A correspondence between fluctuations of non-minimally coupled scalar fields and that of an effective fluid with heat flux and anisotropic stresses, is shown. Though the correspondence between respective stress tensors of scalar fields and fluids is known and widely used in literature, the fluctuations in the two cases still await a formal correspondence and are open to investigation in all details. Using results obtained in the newly established theory of semiclassical stochastic gravity which focuses on the fluctuations of the quantum stress tensor, we show new relations in this regard. This development is expected to give insight to the mesoscopic phenomena for gravitating systems, and enable backreaction studies of the fluctuations on the perturbations of astrophysical objects. Such a development is aimed to enhance the perturbative analysis for cosmological spacetimes and astrophysical objects specifically in the decoherence limit. A kinetic theory, which can be based on stochastic fluctuations vs particle picture in curved spacetime may find useful insights from such correspondences in future work.
I briefly review some of the recent progress in quantum field theory in curved spacetime and other aspects of semiclassical gravity, as reported at the D3 Workshop at GR15.
A correspondence between fluctuations of conformally invariant quantum fields and that of classical fields finally reducing to perfect fluid matter content is shown to exist. Previously a similar correspondence between the stress tensors was known and well established in the 70's. Using recent results obtained in semiclassical stochastic gravity regarding exact definition and significance given to quantum fluctuations of the stress tensor, we obtain this correspondence, which is fundamental to statistical analysis of systems in curved spacetime. This is of immense importance, in that the fluct…
We present a method to semiclassically compute the pair creation rate of bosons and fermions in de Sitter spacetime. The results in the bosonic case agree with the ones in the literature. We find that for the constant electric field the fermionic and bosonic pair creation rate are the same. This analogy of bosons and fermions in the semiclassical limit is known from several flat spacetime examples.
de Sitter spacetime and Bunch-Davies vacuum are a solution to the semiclassical Einstein-Schroedinger equations describing the evolution of spacetime geometry and a massive scalar quantum field with arbitrary coupling to curvature. The stability of this solution is proven by calculating the renormalized energy momentum tensor expectation value for small spatially homogeneous deviations from the de Sitter - Bunch-Davies system and solving the linearized backreaction problem. A renormalization scheme is developed. All momentum integrations are carried out analytically. The general solution is gi…
A spatially flat Robertson-Walker spacetime driven by a cosmological constant is non-conformally coupled to a massless scalar field. The equations of semiclassical gravity are explicitly solved for this case, and a self-consistent de Sitter solution associated with the Bunch-Davies vacuum state is found (the effect of the quantum field is to shift slightly the effective cosmological constant). Furthermore, it is shown that the corrected de Sitter spacetime is stable under spatially-isotropic perturbations of the metric and the quantum state. These results are independent of the free renormaliz…
The thermodynamical description of black holes is reviewed and critiqued. We present an alternative, microcanonical description of black holes and discuss the major differences. In particular the decay rates of black holes are compared in the two different pictures.
We derive the functional Schrodinger equation for quantum fields in curved spacetime in the semiclassical limit of quantum geometrodynamics with a Gaussian incoherent dust acting as a clock field. We perform the semiclassical limit using a WKB-type expansion of the wave functional in powers of the squared Planck mass. The functional Schrodinger equation that we obtain exhibits a functional time derivative that completes the usual definition of WKB time for curved spacetime, and the usual Schrodinger-type evolution is recovered in Minkowski spacetime.
This article will summarize selected aspects of the semiclassical theory of gravity, which involves a classical gravitational field coupled to quantum matter fields. Among the issues which will be discussed are the role of quantum effects in black hole physics and in cosmology, the effects of quantum violations of the classical energy conditions, and inequalities which constrain the extent of such violations. We will also examine the first steps beyond semiclassical gravity, when the effects of spacetime geometry fluctuations start to appear.
