Semiclassical Einstein Equations Research Articles

Self-gravity and Bekenstein-Hawking entropy

We study the effect of self-gravity on entropy by directly solving the 4D semi-classical Einstein equation. In particular, we focus on whether the Bekenstein-Hawking formula holds when self-gravity is extremely strong. As an example, we consider a simple spherically symmetric static configuration consisting of many quanta and construct a self-consistent non-perturbative solution for ħ in which the entropy exactly follows the area law for many local degrees of freedom of any kind. This can be a candidate for black holes in quantum theory. It represents a compact dense configuration with near-Planckian curvatures, and the interior typically behaves like a local thermal state due to particle creation inside. Here, the information content is stored in the interior bulk, and the self-gravity plays an essential role in changing the entropy from the volume law to the area law. We finally discuss implications to black holes in quantum gravity and a speculative view of entropy as a gravitational charge.

Infinite Quantum Twisting at the Cauchy Horizon of Rotating Black Holes.

We present a numerical calculation of the expectation value of the quantum angular-momentum current flux density for a scalar field in the Unruh state near the inner horizon of a Kerr-de Sitter black hole. Our results indicate that this flux diverges as V_{-}^{-1} in a suitable Kruskal coordinate such that V_{-}=0 at the inner horizon. Depending on the parameter values of the scalar field and black hole that we consider, and depending on the polar angle (latitude), this flux can have different signs. In the near extremal cases considered, the angle average of the expectation value of the quantum angular momentum current flux is of the opposite sign as the angular momentum of the background itself, suggesting that, in the cases considered, quantum effects tend to decrease the total angular momentum of the spheres away from the extremal value. We also numerically calculate the energy flux component, which provides the leading order divergence of the quantum stress energy tensor, dominant over the classical stress energy tensor, at the inner horizon. Taking our expectation value of the quantum stress tensor as the source in the semiclassical Einstein equation, our analysis suggests that the spheres approaching the inner horizon can undergo an infinite twisting due to quantum effects along latitudes separating regions of infinite expansion and contraction.

Cosmological de Sitter Solutions of the Semiclassical Einstein Equation

Exponentially expanding space–times play a central role in contemporary cosmology, most importantly in the theory of inflation and in the dark energy driven expansion in the late universe. In this work, we give a complete list of de Sitter solutions of the semiclassical Einstein equation (SCE), where classical gravity is coupled to the expected value of a renormalized stress–energy tensor of a free quantum field in the Bunch–Davies state. To achieve this, we explicitly determine the stress–energy tensor associated with the Bunch–Davies state using the recently proposed “moment approach” on the cosmological coordinate patch of de Sitter space. From the energy component of the SCE, we thus obtain an analytic consistency equation for the model’s parameters which has to be fulfilled by solutions to the SCE. Using this equation, we then investigate the number of solutions and the structure of the solution set in dependency on the coupling parameter of the quantum field to the scalar curvature and renormalization constants using analytic arguments in combination with numerical evidence. We also identify parameter sets where multiple expansion rates separated by several orders of magnitude are possible. Potentially for such parameter settings, a fast (semi-stable) expansion in the early universe could be compatible with a late-time “Dark Energy-like” behavior of the universe.

Quantum corrections to the Schwarzschild metric from vacuum polarization

We explore static and spherically symmetric solutions of the 4-dimensional semiclassical Einstein's equations using the quantum vacuum polarization of a conformal field as a source. These solutions may be of interest for the study of exotic compact objects (ECOs). The full backreaction problem is addressed by solving the semiclassical Tolman-Oppenheimer-Volkoff (TOV) equations making use of effective equations of state inspired by the trace anomaly and an extra simplifying and reasonable assumption. We combine analytical and numerical techniques to solve the resulting differential equations, both perturbatively and nonperturbatively in $\ensuremath{\hbar}$. In all cases the solution is similar to the Schwarzschild metric up to the vicinity of the classical horizon $r=2M$. However, at $r=2M+ϵ$, with $ϵ\ensuremath{\sim}O(\sqrt{\ensuremath{\hbar}})$, we find a coordinate singularity. In the case of matching with a static star, this leads to an upper bound in the compactness, and sets a constraint on the family of stable ECOs. We also study the corrections that the quantum-vacuum polarization induces on the propagation of waves, and discuss the implications. For the pure vacuum case, we can further extend the solution by using appropriate coordinates until we reach another singular point, where this time a null curvature singularity arises and prevents extending beyond. This picture qualitatively agrees with the results obtained in the effective two-dimensional approach, and reinforces the latter as a reasonable method.

