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Covariant Schwarzschild perturbations I: Initial value formulation for scalars of spin-weight -+ 2

R. B. Burston·A. W. C. Lun
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Abstract

We consider full perturbations to a covariantly defined Schwarzschild spacetime. By constructing complex quantities, we derive two decoupled, covariant and gauge-invariant, wave-like equations for spin-weighted scalars. These arise naturally from the Bianchi identities and comprise a covariant representation of the Bardeen-Press equations for scalars with spin-weight $\pm2$. Furthermore, the covariant and gauge-invariant 1+1+2 formalism is employed, and consequently, the physical interpretation of the energy-momentum perturbations is transparent. They are written explicitly in terms of the energy-momentum specified on spacelike three-slices. Ultimately, a Cauchy problem is constructed whereby, an initial three-slice may be perturbed by an energy-momentum source, which induces resultant gravitational fields.

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R. B. Burston, A. W. C. Lun. 2006-11-08. Covariant Schwarzschild perturbations I: Initial value formulation for scalars of spin-weight -+ 2. https://arxiv.org/abs/gr-qc/0611052

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