Stationary Event Horizons Research Articles

Self-gravitating matter in stationary and axisymmetric black hole spacetimes

Abstract All black holes (BHs) in nature are expected to be described by the Kerr vacuum solution of general relativity. However, the Kerr BH interior contains several problematic features such as a Cauchy horizon, a curvature singularity, and a causality-violating region. Non-Kerr BH models, which are used to examine the genericity of these features, typically contain nontrivial matter content. When such self-gravitating matter is minimally-coupled to Einstein–Hilbert gravity, the Einstein equations can be directly used to investigate its physical properties. We examine here the properties of matter in a broad class of stationary and axisymmetric, geodesically-integrable BH spacetimes, and how they are linked to various features of the spacetime geometry. In these spacetimes, we find the matter to typically flow along timelike Killing orbits in the BH exterior, usually exhibiting differential rotation but sometimes additionally also non-rigid rotation. At a horizon, the matter rest-frame energy density, ε, and principal normal pressure, pn , are shown to necessarily satisfy p n = − ϵ , implying that only specific types of matter can thread stationary event horizons (e.g. electromagnetic fields but not massless real scalar fields). Furthermore, we introduce Boyer–Lindquist-like coordinates for the nonstationary regions in the BH interior, which show the matter to be comoving with the interior cosmology. We also obtain simple expressions for the expansions of the ingoing and outgoing zero angular momentum null congruences and comment on the light-focussing behavior of the cosmology. Finally, we verify above results explicitly by working with a representative set of well-known BH spacetimes which contain various types of matter—scalar fields, electromagnetic fields, anisotropic fluids. Some spacetimes have singularities while others have regular interiors. In the exterior, the matter satisfies the weak energy condition. The framework developed here can be extended to cover more general spacetimes.

Isolated horizons: Hamiltonian evolution and the first law

A framework was recently introduced to generalize black hole mechanics by replacing stationary event horizons with isolated horizons. That framework is significantly extended. The extension is nontrivial in that not only do the boundary conditions now allow the horizon to be distorted and rotating, but also the subsequent analysis is based on several new ingredients. Specifically, although the overall strategy is closely related to that in the previous work, the dynamical variables, the action principle and the Hamiltonian framework are all quite different. More importantly, in the nonrotating case, the first law is shown to arise as a necessary and sufficient condition for the existence of a consistent Hamiltonian evolution. Somewhat surprisingly, this consistency condition in turn leads to new predictions even for static black holes. To complement the previous work, the entire discussion is presented in terms of tetrads and associated (real) Lorentz connections.

Interface of General Relativity, Quantum Physics and Statistical Mechanics: Some Recent Developments

The arena normally used in black holes thermodynamics was recently generalized to incorporate a broad class of physically interesting situations. The key idea is to replace the notion of stationary event horizons by that of `isolated horizons.' Unlike event horizons, isolated horizons can be located in a space-time quasi-locally. Furthermore, they need not be Killing horizons. In particular, a space-time representing a black hole which is itself in equilibrium, but whose exterior contains radiation, admits an isolated horizon. In spite of this generality, the zeroth and first laws of black hole mechanics extend to isolated horizons. Furthermore, by carrying out a systematic, non-perturbative quantization, one can explore the quantum geometry of isolated horizons and account for their entropy from statistical mechanical considerations. After a general introduction to black hole thermodynamics as a whole, these recent developments are briefly summarized.

Interface of General Relativity, Quantum Physics and Statistical Mechanics: Some Recent Developments

AbstractThe arena normally used in black holes thermodynamics was recently generalized to incorporate a broad class of physically interesting situations. The key idea is to replace the notion of stationary event horizons by that of ‘isolated horizons.’ Unlike event horizons, isolated horizons can be located in a space‐time quasi‐locally. Furthermore, they need not be Killing horizons. In particular, a space‐time representing a black hole which is itself in equilibrium, but whose exterior contains radiation, admits an isolated horizon. In spite of this generality, the eroth and first laws of black hole mechanics extend to isolated horizons. Furthermore, by carrying out a systematic, non‐perturbative quantization, one can explore the quantum geometry of isolated horizons and account for their entropy from statistical mechanical considerations. After a general introduction to black hole thermodynamics as a whole, these recent developments are briefly summarized.

On further generalization of the rigidity theorem for spacetimes with a stationary event horizon or a compact Cauchy horizon

A rigidity theorem that applies to smooth electrovacuum spacetimes which represent either (A) an asymptotically flat stationary black hole or (B) a cosmological spacetime with a compact Cauchy horizon ruled by closed null geodesics was given in a recent paper by Friedrich et al (1999 Commun. Math. Phys. 204 691-707). Here we enlarge the framework of the corresponding investigations by allowing the presence of other types of matter fields. In the first part the matter fields are involved merely implicitly via the assumption that the dominant energy condition is satisfied. In the second part Einstein-Klein-Gordon (EKG), Einstein-[non-Abelian]-Higgs (E[nA]H), Einstein-[Maxwell]-Yang-Mills-dilaton (E[M]YMd) and Einstein-Yang-Mills-Higgs (EYMH) systems are studied. The black hole event horizon or, respectively, the compact Cauchy horizon of the considered spacetimes is assumed to be a smooth non-degenerate null hypersurface. It is proved that there exists a Killing vector field in a one-sided neighbourhood of the horizon in EKG, E[nA]H, E[M]YMd and EYMH spacetimes. This Killing vector field is normal to the horizon, moreover, the associated matter fields are also shown to be invariant with respect to it. The presented results provide generalizations of the rigidity theorems of Hawking (for case A) and of Moncrief and Isenberg (for case B) and, in turn, they strengthen the validity of both the black hole rigidity scenario and the strong cosmic censor conjecture of classical general relativity.

On the Rigidity Theorem for Spacetimes with a Stationary Event Horizon or a Compact Cauchy Horizon

We consider smooth electrovac spacetimes which represent either (A) an asymptotically flat, stationary black hole or (B) a cosmological spacetime with a compact Cauchy horizon ruled by closed null geodesics. The black hole event horizon or, respectively, the compact Cauchy horizon of these spacetimes is assumed to be a smooth null hypersurface which is non-degenerate in the sense that its null geodesic generators are geodesically incomplete in one direction. In both cases, it is shown that there exists a Killing vector field in a one-sided neighborhood of the horizon which is normal to the horizon. We thereby generalize theorems of Hawking (for case (A)) and Isenberg and Moncrief (for case (B)) to the non-analytic case.