The Hadamard condition on a Cauchy surface and the renormalized stress-energy tensor
arXiv:2406.01498 · doi:10.1088/1475-7516/2024/10/002
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 `' and `' 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.
65 pages, discussion improved
References in corpus (8)
- Dynamical Backreaction in Robertson-Walker Spacetime
- On the initial value problem for semiclassical gravity without and with quantum state collapses
- Evaporation of four-dimensional dynamical black holes sourced by the quantum trace anomaly
- Special cosmological models derived from the semiclassical Einstein equation on flat FLRW space-times
- Linear stability of semiclassical theories of gravity
- Semiclassical gravity with a conformally covariant field in globally hyperbolic spacetimes
- Hadamard states on spherically symmetric characteristic surfaces, the semi-classical Einstein equations and the Hawking effect
- Quantum strong cosmic censorship and black hole evaporation