Abstract
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.
Highlights
There is as yet no full theoretical description of quantum gravity, it is widely accepted that such a theory should admit a semi-classical limit, where the quantum aspects of the gravitational field become negligible
Using the simplified stress tensor we see that E(φ) > 0 for all nonzero solutions φ with space-like compact support as soon as m2 + ξR ≥ 0 everywhere and either Σ is non-compact or m2 + ξR > 0 somewhere. (Note that [29] uses the simplified formula for the total energy, but without the justification that we provide in Appendix A.)
Systems in semi-classical gravity, like the Einstein–Klein–Gordon system that we studied here, typically involve renormalisation parameters that cannot be determined without further input, either from observations or from an underlying theory of quantum gravity
Summary
There is as yet no full theoretical description of quantum gravity, it is widely accepted that such a theory should admit a semi-classical limit, where the quantum aspects of the gravitational field become negligible. For free fields the quantum stress tensor can be defined rigorously, using a local and generally covariant renormalisation scheme, but even in this case there are ambiguities in its definition, parametrised by renormalisation constants. In that section we will study the ground states in an Einstein static universe in some detail and Eq (24) gives an explicit expression for the two-point distribution in the conformally coupled case. 3 we will analyse all quasi-free states that respect all the symmetries of the spacetime and we characterise their two-point distributions in terms of Gegenbauer polynomials on S3 This will allow us to determine the set of solutions to the semi-classical Einstein Eq (1) rather explicitly.
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