Canonical energy is a valuable tool for analyzing the linear stability of black hole spacetimes; positivity of canonical energy for all perturbations implies mode stability, whereas the failure of positivity for any perturbation implies instability. Nevertheless, even in the case of $4$-dimensional Schwarzschild spacetime --- which is known to be stable --- manifest positivity of the canonical energy is difficult to establish, due to the presence of constraints on the initial data as well as the gauge dependence of the canonical energy integrand. Consideration of perturbations generated by a Hertz potential would appear to be a promising way to improve this situation, since the constraints and gauge dependence are eliminated when the canonical energy is expressed in terms of the Hertz potential. We prove that the canonical energy of a metric perturbation of Schwarzschild that is generated by a Hertz potential is positive. We relate the energy quantity arising in the linear stability proof of Dafermos, Holzegel and Rodnianski (DHR) to the canonical energy of an associated metric perturbation generated by a Hertz potential. We also relate the Regge-Wheeler variable of DHR to the ordinary Regge-Wheeler and twist potential variables of the associated perturbation. Since the Hertz potential formalism can be generalized to a Kerr black hole, our results may be useful for the analysis of the linear stability of Kerr.
The linear perturbation of the Kerr black hole has been discussed by using the Newman--Penrose and the perturbed Weyl scalars, $ψ_0$ and $ψ_4$ can be obtained from the Teukolsky equation. In order to obtain the other Weyl scalars and the perturbed metric, a formalism was proposed by Chrzanowski and by Cohen and Kegeles (CCK) to construct these quantities in a radiation gauge via the Hertz potential. As a simple example of the construction of the perturbed gravitational field with this formalism, we consider the gravitational field produced by a rotating circular ring around a Schwarzschild bla…
The canonical quantization of a Schwarzschild black hole yields a picture of the black hole that is shown to be equivalent to a collection of oscillators whose density of levels is commensurate with that of the statistical bootstrap model. Energy eigenstates of definite parity exhibit the Bekenstein mass spectrum, $M \sim \sqrt{\cal N} M_p$, where ${\cal N} \in {\bf N}$. From the microcanonical ensemble, we derive the statistical entropy of the black hole by explicitly counting the microstates corresponding to a macrostate of fixed total energy.
For long black holes have been considered as endowed with a definite temperature. Yet when the Schwarzschild black hole is treated as a canonical ensemble three problems arise: incompatibility with the Hawking radiation, divergence of the partition function, and a formally negative mean-square fluctuation of the energy. We solve all three problems by considering the Schwarzschild black hole as a grand canonical ensemble, with the Hamiltonian (the ADM mass) and the horizon surface area, separately, as observable parameters. The horizon area simulates the number of particles in statistical mecha…
We derive a transformation from the usual ADM metric-extrinsic curvature variables on the phase space of Schwarzschild black holes, to new canonical variables which have the interpretation of Kruskal coordinates. We explicitly show that this transformation is non-singular, even at the horizon. The constraints of the theory simplify in terms of the new canonical variables and are equivalent to the vanishing of the canonical momenta. Our work is based on earlier seminal work by Kuchar in which he reconstructed curvature coordinates and a mass function from spherically symmetric canonical data. T…
Many authors - beginning with Bekenstein - have suggested that the energy levels E_n of a quantized isolated Schwarzschild black hole have the form E_n = sigma sqrt{n} E_P, n=1,2,..., sigma =O(1), with degeneracies g^n. In the present paper properties of a system with such a spectrum, considered as a quantum canonical ensemble, are discussed: Its canonical partition function Z(g,beta=1/kT), defined as a series for g<1, obeys the 1-dimensional heat equation. It may be extended to values g>1 by means of an integral representation which reveals a cut of Z(g,beta) in the complex g-plane from g=1 t…
Recently is was shown that the imaginary part of the canonical partition function of Schwarzschild black holes with an energy spectrum E_n = σ\sqrt{n} E_P, n= 1,2, ..., has properties which - naively interpreted - leads to the expected unusual thermodynamical properties of such black holes (Hawking temperature, Bekenstein-Hawking entropy etc). The present paper interprets the same imaginary part in the framework of droplet nucleation theory in which the rate of transition from a metastable state to a stable one is proportional to the imaginary part of the canonical partition function. The conc…
