Semiclassical Einstein Equations Research Articles

QFT, string temperature, and the string phase of de Sitter space-time

The density of mass levels $\ensuremath{\rho}(m)$ and the critical temperature for strings in de Sitter space-time are found. QFT and string theory in de Sitter space are compared. A ``dual'' transform is introduced which relates classical to quantum string lengths, and more generally, QFT and string domains. Interestingly, the string temperature in de Sitter space turns out to be the dual transform of the QFT Hawking-Gibbons temperature. The back reaction problem for strings in de Sitter space is addressed self-consistently in the framework of the ``string analogue'' model (or thermodynamical approach), which is well suited to combine QFT and string study. We find de Sitter space-time is a self-consistent solution of the semiclassical Einstein equations in this framework. Two branches for the scalar curvature $R(\ifmmode\pm\else\textpm\fi{})$ show up: a classical, low curvature solution $(\ensuremath{-}),$ and a quantum high curvature solution $(+),$ entirely sustained by the strings. There is a maximal value for the curvature ${R}_{\mathrm{max}}$ due to the string back reaction. Interestingly, our dual relation manifests itself in the back reaction solutions: the $(\ensuremath{-})$ branch is a classical phase for the geometry with the intrinsic temperature given by the QFT Hawking-Gibbons temperature. The $(+)$ is a stringy phase for the geometry with a temperature given by the intrinsic string de Sitter temperature. $2+1$ dimensions are considered, but conclusions hold generically in D dimensions.

Nonperturbative effects of vacuum energy on the recent expansion of the universe

We show that the vacuum energy of a free quantized field of very low mass can significantly alter the recent expansion of the universe. The effective action of the theory is obtained from a non-perturbative sum of scalar curvature terms in the propagator. We numerically investigate the semiclassical Einstein equations derived from it. As a result of non-perturbative quantum effects, the scalar curvature of the matter-dominated universe stops decreasing and approaches a constant value. The universe in our model evolves from an open matter-dominated epoch to a mildly inflating de Sitter expansion. The Hubble constant during the present de Sitter epoch, as well as the time at which the transition occurs from matter-dominated to de Sitter expansion, are determined by the mass of the field and by the present matter density. The model provides a theoretical explanation of the observed recent acceleration of the universe, and gives a good fit to data from high-redshift Type Ia supernovae, with a mass of about 10^{-33} eV, and a current ratio of matter density to critical density, Omega_0 <0.4 . The age of the universe then follows with no further free parameters in the theory, and turns out to be greater than 13 Gyr. The model is spatially open and consistent with the possibility of inflation in the very early universe. Furthermore, our model arises from the standard renormalizable theory of a free quantum field in curved spacetime, and does not require a cosmological constant or the associated fine-tuning.

Quantum scalar field on the massless(2+1)-dimensional black hole background

The behavior of a quantum scalar field is studied in the metric ground state of the (2+1)-dimensional black hole of Ba\~nados, Teitelboim and Zanelli which contains a naked singularity. The one-loop BTZ partition function and the associate black hole effective entropy, the expectation value of the quantum fluctuation as well as the renormalized expectation value of the stress tensor are explicitly computed in the framework of the $\zeta$-function procedure. This is done for all values of the coupling with the curvature, the mass of the field and the temperature of the quantum state. In the massless conformally coupled case, the found stress tensor is used for determining the quantum back reaction on the metric due to the scalar field in the quantum vacuum state, by solving the semiclassical Einstein equations. It is finally argued that, within the framework of the 1/N expansion, the Cosmic Censorship Hypothesis is implemented since the naked singularity of the ground state metric is shielded by an event horizon created by the back reaction.

Time machines constructed from anti–de Sitter space

In this paper time machines are constructed from anti--de Sitter space. One is constructed by identifying points related via boost transformations in the covering space of anti--de Sitter space and it is shown that this Misner-like anti--de Sitter space is just the Lorentzian section of the complex space constructed by Li, Xu, and Liu [Phys. Rev. D 48, 4735 (1993)]. The others are constructed by gluing an anti--de Sitter space to a de Sitter space, which could describe an anti--de Sitter phase bubble living in a de Sitter phase universe. Self-consistent vacua for a massless conformally coupled scalar field are found for these time machines, whose renormalized stress-energy tensors are finite and solve the semiclassical Einstein equations. The extensions to electromagnetic fields and massless neutrinos are discussed. It is argued that, in order to make the results consistent with Euclidean quantization, a new renormalization procedure for quantum fields in Misner-type spaces (Misner space, Misner-like de Sitter space, and Misner-like anti--de Sitter space) is required. Such a ``self-consistent'' renormalization procedure is proposed. With this renormalization procedure, self-consistent vacua exist for massless conformally coupled scalar fields, electromagnetic fields, and massless neutrinos in these Misner-type spaces.

