Static Black Hole Spacetimes Research Articles

Massive tensor field perturbations on extremal and near-extremal static black holes

We develop a new perturbation method to study the dynamics of massive tensor fields on extremal and near-extremal static black hole spacetimes in arbitrary dimensions. On such backgrounds, one can classify the components of massive tensor fields into the tensor, vector, and scalar-type components. For the tensor-type components, which arise only in higher dimensions, the massive tensor field equation reduces to a single master equation, whereas the vector and scalar-type components remain coupled. We consider the near-horizon expansion of both the geometry and the field variables with respect to the near-horizon scaling parameter. By doing so, we reduce, at each order of the expansion, the equations of motion for the vector and scalar-type components to a set of five mutually decoupled wave equations with source terms consisting only of the lower-order variables. Thus, together with the tensor-type master equation, we obtain the set of mutually decoupled equations at each order of the expansion that govern all dynamical degrees of freedom of the massive tensor field on the extremal and near-extremal static black hole background.

Classification of shift-symmetric Horndeski theories and hairy black holes

No-hair theorems for scalar-tensor theories imply that the trivial scalar field configuration is the unique configuration around stationary black hole spacetimes. The most basic assumption in these theorems is that a constant scalar configuration is actually admissible. In this paper, we classify shift-symmetric Horndeski theories according to whether or not they admit the trivial scalar configuration as a solution and under which conditions. Local Lorentz symmetry and the presence of a linear coupling between the scalar field and Gauss-Bonnet invariant plays feature prominently in this classification. We then use the classification to show that any theory without linear Gauss-Bonnet coupling that respects Local Lorentz symmetry admits all GR solutions. We also study the scalar hair configuration around black hole spacetimes in theories where the linear Gauss-Bonnet coupling is present. We show that the scalar hair of the configuration is secondary, fixed by the regularity of the horizon, and is determined by the black hole horizon properties.

P-Wave holographic superconductor in scalar hairy black holes

We study the properties of the p-wave holographic superconductor for the scalar hairy black holes in the probe limit. The black hole solutions in question possess planar topology, which are derived from the Einstein gravity theory minimally coupled to a scalar field with a generic scalar potential. These solutions can be viewed as characterized by two independent parameters, namely, α and k0, where AdS vacuum is manifestly restored when α→∞. Consequently, the p-wave holographic superconductor is investigated by employing the above static planar black hole spacetime as the background metric, where a Maxwell field is introduced to the model by nonminimally coupling it to a complex vector field. The latter is shown to condensate and furnish the superconducting phase when the temperature is below a critical value. By numerical calculations, we examine in detail how the scalar field in the background affects the properties of the superconductivity. It is found that the critical temperature depends crucially on the parameters α and k0, which subsequently affects the condensation process. By employing the Kubo formula, the real, as well as imaginary parts of the conductivity, are calculated and presented as functions of frequency. The results are discussed regarding the poles of the Green function, and the typical values of the BCS theory.

Isolated black holes without Z2 isometry

A mechanism to construct asymptotically flat, isolated, stationary black hole (BH) spacetimes with no $\mathbb{Z}_2$ (No$\mathbb{Z}$) isometry is described. In particular, the horizon geometry of such No$\mathbb{Z}$ BHs does not have the usual north-south (reflection) symmetry. We discuss two explicit families of models wherein No$\mathbb{Z}$ BHs arise. In one of these families, we exhibit the intrinsic horizon geometry of an illustrative example by isometrically embedding it in Euclidean 3-space, resulting in an "egg-like" shaped horizon. This asymmetry leaves an imprint in the No$\mathbb{Z}$ BH phenomenology, for instance in its lensing of light; but it needs not be manifest in the BH shadow, which in some cases can be analytically shown to retain a $\mathbb{Z}_2$ symmetry. Light absorption and scattering due to an isotropic source surrounding a No$\mathbb{Z}$ BH endows it with a non-zero momentum, producing an asymmetry triggered BH rocket effect.

Localization of Energy-Momentum for a Black Hole Spacetime Geometry with Constant Topological Euler Density

The evaluation of the energy-momentum distribution for a new four-dimensional, spherically symmetric, static and charged black hole spacetime geometry with constant nonzero topological Euler density is performed by using the energy-momentum complexes of Einstein and Møller. This black hole solution was recently developed in the context of the coupled Einstein–nonlinear electrodynamics of the Born-Infeld type. The energy is found to depend on the mass M and the charge q of the black hole, the cosmological constant Λ, and the radial coordinate r, while in both prescriptions all the momenta vanish. Some limiting and particular cases are analyzed and discussed, illustrating the rather extraordinary character of the spacetime geometry considered.

