes

It is well-known that the mass of a non-asymptotically flat spacetime cannot be uniquely defined. Some mass formulas for the Kerr-AdS black hole have been found and used in studying black hole thermodynamics. However, the derivations usually need a background subtraction to eliminate the divergence at infinity. It is also unknown whether the mass depends on the choice of coordinates. In this paper, we provide a more straightforward derivation for the mass formula, only demanding that the first law of black hole thermodynamics and Smarr formula are satisfied. We first make use of the Iyer-Wald formalism to derive a first law which avoids the divergence at infinity. Then we apply this formula to charged Kerr-AdS black hole expressed in the coordinates rotating at infinity. However, the first law associated with the timelike Killing vector field ∂∂t is not integrable. Then, by making use of the gauge freedom of t, we find a favorite parameter t′ which just makes the mass integrable. Applying the scaling argument, we show that the mass satisfies the Smarr formula and takes the form M/Ξ3/2. Moreover, applying the conformal method with ∂∂t′ , we obtain the same mass. By applying the first law to the coordinates which is not rotating at infinity, we find a preferred time T that makes the first law integrable and the mass is just the familiar mass M/Ξ2 in the literature. This mass is also confirmed by the conformal method. We find that the two mass formulas correspond to different families of observers and the preferred Killing times. So our work clarifies the ambiguities of mass in Kerr-AdS spacetimes.

Tidal forces in Kerr-AdS and Grey galaxies

In a recent paper (Kim et al 2023 arXiv:2305.08922 [hep-th]), it has been proposed that the endpoint of the Kerr-AdS superradiant instability is a Grey Galaxy. The conjectured solutions are supposed to be made up of a black hole with critical angular velocity in the centre of AdS, surrounded by a large flat disk of thermal bulk gas that revolves around the black hole. In the analysis of the proposed solutions so far, gravitational effects due to the black hole on the thermal gas have been neglected. A way to estimate these effects is via computing tidal forces. With this motivation, we study tidal forces on objects moving in the Kerr-AdS spacetime. To do so, we construct a parallel-transported orthonormal frame along an arbitrary timelike or null geodesic. We then specialise to the class of fast rotating geodesics lying in the equatorial plane, and estimate tidal forces on the gas in the Grey galaxies, modelling it as a collection of particles moving on timelike geodesics. We show that the tidal forces are small (and remain small even in the large mass limit), thereby providing additional support to the idea that the gas is weakly interacting with the black hole.

Quasinormal modes of Proca fields in a Schwarzschild-AdS spacetime

We present new results concerning the Proca massive vector field in a Schwarzschild-AdS black hole geometry. We provide a first principles analysis of Proca vector fields in this geometry using both the vector spherical harmonic (VSH) separation method and the Frolov-Krtou\v{s}-Kubiz\v{n}\'{a}k-Santos (FKKS) method that separates the relevant equations in spinning geometries. The analysis in the VSH method shows, on one hand, that it is arduous to separate the scalar-type from the vector-type polarizations of the electric sector of the Proca field, and on the other hand, it displays clearly the electric and the magnetic mode sectors. The analysis in the FKKS method is performed by taking the nonrotating limit of the Kerr-AdS spacetime, and shows that the ansatz decouples the polarizations in the electric mode sector even in the nonrotating limit. On the other hand, it captures only two of the three possible polarizations, the magnetic mode sector is missing. The reason for the absence of this polarization is related to the degeneracy of the principal tensor in static spherical symmetric spacetimes. The degrees of freedom and quasinormal modes in both separation methods of the Proca field are found. The frequencies of the quasinormal modes are also computed. For the electric mode sector in the VSH method the frequencies are found through an extension, which substitutes number coefficients by matrix coefficients, of the Horowitz-Hubeny numerical procedure, whereas for the magnetic mode sector in the VSH method and the electric sector of the FKKS method it is shown that a direct use of the procedure can be made. The values of the quasinormal mode frequencies obtained for each method are compared and showed to be in good agreement with each other. This further supports the analytical approaches presented here for the behavior of the Proca field in a Schwarzschild-AdS black hole background.

