Schwarzschild metric | EPFL Graph Search

In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, angular momentum of the mass, and universal cosmological constant are all zero. The solution is a useful approximation for describing slowly rotating astronomical objects such as many stars and planets, including Earth and the Sun. It was found by Karl Schwarzschild in 1916, and around the same time independently by Johannes Droste, who published his more complete and modern-looking discussion four months after Schwarzschild. According to Birkhoff's theorem, the Schwarzschild metric is the most general spherically symmetric vacuum solution of the Einstein field equations. A Schwarzschild black hole or static black hole is a black hole that has neither electric charge nor angular momentum. A Schwarzsch

On pointwise decay of linear waves on a Schwarzschild black hole background

Roland Donninger

We prove sharp pointwise

t^{-3}

decay for scalar linear perturbations of a Schwarzschild black hole without symmetry assumptions on the data. We also consider electromagnetic and gravitational perturbations for which we obtain decay rates

t^{-4}

, and

t^{-6}

, respectively. We proceed by decomposition into angular momentum

\ell

and summation of the decay estimates on the Regge-Wheeler equation for fixed

\ell

. We encounter a dichotomy: the decay law in time is entirely determined by the asymptotic behavior of the Regge-Wheeler potential in the far field, whereas the growth of the constants in

\ell

is determined entirely by the behavior of the Regge-Wheeler potential in a small neighborhood around the photon sphere. In other words, the tails are controlled by small energies, whereas the number of angular derivatives needed on the data is determined by energies close to the top of the Regge-Wheeler potential. This dichotomy corresponds to the well-known principle that for initial times the decay reflects the presence of complex resonances generated by the potential maximum, whereas for later times the tails are determined by the far field. However, we do not invoke complex resonances at all, but rely instead on semiclassical Sigal-Soffer type propagation estimates based on a Mourre bound near the top energy.

Springer Verlag2012

Decay Estimates for the One-dimensional Wave Equation with an Inverse Power Potential

Roland Donninger

We study the wave equation on the real line with a potential that falls off like vertical bar x vertical bar(-alpha) for vertical bar x vertical bar -> infinity where 2 < alpha infinity provided that there are no resonances at zero energy and no bound states. As an application, we consider the l = 0 Price Law for Schwarzschild black holes. This paper is part of our investigations into decay of linear waves on a Schwarzschild background, see [5, 6].

Oxford University Press2010

A proof of Price's Law on Schwarzschild black hole manifolds for all angular momenta

Roland Donninger

Price's Law states that linear perturbations of a Schwarzschild black hole fall off as

t^{-2\ell-3}

for

t \to \infty

provided the initial data decay sufficiently fast at spatial infinity. Moreover, if the perturbations are initially static (i.e., their time derivative is zero), then the decay is predicted to be

t^{-2\ell-4}

. We give a proof of

t^{-2\ell-2}

decay for general data in the form of weighted

L^1

L^\infty

bounds for solutions of the Regge--Wheeler equation. For initially static perturbations we obtain

t^{-2\ell-3}

. The proof is based on an integral representation of the solution which follows from self--adjoint spectral theory. We apply two different perturbative arguments in order to construct the corresponding spectral measure and the decay bounds are obtained by appropriate oscillatory integral estimates.

2011

On pointwise decay of linear waves on a Schwarzschild black hole background

Roland Donninger

We prove sharp pointwise

t^{-3}

t^{-4}

, and

t^{-6}

, respectively. We proceed by decomposition into angular momentum

\ell

and summation of the decay estimates on the Regge-Wheeler equation for fixed

\ell

. We encounter a dichotomy: the decay law in time is entirely determined by the asymptotic behavior of the Regge-Wheeler potential in the far field, whereas the growth of the constants in

\ell

Springer Verlag2012

A proof of Price's Law on Schwarzschild black hole manifolds for all angular momenta

Roland Donninger

Price's Law states that linear perturbations of a Schwarzschild black hole fall off as

t^{-2\ell-3}

for

t \to \infty

provided the initial data decay sufficiently fast at spatial infinity. Moreover, if the perturbations are initially static (i.e., their time derivative is zero), then the decay is predicted to be

t^{-2\ell-4}

. We give a proof of

t^{-2\ell-2}

decay for general data in the form of weighted

L^1

L^\infty

bounds for solutions of the Regge--Wheeler equation. For initially static perturbations we obtain

t^{-2\ell-3}

2011