The reliability horizon for semi-classical quantum gravity: metric fluctuations are often more important than back-reaction

In this note I introduce the notion of the “reliability horizon” for semi-classical quantum gravity. This reliability horizon is an attempt to quantify the extent to which we should trust semi-classical quantum gravity, and to get a better handle on just where the “Plack regime” resides. I point out that the key obstruction to pushing semi-classical quantum gravity into the Planck regime is often the existence of large metric fluctuations, rather than a large back-reaction. There are many situations where the metric fluctuations become large long before the back-reaction is significant. Issues of this type are fundamental to any attempt at proving Hawking's chronology protection conjecture from first principles, since I shall prove that the onset of chronology violation is always hidden behind the reliability horizon.     

Quantum fluctuations and nonavoidance of the singularity in Bianchi type I cosmology

An effective metric is defined and used for analyzing the quantum fluctuations in a classical geometry. Earlier work showing that quantum (conformal) fluctuations avoid the classical singularity in the case of spherically symmetric collapse is briefly reviewed. It is shown that this result doesnot extend to anisotropic Bianchi type I cosmology. Here the dispersion in the fluctuations increases too slowly to quench the classical singularity. The singularity persists in the space-time described by the effective metric.     

Scale-invariant scalar metric fluctuations during inflation: non-perturbative formalism from a 5D vacuum

We extend to 5D an approach of a 4D non-perturbative formalism to study scalar metric fluctuations of a 5D Riemann-flat de Sitter background metric. In contrast with the results obtained in 4D, the spectrum of cosmological scalar metric fluctuations during inflation can be scale invariant and the background inflaton field can take sub-Planckian values.     

Warm Inflation and Scalar Perturbations of the Metric

A second-order expansion for the quantum fluctuations of the matter field was considered in the framework of the warm inflation scenario. The friction and Hubble parameters were expanded by means of a semiclassical approach. The fluctuations of the Hubble parameter generates fluctuations of the metric. These metric fluctuations produce an effective term of curvature. The power spectrum for the metric fluctuations can be calculated on the infrared sector.     

Decoherence and Bare Mass Induced by Nonconformal Metric Fluctuations

The effects, upon the Klein–Gordon field, of nonconformal stochastic metric fluctuations, are analyzed. It will be shown that these fluctuations allow us to consider an effective mass, i.e., the mass detected in a laboratory is not the parameter appearing in the Klein–Gordon equation, but a function of this parameter and of the fluctuations of the metric. In other words, in analogy to the case of a nonrelativistic electron in interaction with a quantized electromagnetic field, we may speak of a bare mass, where the observed mass shows a dependence upon the stochastic terms included in the metric. Afterwards, we prove, resorting to the influence functional, that the energy–momentum tensor of the Klein–Gordon field inherites this stochastic behavior, and that this feature provokes decoherence upon a particle immersed in the region where this tensor is present.     

Instability of flatspace and the origin of conformal fluctuations

It is shown that conformal fluctuations in the metric can be initiated by the vacuum fluctuations of a scalar field with mass greater than the Planck mass. Flatspace is unstable against such 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.     

A Naturally Renormalized Quantum Field Theory

It was shown that quantum metric fluctuations smear out the singularities of Green functions on the light cone (Ford, ), but it does not remove other ultraviolet divergences of the quantum field theory (QFT). We have proved that quantization in indefinite metric, i.e. QFT in Krein space, removes all divergences of the theory except light cone singularity (Gazeau, et al., Class. Quantum Gravity, 17:1415, 2000, ; Takook, Int. J. Mod. Phys. E, 11:509, 2002, ). In this paper, by considering the QFT in Krein space and the quantum metric fluctuations, it is shown that all divergences can be removed.     

Fluctuations of an Evaporating Black Hole from Back Reaction of Its Hawking Radiation: Questioning a Premise in Earlier Work

This paper delineates the first steps in a systematic quantitative study of the spacetime fluctuations induced by quantum fields in an evaporating black hole. We explain how the stochastic gravity formalism can be a useful tool for that purpose within a low-energy effective field theory approach to quantum gravity. As an explicit example we apply it to the study of the spherically-symmetric sector of metric perturbations around an evaporating black hole background geometry. For macroscopic black holes we find that those fluctuations grow and eventually become important when considering sufficiently long periods of time (of the order of the evaporation time), but well before the Planckian regime is reached. In addition, the assumption of a simple correlation between the fluctuations of the energy flux crossing the horizon and far from it, which was made in earlier work on spherically-symmetric induced fluctuations, is carefully analyzed and found to be invalid. Our analysis suggests the existence of an infinite amplitude for the fluctuations of the horizon as a three-dimensional hypersurface. We emphasize the need for understanding and designing operational ways of probing quantum metric fluctuations near the horizon and extracting physically meaningful information. Dedicated to Rafael Sorkin on the occasion of his 60th birthday.     

Cosmological perturbation theory in slow-roll spacetimes

We present a gauge-invariant argument that a nonlocal measure of second-order metric and matter perturbations dominates that of linear fluctuations in its effect on the gravitational field in "slow-rolling" spacetimes.     

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.     

