Stability of Semiclassical Gravity Solutions with Respect to Quantum Metric Fluctuations

We discuss the stability of semiclassical gravity solutions with respect to small quantum corrections by considering the quantum fluctuations of the metric perturbations around the semiclassical solution. We call the attention to the role played by the symmetrized 2-point quantum correlation function for the metric perturbations, which can be naturally decomposed into two separate contributions: intrinsic and induced fluctuations. We show that traditional criteria on the stability of semiclassical gravity are incomplete because these criteria based on the linearized semiclassical Einstein equation can only provide information on the expectation value and the intrinsic fluctuations of the metric perturbations. By contrast, the framework of stochastic semiclassical gravity provides a more complete and accurate criterion because it contains information on the induced fluctuations as well. The Einstein–Langevin equation therein contains a stochastic source characterized by the noise kernel (the symmetrized 2-point quantum correlation function of the stress tensor operator) and yields stochastic correlation functions for the metric perturbations which agree, to leading order in the large N limit, with the quantum correlation functions of the theory of gravity interacting with N matter fields. These points are illustrated with the example of Minkowski space-time as a solution to the semiclassical Einstein equation, which is found to be stable under both intrinsic and induced fluctuations.     

Stochastic Gravity

We give a summary of the status of currentresearch in stochastic semiclassical gravity and suggestdirections for further investigations. This theorygeneralizes the semiclassical Einstein equation to an Einstein-Langevin equation with a stochasticsource term arising from the fluctuations of theenergy-momentum tensor of quantum fields. We mentionrecent efforts in applying this theory to the study of black hole fluctuation and backreactionproblems, linear response of hot flat space, andstructure formation in inflationary cosmology. Toexplore the physical meaning and implications of thisstochastic regime in relation to both classical andquantum gravity, we find it useful to take the view thatsemiclassical gravity is mesoscopic physics and thatgeneral relativity is the hydrodynamic limit of certain spacetime quantum substructures. We view theclassical spacetime depicted by general relativity as acollective state and the metric or connection functionsas collective variables. Three basic issues —stochasticity, collectivity, correlations — andthree processes — dissipation, fluctuations,decoherence — underscore the transformation fromquantum microstructure and interaction to the emergenceof classical macrostructure and dynamics. We discuss ways toprobe into the high-energy activity from below and maketwo suggestions: via effective field theory and thecorrelation hierarchy. We discuss how stochastic behavior at low energy in an effective theoryand how correlation noise associated with coarse-grainedhigher correlation functions in an interacting quantumfield could carry nontrivial information about the high-energy sector. Finally, we describeprocesses deemed important at the Planck scale,including tunneling and pair creation, wave scatteringin random geometry, growth of fluctuations and forms, Planck-scale resonance states, and spacetimefoams.     

Semiclassical Cosmological Perturbations Generated During Inflation

Interaction with the environment may induce stochastic semiclassical dynamicsin open quantum systems. In the gravitational context, stress-energy fluctuationsof quantum matter fields give rise to a stochastic behavior in the spacetimegeometry. The Einstein—Langevin equation is a suitable tool to take these effectsinto account when addressing the backreaction problem in semiclassical gravity.We analyze within this framework the generation of gravitational fluctuationsduring inflation, which are of great interest for large-scale structure formationin cosmology.     

Nonsingular Collapse of a Perfect Fluid Sphere Within a Dilaton-Gravity Reformulation of the Oppenheimer Model

A generic four-dimensional dilaton gravity is considered as a basis for reformulating the paradigmatic Oppenheimer–Synder model of a gravitationally collapsing star modelled as a perfect fluid or dust sphere. Initially, the vacuum Einstein scalar-tensor equations are modified to Einstein–Langevin equations which incorporate a noise or micro-turbulence source term arising from Planck scale conformal, dilaton fluctuations which induce metric fluctuations. Coupling the energy-momentum tensor for pressureless dust or fluid to the Einstein–Langevin equations, a modification of the Oppenheimer–Snyder dust collapse model is derived. The Einstein–Langevin field equations for the collapse are of the form of a Langevin equation for a non-linear Brownian motion of a particle in a homogeneous noise bath. The smooth worldlines of collapsing matter become increasingly randomised Brownian motions as the star collapses, since the backreaction coupling to the fluctuations is non-linear; the input assumptions of the Hawking–Penrose singularity theorems are then violated. The solution of the Einstein–Langevin collapse equation can be found and is non-singular with the singularity being smeared out on the correlation length scale of the fluctuations, which is of the order of the Planck length. The standard singular Oppenheimer–Synder model is recovered in the limit of zero dilaton fluctuations.     

