Backreaction Problem Research Articles

Effective Lagrangian and the back-reaction problem in a self-interacting O(N) scalar theory in curved spacetime.

A derivation of the one-loop effective Lagrangian in the self-interacting O(N) scalar theory, in slowly varying gravitational fields, is presented (using \ensuremath{\zeta}-function regularization and heat-kernel techniques). The result is given in terms of the expansion in powers of the curvature tensors (up to quadratic terms) and their derivatives, as well as in derivatives of the background scalar field (up to second derivatives). The renormalization group improved effective Lagrangian is studied, which gives the leading-log approach of the whole perturbation theory. An analysis of the effective equations (back-reaction problem) on the static hyperbolic spacetime ${\mathit{openR}}^{2}$\ifmmode\times\else\texttimes\fi{}${\mathit{H}}^{2}$/\ensuremath{\Gamma} is carried out for the simplest version of the theory: ${\mathit{m}}^{2}$=0 and N=1. The possibility of the existence of the solution ${\mathit{openR}}^{2}$\ifmmode\times\else\texttimes\fi{}${\mathit{H}}^{2}$/\ensuremath{\Gamma}, induced by purely quantum effects, is discussed.

On 4D Hawking radiation from the effective action

We determine the s-waves contribution of a scalar field to the four-dimensional (4D) effective action for arbitrary spherically symmetric external gravitational fields. The result is applied to 4D black holes and it is shown that the energy-momentum tensor derived from the (nonlocal) effective action contains the Hawking radiation. The luminosity is close to the expected one in the s-channel. The energy-momentum tensor may be used as starting point to study the backreaction problem.

Noise and fluctuations in semiclassical gravity.

We continue our earlier investigation of the backreaction problem in semiclassical gravity with the Schwinger-Keldysh or closed-time-path (CTP) functional formalism using the language of the decoherent history formulation of quantum mechanics. Making use of its intimate relation with the Feynman-Vernon influence functional (IF) method, we examine the statistical mechanical meaning and show the interrelation of the many quantum processes involved in the backreaction problem, such as particle creation, decoherence and dissipation. We show how noise and fluctuation arise naturally from the CTP formalism. We derive an expression for the CTP effective action in terms of the Bogolubov coefficients and show how noise is related to the fluctuations in the number of particles created. In so doing we have extended the old framework of semiclassical gravity, based on the mean field theory of Einstein equation with a source given by the expectation value of the energy-momentum tensor, to that based on a Langevin-type equation, where the dynamics of fluctuations of spacetime is driven by the quantum fluctuations of the matter field. This generalized framework is useful for the investigation of quantum processes in the early universe involving fluctuations, vacuum stability and phase transtion phenomena and the non-equilibrium thermodynamics of black holes. It is also essential to an understanding of the transition from any quantum theory of gravity to classical general relativity. \pacs{pacs numbers: 04.60.+n,98.80.Cq,05.40.+j,03.65.Sq}

The thermodynamical approach to the back reaction problem

A new approach to the back reaction of Hawking radiation on a Schwarzschild black hole (SBH), based on thermodynamics, is proposed, in which the thermodynamical system composed of SBH and thermal radiation in curved space is treated as a thermodynamical system composed of SBH, thermal radiation and a two-dimensional thermodynamical membrane (i.e. event horizon) in flat space.

Pair creation in transport equations using the equal-time Wigner function.

Based on the equal-time Wigner function for the Klein-Gordon field, we discuss analytically the mechanism of pair creation in a classical electromagnetic field including back-reaction. It is shown that the equations of motion for the Wigner function can be reduced to a variable-frequency oscillator. The pair-creation rate results then from a calculation analogous to barrier penetration in nonrelativistic quantum mechanics. The Wigner function allows one to utilize this treatment for the formulation of an effective transport theory for the back-reaction problem with a pair-creation source term including Bose enhancement.

A Comparison of Various Approaches to the Back-Reaction Problem

We compare three possible prescriptions for the back-reaction of a quantum mechanical source on the classical gravitational field to which it is coupled. These prescriptions are (i) the transition element 〈out| Tik |in〉 of the energy-momentum operator between asymptotic vacuum states, (ii) the expectation value 〈Ψ| Tik |Ψ〉, and (iii) the Born-Oppenheimer type approximation for back-reaction as used in molecular physics. It is shown that the three approaches match only when the gravitational field is varying adiabatically, and of the three, the use of expectation value provides the most accurate description of the validity of semiclassical Einstein equations. The analysis is carried Out by studying the model of a quantized time-dependent oscillator coupled to a classical particle.

Evaporation of a Collapsing Shell with Scalar Field Production

We consider gravitational collapse of a spherically symmetric dust shell coupled with a massless scalar field. The rest mass can decrease as a result of the scalar wave production. Our purpose is to discuss the backreaction problem in the context of classical general relativity. We construct junction conditions at the shell including the scalar field. Possible final fates of the gravitational collapse are found by solving them when the shell is close to the apparent horizon. Interestingly, as the collapsing shell shrinks to a point, it can completely evaporate without forming a black hole and naked singularity. The mass-loss rate measured by a static observer turns out to remain finite even at the final stage of the shell evaporation.