Quantum backreaction for overspinning BTZ geometries

We examine the semiclassical backreaction of a conformally coupled scalar field on an overspinning BTZ geometry. This extends the work done on a similar problem for ($2+1$)- AdS geometries of the BTZ family with $|M|>|J|$. The overspinning classical solutions corresponds to $|M|<|J|$ and possess a naked singularity at $r=0$. Using the renormalized quantum stress-energy tensor for a conformally coupled scalar field on such a spacetime, we obtain the semiclassical Einstein equations, which we attempt to solve perturbatively. We show that the stress-energy tensor is non-renormalizable in this approach, and consequently the perturbative solution to the semiclassical equations in the overspinning case does not exist. This could be an indication of the fact that the naked singularity at the center of an overspinning geometry is of a more severe nature than the conical singularity found in the same family of BTZ geometries.

Asymptotically flat vacuum solutions in order-reduced semiclassical gravity

We investigate the effects of quantum backreaction on the Schwarzschild geometry in the semiclassical approximation. The renormalized stress-energy tensor (RSET) of a scalar field is modelled via an order reduction of the analytical approximation derived by Anderson, Hiscock and Samuel (AHS). As the resulting AHS semiclassical Einstein equations are of fourth-derivative order in the metric, we follow a reduction of order prescription to shrink the space of solutions. Motivated by this prescription, we develop a method that allows to obtain a novel analytic approximation for the RSET that exhibits all the desired properties for a well-posed RSET: conservation, regularity, and correct estimation of vacuum-state contributions. We derive a set of semiclassical equations sourced by the order-reduced AHS-RSET in the Boulware state. We classify the self-consistent solutions to this set of field equations, discuss their main features and address how well they resemble the solutions of the higher-order semiclassical theory. Finally, we establish a comparison with previous results in the literature obtained through the Polyakov approximation for minimally coupled scalar fields.

Hadamard states on spherically symmetric characteristic surfaces, the semi-classical Einstein equations and the Hawking effect

We investigate quasi-free Hadamard states defined via characteristic initial data on null cones centred at the axis of symmetry in spherically symmetric space-times. We characterize the necessary singular behaviour of null boundary two-point functions such that one can define non-linear observables at this null boundary and give formulas for the calculation of these observables. These results extend earlier characterizations of null boundary states defining Hadamard states in the bulk of the null cone. As an application of our derived formulas, we consider their implications for the semi-classical Einstein equations and calculate the vacuum polarization associated with Hawking radiation near a collapsing body.

On the initial value problem for semiclassical gravity without and with quantum state collapses