Stress Tensor Fluctuations and Passive Quantum Gravity

The quantum fluctuations of the stress tensor of a quantum field are discussed, as are the resulting space-time metric fluctuations. Passive quantum gravity is an approximation in which gravity is not directly quantized, but fluctuations of the space-time geometry are driven by stress tensor fluctuations. We discuss a decomposition of the stress tensor correlation function into three parts, and consider the physical implications of each part. The operational significance of metric fluctuations and the possible limits of validity of semiclassical gravity are discussed.     

Instability of flat spacetime in semiclassical gravity

The semiclassical theory of gravity is considered in which an asymptotically flat background metric is coupled to quantized matter. We show that, in general, there are modes with spacelike wave vectors for small metric fluctuations around flat spacetime. Besides the usual axioms of quantum field theory in flat spacetime, the proof rests on the existence of a hard trace anomaly in the energy-momentum tensor due to matter self-couplings. Two possible interpretations of the result are discussed.     

<Emphasis Type="Italic">z</Emphasis> = 3 Lifshitz-Hořava model and fermi-point scenario of emergent gravity

Recently Hořava proposed a model for gravity which is described by the Einstein action in the infrared, but lacks the Lorentz invariance in the high-energy region where it experiences the anisotropic scaling. We test this proposal using two condensed matter examples of emergent gravity: acoustic gravity and gravity emerging in the fermionic systems with Fermi points. We suggest that quantum hydrodynamics, which together with the quantum gravity is the non-renormalizable theory, may exhibit the anisotropic scaling in agreement with the proposal. The Fermi point scenario of emergent general relativity demonstrates that under general conditions, the infrared Einstein action may be distorted, i.e., the Hořava parameter λ is not necessarily equal 1 even in the low energy limit. The consistent theory requires special hierarchy of the ultra-violet energy scales and the fine-tuning mechanism for the Newton constant. The article is published in the original.     

Particle Creation in the Oscillatory Phase of Inflaton

A thermal squeezed state representation of inflaton is constructed for a flat Friedmann–Robertson–Walker (FRW) background metric and the phenomenon of particle creation is examined during the oscillatory phase of inflaton, in the semiclassical theory of gravity. An approximate solution to the semiclassical Einstein equation is obtained in thermal squeezed state formalism perturbatively and is found obey the same power-law expansion as that of classical Einstein equation. In addition to that the solution shows oscillatory in nature except on a particular condition. It is also noted that, the coherently oscillating nonclassical inflaton, in thermal squeezed vacuum state, thermal squeezed state, and thermal coherent state, suffers particle production and the created particles exhibit oscillatory behavior. The present study can account for the postinflation particle creation due to thermal and quantum effects of inflation in a flat FRW universe.     

A Semiclassical Theory of a Dissipative Henon—Heiles System

A semiclassical theory of a dissipative Henon—Heiles system is proposed. Based on -scaling of an equation for the evolution of the Wigner quasiprobability distribution function in the presence of dissipation and thermal diffusion, we derive a semiclassical equation for quantum fluctuations, governed by the dissipation and the curvature of the classical potential. We show how the initial quantum noise gets amplified by classical chaotic diffusion, which is expressible in terms of a correlation of stochastic fluctuations of the curvature of the potential due to classical chaos, and ultimately settles down to equilibrium under the influence of dissipation. We also establish that there exists a critical limit to the expansion of phase space. The limit is set by chaotic diffusion and dissipation. Our semiclassical analysis is corroborated by numerical simulation of a quantum operator master equation.     

Nonlinear Connections and Nearly Autoparallel Maps in General Relativity

We apply the method of moving anholonomic frames with associated nonlinear connections to the (pseudo) Riemannian space geometry and examine the conditions when locally anisotropic structures (Finsler like and more general ones) could be modeled in the general relativity theory and/or Einstein–Cartan–Weyl extensions [1]. New classes of solutions of the Einstein equations with generic local anisotropy are constructed. We formulate the theory of nearly autoparallel (na) maps generalizing the conformal transforms and formulate the Einstein gravity theory on na–backgrounds provided with a set of na–map invariant conditions and local conservation laws. There are illustrated some examples when vacuum Einstein fields are generated by Finsler like metrics and chains of na–maps.     

Stochastic and quantum space-time metrics and the weak-field limit

In this review we present a simple method of introducing stochastic and quantum metrics into gravitational theory at short distances in terms of small fluctuations around a classical background space-time. We consider only residual effects due to the stochastic (or quantum) theory of gravity and use a perturbative stochastization (or quantization) method. By using the general covariance and correspondence principles, we reconstruct the theory of gravitational, mechanical, electromagnetic, and quantum mechanical processes and tensor algebra in the space-time with stochastic and quantum metrics. Some consequences of the theory are also considered, in particular, it indicates that the value of the fundamental lengthl lies in the interval 10^–23l10^–22 cm.     

