Abstract
In a nonlinear theory, such as General Relativity, linearized field equations around an exact solution are necessary but not sufficient conditions for linearized solutions. Therefore, the linearized field equations can have some solutions which do not come from the linearization of possible exact solutions. This fact can make the perturbation theory ill-defined, which would be a problem both at the classical and semiclassical quantization level. Here we study the first and second order perturbation theory in cosmological Einstein gravity and give the explicit form of the integral constraint, which is called the Taub charge, on the first order solutions for spacetimes with a Killing symmetry and a compact hypersurface without a boundary.
Highlights
Let us consider a generic gravity theory defined by the nonlinear field equationsEμνðgÞ 1⁄4 0; ð1Þ in some local coordinates
If (9) is not satisfied, one speaks of a linearization instability. This issue was studied in various aspects in [7,8,9,10,11,12,13] for Einstein’s theory and summarized in [14,15]; and extended to generic gravity theories more recently [1,2,16]. From these works two main conclusions follow: first, in Einstein’s theory, a solution set to the constrained initial data on a compact Cauchy surface without a boundary may not have nearby solutions, they can be isolated and perturbations are not allowed; second, for generic gravity theories in asymptotically de Sitter spacetimes, linearization instability arises for certain combinations of the parameters defining the theory
If the background metric g, about which perturbation theory is performed, has Killing symmetries, there are constraints to the first order perturbation theory coming from the second order perturbation theory
Summary
Let us consider a generic gravity theory defined (in a vacuum) by the nonlinear field equations. This issue was studied in various aspects in [7,8,9,10,11,12,13] for Einstein’s theory and summarized in [14,15]; and extended to generic gravity theories more recently [1,2,16] From these works two main conclusions follow: first, in Einstein’s theory, a solution set to the constrained initial data on a compact Cauchy surface without a boundary may not have nearby solutions, they can be isolated and perturbations are not allowed; second, for generic gravity theories in asymptotically (anti) de Sitter spacetimes, linearization instability arises for certain combinations of the parameters defining the theory. IV we discuss the gauge invariance issue and relegate some of the computations to the Appendices
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