Computation of the renormalized stress-energy tensor is the most serious obstacle in studying the dynamical, self-consistent, semiclassical evaporation of a black hole in 4D. The difficulty arises from the delicate regularization procedure for the stress-energy tensor, combined with the fact that in practice the modes of the field need be computed numerically. We have developed a new method for numerical implementation of the point-splitting regularization in 4D, applicable to the renormalized stress-energy tensor as well as to $\left\langle ϕ^{2}\right\rangle _{ren}$, namely the renormalized $\left\langle ϕ^{2}\right\rangle$. So far we have formulated two variants of this method: t-splitting (aimed for stationary backgrounds) and angular splitting (for spherically-symmetric backgrounds). In this paper we introduce our basic approach, and then focus on the t-splitting variant, which is the simplest of the two (deferring the angular-splitting variant to a forthcoming paper). We then use this variant, as a first stage, to calculate $\left\langle ϕ^{2}\right\rangle _{ren}$ in Schwarzschild spacetime, for a massless scalar field in the Boulware state. We compare our results to previous ones, obtained by a different method, and find full agreement. We discuss how this approach can be applied (using the angular-splitting variant) to analyze the dynamical self-consistent evaporation of black holes.
The point-splitting renormalization method offers a prescription to calculate finite expectation values of quadratic operators constructed from quantum fields in a general curved spacetime. It has been recently shown by Levi and Ori that when the background metric possesses an isometry, like stationary or spherically symmetric black holes, the method can be upgraded into a pragmatic procedure of renormalization that produces efficient numerical calculations. In this note we show that when the background enjoys three-dimensional spatial symmetries, like homogeneous expanding universes, the abov…
We study the relativistic orbit of binary black holes in systems with small mass ratio. The trajectory of the smaller object (another black hole or a neutron star), represented as a particle, is determined by the geodesic equation on the perturbed massive black hole spacetime. The particle itself generates the gravitational perturbations leading to a problem that needs regularization. Here we study perturbations around a Schwarzschild black hole using Moncrief's gauge invariant formalism. We decompose the perturbations into $\ell-$multipoles to show that all $\ell-$metric coefficients are $C^0…
We continue the presentation of the pragmatic mode-sum regularization (PMR) method for computing the renormalized stress-energy tensor (RSET). We show in detail how to employ the $t$-splitting variant of the method, which was first presented for $\left\langleϕ^{2}\right\rangle_{ren}$, to compute the RSET in a stationary, asymptotically-flat background. This variant of the PMR method was recently used to compute the RSET for an evaporating spinning black hole. As an example for regularization, we demonstrate here the computation of the RSET for a minimally-coupled, massless scalar field on Schw…
From calculations of the variance of fluctuations and of the mean of the energy density of a massless scalar field in the Minkowski vacuum as a function of an intrinsic scale defined by the world function between two nearby points (as used in point separation regularization) we show that, contrary to prior claims, the ratio of variance to mean-squared being of the order unity does not imply a failure of semiclassical gravity. It is more a consequence of the quantum nature of the state of matter field than any inadequacy of the theory of spacetime with quantum matter as source.
Quantum corrections coming from massless fields conformally coupled with gravity are studied, in order to see if they can lead to avoidance of the annoying Big Rip singularity which shows up in a flat Friedmann-Robertson-Walker universe filled with dark energy and modeled by a scalar phantom field. The dynamics of the model are discussed for all values of the two parameters, named $α>0$ and $β<0$, corresponding to the regularization process. The new results are compared with the ones obtained in \cite{hae11} previously, where dark energy was modeled by means of a phantom fluid with equation of…
The renormalization of a scalar field theory with a quartic self-coupling (a $λϕ^4$ theory) via adiabatic regularization in a general Robertson-Walker spacetime is discussed. The adiabatic counterterms are presented in a way that is most conducive to numerical computations. A variation of the adiabatic regularization method is presented which leads to analytic approximations for the energy-momentum tensor of the field and the quantum contribution to the effective mass of the mean field. Conservation of the energy-momentum tensor for the field is discussed and it is shown that the part of the e…
We consider free and self-interacting quantum scalar fields satisfying modified dispersion relations in the framework of Einstein-Aether theory. Using adiabatic regularization, we study the renormalization of the equation for the mean value of the field in the self-interacting case, and the renormalization of the semiclassical Einstein-Aether equations for free fields. In both cases we consider Bianchi type I background spacetimes. Contrary to what happens for {\it free} fields in {\it flat} Robertson-Walker spacetimes, the self-interaction and/or the anisotropy produce non-purely geometric te…
We use a perturbative method to evaluate the effective action of a free scalar field propagating in the Bianchi type I spacetime with large space anisotropy. The zeta- function regularization method is used to evaluate the action to the second order in the Schwinger perturbative formula. As the quantum corrections contain fourth derivative in the metric we apply the method of iterative reduction to reduce it to the second-order form to obtain the self-consistent solution of the semiclassical gravity theory, The reduced Einstein equation shows that the space anisotropy, which will be smoothed o…
We extend the adiabatic regularization method for an expanding universe to include the Yukawa interaction between quantized Dirac fermions and a homogeneous background scalar field. We give explicit expressions for the renormalized expectation values of the stress-energy tensor $\langle T_{μν} \rangle$ and the bilinear $\langle \barψψ\rangle$ in a spatially flat FLRW spacetime. These are basic ingredients in the semiclassical field equations of fermionic matter in curved spacetime interacting with a background scalar field. The ultraviolet subtracting terms of the adiabatic regularization can …
We construct an exact quantum gravitational state describing the collapse of an inhomogeneous spherical dust cloud using a lattice regularization of the Wheeler-DeWitt equation. In the semiclassical approximation around a black hole, this state describes Hawking radiation. We show that the leading quantum gravitational correction to Hawking radiation renders the spectrum non-thermal.
