Decay of the Maxwell Field on the Schwarzschild Manifold

Abstract

We study solutions of the decoupled Maxwell equations in the exterior region of a Schwarzschild black hole. In stationary regions, where the Schwarzschild coordinate r ranges over 2M < r1 < r < r2, we obtain a decay rate of t −1 for all components of the Maxwell field. We use vector field methods and do not require a spherical harmonic decomposition.In outgoing regions, where the Regge-Wheeler tortoise coordinate is large, r * > ǫt, we obtain decay for the null components with rates of |φ+| ∼ |α| < Cr −5/2 , |φ0| ∼ |ρ| + |σ| < Cr −2 |t − r * | −1/2 , and |φ−1| ∼ |α| < Cr −1 |t − r * | −1 . Along the event horizon and in ingoing regions, where r * < 0, and when t + r * > 1, all components (normalized with respect to an ingoing null basis) decay at a rate of Cu+ −1 with u+ = t + r * in the exterior region.In Einstein's equations, the energy-momentum tensor of the matter fields should influence the curvature. By decoupled, we mean that the electromagnetic field does not influence the Schwarzschild solution, which is taken as a fixed background manifold. We call the Schwarzschild solution the Schwarzschild manifold and use the word solution to refer to solutions of the Maxwell equations (1)-(2).Since F is a tensor, there is no coordinate independent norm with which to measure it (or, at least, not all components of it). To discuss the decay of F , we make a choice of basis and show that the corresponding components decay. A simple choice of basis consists of the coordinate vector fields rescaled so that they have unit length (|g(X, X)| = 1). The rescaled vectors arêGiven a time-like vector, there is a natural decomposition of the Maxwell field into electric and magnetic components. Since the Schwarzschild manifold has a time-translation symmetry, this provides a natural choice of time-like direction,T . The corresponding electric and magnetic components are

Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior

“…Our earlier works [2,8] provide a more complete description of this. Here we briefly summarise some of the major results.…”

“…Energy and Morawetz estimates have also been used to prove pointwise decay in the full exterior region [8]. Energy and Morawetz estimates have also been proved on arbitrary spherically symmetric black hole space-times, independently of whether they satisfy the Einstein equation [25].…”

“…Here, the components are indexed by spin weight, following the convention of Price [22] and used in [8]. This differs from the indexing convention used by almost the entire rest of the physical literature, in which, ignoring powers of √ 2, the quantities φ −1 , φ 0 , and φ 1 are denoted −φ N P,2 , φ N P,1 , and φ N P,0 .…”

“…(3) There are stationary solutions. In the Schwarzschild case, the stationary solutions are the only spherically symmetric solutions, so that, after projecting out the stationary solutions, one can use an additional positivity arising from the angular derivatives to counteract the influence of the potential in the Morawetz estimate [8]. Although, the stationary solutions are not exactly spherically symmetric in the a = 0 case, there is still a lower bound for the charge-free solutions in terms of their angular derivatives, which we state in lemma 3.6.…”

Asymptotic Behavior of Massless Dirac Waves in Schwarzschild Geometry

“…Recently, in [17], Finster and Smoller obtained decay in L ∞ loc for the Teukolsky equation for spin s = 1/2, 1 and 2 (corresponding to Dirac equations, Maxwell equations, gravitational perturbations) in the Schwarzschild geometry. The decay of Maxwell's equations in the Schwarzschild geometry was studied in [6]. In this paper, we consider massless Dirac waves in the Schwarzschild geometry, and we prove that the solutions decay to zero at a rate t −2λ where λ = 1, 2, · · · is the angular momentum.…”