The semiclassical Einstein-Langevin equations which describe the dynamics of stochastic perturbations of the metric induced by quantum stress-energy fluctuations of matter fields in a given state are considered on the background of the ground state of semiclassical gravity, namely, Minkowski spacetime and a scalar field in its vacuum state. The relevant equations are explicitly derived for massless and massive fields arbitrarily coupled to the curvature. In doing so, some semiclassical results, such as the expectation value of the stress-energy tensor to linear order in the metric perturbation…
We consider the linearized semiclassical Einstein equations for small deviations around de Sitter spacetime including the vacuum polarization effects of conformal fields. Employing the method of order reduction, we find the exact solutions for general metric perturbations (of scalar, vector and tensor type). Our exact (nonperturbative) solutions show clearly that in this case de Sitter is stable with respect to small metric deviations and a late-time attractor. Furthermore, they also reveal a breakdown of perturbative solutions for a sufficiently long evolution inside the horizon. Our results …
We prove that semiclassical gravity in conformally static, globally hyperbolic spacetimes with a massless, conformally coupled Klein-Gordon field is well posed, when viewed as a coupled theory for the dynamical conformal factor of the metric and the Klein-Gordon theory. Namely, it admits unique and stable solutions whenever constrained fourth-order initial data for the conformal factor and suitably defined Hadamard initial data for the Klein-Gordon state are provided on a spacelike Cauchy surface. As no spacetime symmetries are imposed on the conformal factor, the present result implies that, …
The semiclassical gravity describes gravitational back-reactions of the classical spacetime interacting with quantum matter fields but the quantum effects on the background is formally defined as higher derivative curvatures. These induce catastrophic instabilities and classic solutions become unstable under small perturbations or their evolutions. In this paper we discuss validity of the semiclassical gravity from the perspective of the spacetime instabilities and consider cosmological dynamics of the Universe in this theory. We clearly show that the homogenous and isotropic flat Universe is …
We study the semiclassical Einstein field equations with a Klein-Gordon field in ultrastatic and static spacetimes. In both cases, the equations for the spacetime metric become constraint equations. In the ultrastatic case, the Hadamard singular structure can be characterised explicitly, which allows one to in principle give initial data for the Wightman function that has correct distributional singularities, such that the expectation value of the renormalised stress-energy tensor of the solution can be defined, and that hence the semiclassical Einstein equations make sense. Assuming a "positi…
We study the solutions of the semiclassical Einstein equation in flat cosmological spacetimes driven by a massive conformally coupled scalar field. In particular, we show that it is possible to give initial conditions at finite time to get a state for the quantum field which gives finite expectation values for the stress-energy tensor. Furthermore, it is possible to control this expectation value by means of a global estimate on regular cosmological spacetimes. The obtained estimates permit to write a theorem about the existence and uniqueness of the local solutions encompassing both the space…
We analyze the gravitational radiation emitted from a particle in circular motion around a Schwarzschild black hole using the framework of quantum field theory in curved spacetime at tree level. The gravitational perturbations are written in a gauge-invariant formalism for spherically symmetric spacetimes. We discuss the results, comparing them to the radiation emitted by a particle when it is assumed to be orbiting a massive object due to a Newtonian force in flat spacetime.
We consider the semiclassical Einstein equations (SEE) in the presence of a quantum scalar field with self-interaction $λϕ^4$. Working in the Hartree truncation of the two-particle irreducible (2PI) effective action, we compute the vacuum expectation value of the energy-momentum tensor of the scalar field, which act as a source of the SEE. We obtain the renormalized SEE by implementing a consistent renormalization procedure. We apply our results to find self-consistent de Sitter solutions to the SEE in situations with or without spontaneous breaking of the $Z_2$-symmetry.
The theory of gravitational lensing has revealed many generic and fundamental properties of compact objects like black holes and wormholes. In this article, we utilize a recent formulation to compute the quantum effects on the deflection angle of a light ray, namely, the Gauss-Bonnet theorem (GBT) to explore the semiclassical gravitational effects in the spacetime of a point-like global monopole and a cosmic string. Previously, the Gauss-Bonnet theorem [Class. Quant. Grav. 25, 235009 (2008)] was proposed as an alternative way to compute the deflection angle of light in a static, spherically sy…
We present a semiclassical analysis for Dirac fields on an arbitrary spacetime background and in the presence of a fixed electromagnetic field. Our approach is based on a Wentzel-Kramers-Brillouin approximation, and the results are analyzed at leading and next-to-leading order in the small expansion parameter $\hbar$. Taking into account the spin-orbit coupling between the internal and external degrees of freedom of wave packets, we derive effective ray equations with spin-dependent terms. These equations describe the gravitational spin Hall effect of localized Dirac wave packets. We treat bot…
It is well-known that some physical effects may arise in the spacetime of a straigth cosmic string due to its global conic properties. Among these effects, the vacuum polarization effect has been extensively studied in the litterature. In papers of reference [4] a more general situation has been considered in which the cosmic string carries a magnetic flux $Φ$ and interacts with a charged scalar field. In this case, the vacuum polarization arises both via non-trivial gravitational interaction (i.e, the conical structure) and via Aharonov-Bohm interaction. In papers [4] the non vanishing VEV of…
We present a new solution in Einstein-Maxwell theory which can be considered as the magnetized version of Kerr-Taub-NUT solution. Some properties of the spacetime are discussed. We also compute the entropy of extremal black hole in the spacetime by using the Kerr/CFT correspondence approach.