Semiclassical gravity is the theory in which the classical Einstein tensor of a spacetime is coupled to quantum matter fields propagating on the spacetime via the expectation value of their renormalized stress-energy tensor in a quantum state. We explore two issues, taking the Klein Gordon equation as our model quantum field theory. The first is the provision of a suitable initial value formulation for the theory. Towards this, we address the question, for given initial data consisting of the classical metric and its first three 'time' derivatives off the surface together with a choice of initial quantum state, of what is an appropriate 'surface Hadamard' condition such that, for initial data for which it is satisfied it is reasonable to conjecture that there will be a Cauchy development whose quantum state is Hadamard. This requires dealing with the fact that, given two points on an initial surface, the spacetime geodesic between them does not, in general, lie on that surface. So the (squared) geodesic distance that occurs in the Hadamard subtraction differs from that intrinsic to the initial surface. We handle this complication by expanding the former as a suitable 3-dimensional covariant Taylor expansion in the latter. Moreover the renormalized expectation value of the stress-energy tensor in the initial surface depends explicitly on the fourth, 'time', derivative of the metric, which is not part of the initial data, but which we argue is given, implicitly, by the semiclassical Einstein equations on the initial surface. (The rôle played by those equations also entails that the surface Hadamard condition subsumes the constraints.) We also introduce the notion of physical solutions, which, inspired by a 1993 proposal of Parker and Simon, we define to be solutions which are smooth in ħ at ħ = 0. We conjecture that for these solutions the second and third time derivatives of the metric will be determined once the first and second time derivatives are specified. We point out that a simpler treatment of the initial value problem can be had if we adopt yet more of the spirit of Parker and Simon and content ourselves with solutions to order ħ which are Hadamard to order ħ. A further simplification occurs if we consider semiclassical gravity to order ħ 0. This resembles classical general relativity in that it is free from the complications of higher derivative terms and does not require any Hadamard condition. But it can still incorporate nontrivial quantum features such as superpositions of classical-like quantum states of the matter fields. Our second issue concerns the prospects for combining semiclassical gravity with theories of spontaneous quantum state collapse. We will focus our attention on proposals involving abrupt changes in the quantum field state which occur on certain (random, non-intersecting) Cauchy surfaces according to some — yet to be developed — generally covariant objective collapse model but that, in between such collapse surfaces, we have a physical semiclassical solution (or a solution of order O(ħ) or a solution of order O(ħ 0)). On each collapse surface, the semiclassical gravity equations will necessarily be violated and, as Page and Geilker pointed out in 1981, there will therefore necessarily be a discontinuity in the expectation value of the renormalized stress-energy tensor. Nevertheless, we argue, based on our conjecture about the well-posedness of the initial value problem for physical solutions, that, with a suitable rule for the jump in the metric and/or the extrinsic curvature, the time evolution will still be uniquely determined. We tentatively argue that a natural jump rule would be one in which the metric itself and the transverse traceless part of the extrinsic curvature will be continuous and the jump will be confined to the remaining parts of the extrinsic curvature. We aid and complement our discussion by studying our two issues also in the simpler cases of a semiclassical scalars model and semiclassical electrodynamics.

Squeezed Number State Representation of the Inflaton and Particle Production in the FRW Universe

In this study, we use a single-mode squeezed thermal vacuum state formalism and examine the nature of a massive homogeneous scalar field, minimally coupled to the gravity in the framework of semiclassical gravity in the Bianchi type-I universe. We have obtained an estimate leading solution to the semiclassical Einstein equation for the Bianchi type-I universe shows, each scale factor in its respective direction obeys t^(2/3) power-law expansion. The mechanism of the nonclassical thermal cosmological particle production is also analyzed in the Bianchi type-I universe.

Linear Stability of Semiclassical Theories of Gravity

The linearization of semiclassical theories of gravity is investigated in a toy model, consisting of a quantum scalar field in interaction with a second classical scalar field which plays the role of a classical background. This toy model mimics also the evolution induced by semiclassical Einstein equations, such as the one which describes the early universe in the cosmological case. The equations governing the dynamics of linear perturbations around simple exact solutions of this toy model are analyzed by constructing the corresponding retarded fundamental solutions, and by discussing the corresponding initial value problem. It is shown that, if the quantum field which drives the back-reaction to the classical background is massive, then there are choices of the renormalization parameters for which the linear perturbations with compact spatial support decay polynomially in time for large times, thus indicating stability of the underlying semiclassical solution.

Local solutions of general relativity in the presence of the trace anomaly

Local solutions are a vivid and long-living topic in gravitational physics. We consider exact static pseudo-spherically symmetric solutions of semi-classical Einstein's equations in presence of the trace anomaly contribution. We investigate black hole solutions and propose new metrics describing traversable wormholes. Thanks to the trace anomaly, wormholes are realized in the vacuum and, nevertheless, violate the null energy condition.