We solve Wheeler-De Witt (WDW) metric probability wave equation on the apparent horizon hypersurface of the Schwarzschild de Sitter (SdS) black hole. To do so we choose radial dependent mass function $M(r)$ for its internal regions in the presence of a dynamical massless quantum matter scalar field $ψ(r)$ and calculate canonical supper hamiltonian constraint on $t-$constant hypersurface near the horizon $r=M(r)$. In this case $M(r)$ become geometrical degrees of freedom while $ψ(r)$ is matter degrees of freedom of the apparent horizon. However our solution is obtained versus the quantum harmon…
We propose a scheme for quantizing a scalar field over the Schwarzschild manifold including the interior of the horizon. On the exterior, the timelike Killing vector and on the horizon the isometry corresponding to restricted Lorentz boosts can be used to enforce the spectral condition. For the interior we appeal to the need for CPT invariance to construct an explicitly positive definite operator which allows identification of positive and negative frequencies. This operator is the translation operator corresponding to the inexorable propagation to smaller radii as expected from the classical …
Analytic semiclassical backreaction of a Schwarzschild black hole in a finite cavity: horizon shift, temperature renormalization, and canonical stability in the Hartle-Hawking State
Centre for Theoretical Physics, The British University（英国大学理论物理中心）；Department of Mathematics and Statistics, Imam Mohammad Ibn Saud Islamic University (IMSIU)（伊马姆穆罕默德·本·沙特伊斯兰大学数学与统计系）；Faculty of Physics, Babeş-Bolyai University（巴比什-博雅依大学物理学院）；Astronomical Observatory（天文台）
We discuss the problem of canonical quantization of a free real massive scalar field in the Schwarzschild spacetime. It is shown that a consistent procedure of canonical quantization of the field can be carried out without taking into account the black hole interior, so that in the resulting theory the canonical commutation relations are satisfied exactly, and the Hamiltonian has the standard form. However, unlike some papers, in which the expansion of the quantum field in spherical harmonics is used, here we use an expansion in scatteringlike states for energies larger than the mass of the fi…
A conditional symmetry is defined, in the phase-space of a quadratic in velocities constrained action, as a simultaneous conformal symmetry of the supermetric and the superpotential. It is proven that such a symmetry corresponds to a variational (Noether) symmetry.The use of these symmetries as quantum conditions on the wave-function entails a kind of selection rule. As an example, the minisuperspace model ensuing from a reduction of the Einstein - Hilbert action by considering static, spherically symmetric configurations and r as the independent dynamical variable, is canonically quantized. T…
This paper argues that the weak cosmic censorship hypothesis implies that the Schwarzschild black hole has maximal entropy in all stationary black holes of fixing temperature, or equivalently, to store a same amount of information the Schwarzschild black hole has highest temperature. It then gives the independent mathematical proofs for 4-dimensional general static black holes and stationary-axisymmetric black holes which have "$t$-$ϕ$" reflection isometry. This result does not only provide a new universal bound between temperature and entropy of black holes but also offers us new evidence to …
We study the thermodynamics of a $d$-dimensional Schwarzschild black hole in the canonical ensemble. This generalizes York's formalism to any number $d$ of dimensions. The canonical ensemble, characterized by a cavity of fixed radius $r$ and fixed temperature $T$ at the boundary, allows for two possible solutions in thermal equilibrium, a small and a large black hole. From the Euclidean action and the path integral approach, we obtain the free energy, the thermodynamic energy, the pressure, and the entropy, of the black hole plus cavity system. The entropy is given by the Bekenstein-Hawking ar…
We present a consistent canonical formulation of the flat Oppenheimer-Snyder model, including the Schwarzschild exterior. The switching between comoving and stationary observer is realized by promoting the coordinate transformation between dust proper time and Schwarzschild-Killing time to a canonical one. This leads to two different forms of the Hamiltonian constraint, both (almost) deparameterizable with regard to one of these times. A preliminary quantization of these constraints reveals a consistent picture for both observers: the singularity is avoided by a bounce.