Null hypersurface caustics, closed null curves, and super entropy

Recently it was discovered that null hypersurfaces can develop caustics outside the event horizon of super-entropic Kerr-AdS black holes, in contrast to the usual Kerr-AdS case. In this work we explore a few more examples of black hole spacetimes in which such exterior caustics can develop. If a closed null curve is present, e.g., in the case of Taub-NUT and the "transunital" Kerr-AdS spacetimes, then it coincides with a null hypersurface caustic (NHC) of a minimal separation parameter. Thus a spacetime on the verge of forming closed timelike curves could develop a caustic. Known examples of super-entropic black holes also have exterior NHC, although such spacetimes are free of closed null/timelike curves. Nevertheless the relationship between closed causal curves, NHC, and super-entropy is not straightforward. This is best illustrated with the BTZ black string, which for some choices of the warp factor in the extra dimension and the value of the charge, can be super-entropic. However, even those that are not super-entropic can admit NHC outside the horizon.

Entropy in Poincaré gauge theory: Kerr-AdS solution

Using a Hamiltonian approach, we introduce black hole entropy for Kerr-AdS spacetimes with torsion as the canonical charge on horizon. In spite of a completely different geometric setting with respect to GR, the resulting thermodynamic variables, energy, angular momentum and entropy, are shown to be proportional to the corresponding GR expressions. The validity of the first law is confirmed.

Entropy in three-dimensional general relativity: Kerr-AdS black hole

Black hole thermodynamics of the Kerr-AdS spacetime in three-dimensional general relativity is analyzed using a Hamiltonian approach. The values of the conserved charges and entropy, obtained by a proper treatment of the AdS asymptotic conditions, are shown to satisfy the first law of black hole dynamics.

The Quantization of a Kerr-AdS Black Hole

We apply our model of quantum gravity to a Kerr-AdS space-time of dimension2m+1,m≥2, where all rotational parameters are equal, resulting in a wave equation in a quantum space-time which has a sequence of solutions that can be expressed as a product of stationary and temporal eigenfunctions. The stationary eigenfunctions can be interpreted as radiation and the temporal ones as gravitational waves. The event horizon corresponds in the quantum model to a Cauchy hypersurface that can be crossed by causal curves in both directions such that the information paradox does not occur. We also prove that the Kerr-AdS space-time can be maximally extended by replacing in a generalized Boyer-Lindquist coordinate system thervariable byρ=r2such that the extended space-time has a timelike curvature singularity inρ=-a2.

Existence of Quasinormal Modes for Kerr–AdS Black Holes

This paper establishes the existence of quasinormal frequencies converging exponentially to the real axis for the Klein--Gordon equation on a Kerr-AdS spacetime when Dirichlet boundary conditions are imposed at the conformal boundary. The proof is adapted from results in Euclidean scattering about the existence of scattering poles generated by time-periodic approximate solutions to the wave equation.

Unstable Mode Solutions to the Klein\u2013Gordon Equation in Kerr-anti-de Sitter Spacetimes

For any cosmological constant {Lambda = -3/ell^{2} < 0} and any {alpha < 9/4}, we find a Kerr-AdS spacetime {({mathcal{M}}, g_{{rm KAdS}})}, in which the Klein–Gordon equation {Box_{g_{{rm KAdS}}}psi + alpha/ell^{2}psi = 0} has an exponentially growing mode solution satisfying a Dirichlet boundary condition at infinity. The spacetime violates the Hawking–Reall bound {r_{+}^{2} > |a|ell}. We obtain an analogous result for Neumann boundary conditions if {5/4 < alpha < 9/4}. Moreover, in the Dirichlet case, one can prove that, for any Kerr-AdS spacetime violating the Hawking–Reall bound, there exists an open family of masses {alpha} such that the corresponding Klein–Gordon equation permits exponentially growing mode solutions. Our result adopts methods of Shlapentokh-Rothman developed in (Commun. Math. Phys. 329:859–891, 2014) and provides the first rigorous construction of a superradiant instability for negative cosmological constant.

