Master functions and hybrid quantization of perturbed nonrotating black hole interiors

Perturbation theory of vacuum spherically-symmetric spacetimes is a crucial tool to understand the dynamics of black hole perturbations. Spherical symmetry allows for an expansion of the perturbations in scalar, vector, and tensor harmonics. The resulting perturbative equations are decoupled for modes with different parity and different harmonic numbers. Moreover, for each harmonic and parity, the equations for the perturbations can be decoupled in terms of (gauge-invariant) master functions that satisfy 1+1 wave equations. By working in a completely general perturbative gauge, in this paper we study what is the most general master function that is linear in the metric perturbations and their first-order derivatives and satisfies a wave equation with a potential. The outcome of the study is that for each parity we have two branches of solutions with similar features. One of the branches includes the known results: In the odd-parity case, the most general master function is an arbitrary linear combination of the Regge-Wheeler and the Cunningham-Price-Moncrief master functions whereas in the even-parity case it is an arbitrary linear combination of the Zerilli master function and another master function that is new to our knowledge. The other branch is very different since it includes an infinite collection of potentials which in turn lead to an independent collection master of functions which depend on the potential. The allowed potentials satisfy a non-linear ordinary differential equation. Finally, all the allowed master functions are gauge invariant and can be written in a fully covariant form.

We revisit the problem of perturbations of Schwarzschild-AdS$_4$ black holes\nby using a combination of the Martel-Poisson formalism for perturbations of\nfour-dimensional spherically symmetric spacetimes and the Kodama-Ishibashi\nformalism. We clarify the relationship between both formalisms and express the\nBrown-York-Balasubramanian-Krauss boundary stress-energy tensor,\n$\\bar{T}_{\\mu\\nu}$, on a finite-$r$ surface purely in terms of the even and odd\nmaster functions. Then, on these surfaces we find that the spacelike components\nof the conservation equation $\\bar{\\mathcal{D}}^\\mu \\bar{T}_{\\mu\\nu} =0$ are\nequivalent to the wave equations for the master functions. The renormalized\nstress-energy tensor at the boundary $\\displaystyle \\frac{r}{L} \\lim_{r\n\\rightarrow \\infty} \\bar{T}_{\\mu\\nu}$ is calculated directly in terms of the\nmaster functions.\n

Gravitational perturbations due to a point particle moving on a static black\nhole background are naturally described in Regge-Wheeler gauge. The first-order\nfield equations reduce to a single master wave equation for each radiative\nmode. The master function satisfying this wave equation is a linear combination\nof the metric perturbation amplitudes with a source term arising from the\nstress-energy tensor of the point particle. The original master functions were\nfound by Regge and Wheeler (odd parity) and Zerilli (even parity). Subsequent\nwork by Moncrief and then Cunningham, Price and Moncrief introduced new master\nvariables which allow time domain reconstruction of the metric perturbation\namplitudes. Here I explore the relationship between these different functions\nand develop a general procedure for deriving new higher-order master functions\nfrom ones already known. The benefit of higher-order functions is that their\nsource terms always converge faster at large distance than their lower-order\ncounterparts. This makes for a dramatic improvement in both the speed and\naccuracy of frequency domain codes when analyzing unbound motion.\n

Gravitational wave emission from extreme mass ratio binaries (EMRBs) should be detectable by the joint NASA–ESA LISA project, spurring interest in analytical and numerical methods for investigating EMRBs. We describe a discontinuous Galerkin (dG) method for solving the distributionally forced 1+1 wave equations which arise when modeling EMRBs via the perturbation theory of Schwarzschild black holes. Despite the presence of jump discontinuities in the relevant polar and axial gravitational ‘master functions’, our dG method achieves global spectral accuracy, provided that we know the instantaneous position, velocity and acceleration of the small particle. Here these variables are known, since we assume that the particle follows a timelike geodesic of the Schwarzschild geometry. We document the results of several numerical experiments testing our method, and in our concluding section discuss the possible inclusion of gravitational self-force effects.