We study the thermodynamics of a five-dimensional Schwarzschild black hole in the canonical ensemble using York's formalism. Inside a cavity of fixed size $r$ and fixed temperature $T$, there is a threshold at $πr T = 1$ above which a black hole can be in thermal equilibrium. This thermal equilibrium can be achieved for two specific black holes, a small black hole of horizon radius $r_{+1}$, and a large black hole of radius $r_{+2}$. In five dimensions, the radii $r_{+1}$ and $r_{+2}$ have an exact expression. Through the path integral formalism and the partition function, one obtains the acti…
Using the canonical quantum theory of spherically symmetric pure gravitational systems, we present a direct correspondence between the Friedmann-Robertson-Walker (FRW) cosmological model in the interior of a Schwarzschild black hole and the nth energy eigenstate of a linear harmonic oscillator. Such type of universe has a quantized mass of the order of the Planck mass and harmonic oscillator wave functions
From the curved spacetime Lagrangian the first approximation scalar particle quantum equation was obtained following the canonical formalism. The roots of this equation in Schwarzschild's pseudo flat space were found. As it was shown in a more general curved spacetime it is tedious to find equation's roots. The massless particle limiting case was considered. The Maxwell set of equations generalized to Schwarzschild's space was reproduced.
A phase-space noncommutativity in the context of a Kantowski-Sachs cosmological model is considered to study the interior of a Schwarzschild black hole. Due to the divergence of the probability of finding the black hole at the singularity from a canonical noncommutativity, one considers a non-canonical noncommutativity. It is shown that this more involved type of noncommutativity removes the problem of the singularity in a Schwarzschild black hole.
Self-force methods can be applied in calculations of the scatter angle in two-body hyperbolic encounters, working order by order in the mass ratio (assumed small) but with no recourse to a weak-field approximation. This, in turn, can inform ongoing efforts to construct an accurate model of the general-relativistic binary dynamics via an effective-one-body description and other semi-analytical approaches. Existing self-force methods are to a large extent specialised to bound, inspiral orbits. Here we develop a technique for (numerical) self-force calculations that can efficiently tackle scatter…
We use Fermi coordinates to calculate the canonical partition function for an ideal gas in a circular geodesic orbit in Schwarzschild spacetime. To test the validity of the results we prove theorems for limiting cases. We recover the Newtonian gas law subject only to tidal forces in the Newtonian limit. Additionally we recover the special relativistic gas law as the radius of the orbit increases to infinity. We also discuss how the method can be extended to the non ideal gas case.
The physical interpretation and eventual fate of gravitational singularities in a theory surpassing classical general relativity are puzzling questions that have generated a great deal of interest among various quantum gravity approaches. In the context of loop quantum gravity (LQG), one of the major candidates for a non-perturbative background-independent quantisation of general relativity, considerable effort has been devoted to construct effective models in which these questions can be studied. In these models, classical singularities are replaced by a "bounce" induced by quantum geometry c…
Generalizing previous quantum gravity results for Schwarzschild black holes from 4 to D > 3 space-time dimensions yields an energy spectrum E_n = alpha n^{(D-3)/(D-2)} E_P, n=1,2,..., alpha = O(1), where E_P is the Planck energy in that space-time. This spectrum means that the quantized area A_{D-2}(n) of the D-2 dimensional horizon has universally the form A_{D-2} = n a_{D-2}, where a_{D-2} is essentially the (D-2)th power of the D-dimensional Planck length. Assuming that the basic area quantum has a Z(2)-degeneracy according to its two possible orientation degrees of freedom implies a degene…
We consider the possibility of deriving a decoupled equation in terms of Weyl tensor components for gravitational perturbations of the Schwarzschild-Tangherlini solution. We find a particular gauge invariant component of the Weyl tensor does decouple and argue that this corresponds to the vector modes of Ishibashi and Kodama. Also, we construct a Hertz potential map for solutions of the electromagnetic and gravitational perturbation equations of a higher dimensional Kundt background using the decoupled equation of Durkee and Reall. Motivated by recent work of Guica and Strominger, we use this …
We present spherically symmetric solutions to Einstein's equations which are equivalent to canonical Schwarzschild and Reissner-Nordstrom black holes on the exterior, but with singular (Planck-density) shells at their respective event and inner horizons. The locally measured mass of the shell and the singularity are much larger than the asymptotic ADM mass. The area of the shell is equal to that of the corresponding canonical black hole, but the physical distance from the shell to the singularity is a Planck length, suggesting a natural explanation for the scaling of the black hole entropy wit…
