# The wave equation on Schwarzschild-de Sitter spacetimes

@article{Dafermos2007TheWE, title={The wave equation on Schwarzschild-de Sitter spacetimes}, author={Mihalis Dafermos and I. Rodnianski}, journal={arXiv: General Relativity and Quantum Cosmology}, year={2007} }

We consider solutions to the linear wave equation $\Box_g\phi=0$ on a non-extremal maximally extended Schwarzschild-de Sitter spacetime arising from arbitrary smooth initial data prescribed on an arbitrary Cauchy hypersurface. (In particular, no symmetry is assumed on initial data, and the support of the solutions may contain the sphere of bifurcation of the black/white hole horizons and the cosmological horizons.) We prove that in the region bounded by a set of black/white hole horizons and… Expand

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#### References

SHOWING 1-10 OF 29 REFERENCES

Linear stability of Schwarzschild under perturbations which are non-vanishing on the bifurcation 2-sphere

- Physics
- 1987

The authors prove boundedness on an exterior Schwarzschild wedge for Cinfinity solutions of the covariant Klein-Gordon equation which have compact support on Cauchy surfaces in Kruskal spacetime.… Expand

Telling tails in the presence of a cosmological constant

- Physics
- 1997

We study the evolution of massless scalar waves propagating on spherically symmetric spacetimes with a nonzero cosmological constant. Considering test fields on both Schwarzschild\char21{}de Sitter… Expand

Nonspherical perturbations of relativistic gravitational collapse

- Physics
- 1971

It is known that there can be no gravitational, electromagnetic, or scalar field perturbations (except angular momentum) of a Schwarzschild black hole. A gravitationally collapsing star with… Expand

Late-time behavior of stellar collapse and explosions. I. Linearized perturbations.

- Physics, Medicine
- Physical review. D, Particles and fields
- 1994

Analysis of the power-law tails in the evolution of massless fields around a fixed background geometry corresponding to a black hole confirms their existence and suggests the behavior of the full nonlinear dynamics, which is studied numerically in a companion paper. Expand

A proof of Price’s law for the collapse of a self-gravitating scalar field

- Mathematics, Physics
- 2005

A well-known open problem in general relativity, dating back to 1972, has been to prove Price’s law for an appropriate model of gravitational collapse. This law postulates inverse-power decay rates… Expand

Cosmological Event Horizons, Thermodynamics, and Particle Creation

- Physics
- 1977

It is shown that the close connection between event horizons and thermodynamics which has been found in the case of black holes can be extended to cosmological models with a repulsive cosmological… Expand

Decay and Non-Decay of the Local Energy for the Wave Equation on the De Sitter–Schwarzschild Metric

- Mathematics, Physics
- 2007

We describe an expansion of the solution of the wave equation on the De Sitter–Schwarzschild metric in terms of resonances. The principal term in the expansion is due to a resonance at 0. The error… Expand

Uniform Decay of Local Energy and the Semi-Linear Wave Equation on Schwarzschild Space

- Mathematics, Physics
- 2006

AbstractWe provide a uniform decay estimate for the local energy of general solutions to the inhomogeneous wave equation on a Schwarzschild background. Our estimate implies that such solutions have… Expand

Effects of a nonvanishing cosmological constant on the spherically symmetric vacuum manifold

- Physics
- 1977

The conformal structure of timelike 2-surfaces of the vacuum manifold is examined in detail in the cases when the cosmological constant, ..lambda.., is not assumed to be zero. For ..lambda.. > 0 with… Expand

CMC-Slicings of Kottler-Schwarzschild-de Sitter Cosmologies

- Physics
- 2005

There is constructed, for each member of a one-parameter family of cosmological models, which is obtained from the Kottler-Schwarzschild-de Sitter spacetime by identification under discrete… Expand