Planckian AdS2× S2 space is an exact solution of the semiclassical Einstein equations

The product space configuration AdS 2× S 2 (with l and r being radiuses of the components) carrying the electric charge Q is demonstrated to be an exact solution of the semiclassical Einstein equations in presence of the Maxwell field. If the logarithmic UV divergences are absent in the four-dimensional theory the solution we find is identical to the classical Bertotti-Robinson space ( r= l= Q) with no quantum corrections added. In general, the analysis involves the quadratic curvature coupling λ appearing in the effective action. The solutions we find are of the following types: (i) (for arbitrary λ) charged configuration which is quantum deformation of the Bertotti-Robinson space; (ii) ( λ> λ cr ) Q=0 configuration with l and r being of the Planck order; (iii) ( λ< λ cr ) Q≠0 configuration ( l and r are of the Planck order) not connected analytically to the Bertotti-Robinson space. The interpretation of the solutions obtained and an indication on the internal structure of the Schwarzschild black hole are discussed.

Stochastic Gravity

We give a summary of the status of current research in stochastic semiclassical gravity and suggest directions for further investigations. This theory generalizes the semiclassical Einstein equation to an Einstein-Langevin equation with a stochastic source term arising from the fluctuations of the energy-momentum tensor of quantum fields. We mention recent efforts in applying this theory to the study of black hole fluctuations and backreaction problems, linear response of hot flat space, and structure formation in inflationary cosmology. To explore the physical meaning and implications of this stochastic regime in relation to both classical and quantum gravity, we find it useful to take the view that semiclassical gravity is mesoscopic physics and that general relativity is the hydrodynamic limit of certain spacetime quantum substructures. Three basic issues - stochasticity, collectivity, correlations- and three processes - dissipation, fluctuations, decoherence- underscore the transformation from quantum micro structure and interaction to the emergence of classical macro structure and dynamics. We discuss ways to probe into the high energy activity from below and make two suggestions: via effective field theory and the correlation hierarchy. We discuss how stochastic behavior at low energy in an effective theory and how correlation noise associated with coarse-grained higher correlation functions in an interacting quantum field could carry nontrivial information about the high energy sector. Finally we describe processes deemed important at the Planck scale, including tunneling and pair creation, wave scattering in random geometry, growth of fluctuations and forms, Planck scale resonance states, and spacetime foams.

Inflation in Kaluza-Klein theory: Relation between the fine-structure constant and the cosmological constant

In this paper we investigate a model of an inflationary universe in Kaluza-Klein theory, which is a four-dimensional de Sitter space plus a one-dimensional compactified internal space. We find that the energy scale for inflation can be predicted from the fine-structure constant in a self-consistent solution of the semi-classical Einstein equations including the Casimir effect. From the observed value of the fine-structure constant, we obtain an energy scale for inflation of $\ensuremath{\epsilon}=1.84\ifmmode\times\else\texttimes\fi{}{10}^{16}{g}_{*}^{1/4}\mathrm{GeV},$ where ${g}_{*}$ is a dimensionless number depending on the spin and number of matter fields existing in the early universe. This value is consistent with the values often discussed for inflation and grand unification. The wave function for this model predicts a high probability for forming such universes, independent of the value of the cosmological constant. The tunneling probability favors the creation of inflationary universes with a compactified dimension, over those with all macroscopic dimensions.

Rotating vacuum wormhole

We investigate whether self-maintained vacuum traversible wormhole can exist described by stationary but nonstatic metric. We consider metric being the sum of static spherically symmetric one and a small nondiagonal component which describes rotation sufficiently slow to be taken into account in the linear approximation. We study semiclassical Einstein equations for this metric with vacuum expectation value of stress-energy of physical fields as the source. In suggestion that the static traversible wormhole solution exists we reveal possible azimuthal angle dependence of angular velocity of the rotation (angular velocity of the local inertial frame) that solves semiclassical Einstein equations. We find that in the macroscopic (in the Plank scale) wormhole case a rotational solution exists but only such that, first, angular velocity depends on radial coordinate only and, second, the wormhole connects the two asymptotically flat spacetimes rotating with angular velocities different in asymptotic regions.

THE REACTION TO THE SCHWARZSCHILD SPACE-TIME BY THE HAWKING EVAPORATION

For the black hole having Hawking radiation, it must have effect on the space-time metric (the so-called back-reaction). By combination of the result from themodynamical approach with the back-reaction of the black hole and the semiclassical Einstein equation, we get the modified Schwarzschild space-time metric. This method is much more simple and convenient than that of Jork's.

Coherence and emergence of classical spacetime

Using the coherent-state representation we show that the classical Einstein equation for the FRW cosmological model with a general minimal scalar field can be derived from the semiclassical quantum Einstein equation.