Inaccessibility of traversable wormholes

Wormhole solutions to the equations of general relativity have some spectacular local and global properties. As these unusual features are not explicitly forbidden by known physics, wormholes are considered in various astrophysical and cosmological scenarios. The paradigmatic traversable wormhole models are described by static spherically-symmetric Ellis--Morris--Thorne and Simpson--Visser metrics. We show that no dynamical solution of the semiclassical Einstein equations can have these metrics as their static limit. On the other hand, possible static limits of the dynamical solutions are not traversable. Moreover, they lead to violation of a quantum energy inequality that bounds violations of the null energy condition by quantum fields. This conclusion does not depend on specific properties of fields that may be proposed for wormhole construction. As a result, spherically-symmetric wormholes cannot exist in semiclassical gravity.

Special cosmological models derived from the semiclassical Einstein equation on flat FLRW space-times

This article presents numerical work on a special case of the cosmological semiclassical Einstein equation (SCE). The SCE describes the interaction of relativistic quantum matter by the expected value of the renormalized stress–energy tensor of a quantum field with classical gravity. Here, we consider a free, massless scalar field with general (not necessarily conformal) coupling to curvature. In a cosmological scenario with flat spatial sections for special choices of the initial conditions, we observe a separation of the dynamics of the quantum degrees of freedom from the dynamics of the scale factor, which extends a classical result by Starobinski (1980 Phys. Lett. B 91 99–102) to general coupling. For this new equation of fourth order governing the dynamics of the scale factor, we study numerical solutions. Typical solutions show a radiation-like Big Bang for the early Universe and de Sitter-like expansion for the late Universe. We discuss a specific solution to the cosmological horizon problem that can be produced by tuning parameters in the given equation. Although the model proposed here only contains massless matter, we give a preliminary comparison of the obtained cosmology with the ΛCDM standard model of cosmology and investigate parameter ranges in which the new models, to a certain extent, is capable of assimilating standard cosmology.

Validity of semiclassical limit to quantum gravity in two-mode oscillating quantized massive scalar field quantum cosmology

Semiclassical Einstein equations are used to describe the interaction of the back-reaction of the classical gravitational field with quantum matter fields in semiclassical gravity. We in our previous studies have made use of the semiclassical approximation to demonstrate the phenomenon of particle production, often called preheating/reheating of the universe, which occurs after the inflationary epoch during the oscillatory phase of two-mode quantized scalar field of chaotic inflationary model. During this oscillatory phase, back-reaction effects from the created particles, on account of the quantum nature of the states considered, could be significant and one might be concerned about the validity of the semiclassical approximation in these two-mode quantum optical states. The validity of the semiclassical approximation in these states is examined and it is presented how the magnitude of states parameter draws limit on the applicability and reliability of semiclassical theory of gravity. It is argued that semiclassical theory to gravity is a good approximation for states which are closer to coherent states i.e., with coherent parameters greater than unity and with squeezed parameter much smaller than unity.

Semiclassical black holes and horizon singularities

In spherical symmetry, solutions of the semiclassical Einstein equations belong to one of two possible classes. Both classes contain solutions that—depending on the dynamic behavior of the horizon—describe evaporating physical black holes or expanding white holes (trapped/anti-trapped regions that form in finite time of a distant observer). These solutions are real-valued only if the null energy condition (NEC) is violated in the vicinity of the Schwarzschild sphere. We review their properties and describe the only consistent black hole formation scenario. While the curvature scalars are finite on the outer apparent/anti-trapping horizon, it is still a weakly singular surface. This singularity manifests itself in a mild firewall. Near the inner apparent horizon, the NEC is satisfied. Models of static regular black holes are known to be unstable, but since dynamic models of regular black holes are severely constrained by self-consistency requirements, their stability requires further investigation.

