1. The Hadamard condition on a Cauchy surface and the renormalized stress-energy tensor
- Author
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Juárez-Aubry, Benito A., Kay, Bernard S., Miramontes, Tonatiuh, and Sudarsky, Daniel
- Subjects
General Relativity and Quantum Cosmology ,High Energy Physics - Theory ,Mathematical Physics - 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., Comment: 65 pages, discussion improved
- Published
- 2024
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