Massive vector field perturbations on extremal and near-extremal static black holes

We discuss a new perturbation method to study the dynamics of massive vector fields on (near-)extremal static black hole spacetimes. We start with, as our background, a rather generic class of warped product metrics, and classify the field variables into the vector(axial)- and scalar(polar)-type components. On this generic background, we show that for the vector-type components, the Proca equation reduces to a single master equation, whereas the scalar-type components remain to be coupled. Then, focusing on the case of (near-)extremal static black holes in four-dimensions, we consider the near-horizon expansion of both the background geometry and massive vector field by a scaling parameter $\lambda$ with the leading-order geometry being the so called near-horizon geometry. We show that on the near-horizon geometry, thanks to its enhanced symmetry, the Proca equation for the scalar-type components also reduces to a set of two mutually decoupled homogeneous wave equations for two variables, plus a coupled equation through which the remaining variable is determined. Therefore, together with the vector-type master equation, we obtain the set of three decoupled master wave equations, which govern the three independent dynamical degrees of freedom of the massive vector field in four-dimensions. We further expand the geometry and massive vector field with respect to $\lambda$ and show that at each order, the Proca equation for the scalar-type components can reduce to a set of decoupled inhomogeneous wave equations whose source terms consist only of the lower-order variables, plus one coupled equation that determins the remaining variable. Therefore, if one solves the master equations on the leading-order near-horizon geometry, then in principle one can successively solve the Proca equation at any order.

Bespoke analogue space-times: meta-material mimics

Modern meta-materials allow one to construct electromagnetic media with almost arbitrary bespoke permittivity, permeability, and magneto-electric tensors. If (and only if) the permittivity, permeability, and magneto-electric tensors satisfy certain stringent compatibility conditions, can the meta-material be fully described (at the wave optics level) in terms of an effective Lorentzian metric—an analogue spacetime. We shall consider some of the standard black-hole spacetimes of primary interest in general relativity, in various coordinate systems, and determine the equivalent meta-material susceptibility tensors in a laboratory setting. In static black hole spacetimes (Schwarzschild and the like) certain eigenvalues of the susceptibility tensors will be seen to diverge on the horizon. In stationary black hole spacetimes (Kerr and the like) certain eigenvalues of the susceptibility tensors will be seen to diverge on the ergo-surface.

Black hole shadow in an asymptotically flat, stationary, and axisymmetric spacetime: The Kerr-Newman and rotating regular black holes

The shadow of a black hole can be one of the strong observational evidences for stationary black holes. If we see shadows at the center of galaxies, we would say whether the observed compact objects are black holes. In this paper, we consider a formula for the contour of a shadow in an asymptotically-flat, stationary, and axisymmetric black hole spacetime. We show that the formula is useful for obtaining the contour of the shadow of several black holes such as the Kerr-Newman black hole and rotating regular black holes. Using the formula, we can obtain new examples of the contour of the shadow of rotating black holes if assumptions are satisfied.

Vacuum Polarization of Massive Fields in the Spacetime of the Higher-dimensional Black Holes

We construct and study the vacuum polarization, $\langle \phi^{2}\rangle_{D},$ of the quantized massive scalar field with a general curvature coupling parameter in higher-dimensional static and spherically-symmetric black hole spacetimes, with a special emphasis put on the electrically charged Tangherlini solutions and the extremal and ultraextremal configurations. For $4 \leq D \leq 7$ the explicit analytic expressions for the vacuum polarization are given. For the conformally coupled fields the relation between the trace of the stress-energy tensor and the vacuum polarization is examined, which requires knowledge of the higher-order terms in the Schwinger-DeWitt expansion.

Mode-sum prescription for vacuum polarization in black hole spacetimes in even dimensions

We present a mode-sum regularization prescription for computing the vacuum polarization of a scalar field in static spherically-symmetric black hole spacetimes in even dimensions. This is the first general and systematic approach to regularized vacuum polarization in higher even dimensions, building upon a previous scheme we developed for odd dimensions. Things are more complicated here since the even-dimensional propagator possesses logarithmic singularities which must be regularized. However, in spite of this complication, the regularization parameters can be computed in closed form in arbitrary even dimensions and for arbitrary metric function $f(r)$. As an explicit example of our method, we show plots for vacuum polarization of a massless scalar field in the Schwarzschild-Tangherlini spacetime for even $d=4,...,10$. However, the method presented applies straightforwardly to massive fields or to nonvacuum spacetimes.