The Kottler spacetime in isotropic coordinates is known where the metric is time-dependent. In this paper, the Kottler spacetime is given in isotropic static coordinates (i.e., the metric components are time-independent). The metric is found in terms of the Jacobian elliptic functions through coordinate transformations from the Schwarzschild-(anti-)de Sitter metric. In canonical coordinates, it is known that the unparameterized spatially projected null geodesics of the Kottler and Schwarzschild spacetimes coincide. We show that in isotropic static coordinates, the refractive indices of Kottler…
An analysis of conformal geodesics in the Schwarzschild-de Sitter and Schwarzschild-anti de Sitter families of spacetimes is given. For both families of spacetimes we show that initial data on a spacelike hypersurface can be given such that the congruence of conformal geodesics arising from this data cover the whole maximal extension of canonical conformal representations of the spacetimes without forming caustic points. For the Schwarzschild-de Sitter family, the resulting congruence can be used to obtain global conformal Gaussian systems of coordinates of the conformal representation. In the…
We study perturbations of a Schwarzschild black hole in Chern-Simons modified gravity. We begin by showing that Birkhoff's theorem holds for a wide family of Chern-Simons coupling functions, a scalar field present in the theory that controls the strength of the Chern-Simons correction to the Einstein-Hilbert action. After decomposing the perturbations in spherical harmonics, we study the linearized modified field equations and find that axial and polar modes are coupled, in contrast to general relativity. The divergence of the modified equations leads to the Pontryagin constraint, which forces…
We revisit the stability of black hole saddles for the Euclidean path integral describing the canonical partition function $Z(β)$ for gravity inside a spherical reflecting cavity. The boundary condition at the cavity wall couples the transverse-traceless (TT) and pure-trace modes that are traditionally used to describe fluctuations about Euclidean Schwarzschild black holes in infinite-volume asymptotically flat and asymototically AdS spacetimes. This coupling obstructs the familiar Gibbons-Hawking-Perry treatment of the conformal factor problem, as Wick rotation of the pure-trace modes would r…
In this work, in the context of modified gravity, a curved spacetime analogous to the "canonical acoustic black hole" is constructed. The source is a self-interacting scalar field which is non-minimally coupled to gravity through the Gauss-Bonnet invariant. The scalar-Gauss-Bonnet coupling function is characterized by three positive parameters: $σ$ with units of $(length)$, $μ$ with units of $(length)^{4}$, and a dimensionless parameter $s$, thus defining a three-parameter model for which the line element of canonical acoustic black hole is a solution. The spacetime is equipped with spherical …
We present a canonical Hamiltonian formulation of GR in which $τ$, the parameter of system evolution, is external to spacetime, playing a role similar to what we call time in nonrelativistic mechanics. This approach, known as Stueckelberg-Horwitz-Piron (SHP) theory, inherits the full computational power of classical analytical mechanics while maintaining manifest covariance throughout and eliminating possible conflict with general diffeomorphism invariance. In particular, SHP simplifies the initial value problem with potential applications in highly dynamical interactions, such as black hole c…
We formulate a spherical harmonically decomposed 1+1 scheme to self-consistently evolve the trajectory of a point particle and its gravitational metric perturbation to a Schwarzschild background spacetime. Following the work of Moncrief, we write down an action for perturbations in space-time geometry, combine that with the action for a point-particle, and then obtain Hamiltonian equations of motion for metric perturbations, the particle's coordinates, as well as their canonical momenta. Hamiltonian equations for the metric-perturbation and their conjugate momenta reduce to Zerilli-Moncrief an…
In this paper, we have constructed a model for well behaved isotropic compact star in the presence of charged perfect fluid, by considering a static and spherically symmetric metric in Schwarzschild's canonical coordinate system. To put the resulting differential equations into a closed system, we have employed the Vaidya & Tikekar (J. Astrophys. Astron. 3:325, 1982) form of the metric potential grr. The resulting energy-momentum components, i.e., energy density and pressure contain six constants; two of these are determined through the junction condition (matching the interior with the exteri…
