The Hadamard condition on a Cauchy surface and the renormalized stress-energy tensor

TL;DR

This paper demonstrates that the renormalized stress-energy tensor in curved spacetime can be fully determined by the geometry of a Cauchy surface and the first four time derivatives of the metric off the surface, based on Hadamard state conditions.

Contribution

It provides a detailed covariant expansion showing the dependence of the stress-energy tensor on geometric data and derivatives up to fourth order, clarifying initial value problem aspects in semiclassical gravity.

Findings

The expectation value depends on the surface geometry and first four derivatives of the metric.

The fourth derivative of the metric can be derived from field equations.

The result supports initial value formulations in semiclassical gravity.

Abstract

Given a Cauchy surface in a curved spacetime and a suitably defined quantum state on the CCR algebra of the Klein-Gordon quantum field on that surface, we show, by expanding the squared spacetime geodesic distance and the `$U$' and `$V$' Hadamard coefficients (and suitable derivatives thereof) in sufficiently accurate covariant Taylor expansions on the surface that the renormalized expectation value of the quantum stress-energy tensor on the surface is determined by the geometry of the surface and the first 4 time derivatives of the metric off the surface, in addition to the Cauchy data for the field's two-point function. This result has been anticipated in and is motivated by a previous investigation by the authors on the initial value problem in semiclassical gravity, for which the geometric initial data corresponds {\it a priori} to the metric on the surface and up to 3 time derivatives off the surface, but where it was argued that the fourth derivative can be obtained with aid of the field equations on the initial surface.