Recent efforts have shown that Kantowski-Sachs spacetime provides a useful framework for analyzing perturbations inside a Schwarzschild black hole (BH). In these studies, the adoption of a Hamiltonian formulation offers an insightful perspective. The aim of this work is twofold. First, we revisit and elaborate the results obtained so far in Kantowski-Sachs, with the focus placed on axial perturbations. In particular, by exploiting the relation between this spacetime and the interior of a nonrotating BH, we consider the extension of those results to the exterior geometry of the BH. In this way, we clarify the relation between the axial perturbative gauge invariants emerging from the canonical analysis and the already well-established axial BH invariants, often referred to as master functions. We do so by providing a unified picture of the Hamiltonian formalism, which does not distinguish, formally, between exterior and interior geometries. The second objective is to explore the role of Darboux transformations, which were found as hidden symmetries in the context of BH perturbations, and their appearance in the Hamiltonian setting. Within this framework, the Hamiltonian formulation provides a clear geometric interpretation and characterization of Darboux transformations within the axial sector, viewing them as the set of canonical transformations between Hamiltonians for axial master functions.

We calculate the gravitational perturbations produced by a small mass in\neccentric orbit about a much more massive Schwarzschild black hole and use the\nnumerically computed perturbations to solve for the metric. The calculations\nare initially made in the frequency domain and provide Fourier-harmonic modes\nfor the gauge-invariant master functions that satisfy inhomogeneous versions of\nthe Regge-Wheeler and Zerilli equations. These gravitational master equations\nhave specific singular sources containing both delta function and\nderivative-of-delta function terms. We demonstrate in this paper successful\napplication of the method of extended homogeneous solutions, developed recently\nby Barack, Ori, and Sago, to handle source terms of this type. The method\nallows transformation back to the time domain, with exponential convergence of\nthe partial mode sums that represent the field. This rapid convergence holds\neven in the region of $r$ traversed by the point mass and includes the\ntime-dependent location of the point mass itself. We present numerical results\nof mode calculations for certain orbital parameters, including highly accurate\nenergy and angular momentum fluxes at infinity and at the black hole event\nhorizon. We then address the issue of reconstructing the metric perturbation\namplitudes from the master functions, the latter being weak solutions of a\nparticular form to the wave equations. The spherical harmonic amplitudes that\nrepresent the metric in Regge-Wheeler gauge can themselves be viewed as weak\nsolutions. They are in general a combination of (1) two differentiable\nsolutions that adjoin at the instantaneous location of the point mass (a result\nthat has order of continuity $C^{-1}$ typically) and (2) (in some cases) a\ndelta function distribution term with a computable time-dependent amplitude.\n

Bound inside rotating or charged black holes, there are stable periodic planetary orbits, which neither come out nor terminate at the central singularity. Stable periodic orbits inside black holes exist even for photons. These bound orbits may be defined as orbits of the third kind, following the Chandrasekhar classification of particle orbits in the black hole gravitational field. The existence domain for the third-kind orbits is rather spacious, and thus there is place for life inside supermassive black holes in the galactic nuclei. Interiors of the supermassive black holes may be inhabited by civilizations, being invisible from the outside. In principle, one can get information from the interiors of black holes by observing their white hole counterparts.

In this paper, we have examined the validity of a proposed definition of gravitational entropy in the context of accelerating black hole solutions of the Einstein field equations, which represent the realistic black hole solutions. We have adopted a phenomenological approach proposed in Rudjord et al. [Phys. Scr. 77, 055901 (2008)] and expanded by Romero et al. [Int. J. Theor. Phys. 51, 925 (2012)], in which the Weyl curvature hypothesis is tested against the expressions for the gravitational entropy. Considering the [Formula: see text]-metric for the accelerating black holes, we have evaluated the gravitational entropy and the corresponding entropy density for four different types of black holes, namely, nonrotating black hole, nonrotating charged black hole, rotating black hole and rotating charged black hole. We end up by discussing the merits of such an analysis and the possible reason of failure in the particular case of rotating charged black hole and comment on the possible resolution of the problem.

We consider test planet and photon orbits of the third kind inside a black hole, which are stable, periodic and neither come out of the black hole nor terminate at the singularity. Interiors of supermassive black holes may be inhabited by advanced civilizations living on planets with the third-kind orbits. In principle, one can get information from the interiors of black holes by observing their white hole counterparts.