Interior metric of slowly formed black holes in a heat bath

We study a spherical black hole formed slowly in a heat bath in the context of ordinary field theory, which we expect to have the typical properties of black holes. We assume that the matter field is conformal and that the metric satisfies the semi-classical Einstein equation $G_{\mu\nu}=8\pi G \langle \psi| T_{\mu\nu}|\psi \rangle$, where $|\psi \rangle$ is the wave function of the matter field. Then as a necessary condition, its trace part must be satisfied, $G^\mu{}_\mu=8\pi G \langle \psi| T^\mu{}_\mu|\psi \rangle$, whose right-hand side is independent of $|\psi \rangle$ and is determined only by the metric through the 4-dimensional Weyl anomaly. With some physically reasonable assumptions, this equation restricts the interior metric to a certain class. Such metrics are approximately warped products of $AdS_2$ and $S^2$ with almost Planckian curvature. Among them, we find one that is consistent with Hawking radiation and is smoothly connected to the exterior Schwarzschild metric slightly outside the Schwarzschild radius. This leads to a picture that the black hole is a dense object with a surface (not a horizon), which evaporates due to Hawking-like radiation when taken out of the bath. Other solutions represent objects that are larger in size than a black hole, which may be useful for analyzing ultra-dense stars.

Semiclassical Gravity in Static Spacetimes as a Constrained Initial Value Problem

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 “positive energy” condition for the Klein–Gordon operator, we characterise the states for which, if the constraints hold for initial data, they hold everywhere in spacetime. These turn out to be time-translation invariant states. The static case is analysed by conformal techniques, effectively reducing the problem to an ultrastatic one.

Static Symmetric Solutions of the Semi-Classical Einstein\u2013Klein\u2013Gordon System

We consider solutions of the semi-classical Einstein–Klein–Gordon system with a cosmological constant Lambda in mathbb {R}, where the spacetime is given by Einstein’s static metric on mathbb {R}times mathbb {S}^3 with a round sphere of radius a>0 and the state of the scalar quantum field has a two-point distribution omega _2 that respects all the symmetries of the metric. We assume that the mass mge 0 and scalar curvature coupling xi in mathbb {R} of the field satisfy m^2+xi R>0, which entails the existence of a ground state. We do not require states to be Hadamard or quasi-free, but the quasi-free solutions are characterised in full detail. The set of solutions of the semi-classical Einstein–Klein–Gordon system depends on the choice of the parameters (a,Lambda ,m,xi ) and on the renormalisation constants in the renormalised stress tensor of the scalar field. We show that the set of solutions is either (i) the empty set, or (ii) the singleton set containing only the ground state, or (iii) a set with infinitely many elements. We characterise the ranges of the parameters and renormalisation constants where each of these alternatives occur. We also show that all quasi-free solutions are given by density matrices in the ground state representation and we show that in cases (ii) and (iii) there is a unique quasi-free solution which minimises the von Neumann entropy. When m=0 this unique state is a beta -KMS state. We argue that all these conclusions remain valid in the reduced order formulation of the semi-classical Einstein equation.

Evaporation of four-dimensional dynamical black holes sourced by the quantum trace anomaly

We study the evaporation of a four-dimensional spherically symmetric black hole formed in a gravitational collapse. We analyze the back-reaction of a massless quantum scalar field conformally coupled to the scalar curvature by means of the semiclassical Einstein equations. We show that the evaporation is linked to an ingoing negative energy flux at the dynamical horizon and that this flux is induced by the quantum matter trace anomaly outside the black hole horizon whenever a suitable averaged energy condition is satisfied. For illustrative purposes, we evaluate the negative ingoing flux and the corresponding rate of evaporation in the case of a null radiating star described by the Vaidya spacetime.

Firewall from Effective Field Theory

For an effective field theory in the background of an evaporating black hole with spherical symmetry, we consider non-renormalizable interactions and their relevance to physical effects. The background geometry is determined by the semi-classical Einstein equation for an uneventful horizon where the vacuum energy–momentum tensor is small for freely falling observers. Surprisingly, after Hawking radiation appears, the transition amplitude from the Unruh vacuum to certain multi-particle states grows exponentially with time for a class of higher-derivative operators after the collapsing matter enters the near-horizon region, despite the absence of large curvature invariants. Within the scrambling time, the uneventful horizon transitions towards a firewall, and eventually the effective field theory breaks down.