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Linear Response, Validity of Semi-Classical Gravity, and the Stability of Flat Space

Paul R. Anderson·Carmen Molina-Paris·Emil Mottola
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Abstract

A quantitative test for the validity of the semi-classical approximation in gravity is given. The criterion proposed is that solutions to the semi-classical Einstein equations should be stable to linearized perturbations, in the sense that no gauge invariant perturbation should become unbounded in time. A self-consistent linear response analysis of these perturbations, based upon an invariant effective action principle, necessarily involves metric fluctuations about the mean semi-classical geometry, and brings in the two-point correlation function of the quantum energy-momentum tensor in a natural way. This linear response equation contains no state dependent divergences and requires no new renormalization counterterms beyond those required in the leading order semi-classical approximation. The general linear response criterion is applied to the specific example of a scalar field with arbitrary mass and curvature coupling in the vacuum state of Minkowski spacetime. The spectral representation of the vacuum polarization function is computed in n dimensional Minkowski spacetime, and used to show that the flat space solution to the semi-classical Einstein equations for n=4 is stable to all perturbations on distance scales much larger than the Planck length.

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Paul R. Anderson, Carmen Molina-Paris, Emil Mottola. 2002-09-21. Linear Response, Validity of Semi-Classical Gravity, and the Stability of Flat Space. https://doi.org/10.1103/physrevd.67.024026

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