A note on energy currents and decay for the wave equation on a Schwarzschild background

  • Mihalis Dafermos, Igor Rodnianski
  • Published 2008

Abstract

In recent work, we have proven uniform decay bounds for solutions of the wave equation 2gφ = 0 on a Schwarzschild exterior, in particular, the uniform pointwise estimate |φ| ≤ Cv + which holds throughout the domain of outer communications, where v is an advanced EddingtonFinkelstein coordinate, v+ . = max{v, 1}, and C is a constant depending on a Sobolev norm of initial data. A crucial estimate in the proof required a decomposition into spherical harmonics. We here give an alternative proof of this estimate not requiring such a decomposition. In [3], we studied the problem of decay for general solutions φ of the equation 2gφ = 0 (1) on a Schwarzschild background. The estimates of [3] were obtained by exploiting compatible currents associated with vector field mutipliers applied to the energy momentum tensor. (See [2] for a general discussion of such currents.) Understanding the decay properties of solutions of (1) in terms of such energy estimates appears to be a fundamental first step, if one is ever to address the problem of non-linear stability of black hole solutions of the Einstein equations of general relativity. A crucial role in the results of [3] is played by an energy current Jμ related to vector fields of the form f(r)∂r∗ , where r ∗ is a Regge-Wheeler coordinate. For the current Jμ constructed in [3], the divergence K = ∇Jμ was shown to be nonnegative upon integration over spheres of symmetry. (The integral of this current over an arbitrary spacetime region R was denoted in [3] by IX(R).) The construction of Jμ was quite elaborate. In particular, a decomposition into spherical harmonics was required, and a separate definition of fl was made for each spherical harmonic, characterized by a non-negative integer l. These currents were then summed to obtain a total current. As l → ∞, the unique University of Cambridge, Department of Pure Mathematics and Mathematical Statistics, Wilberforce Road, Cambridge CB3 0WB United Kingdom Princeton University, Department of Mathematics, Fine Hall, Washington Road, Princeton, NJ 08544 United States

Cite this paper

@inproceedings{Dafermos2008ANO, title={A note on energy currents and decay for the wave equation on a Schwarzschild background}, author={Mihalis Dafermos and Igor Rodnianski}, year={2008} }