Abstract
In general relativity, perturbation theory about a background solution fails if the background spacetime has a Killing symmetry and a compact spacelike Cauchy surface. This failure, dubbed as {\it linearization instability}, shows itself as non-integrability of the perturbative infinitesimal deformation to a finite deformation of the background. Namely, the linearized field equations have spurious solutions which cannot be obtained from the linearization of exact solutions. In practice, one can show the failure of the linear perturbation theory by showing that a certain quadratic (integral) constraint on the linearized solutions is not satisfied. For non-compact Cauchy surfaces, the situation is different and for example, Minkowski space having a non-compact Cauchy surface, is linearization stable. Here we study, the linearization instability in generic metric theories of gravity where Einstein's theory is modified with additional curvature terms. We show that, unlike the case of general relativity, for modified theories even in the non-compact Cauchy surface cases, there are some theories which show linearization instability about their anti-de Sitter backgrounds. Recent $D$ dimensional critical and three dimensional chiral gravity theories are two such examples. This observation sheds light on the paradoxical behavior of vanishing conserved charges (mass, angular momenta) for non-vacuum solutions, such as black holes, in these theories.
Highlights
There is a very interesting conundrum in nonlinear theories, such as Einstein’s gravity or its modifications with higher curvature terms: exact solutions without symmetries are hard to find, one resorts to symmetric “background” solutions and develops a perturbative expansion about them
Our goal here is to extend the discussion to generic gravity theories: we show that if the field equations of the theory are defined by the Einstein tensor plus a covariantly conserved two tensor, a new source of linearization instability that does not exist in general relativity (GR) arises, especially in de Sitter or Anti-de Sitter backgrounds, with noncompact Cauchy surfaces
As discussed in the previous section, vanishing of the constant c leads to two strong constraints (41) and (42) on the linearized solution h which is a statement of the instability of the background solution
Summary
There is a very interesting conundrum in nonlinear theories, such as Einstein’s gravity or its modifications with higher curvature terms: exact solutions without symmetries (which are physically interesting) are hard to find, one resorts to symmetric “background” solutions and develops a perturbative expansion about them. Our goal here is to extend the discussion to generic gravity theories: we show that if the field equations of the theory are defined by the Einstein tensor plus a covariantly conserved two tensor, a new source of linearization instability that does not exist in GR arises, especially in de Sitter or Anti-de Sitter backgrounds, with noncompact Cauchy surfaces. This happens because in these backgrounds there are special critical points in the space of parameters of the theory which conspire to cancel the conserved charge (mass, angular momentum etc) of nonperturbative objects (black holes) or the energies of the perturbative excitations. In a forthcoming paper, we shall give a more detailed analysis of the chiral gravity discussion in the initial value formulation context
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