Ashtekar's formulation for canonical quantum gravity is known to possess the topological solutions which have their supports only on the moduli space $\CN$ of flat $SL(2,C)$ connections. We show that each point on the moduli space $\CN$ corresponds to a geometric structure, or more precisely the Lorentz group part of a family of Lorentzian structures, on the flat (3+1)-dimensional spacetime. A detailed analysis is given in the case where the spacetime is homeomorphic to $R\times T^{3}$. Most of the points on the moduli space $\CN$ yield pathological spacetimes which suffers from singularities …
Improved semiclassical techniques are developed and applied to a treatment of a real scalar field in a $D$-dimensional gravitational background. This analysis, leading to a derivation of the thermodynamics of black holes, is based on the simultaneous use of: (i) a near-horizon description of the scalar field in terms of conformal quantum mechanics; (ii) a novel generalized WKB framework; and (iii) curved-spacetime phase-space methods. In addition, this improved semiclassical approach is shown to be asymptotically exact in the presence of hierarchical expansions of a near-horizon type. Most imp…
The subject of this thesis is the coupling of quantum fields to a classical gravitational background in a semiclassical fashion. It contains a thorough introduction into quantum field theory on curved spacetime with a focus on the stress-energy tensor and the semiclassical Einstein equation. Basic notions of differential geometry, topology, functional and microlocal analysis, causality and general relativity will be summarised, and the algebraic approach to QFT on curved spacetime will be reviewed. Apart from these foundations, the original research of the author and his collaborators will be …
The vacuum polarization of the quantized scalar field with non-conformal coupling $ξ$ on the long throat of a wormhole background is calculated. It is shown that the stress-energy tensor of vacuum fluctuations in considered spacetime is determined by the local geometry of spacetime only. The self-consistent solutions of the semiclassical Einstein field equations describing a long throat of a traversable wormhole are obtained.
We present a coherent picture of the quantum mechanics of black holes. The picture does not require the introduction of any drastically new physical effect beyond what is already known; it arises mostly from synthesizing and (re)interpreting existing results in appropriate manners. We identify the Bekenstein-Hawking entropy as the entropy associated with coarse-graining performed to obtain semiclassical field theory from a fundamental microscopic theory of quantum gravity. This clarifies the issues around the unitary evolution, the existence of the interior spacetime, and the thermodynamic nat…
Sonic point/photon sphere (SP/PS) correspondence is a theoretical phenomenon which appears in fluid dynamics on curved spacetime and its existence has been recently proved in quite wide situations as theorems. The theorems state that a sonic point (SP) of radiation fluid flow must be on an unstable photon sphere (PS) when the fluid flows radially or rotationally on an equatorial plane in spherically symmetric spacetime of arbitrary dimensions. In this paper, we investigate SP/PS correspondence in spherically, planarly and hyperbolically symmetric spacetime. As the corresponding objects of phot…
The first and second laws of black hole thermodynamics are verified to emerge from a generic semiclassical theory of gravity for which a Hamiltonian can be defined. The first law is established for stationary spacetimes, and the second law is established in the here defined "piecewise stationary" spacetimes. Black hole entropy is defined in the Noether charge approach, and the entropy for the matter in its exterior is the von Neumann entropy of the quantum fields. These results strongly support the view of black hole entropy as an account of the information that is hidden behind the horizon.
In the semiclassical quantum gravity derived from the Wheeler-DeWitt equation, the energy density of a matter field loses quantum coherence due to the induced gauge potential from the parametric interaction with gravity in a non-static spacetime. It is further shown that the energy density takes only positive values and makes superposition principle hold true. By studying a minimal massive scalar field in a FRW spacetime background, we illustrate the positivity of energy density and obtain the classical Hamiltonian of a complex field from the energy density in coherent states.
Bargmann-Wigner equations and their solutions are studied in (3+1)-dimensional curved spacetime. Fermion-Boson correspondence for bi-spinor case is studied through the Bargmann-Wigner equations and solutions over curved spacetime. As an application to scattering phenomena of massive Fermions and Bosons on the rotating black holes, the superradiance with negative energy $(ω<0)$ and positive effective energy in co-rotating coordinate system near horizon $(ω-mΩ_{H}>0)$ is possible to occur as stable physical states in Kerr spacetime.