Evaporation of a Collapsing Shell with Scalar Field Production

We consider gravitational collapse of a spherically symmetric dust shell coupled with a massless scalar field. The rest mass can decrease as a result of the scalar wave production. Our purpose is to discuss the backreaction problem in the context of classical general relativity. We construct junction conditions at the shell including the scalar field. Possible final fates of the gravitational collapse are found by solving them when the shell is close to the apparent horizon. Interestingly, as the collapsing shell shrinks to a point, it can completely evaporate without forming a black hole and naked singularity. The mass-loss rate measured by a static observer turns out to remain finite even at the final stage of the shell evaporation.

A comparison of various approaches to the back-reaction problem: T. Padmanabhan and T. P. Singh. Theoretical Astrophysics Group, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India

A comparison of various approaches to the back-reaction problem: T. Padmanabhan and T. P. Singh. Theoretical Astrophysics Group, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India

Dynamics of extremal black holes.

Particle scattering and radiation by a magnetically charged, dilatonic black hole is investigated near the external limit at which the mass is a constant times the charge. Near this limit a neighborhood of the horizon of the black hole is closely approximated by a trivial product of a two-dimensional black hole with a sphere. This is shown to imply that the scattering of long-wavelength particles can be described by a (previously analyzed) two-dimensional effective field theory, and is related to the formation and/or evaporation of two-dimensional black holes. The scattering proceeds via particle capture followed by Hawking reemission, and naively appears to violate unitarity. However this conclusion can be altered when the effects of back reaction are included. Particle-hole scattering is discussed in the light of a recent analysis of the two-dimensional back-reaction problem. It is argued that the quantum-mechanical possibility of scattering off of extermal black holes implies the potential existence of additional quantum numbers, referred to as "quantum whiskers," characterizing the black hole.

Self-consistent solution to the back-reaction problem for vector fields in two dimensions

We describe vector fields propagating in a two-dimensional spatially flat Robertson-Walker background using a curved space-time generalization of the Stueckelberg formalism. We prove that the energy-momentum tensor expectation value in the vacuum defined through energy minimization is renormalizable and yields the usual anomalous trace in the massless limit of vector mesons. Further on we study the back-reaction problem using the semiclassical Einstein equations. In the massive case we found that the physical solution requires the cosmological constant to vanish but not necessarily with a vanishing curvature. However, for certain initial conditions of the scale factor, the de Sitter metric is a consistent solution with the cosmological constant depending on powers of the curvature scalar greater than one. In addition, matter is continuously being created at a steady state.

Quantum back reaction in scalar QED as an initial-value problem.

An initial-value formulation of the quantum back-reaction problem in scalar QED is presented. The semiclassical, mean-field limit of the theory for spatially homogeneous initial states yields (after renormalization) a tractable set of finite, coupled ordinary differential equations for the time evolution of the system. These equations describe the effects of particle creation and the associated induced electric current on the decay of a uniform background electric field, in a form suitable for implementation on a computer. The renormalization techniques and formulation of the initial-value problem employed provide a model for the numerical attack on the analogous back-reaction problem, involving fully interacting fields, in early Universe quantum cosmology.

Renormalized evolution equations for the back-reaction problem with a self-interacting scalar field.

We present the renormalized equations which rule the evolution of the mean value of the field and the metric of the spacetime for the lambdaphi/sup 4/ theory. The calculations are done in the one-loop approximation and the classical background gravitational field is a general one. For the particular example of a Robertson-Walker metric, we show that the numerical method used to solve the back-reaction problem in the free-field case can also be applied here without additional complications. We discuss the possible generalizations of our formalism and its relevance in the study of phase transitions in the early Universe.

Closed-time-path functional formalism in curved spacetime: Application to cosmological back-reaction problems.

We discuss the generalization to curved spacetime of a path-integral formalism of quantum field theory based on the sum over paths first going forward in time in the presence of one external source from an in vacuum to a state defined on a hypersurface of constant time in the future, and then backwards in time in the presence of a different source to the same in vacuum. This closed-time-path formalism which generalizes the conventional method based on in-out vacuum persistence amplitudes yields real and causal effective actions, field equations, and expectation values. We apply this method to two problems in semiclassical cosmology. First we study the back reaction of particle production in a radiation-filled Bianchi type-I universe with a conformal scalar field. Unlike the in-out formalism which yields complex geometries the real and causal effective action here yields equations for real effective geometries, with more readily interpretable results. It also provides a clear identification of particle production as a dissipative process in semiclassical theories. In the second problem we calculate the vacuum expectation value of the stress-energy tensor for a nonconformal massive \ensuremath{\lambda}${\ensuremath{\varphi}}^{4}$ theory in a Robertson-Walker universe. This study serves to illustrate the use of Feynman diagrams and higher-loop calculations in this formalism. It also demonstrates the economy of this method in the calculation of expectation values over the mode-sum Bogolubov transformation methods ordinarily applied to matrix elements calculated in the conventional in-out approach. The capability of the closed-time-path formalism of dealing with Feynman, causal, and correlation functions on the same footing makes it a potentially powerful and versatile technique for treating nonequilibrium statistical properties of dynamical systems as in early-Universe quantum processes.