Exact self-consistent gravitational shock wave in semiclassical gravity

We find a self-consistent $\mathrm{pp}$-gravitational shock wave solution to the semiclassical Einstein equations resulting from the $1/N$ approach to the effective action. We model the renormalized matter stress-energy-momentum tensor by $N$ massless scalar fields in the Minkowski vacuum plus a classical particle. We show that quantum effects generate a milder singularity at the position of the particle than the classical solution, but the singularity does not disappear. At large distances from the particle, the quantum correction decreases slowly, as $1/{\ensuremath{\rho}}^{2}$ $(\ensuremath{\rho}$ being the distance to the particle in the shock wave plane). We argue that this large distance correction is a necessary consequence of quantum gravity.

Problem of unitarity and quantum corrections in semiclassical quantum gravity

Using both the Born-Oppenheimer idea and the de Broglie-Bohm interpretation of wavefunction we represent in a different way the semiclassical quantum gravity from the Wheeler-DeWitt equation in an oscillating regime which can preserve completely the unitary quantum evolution of a matter field at the expense of a nonlinear gravitational field equation, but has the same asymptotic limit as the others. We apply the de Broglie-Bohm interpretation to the nonlinear gravitational field equation to develop a perturbation method to find the quantum corrections of a matter field to the gravity. The semiclassical Einstein equation with the quantum corrections is found for a minimal quantum FRW cosmological model.

Quantum fields interacting with colliding plane waves: the stress-energy tensor and backreaction

Following a previous work on the quantization of a massless scalar field in a space-time representing the head on collision of two plane waves which focus into a Killing-Cauchy horizon, we compute the renormalized expectation value of the stress-energy tensor of the quantum field near that horizon in the physical state which corresponds to the Minkowski vacuum before the collision of the waves. It is found that for minimally coupled and conformally coupled scalar fields the respective stress-energy tensors are unbounded in the horizon. The specific form of the divergences suggests that when the semiclassical Einstein equations describing the backreaction of the quantum fields on the space-time geometry are taken into account, the horizon will acquire a curvature singularity. Thus the Killing-Cauchy horizon which is known to be unstable under “generic” classical perturbations is also unstable by vacuum polarization. The calculation is done following the point-splitting regularization technique. The dynamical colliding wave space-time has four quite distinct space-time regions, namely, one flat region, two single plane wave regions,. and one interaction region. Exact mode solutions of the quantum field equation cannot be found exactly, but the blueshift suffered by the initial modes in the plane wave and interaction regions makes the use of the WKB expansion a suitable method of solution. To ensure the correct regularization of the stress-energy tensor, the initial flat modes propagated into the interaction region must be given to a rather high adiabatic order of approximation.

Semi-Classical Cosmological Models with Scalar, Dirac and Electromagnetic Fields

Exact solutions of the semi-classical Einstein equations for conformally invariant free quantum fields in an homogeneous and isotropic space-time, with cosmological constant and containing a classical scalar field, dust matter, an unquantised Dirac field and electromagnetic radiation are found. The initial behaviour of the semiclassical models is investigated. Some of the solutions found avoid the singularity and do not have particle horizons.

Does back reaction enforce the averaged null energy condition in semiclassical gravity?

The expectation value $〈{T}_{\mathrm{ab}}〉$ of the renormalized stress-energy tensor of quantum fields generically violates the classical, local positive energy conditions of general relativity. Nevertheless, it is possible that $〈{T}_{\mathrm{ab}}〉$ may still satisfy some nonlocal positive energy conditions. Most prominent among these nonlocal conditions is the averaged null energy condition (ANEC), which states that $\ensuremath{\int}〈{T}_{\mathrm{ab}}〉{k}^{a}{k}^{b}d\ensuremath{\lambda}&gt;~0$ along any complete null geodesic, where ${k}^{a}$ denotes the geodesic tangent, with affine parameter $\ensuremath{\lambda}$. If the ANEC holds, then traversable wormholes cannot occur. However, although the ANEC holds in Minkowski spacetime, it is known that the ANEC can be violated in curved spacetimes if one is allowed to choose the spacetime and quantum state arbitrarily, without imposition of the semiclassical Einstein equation, ${G}_{\mathrm{ab}}=8\ensuremath{\pi}〈{T}_{\mathrm{ab}}〉$. In this paper, we investigate whether the ANEC holds for self-consistent solutions of the semiclassical Einstein equation. We study a free, linear, massless scalar field with arbitrary curvature coupling in the context of perturbation theory about the flat spacetime/vacuum solution, and we modify the perturbed semiclassical equations by the "reduction of order" procedure to eliminate spurious solutions. We also restrict attention to the limit in which the length scales determined by the state and metric are much larger than the Planck length. At first order in the metric and state perturbations, and for pure states of the scalar field, we find that the ANEC integral vanishes, as it must for any positivity result to hold. For mixed states, the ANEC integral can be negative. However, we prove that if we average the ANEC integral transverse to the geodesic, using a suitable Planck scale smearing function, a strictly positive result is obtained in all cases except for the trivial flat spacetime/vacuum solution. Similar results hold for pure states at second order in perturbation theory, when we additionally specialize to the situation where incoming classical gravitational radiation does not dominate the first-order metric perturbation. These results suggest---in agreement with conclusions drawn by Ford and Roman from entirely independent arguments---that if traversable wormholes do exist as self-consistent solutions of the semiclassical equations, they cannot be macroscopic but must be "Planck scale." In the course of our analysis, we investigate a number of more general issues relevant to doing perturbative expansions of the semiclassical equations off of flat spacetime, including an analysis of the nature of the semiclassical Einstein equation and of prescriptions for extracting physically relevant solutions. A large portion of our paper is devoted to the treatment of these more general issues.