Semiclassical gravity is investigated in a large class of asymptotically flat, static, spherically symmetric spacetimes including those containing static stars, black holes, and wormholes. Specifically the stress-energy tensors of massless free spin 0 and spin 1/2 fields are computed to leading order in the asymptotic regions of these spacetimes. This is done for spin 0 fields in Schwarzschild spacetime using a WKB approximation. It is done numerically for the spin 1/2 field in Schwarzschild, extreme Reissner-Nordstrom, and various wormhole spacetimes. And it is done by finding analytic soluti…
We argue that classical spacetimes represent amplified information in the holographic theory of quantum gravity. In general, classicalization of a quantum system involves amplification of information at the cost of exponentially reducing the number of observables. In quantum gravity, the geometry of spacetime must be the analogously amplified information. Bulk local semiclassical operators probe this information without disturbing it; these correspond to logical operators acting on code subspaces of the holographic theory. From this viewpoint, we study how bulk local operators may be realized …
It has been proposed recently to consider in the framework of cosmology an extension of the semiclassical Einstein's equations in which the Einstein tensor is considered as a random function. This paradigm yields a hierarchy of equations between the $n$-point functions of the quantum, normal ordered, stress energy-tensor and those associated to the stochastic Einstein tensor. Assuming that the matter content is a conformally coupled massive scalar field on de Sitter spacetime, this framework has been applied to compute the power spectrum of the quantum fluctuations and to show that it is almos…
Motivated by the understanding of holography as realized in tensor networks, we develop a bulk procedure that can be interpreted as generating a sequence of coarse-grained holographic states. The coarse-graining procedure involves identifying degrees of freedom entangled at short distances and disentangling them. This is manifested in the bulk by a flow equation that generates a codimension-1 object, which we refer to as the holographic slice. We generalize the earlier classical construction to include bulk quantum corrections, which naturally involves the generalized entropy as a measure of t…
We study the Hilbert space structure of classical spacetimes under the assumption that entanglement in holographic theories determines semiclassical geometry. We show that this simple assumption has profound implications; for example, a superposition of classical spacetimes may lead to another classical spacetime. Despite its unconventional nature, this picture admits the standard interpretation of superpositions of well-defined semiclassical spacetimes in the limit that the number of holographic degrees of freedom becomes large. We illustrate these ideas using a model for the holographic theo…
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 …
In this paper we study the correspondence between the real part of quasinormal modes (QNMs) and the shadow radius in a wormhole spacetime. Firstly we consider the above correspondence in a static and spherically symmetric wormhole spacetime and then explore this correspondence numerically by considering different wormhole models having specific redshift functions. To this end, we generalize this correspondence to the rotation wormhole spacetime and calculate the typical shadow radius of the rotating wormhole when viewed from the equatorial plane. We argue that due to the rotation and depending…
Motivated by the quest for black holes in AdS braneworlds, and in particular by the holographic conjecture relating 5D classical bulk solutions with 4D quantum corrected ones, we numerically solve the semiclassical Einstein equations (backreaction equations) with matter fields in the (zero temperature) Boulware vacuum state. In the absence of an exact analytical expression for <T_(mu nu)> in four dimensions we work within the s-wave approximation. Our results show that the quantum corrected solution is very similar to Schwarzschild till very close to the horizon, but then a bouncing surface fo…
We study the internal structure of a two-dimensional dilatonic evaporating black hole, based on the CGHS model. At the semiclassical level, a (weak) spacelike singularity was previously found to develop inside the black hole. We employ here a simplified quantum formulation of spacetime dynamics in the neighborhood of this singularity, using a minisuperspace-like approach. Quantum evolution is found to be regular and well-defined at the semiclassical singularity. A well-localized initial wave-packet propagating towards the singularity bounces off the latter and retains its well-localized form. …
The phenomenon of quantum vacuum polarization in the presence of a gravitational field is well understood and is expected to have a physical reality, but studies of its back-reaction on the dynamics of spacetime are practically non-existent outside the specific context of homogeneous cosmologies. Building on previous results of quantum field theory in curved spacetimes, in this letter we first derive the semiclassical equations of stellar equilibrium in the s-wave Polyakov approximation. It is highlighted that incorporating the polarization of the quantum vacuum leads to a generalization of th…
We elaborate the recently introduced asymptotically exact semiclassical quantum gravity derived from the Wheeler-DeWitt equation by finding a particular coherent state representation of a quantum scalar field in which the back-reaction of the scalar field Hamiltonian exactly gives rise to the classical one. In this coherent state representation classical spacetime emerges naturally from semiclassical quantum gravity.