Semiclassical quantum gravity and Liouville theory a complete solution to the back-reaction problem in two dimensions

We consider semiclassical Einstein equations in two dimensions. The solution to the back-reaction problem is given by a constant curvature metric parametrized by solutions (“wave functions”) of a zero-energy Schrödinger equation. The potential of this equation determines the quantum state of matter fields. We classify the geometry configurations and analyze the global properties of the space-time as a function of the quantum matter content both at zero and finite temperature. For a given sign of the cosmological constant Λ, it depends on the sign of the trace anomaly coefficient. If the number of matter fields goes to ∞, the Hawking temperature vanishes of the theory.

A model of spontaneous symmetry breakdown in spatially flat cosmological spacetimes

This paper is an elaboration of a previous short exposition of a theory of spontaneous symmetry breaking in a conformally coupled, massless λø 4 model in a spatially flat Robertson-Walker spacetime. Under the weakened global boundary condition allowing the physical spacetime to be conformal to only a portion of the Minkowski spacetime, the model admits a pair of degenerate vacua in which the ø → − ø symmetry is spontaneously broken. The model is formulated as a euclidean field theory in a space with a positive-definite metric obtained by analytically continuing the conformal time coordinate. An appropriate time-dependent zero energy solution of the euclidean equation of motion representing the field configuration in the asymmetric vacuum is considered and the corresponding quantum trace anomaly 〈 T μ μ 〉 is computed in the one-loop approximation. The nontrivial infrared behavior of the model due to the singular nature of the classical background field forces a modification of the boundary conditions on the propagator. A general form for an “improved|DD one-loop trace anomaly is found by a simple argument based on renormalization group invariance. Via the Einstein equation, the trace anomaly leads to a self-consistent dynamical equation for the cosmic expansion scale factor. Some physical aspects of the back-reaction problem based on a simple power law model of the expansion scale factor are discussed.

Effects of quantum fields on singularities and particle horizons in the early universe. II

The back-reaction problem for conformally invariant free quantum fields in homogeneous and isotropic spacetimes containing classical radiation is solved for spacetimes with nonzero spatial curvatures and/or nonzero cosmological constants. The solutions depend upon two regularization parameters which we call $\ensuremath{\alpha}$ and $\ensuremath{\beta}$. Only solutions which at late times approach the appropriate solution to the field equations of general relativity are considered. The results are much the same as those found previously for spatially flat spacetimes with zero cosmological constants. Thus, if $\ensuremath{\beta}>3\ensuremath{\alpha}>0$, there is always one solution which undergoes a "time-symmetric bounce" and which contains no singularities, if $\ensuremath{\alpha}$,$\ensuremath{\beta}>0$ there is a family of solutions with particle horizons and no singularities, and if $\ensuremath{\alpha}>0$ there is always at least one solution with an initial singularity but no particle horizons. The differences caused by the spatial curvature and cosmological constant include the initial behavior of the time-symmetric bounce solution and, if the spatial curvature is nonzero, the initial behavior of many solutions for the cases $\ensuremath{\beta}=3\ensuremath{\alpha}>0$ and $\ensuremath{\beta}\ensuremath{\approx}3\ensuremath{\alpha}<0$.

De Sitter solution of the back reaction problem

De Sitter solution of the back reaction problem

De Sitter metric as a self-consistent solution of the back reaction problem

The de Sitter metric is a solution of the Einstein equation even when quantum effects of matter fields are included. However between the curvature and the cosmological constant Λ changes drastically. For some values of Λ, there are two or three solutions, while there is none in some other cases. There can be even a positive curvature solution for negative Λ. These features depend on mass and spin of matter fields and the matter-curvature coupling.

Effects of quantum fields on singularities and particle horizons in the early universe

The back-reaction problem for conformally invariant free quantum fields in homogeneous and isotropic spacetimes containing classical radiation is solved for spacetimes with nonzero spatial curvatures and/or nonzero cosmological constants. The solutions depend upon two regularization parameters which we call ..cap alpha.. and ..beta... Only solutions which at late times approach the appropriate solution to the field equations of general relativity are considered. The results are much the same as those found previously for spatially flat spacetimes with zero cosmological constants. Thus, if ..beta..>3..cap alpha..>0, there is always one solution which undergoes a ''time-symmetric bounce'' and which contains no singularities, if ..cap alpha..,..beta..>0 there is a family of solutions with particle horizons and no singularities, and if ..cap alpha..>0 there is always at least one solution with an initial singularity but no particle horizons. The differences caused by the spatial curvature and cosmological constant include the initial behavior of the time-symmetric bounce solution and, if the spatial curvature is nonzero, the initial behavior of many solutions for the cases ..beta.. = 3..cap alpha..>0 and ..beta..roughly-equal3..cap alpha..<0.