Lux in obscuro II: photon orbits of extremal AdS black holes revisited

A large class of spherically symmetric static extremal black hole spacetimes possesses a stable null photon sphere on their horizons. For the extremal Kerr–Newman family, the photon sphere only really coincides with the horizon in the sense clarified by Doran. The condition under which a photon orbit is stable on an asymptotically flat extremal Kerr–Newman black hole horizon has recently been clarified; it is found that a sufficiently large angular momentum destabilizes the photon orbit, whereas an electrical charge tends to stabilize it. We investigated the effect of a negative cosmological constant on this observation, and found the same behavior in the case of extremal asymptotically Kerr–Newman–AdS black holes in -dimensions. In -dimensions, in the presence of an electrical charge, the angular momentum never becomes large enough to destabilize the photon orbit. We comment on the instabilities of black hole spacetimes with a stable photon orbit.

Mode-sum prescription for the vacuum polarization in odd dimensions

We present a new mode-sum regularization prescription for computing the vacuum polarization of a scalar field in static spherically-symmetric black hole spacetimes in odd dimensions. This is the first general and systematic approach to regularized vacuum polarization in higher dimensions. Remarkably, the regularization parameters can be computed in closed form in arbitrary dimensions and for arbitrary metric function $f(r)$. In fact, we show that in spite of the increasing severity and number of the divergences to be regularized, the method presented is mostly agnostic to the number of dimensions. Finally, as an explicit example of our method, we show plots for vacuum polarization in the Schwarzschild-Tangherlini spacetime for odd $d=5,...,11$.

Hawking radiation of stationary and non-stationary Kerr–de Sitter black holes

Hawking radiation of the stationary Kerr–de Sitter black hole is investigated using the relativistic Hamilton–Jacobi method. Meanwhile, extending this work to a non-stationary black hole using Dirac equations and generalized tortoise coordinate transformation, we derived the locations, the temperature of the thermal radiation as well as the maximum energy of the non-thermal radiation. It is found that the surface gravity and the Hawking temperature depend on both time and different angles. An extra coupling effect is obtained in the thermal radiation spectrum of Dirac particles which is absent from thermal radiation of scalar particles. Further, the chemical potential derived from the thermal radiation spectrum of scalar particle has been found to be equal to the highest energy of the negative energy state of the scalar particle in the non-thermal radiation for the Kerr–de Sitter black hole. It is also shown that for stationary black hole space time, these two different methods give the same Hawking radiation temperature.

Reflection symmetry in higher dimensional black hole spacetimes

In four spacetime dimensions there is a well known proof that for any asymptotically flat, stationary, and axisymmetric vacuum solution of Einstein’s equation there exists a ‘t-ϕ’ reflection isometry that reverses the direction of the timelike Killing vector field and the direction of the axial Killing vector field. However, this proof does not generalize to higher spacetime dimensions. Here we consider asymptotically flat, stationary, and axisymmetric (i.e., having one or more commuting rotational isometries) black hole spacetimes in vacuum general relativity in spacetime dimensions such that the action of the isometry group is trivial. (Here ‘trivial’ means that if the ‘axes’—i.e., the points where the axial Killing fields are linearly dependent—are removed, the action of the isometry group is that of a trivial principal fiber bundle. This excludes actions like that found in the Sorkin monopole.) We prove that there exists a ‘t-ϕ’ reflection isometry that reverses the direction of the timelike Killing vector field and the direction of each axial Killing vector field. The proof relies in an essential way on the first law of black hole mechanics.

On the Construction of Hartle–Hawking–Israel States Across a Static Bifurcate Killing Horizon

We consider a linear scalar quantum field propagating in a spacetime of dimension d ≥ 2 with a static bifurcate Killing horizon and a wedge reflection. Under suitable conditions (e.g. positive mass), we prove the existence of a pure Hadamard state which is quasi-free, invariant under the Killing flow and restricts to a double β H -KMS state on the union of the exterior wedge regions, where β H is the inverse Hawking temperature. The existence of such a state was first conjectured by Hartle and Hawking (Phys Rev D 13:2188–2203, 1976) and by Israel (Phys Lett 57:107–110, 1976), in the more general case of a stationary black hole spacetime. Jacobson (Phys Rev D 50:R6031–R6032, 1994) has conjectured a similar state to exist even for interacting fields in spacetimes with a static bifurcate Killing horizon. The state can serve as a ground state on the entire spacetime and the resulting situation generalises that of the Unruh effect in Minkowski spacetime. Our result complements a well-known uniqueness result of Kay and Wald (Phys Rep 207:49–136, 1991) and Kay (J Math Phys 34:4519–4539, 1993), who considered a general bifurcate Killing horizon and proved that a certain (large) subalgebra of the free field admits at most one Hadamard state which is invariant under the Killing flow. This state is pure and quasi-free and in the presence of a wedge reflection it restricts to a β H -KMS state on the smaller subalgebra associated to one of the exterior wedge regions. Our result establishes the existence of such a state on the full algebra, but only in the static case. Our proof follows the arguments of Sewell (Ann Phys 141: 201–224, 1982) and Jacobson (Phys Rev D 50:R6031–R6032, 1994), who exploited a Wick rotation in the Killing time coordinate to construct a corresponding Euclidean theory. In particular, we show that for the linear scalar field we can recover a Lorentzian theory by Wick rotating back. Because the Killing time coordinate is ill-defined on the bifurcation surface, we systematically replace it by a Gaussian normal coordinate. A crucial part of our proof is to establish that the Euclidean ground state satisfies the necessary analogues of analyticity and reflection positivity with respect to this coordinate.