Semiclassical states in loop quantum gravity: an introduction

Unifying general relativity and quantum mechanics is a great challenge left to us by Einstein. To face this challenge, considerable progress has been made in non-perturbative canonical (loop) quantum gravity during the past 20 years. The kinematical Hilbert space of the quantum theory is constructed rigorously. However, the semiclassical analysis of the theory is still a crucial and open issue. In this review, we first introduce our work on constructing a semiclassical weave state, using the [ω] operator on the kinematical Hilbert space of loop quantum gravity. Then we give an introduction to the two different approaches currently investigated for constructing coherent states in the kinematical Hilbert space. The current status of semiclassical analysis in loop quantum gravity is then summarized.     

Analogue Models of and for Gravity

Condensed matter systems, such as acoustics in flowing fluids, light in moving dielectrics, or quasiparticles in a moving superfluid, can be used to mimic aspects of general relativity. More precisely these systems (and others) provide experimentally accessible models of curved-space quantum field theory. As such they mimic kinematic aspects of general relativity, though typically they do not mimic the dynamics. Although these analogue models are thereby limited in their ability to duplicate all the effects of Einstein gravity they nevertheless are extremely important—they provide black hole analogues (some of which have already been seen experimentally) and lead to tests of basic principles of curved-space quantum field theory. Currently these tests are still in the realm of gedanken-experiments, but there are plausible candidate models that should lead to laboratory experiments in the not too distant future.     

Planck—Wheeler Quantum Foam as White Noise: Metric Diffusion and Congruence Focussing for Fluctuating Spacetime Geometry

It is expected that quantum effects endow spacetime with stochastic properties near the Planck scale as exemplified by random fluctuations of the metric, usually referred to as spacetime foam or geometrodynamics. In this paper, a methodology is presented for incorporating Planck scale stochastic effects and corrections into general relativity within the ADM formalism, by coupling the Riemann 3-metric to white noise. The ADM—Cauchy evolution of a Riemann 3-metric h _ij (t) induced on spacelike hypersurface C(t) can be interpreted within pure general relativity as a smooth geodesic flow in superspace, whose points consist of equivalence classes of 3-metrics. Coupling white noise to h _ij gives Langevin stochastic differential equations for the Cauchy evolution of h _ij, which is now a Brownian motion or diffusion in superspace. A fluctuation h_ij away from h _ij is considered to be related to h _ij by elements of the diffeomorphism group diff(C). Hydrodynamical Fokker—Planck continuity equations are formulated describing the stochastic Cauchy evolution of h _ij as a probability flow. The Cauchy invariant or equilibrium solution gives a stationary probability distribution of fluctuations peaked around the deterministic metric. By selecting a physically viable ansatz for the scale dependent diffusion coefficient, one reproduces the Wheeler uncertainty relation for the metric fluctuations of quantum geometrodynamics. Treating h _ij as a random variable, a non-linear Raychaudhuri—Langevin equation is derived describing the geometro-hydrodynamics of a congruence of fluid or dust matter propagating on the stochastic spacetime. For an initially converging congruence >0 at s the singularity =– at future proper time s=3/||$, which is expected in general relativity, is now smeared out near the Planck scale. Proper time s can be extended indefinitely (s) so that intrinsic metric fluctuations can restore geodesic completeness although the geodesics remain trapped for all time: although a singularity can be removed the collapsing matter still creates a black hole. A Fokker—Planck formulation also gives zero probability that – for s. Essentially, the short distance stochastic corrections to the deterministic equations of general relativity can remove pathologies such as singularities, conjugate points and geodesic incompleteness.     

Quantum fluctuations of vacuum stress tensors and spacetime curvatures

We analyze the quantum fluctuations of vacuum stress tensors and spacetime curvatures, using the framework of linear response theory which connects these fluctuations to dissipation mechanisms arising when stress tensors and spacetime metric are coupled. Vacuum fluctuations of spacetime curvatures are shown to be a sum of two contributions at lowest orders; the first one corresponds to vacuum gravitational waves and is restricted to light-like wavevectors and vanishing Einstein curvature, while the second one arises from gravity of vacuum stress tensors. From these fluctuations, we deduce noise spectra for geodesic deviations registered by probe fields which determine ultimate limits in length or time measurements. In particular, a relation between noise spectra characterizing spacetime fluctuations and the number of massless neutrino fields is obtained.