Semiclassical gravity with a conformally covariant field in globally hyperbolic spacetimes

TL;DR

This paper proves that semiclassical gravity with a conformally coupled massless field is well-posed in globally hyperbolic, conformally static spacetimes, ensuring unique and stable solutions under appropriate initial conditions, even beyond symmetric cases.

Contribution

It establishes the existence and stability of solutions to semiclassical gravity equations with conformal coupling in a broad class of spacetimes without symmetry assumptions.

Findings

Unique and stable solutions exist for the coupled system.

Solutions are valid beyond symmetric spacetime cases.

Initial data constraints ensure well-posedness.

Abstract

We prove that semiclassical gravity in conformally static, globally hyperbolic spacetimes with a massless, conformally coupled Klein-Gordon field is well posed, when viewed as a coupled theory for the dynamical conformal factor of the metric and the Klein-Gordon theory. Namely, it admits unique and stable solutions whenever constrained fourth-order initial data for the conformal factor and suitably defined Hadamard initial data for the Klein-Gordon state are provided on a spacelike Cauchy surface. As no spacetime symmetries are imposed on the conformal factor, the present result implies that, provided constrained initial data exists, there also exist exact solutions to the semiclassical gravity equations beyond the isotropic, homogeneous or static cases.