Black hole in thermal equilibrium with a spin-2 quantum field.

An approximate form for the vacuum averaged stress-energy tensor of a conformal spin-2 quantum field on a black hole background is employed as a source term in the semiclassical Einstein equations. Analytic corrections to the Schwarzschild metric are obtained to first order in $\epsilon = {\hbar}/M^2$, where $M$ denotes the mass of the black hole. The approximate tensor possesses the exact trace anomaly and the proper asymptotic behavior at spatial infinity, is conserved with respect to the background metric and is uniquely defined up to a free parameter $\hat c_2$, which relates to the average quantum fluctuation of the field at the horizon. We are able to determine and calculate an explicit upper bound on $\hat c_2$ by requiring that the entropy due to the back-reaction be a positive increasing function in $r$. A lower bound for $\hat c_2$ can be established by requiring that the metric perturbations be uniformly small throughout the region $2M \leq r < r_o$, where $r_o$ is the radius of perturbative validity of the modified metric. Additional insight into the nature of the perturbed spacetime outside the black hole is provided by studying the effective potential for test particles in the vicinity of the horizon.

Semiclassical back reaction in the formation of a straight cosmic string.

A simple model for the formation of a straight cosmic string, wiggly or unperturbed is considered. The gravitational field of such string is computed in the linear approximation. The vacuum expectation value of the stress tensor of a massless scalar quantum field coupled to the string gravitational field is computed to the one loop order. Finally, the back-reaction effect on the gravitational field of the string is obtained by solving perturbatively the semiclassical Einstein's equations.

Relevance of induced gauge interactions in decoherence.

Decoherence in quantum cosmology is shown to occur naturally in the presence of induced geometric gauge interactions associated with particle production.A new 'gauge '-variant form of the semiclassical Einstein equations is also presented which makes the non-gravitating character of the vacuum polarisation energy explicit.

Decoherence of Friedmann-Robertson-Walker geometries in the presence of massive vector fields with U(1) or SO(3) global symmetries

Retrieval of classical behaviour in quantum cosmology is usually discussed in the framework of midi superspace models in the presence of scalar fields and the inhomogeneous model to either to gravitational or scalar fields. In this work, we propose an alternative model to study the decoherence of homogeneous and isotropic geometries where the scalar field is replaced by a massive vector field with a global internal symmetry. We study here the cases with U(1) and SO(3) global internal symmetries. The presence of a mass term breaks the conformal invariance and allows for the longitudinal modes of the spin-1 field to be present in the Wheeler-DeWitt equation. In the case of the U(1) global internal symmetry, we have only one single “classical” degree of freedom while in the case of the SO(3) global symmetry, we are led to consider a simple two-dimensional minisuperspace model. These minisuperspaces are shown to be equivalent to a set of coupled harmonic oscillators where the kinetic term of the longitudinal modes has a coefficient proportional to the inverse of the scale factor. The conditions for a suitable decoherence process and correlations between coordinates and momenta are established. The validity of the semi-classical Einstein equations when massive vector fields (Abelian and non-Abelian) are present is also discussed.

Can vacuum energy gravitate?

In this essay we discuss an interesting recent development in semiclassical gravity. Using an improved Born-Oppenheimer approximation, the semiclassical reduction of the Wheeler-DeWitt equation turns out to give important insights into the nature and the level of validity of the semi-classical Einstein equations (SCEE). Back reactions from the quantized matter fields in SCEE are shown to be completely determined by adiabatically induced geometricU(N) gauge potentials. The finite energy from the vacuum polarization, in particular, is found to be intimately related to the ‘magnetic’ type geometric gauge potential. As a result the vacuum energy in a universe emerging from a ‘source-free’ flat simply-connected superspace is gauge equivalent to zero, leading to some dramatic consequences.