We discuss the status of the black hole entropy formula $S_{\rm BH} = A_H /4G$ in low energy effective field theory. The low energy expansion of the black hole entropy is studied in a non-equilibrium situation: the semiclassical decay of hot flat space by black hole nucleation. In this context the entropy can be defined as an enhancement factor in the semiclassical decay rate, which is dominated by a sphaleron-like saddle point. We find that all perturbative divergences appearing in Euclidean calculations of the entropy can be renormalized in low energy couplings. We also discuss some formal a…
Finite expressions for the mean value of the stress tensor corresponding to a scalar field with a generalized dispersion relation in a Friedman--Robertson--Walker universe are obtained using adiabatic renormalization. Formally divergent integrals are evaluated by means of dimensional regularization. The renormalization procedure is shown to be equivalent to a redefinition of the cosmological constant and the Newton constant in the semiclassical Einstein equations.
Hawking radiation is computed in different coordinate systems using the method of complex paths. In this procedure the event horizon of the 2D Schwarzschild stringy black hole is treated as a singularity for the semiclassical action functional. After the regularization of the event horizon's singularity the emission/absorption probabilities and the Hawking temperature in the different coordinate representations are derived. The identical results obtained indicate the covariance of the Hawking radiation.
We study the entropy of the black hole with torsion using the covariant form of the partition function. The regularization of infinities appearing in the semiclassical calculation is shown to be consistent with the grand canonical boundary conditions. The correct value for the black hole entropy is obtained provided the black hole manifold has two boundaries, one at infinity and one at the horizon. However, one can construct special coordinate systems, in which the entropy is effectively associated with only one of these boundaries.
In an ongoing effort to explore quantum effects on the interior geometry of black holes, we explicitly compute the semiclassical flux components $\left\langle T_{uu}\right\rangle _{ren}$ and $\left\langle T_{vv}\right\rangle _{ren}$ ($u$ and $v$ being the standard Eddington coordinates) of the renormalized stress-energy tensor for a minimally-coupled massless quantum scalar field, in the vicinity of the inner horizon (IH) of a Reissner-Nordström black hole. These two flux components seem to dominate the effect of backreaction in the IH vicinity; and furthermore, their regularization procedure …
The emergence of nontrivial symmetries for black holes minisuperspaces has been recently pointed out. These Noether symmetries possess non-null charges and hence map physical solutions to different ones. The symmetry group is isomorphic to the finite-dimensional Poincaré group ISO(2,1), whose irreducible representations are well known. This structure is used to build a consistent quantum theory of black hole minisuperspace. This has, among other consequences, the striking consequence of implying a continuous spectrum for the mass operator. Following loop quantum cosmology, we obtain a regulari…
Following a previous work on the quantization of a massless scalar field in a spacetime representing the head on collision of two plane waves which fucus into a Killing-Cauchy horizon, we compute the renormalized expectation value of the stress-energy tensor of the quantum field near that horizon in the physical state which corresponds to the Minkowski vacuum before the collision of the waves. It is found that for minimally coupled and conformally coupled scalar fields the respective stress-energy tensors are unbounded in the horizon. The specific form of the divergences suggests that when the…
A family of spacetimes suitable for describing the matter conditions of a static, spherically symmetric quantum vacuum is studied, as well as its reliability for describing a regular model for the interior of a semiclassical, static black hole ---without ever invoking a mass shell for the final object. In paper I, this condition was seen to limit the search to only one, distinguished family, that was investigated in detail. Here it will be proven that, aside from being mathematically generic (in its uniqueness), this family exhibits beautiful physical properties, that one would reasonably dema…
We employ a recently developed mode-sum regularization method to compute the renormalized stress-energy tensor of a quantum field in the Kerr background metric (describing a stationary spinning black hole). More specifically, we consider a minimally-coupled massless scalar field in the Unruh vacuum state, the quantum state corresponding to an evaporating black hole. The computation is done here for the case $a=0.7M$, using two different variants of the method: $t$-splitting and $φ$-splitting, yielding good agreement between the two (in the domain where both are applicable). We briefly discuss …
This dissertation presents a semiclassical analysis of conical topology change in $1+1$ spacetime dimensions wherein, to lowest order, the ambient spacetime is classical and fixed while the scalar field coupled to it is quantized. The vacuum expectation value of the scalar field stress-energy tensor is calculated via two different approaches. The first of these involves the explicit determination of the so called Sorkin-Johnston state on the cone and an original regularization scheme, while the latter employs the conformal vacuum and the more conventional point-splitting renormalization. It is…
We propose a new model of the spherical symmetric quantum black hole in the reduced phase space formulation. We deparametrize gravity by coupling to the Gaussian dust which provides the material coordinates. The foliation by dust coordinates covers both the interior and exterior of the black hole. After the spherical symmetry reduction, our model is a 1+1 dimensional field theory containing infinitely many degrees of freedom. The effective dynamics of the quantum black hole is generated by an improved physical Hamiltonian ${\bf H}_Δ$ where the holonomy correction is implemented by the $\barμ$-…
Gravitational wave `echoes' during black-hole merging events have been advocated as possible signals of modifications to gravity in the strong-field (but semiclassical) regime. In these proposals the observable effect comes entirely from the appearance of nonzero reflection probability at the horizon, which vanishes for a standard black hole. We show how to apply EFT reasoning to these arguments, using and extending earlier work for localized systems that relates choices of boundary condition to the action for the physics responsible for these boundary conditions. EFT reasoning applied to this…