A Class of Conformal Curves in the Reissner–Nordström Spacetime

A class of curves with special conformal properties (conformal curves) is studied on the Reissner-Nordstr\"om spacetime. It is shown that initial data for the conformal curves can be prescribed so that the resulting congruence of curves extends smoothly to future and past null infinity. The formation of conjugate points on these congruences is examined. The results of this analysis are expected to be of relevance for the discussion of the Reissner-Nordstr\"om spacetime as a solution to the conformal field equations and for the global numerical evaluation of static black hole spacetimes.

Imaginary part of the gravitational action at asymptotic boundaries and horizons

We study the imaginary part of the Lorentzian gravitational action for bounded regions, as described in arXiv:1301.7041. By comparing to a Euclidean calculation, we explain the agreement between the formula for this imaginary part and the formula for black hole entropy. We also clarify the topological structure of the imaginary part in Lovelock gravity. We then evaluate the action's imaginary part for some special regions. These include cylindrical slabs spanning the exterior of a stationary black hole spacetime, 'maximal diamonds' in various symmetric spacetimes, as well as local near-horizon regions. In the first setup, the black hole's entropy and conserved charges contribute to the action's imaginary and real parts, respectively. In the other two setups, the imaginary part coincides with the relevant entropy.

The imaginary part of the gravity action and black hole entropy

As observed recently in arXiv:1212.2922, the action of General Relativity (GR) in finite spacetime regions has an imaginary part that resembles the Bekenstein entropy. In this paper, we expand on that argument, with attention to different causal types of boundaries. This property of the GR action may open a new approach to the puzzles of gravitational entropy. In particular, the imaginary action can be evaluated for non-stationary finite regions, where the notion of entropy is not fully understood. As a first step in constructing the precise relationship between the imaginary action and entropy, we focus on stationary black hole spacetimes. There, we identify a class of bounded regions for which the action's imaginary part precisely equals the black hole entropy. As a powerful test on the validity of the approach, we also calculate the imaginary action for Lovelock gravity. The result is related to the corresponding entropy formula in the same way as in GR.

Spacetime near isolated and dynamical trapping horizons

We study the near-horizon spacetime for isolated and dynamical trapping horizons (equivalently marginally outer trapped tubes). The metric is expanded relative to an ingoing Gaussian null coordinate and the terms of that expansion are explicitly calculated to second order. For the spacelike case, knowledge of the intrinsic and extrinsic geometry of the (dynamical) horizon is sufficient to determine the near-horizon spacetime, while for the null case (an isolated horizon) more information is needed. In both cases spacetime is allowed to be of arbitrary dimension and the formalism accomodates both general relativity as well as more general field equations. The formalism is demonstrated for two applications. First, spacetime is considered near an isolated horizon and the construction is both checked against the Kerr-Newman solution and compared to the well-known near-horizon limit for stationary extremal black hole spacetimes. Second, spacetime is examined in the vicinity of a slowly evolving horizon and it is demonstrated that there is always an event horizon candidate in this region. The geometry and other properties of this null surface match those of the slowly evolving horizon to leading order and in this approximation the candidate evolves in a locally determined way. This generalizes known results for Vaidya as well as certain spacetimes known from studies of the fluid-gravity correspondence.

Massive spin-2 and spin-1/2no hair theorems for stationary axisymmetric black holes

We present a proof of the no hair theorems corresponding to free massive non-perturbative Pauli-Fierz spin-2 and perturbative massive spin-1/2 fields for stationary axisymmetric de Sitter black hole spacetimes of dimension four with two commuting Killing vector fields. The applicability of these results for asymptotically flat and anti-de Sitter spacetimes are also discussed.