Linearization Instability for Generic Gravity in AdS

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.

arXiv comment31 pages, 1 figure, some grammatical typos are corrected

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• Energy in Higher-Derivative Gravity via Topological Regularization
• Bachian Gravity in Three Dimensions
• Einstein-Yang-Mills Theory: Gauge Invariant Charges and Linearization Instability
• Linearization Instability of Chiral Gravity
• Second Order Perturbation Theory in General Relativity: Taub Charges as Integral Constraints
• Consistency conditions for O-plane unsmearing from second-order perturbation theory
• Weyl Covariant Quadratic Curvature Gravity in 3-Dimensional Riemann-Cartan-Weyl Space-Times
• Linearization (in)stabilities and crossed products
• Second Order Gauge Invariant Perturbation Theory and Conserved Charges in Cosmological Einstein's Gravity
• Topologically Massive Gravity: Anyon Scattering, Weyl-Gauging and Causality