New identities for linearized gravity on the Kerr spacetime

Abstract

In this paper we derive a differential identity for linearized gravity on the Kerr spacetime and more generally on vacuum spacetimes of Petrov type D. We show that a linear combination of second derivatives of the linearized Weyl tensor can be formed into a complex symmetric 2-tensor ${\mathcal{M}}_{ab}$ which solves the linearized Einstein equations. The identity makes this manifest by relating ${\mathcal{M}}_{ab}$ to two terms solving the linearized Einstein equations by construction. The self-dual Weyl curvature of ${\mathcal{M}}_{ab}$ gives a covariant version of the Teukolsky-Starobinsky identities for linearized gravity which, in addition to the two classical identities for linearized Weyl scalars with extreme spin weights, includes three additional equations. In particular, they are not consequences of the classical Teukolsky-Starobinsky identities, but are additional integrability conditions for linearized gravity. The result has direct application in the construction of symmetry operators and also yields a set of non-trivial gauge invariants for linearized gravity.

Operators now without valence indices. 20 pages

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Cited by in corpus (25)

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• Non-commutative Geometry from Perturbative Quantum Gravity
• Metric perturbations of Kerr spacetime in Lorenz gauge: Circular equatorial orbits
• Symmetries of linearized gravity from adjoint operators
• Implementation of a GHZ-Teukolsky puncture scheme for gravitational self-force calculations
• Compatibility complex for black hole spacetimes
• Price's law for spin fields on a Schwarzschild background
• Flux-balance laws for spinning bodies under the gravitational self-force
• Killing Operator for the Kerr Metric
• Almost Price's law in Schwarzschild and decay estimates in Kerr for Maxwell field
• A Mathematical Comparison of the Schwarzschild and Kerr Metrics
• A class of conserved currents for linearized gravity in the Kerr spacetime
• Sourced metric perturbations of Kerr spacetime in Lorenz gauge
• Two-dimensional twistor manifolds and Teukolsky operators
• A space-time calculus based on symmetric 2-spinors
• Minimum Resolution of the Minkowski, Schwarzschild and Kerr Differential Modules
• Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations
• Homogeneous Symmetry Operators in Kerr--NUT--AdS Spacetimes
• Wave and Dirac equations on manifolds
• Metric reconstruction and the Hamiltonian for eccentric, precessing binaries in the small-mass-ratio limit
• Precise asymptotics of the spin $+2$ Teukolsky field in the Kerr black hole interior
• Perturbations of Kerr with approximate Killing spinors -- Wave version