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…
In any quantum theory of gravity, it is of the utmost importance to recover the limit of quantum theory in an external spacetime. In quantum geometrodynamics (quantization of general relativity in the Schrödinger picture), this leads in particular to the recovery of a semiclassical (WKB) time which governs the dynamics of non-gravitational fields in spacetime. Here, we first review this procedure with special emphasis on conceptual issues. We then turn to an alternative theory - Weyl (conformal) gravity, which is defined by a Lagrangian that is proportional to the square of the Weyl tensor. We…
We consider the classical stochastic fluctuations of spacetime geometry induced by quantum fluctuations of massless non-conformal matter fields in the Early Universe. To this end, we supplement the stress-energy tensor of these fields with a stochastic part, which is computed along the lines of the Feynman-Vernon and Schwinger-Keldysh techniques; the Einstein equation is therefore upgraded to a so called Einstein-Langevin equation. We consider in some detail the conformal fluctuations of flat spacetime and the fluctuations of the scale factor in a simple cosmological modelintroduced by Hartle,…
We compute the Renormalized Stress-Energy Tensor (RSET) of a massless minimally coupled scalar field in the Regularized Polyakov approximation, as well as its backreaction, on the classical Reissner-Nordström spacetime. The complete set of solutions of the semiclassical self-consistent equations is obtained and compared with the classical counterparts. The semiclassical Reissner-Nordström family involves three kinds of geometries that depend on the charge-to-mass ratio of the spacetime. In the under-charged regime, the geometry has its external horizon replaced by a wormhole neck that leads to…
We derive the gravitational field and the spacetime metric generated by sources in quantum superposition of different locations. We start by working in a Newtonian approximation, in which the effective gravitational potential is computed as the expectation value of the gravitational potential operator in a Gaussian distribution of width $R$ for the position of the source. The effective gravitational potential is then covariantly uplifted to a fully relativistic metric in general relativity, describing the spacetime generated by averaging over the state of such sources. These results are then r…
This paper is a contribution to the development of a framework, to be used in the context of semiclassical canonical quantum gravity, in which to frame questions about the correspondence between discrete spacetime structures at "quantum scales" and continuum, classical geometries at large scales. Such a correspondence can be meaningfully established when one has a "semiclassical" state in the underlying quantum gravity theory, and the uncertainties in the correspondence arise both from quantum fluctuations in this state and from the kinematical procedure of matching a smooth geometry to a disc…
An algebraic quantization procedure for discretized spacetime models is suggested based on the duality between finitary substitutes and their incidence algebras. The provided limiting procedure that yields conventional manifold characteristics of spacetime structures is interpreted in the algebraic quantum framework as a correspondence principle.
It is shown that the recent corpuscular description of gravity generically excludes de Sitter spacetime in any semiclassical version of f(R) gravity. A phantom phenomenology of the cosmic dynamics is also naturally excluded.
We study the fermionic Schwinger effect in two dimensional de Sitter spacetime. To do so we first present a method to semiclassically compute the number of pairs created per momentum mode for general time dependent fields. In addition the constant electric field is studied in depth. In this case solutions for the Dirac equation can be found and the number of pairs can be computed using the standard Bogoliubov method. This result is shown to agree with the semiclassical one in the appropriate limit. The solutions are also used to compute the expectation value of the induced current. Comparing t…
The geometric optics approximation provides an interpretation for eikonal correspondence that, in black-hole-containing spacetimes, connects high-frequency black hole quasinormal modes with closed photon orbits around said black hole. This correspondence has been identified explicitly for Schwarzschild, Reissner-Nordström, Kerr, and Kerr-Newman black holes, the violation of which can be a potential hint toward physics beyond General Relativity. Notably, the aforementioned black hole spacetimes have sufficient symmetries such that both the geodesic equations and the master wave equations are se…
We consider the holographic principle, in its lightsheet formulation, in the semiclassical context of statistical-mechanical systems in classical Einstein spacetimes. A local condition, in terms of entropy and energy local densities of the material medium under consideration, is discussed, which turns out to be necessary and sufficient for the validity of the closely-related generalized covariant entropy bound. This condition is apparently a general consequence or expression of flat-spacetime quantum mechanics alone, without any reference to gravity. Using it, a lower bound η/s >= 1/4πcan be d…
