Generating exact solutions to Einstein's equation using linearized approximations

Abstract

We show that certain solutions to the linearized Einstein equation can---by the application of a particular type of linearized gauge transformation---be straightforwardly transformed into solutions of the exact Einstein equation. In cases with nontrivial matter content, the exact stress-energy tensor of the transformed metric has the same eigenvalues and eigenvectors as the linearized stress-energy tensor of the initial approximation. When our gauge exists, the tensorial structure of transformed metric perturbations identically eliminates all nonlinearities in Einstein's equation. As examples, we derive the exact Kerr and gravitational plane wave metrics from standard harmonic-gauge approximations.

13 pages, 1 figure. Corrected coefficients in Sect. VC

References in corpus (4)

• The use of Generalised Functions and Distributions in General Relativity
• Kerr-Schild Structure and Harmonic 2-forms on (A)dS-Kerr-NUT Metrics
• Flat deformation theorem and symmetries in spacetime
• The not-so-nonlinear nonlinearity of Einstein's equation

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• Post-Newtonian Waveforms from Spinning Scattering Amplitudes
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• Second-order gravitational self-force in a highly regular gauge: Covariant and coordinate punctures
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• Nonlinear phenomena in general relativity
• Newman-Janis Algorithm from Taub-NUT Instantons