High-energy behavior and second class constraints in quantum gravity

It is proposed that sensible high-energy behavior in a quantum theory of gravity may be achieved in a class of theories in which the connection and metric are independent and unconstrained and where the action is chosen so that no derivatives of the metric appear. This is because in these theories all ten of the metric field equations are realized as second class constraints. These can in principle be solved, expressing the operators g_μν as functions of the operators for the components of the connection and their canonical momenta. Thus, the metric has no independent quantum fluctuations, and the instabilities resulting from the usual curvature squared terms are eliminated. Furthermore, there is no need to assume metric compatibility, as it is automatically restored in the low-energy limit by the dominance of dimension-two terms.In order to explore these ideas a toy model with two degrees of freedom, corresponding to a metric and a connection variable, is quantized and shown to have a sensible high energy limit, while a related model, in which a constraint analogous to metric compatibility is imposed, is found to be unstable at high energies.     

A Kinetic Theory Approach to Quantum Gravity

We describe a kinetic theory approach to quantum gravity by which we mean a theory of the microscopic structure of space-time, not a theory obtained by quantizing general relativity. A figurative conception of this program is like building a ladder with two knotty poles: quantum matter field on the right and space-time on the left. Each rung connecting the corresponding knots represents a distinct level of structure. The lowest rung is hydrodynamics and general relativity; the next rung is semiclassical gravity, with the expectation value of quantum fields acting as source in the semiclassical Einstein equation. We recall how ideas from the statistical mechanics of interacting quantum fields helped us identify the existence of noise in the matter field and its effect on metric fluctuations, leading to the establishment of the third rung: stochastic gravity, described by the Einstein–Langevin equation. Our pathway from stochastic to quantum gravity is via the correlation hierarchy of noise and induced metric fluctuations. Three essential tasks beckon: (1) deduce the correlations of metric fluctuations from correlation noise in the matter field; (2) reconstituting quantum coherence—this is the reverse of decoherence—from these correlation functions; and (3) use the Boltzmann–Langevin equations to identify distinct collective variables depicting recognizable metastable structures in the kinetic and hydrodynamic regimes of quantum matter fields and how they demand of their corresponding space-time counterparts. This will give us a hierarchy of generalized stochastic equations—call them the Boltzmann–Einstein hierarchy of quantum gravity—for each level of space-time structure, from the the macroscopic (general relativity) through the mesoscopic (stochastic gravity) to the microscopic (quantum gravity).     

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.     

An Effective Stochastic Semiclassical Theory for the Gravitational Field

Assuming that the mechanism proposed byGell-Mann and Hartle works as a mechanism fordecoherence and classicalization of the metric field, weformally derive the form of an effective theory for thegravitational field in a semiclassical regime. This effectivetheory takes the form of the usual semiclassical theoryof gravity, based on the semiclassical Einsteinequation, plus a stochastic correction which accounts for the backreaction of the lowest order matterstress-energy fluctuations.     

Black holes: a physical route to the Kerr metric

As a consequence of Birkhoff's theorem, the exterior gravitational field of a spherically symmetric star or black hole is always given by the Schwarzschild metric. In contrast, the exterior gravitational field of a rotating (axisymmetric) star differs, in general, from the Kerr metric, which describes a stationary, rotating black hole. In this paper I discuss the possibility of a quasi–stationary transition from rotating equilibrium configurations of normal matter to rotating bla ck holes.     

Super-exponential inflation from a dynamical foliation of a 5D vacuum state

We introduce super-exponential inflation (ω<−1) from a 5D Riemann-flat canonical metric on which we make a dynamical foliation. The resulting metric describes a super accelerated expansion for the early universe well known as super-exponential inflation that, for very large times, tends to an asymptotic de Sitter (vacuum dominated) expansion. The scalar field fluctuations are analyzed. The important result here obtained is that the spectral index for energy density fluctuations is not scale invariant, and for cosmological scales becomes ns(k<k_?)?1. However, for astrophysical scales this spectrum changes to negative values ns(k>k_?)<0.     

Conformal transformations in scalar-tensor gravitation theories

Conformal transformations in scalar-tensor gravitation theories are investigated in the present paper. It is demonstrated that the field equations of these theories can be expressed in the Vaidya form. A scalar field equation is derived based on the compatibility condition for the field equations. The conformal factor, dilaton, and restrictions on the metric are determined for the diagonal metric of type I in the Bianchi classification. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 10–14, February, 2006.     

Is there the radion in the RS2 model?

We analyse the physical boundary conditions at infinity for metric fluctuations and gauge functions in the RS2 model with matter on the brane. We argue that due to these boundary conditions the radion field cannot be gauged out in this case. Thus, it represents a physical degree of freedom of the model.     

Cylindrically symmetric Brans-Dicke fields. III

In the previous work I and II, we have obtained a class of exact solutions for the Brans-Dicke scalar-tensor theory for Einstein-Rosen nonstatic cylindrically symmetric metric when only scalar field is present and then in presence of source-free electromagnetic field. In the present work we have developed a more general sets of solutions from those given in I and II under the unit transformation by Morganstern [Phys. Rev. D 3 (1971), 2946]. These have been found to be the solutions of the Brans-Dicke scalar-tensor theory for the most general cylindrically symmetric metric of Marder.