We show that the number of horizons of static black holes can be strongly constrained by energy conditions of matter fields. After a careful clarification on the ‘interior’ of a black hole, we prove that if the interior of a static black hole satisfies strong energy condition or null energy condition, there is at most one non-degenerated inner Killing horizon behind the non-degenerated event horizon. Our result offers some universal restrictions on the number of horizons. Interestingly and importantly, it also suggests that matter not only promotes the formation of event horizon but also prevents the appearance of multiple horizons inside black holes. Furthermore, using the geometrical construction, we obtain a radially conserved quantity which is valid for general static spacetimes.

We discuss a solution of the Einstein equations, obtained by gluing the external Kerr metric and the internal Weyl metric, describing an axisymmetric static vacuum distorted black hole. These metrics are glued at the null surfaces representing their horizons. For this purpose we use the formalism of massive thin null shells. The corresponding solution is called a "hybrid" black hole. The massive null shell has an angular momentum which is the origin of the rotation of the external Kerr spacetime. At the same time, the shell distorts the geometry inside the horizon. The inner geometry of the "hybrid" black hole coincides with the geometry of the interior of a non-rotating Weyl-distorted black hole. Properties of the "hybrid" black holes are briefly discussed.

The Bondi solution, which describes the radial inflow of a gas onto a non-rotating black hole, provides a powerful test for numerical relativistic codes. However, the Bondi solution is usually derived in Schwarzschild coordinates, which are not well suited for dynamical spacetime evolutions. Instead, many current numerical relativistic codes adopt moving-puncture coordinates, which render black holes in trumpet geometries. Here we transform the Bondi solution into trumpet coordinates, which result in regular expressions for the fluid flow extending into the black-hole interior. We also evolve these solutions numerically and demonstrate their usefulness for testing and calibrating numerical codes.

Gravity warps space and time into a funnel and generates a black hole when a cosmic body undergoes a catastrophic collapse. What can one say about the interior of a black hole? The important point is that inside a black hole the space radial direction becomes time, and time becomes a space direction. The path into the gravitational abyss of the interior of a black hole is a progression in time. There is a peculiar region inside a black hole where some characteristics of the space-time curvature become singular. We call this region singularity. The colossal tidal gravitational forces near singularity modify physical laws. Space and time are not only strongly curved near the singularity, but they split into quanta. The fall into the singularity is unstoppable for a body inside a black hole. This paper also addresses the following questions: Can one see what happens inside a black hole? Can a falling observer cross the singularity without being crushed? Can new baby universes arise inside a black hole? An answer to all these questions is probably “yes”. We give also a brief review of the modern black hole astrophysics.

We reconsider the black hole firewall puzzle, emphasizing that quantum error-correction, computational complexity, and pseudorandomness are crucial concepts for understanding the black hole interior. We assume that the Hawking radiation emitted by an old black hole is pseudorandom, meaning that it cannot be distinguished from a perfectly thermal state by any efficient quantum computation acting on the radiation alone. We then infer the existence of a subspace of the radiation system which we interpret as an encoding of the black hole interior. This encoded interior is entangled with the late outgoing Hawking quanta emitted by the old black hole, and is inaccessible to computationally bounded observers who are outside the black hole. Specifically, efficient operations acting on the radiation, those with quantum computational complexity polynomial in the entropy of the remaining black hole, commute with a complete set of logical operators acting on the encoded interior, up to corrections which are exponentially small in the entropy. Thus, under our pseudorandomness assumption, the black hole interior is well protected from exterior observers as long as the remaining black hole is macroscopic. On the other hand, if the radiation is not pseudorandom, an exterior observer may be able to create a firewall by applying a polynomial-time quantum computation to the radiation.

We briefly discuss non-singular black hole models, with the main focus on the properties of non-singular evaporating black holes. Such black holes possess an apparent horizon, however the event horizon may be absent. In such a case, the information from the black hole interior may reach the external observer after the complete evaporation of the black hole. This model might be used for the resolution of the information loss puzzle. However, as we demonstrate, in a general case the quantum radiation emitted from the black hole interior, calculated in the given black hole background, is very large. This outburst of the radiation is exponentially large for models with the redshift function α = 1. We show that it can be suppressed by including a non-trivial redshift function. However, even this suppression is not enough to guarantee self-consistency of the model. This problem is a manifestation of a general problem, known as the "mass inflation". We briefly comment on possible ways to overcome this problem in the models of non-singular evaporating black holes.