Boundary conditions for Maxwell fields in Kerr-AdS spacetimes

Perturbative methods are useful to study the interaction between black holes and test fields. The equation for a perturbation itself, however, is not complete to study such a composed system if we do not assign physically relevant boundary conditions. Recently we have proposed a new type of boundary conditions for Maxwell fields in Kerr-anti-de Sitter (Kerr-AdS) spacetimes, from the viewpoint that the AdS boundary may be regarded as a perfectly reflecting mirror, in the sense that energy flux vanishes asymptotically. In this paper, we prove explicitly that a vanishing energy flux leads to a vanishing angular momentum flux. Thus, these boundary conditions may be dubbed as vanishing flux boundary conditions.

Quasimodes and a lower bound on the uniform energy decay rate for Kerr–AdS spacetimes

We construct quasimodes for the Klein-Gordon equation on the black hole exterior of Kerr-Anti-de Sitter (Kerr-AdS) spacetimes. Such quasi-modes are associated with time-periodic approximate solutions of the Klein Gordon equation and provide natural candidates to probe the decay of solutions on these backgrounds. They are constructed as the solutions of a semi-classical non-linear eigenvalue problem arising after separation of variables, with the (inverse of the) angular momentum playing the role of the semi-classical parameter. Our construction results in exponentially small errors in the semi-classical parameter. This implies that general solutions to the Klein Gordon equation on Kerr-AdS cannot decay faster than logarithmically. The latter result completes previous work by the authors, where a logarithmic decay rate was established as an upper bound.

Conserved charges of Kerr—Ads spacetime using Poincaré gauge version

The process of covariant conserved charge of gravitational theory, which is covariant under general coordinate and local Lorentz transformations, has been applied to many tetrad fields, which reproduce Kerr-Ads spacetime, to calculate their conserved charges. It is shown that this process gives an infinite value of the conserved charges for Kerr—Ads spacetime. Therefore, the method of “regularization through relocalization", i.e., modification of the Lagrangian of the gravitational field through a total derivative, is used. This method gaves a finite and a consistent result of total energy and angular momentum for Kerr—Ads spacetime.

On the Massive Wave Equation on Slowly Rotating Kerr-AdS Spacetimes

The massive wave equation $\Box_g \psi - \alpha\frac{\Lambda}{3} \psi = 0$ is studied on a fixed Kerr-anti de Sitter background $(\mathcal{M},g_{M,a,\Lambda})$. We first prove that in the Schwarzschild case (a=0), $\psi$ remains uniformly bounded on the black hole exterior provided that $\alpha < {9/4}$, i.e. the Breitenlohner-Freedman bound holds. Our proof is based on vectorfield multipliers and commutators: The usual energy current arising from the timelike Killing vector field $T$ (which fails to be non-negative pointwise) is shown to be non-negative with the help of a Hardy inequality after integration over a spacelike slice. In addition to $T$, we construct a vectorfield whose energy identity captures the redshift producing good estimates close to the horizon. The argument is finally generalized to slowly rotating Kerr-AdS backgrounds. This is achieved by replacing the Killing vectorfield $T=\partial_t$ with $K=\partial_t + \lambda \partial_\phi$ for an appropriate $\lambda \sim a$, which is also Killing and--in contrast to the asymptotically flat case--everywhere causal on the black hole exterior. The separability properties of the wave equation on Kerr-AdS are not used. As a consequence, the theorem also applies to spacetimes sufficiently close to the Kerr-AdS spacetime, as long as they admit a causal Killing field $K$ which is null on the horizon.

Conserved charges of higher D Kerr–AdS spacetimes

We compute the energy and angular momenta of recent D-dimensional Kerr–AdS solutions to cosmological Einstein gravity, using our invariant charge definitions.