Linearized gravity and gauge conditions

Abstract

In this paper we consider the field equations for linearized gravity and other integer spin fields on the Kerr spacetime, and more generally on spacetimes of Petrov type D. We give a derivation, using the GHP formalism, of decoupled field equations for the linearized Weyl scalars for all spin weights and identify the gauge source functions occuring in these. For the spin weight 0 Weyl scalar, imposing a generalized harmonic coordinate gauge yields a generalization of the Regge-Wheeler equation. Specializing to the Schwarzschild case, we derive the gauge invariant Regge-Wheeler and Zerilli equation directly from the equation for the spin 0 scalar.

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

“…In the a = 0 case, the middle component of the Maxwell field, φ 0 , satisfies the Fackerell-Ipser equation [16], and, the middle component of the linearised curvature satisfies a very similar equation [1]. In addition, the extreme components for both the Maxwell and linearised Einstein equations also satisfy second-order PDEs, known as the Teukolsky equations [27].…”

“…Recall that we consider the exterior region of the Kerr space-time, which is a manifold parameterised by (t, r, ω) ∈ R × (r + , ∞) × S 2 with the metric given in equation (1). As is well known [18], this can be uniquely extended to a maximal analytic extension, which, in turn, has a C 0 conformal compactification.…”

Generalized wave operators, weighted Killing fields, and perturbations of higher dimensional spacetimes

“…acting on type {b = p/2, s = p/2} GHP quantities, and the corresponding modified wave operator [6,1] T p := g αβ D α D β (2.2) (note that T 0 = ). Define the 2-forms…”

“…Note that the usual D'Alembertian is included in T 2b , since T 0 = = g αβ ∇ α ∇ β . The operator T 2b for the 4-dimensional Teukolsky equations on the Kerr spacetime was found in [6]; see also [1] for the treatment of all vacuum type D solutions. Introducing certain "potentials" 2 for the spin-s fields, one can formulate the left hand side of (1.1) in terms of self-adjoint operators, and then following Wald's adjoint operator technique (introduced in [33]), it is possible to take the adjoint identity and reconstruct solutions of the field equations from solutions of the scalar equations; this way, stability questions can be approached by the study of a scalar, wave-like equation instead of the more complex tensor field equations.…”

Symmetries and Hidden Symmetries for Fields Outside Black Holes

“…However, these are secondorder equations, and, when put in the form (∂ α L αβ (s)∂ β + W (s))(φ ±s ) = 0, the matrix of coefficients L is not symmetric, which prevents most of the standard tools of hyperbolic PDE from being applied. The middle Maxwell component φ 0 satisfies the equation [16] (∇ α ∇ α + 2M/p 3 )(pφ 0 ) = 0 with p = r + ia cos θ, and it has recently been shown that, for the linearised Einstein equation (in an appropriate gauge) [17], (∇ α ∇ α + 8M/p 3 )(p 2 φ 0 ) = 0. All of these equations can be separated.…”