New identities for linearized gravity on the Kerr spacetime

Abstract

In this paper we derive a differential identity for linearized gravity on the Kerr spacetime and more generally on vacuum spacetimes of Petrov type D. We show that a linear combination of second derivatives of the linearized Weyl tensor can be formed into a complex symmetric 2-tensor M ab which solves the linearized Einstein equations. The identity makes this manifest by relating M ab to two terms solving the linearized Einstein equations by construction. The self-dual Weyl curvature of M ab gives a covariant version of the Teukolsky-Starobinsky identities for linearized gravity which, in addition to the two classical identities for linearized Weyl scalars with extreme spin weights, includes three additional equations. In particular, they are not consequences of the classical Teukolsky-Starobinsky identities, but are additional integrability conditions for linearized gravity. The result has direct application in the construction of symmetry operators and also yields a set of non-trivial gauge invariants for linearized gravity. * steffen.aksteiner@aei.mpg.de †

All Local Gauge Invariants for Perturbations of the Kerr Spacetime

“…Remark 5. In [11] we derived a covariant version of the Teukosky-Starobinski identities. In these identities a real, gauge invariant vector field ImA c appears naturally.…”

“…al. [10] recently constructed three more real, third order scalar invariants and in [11] we found a third order gauge invariant vector field. Here we take a different perspective by asking: in terms of which variables can the gauge invariant content of the theory be described?…”

“…For perturbations of the Kerr spacetime the two complex Teukolsky scalars (and their derivatives) together with the linearized Ricci tensor are well known gauge invariants. In this letter we add two complex scalar fields, I ξ and I ζ , to this list of local gauge invariants (collectively they can be found in [10], [11], see remarks 5 and 6, but here they are identified as a generating set for the first time). Their construction involves the Killing vectors ξ a , ζ a , see proposition 1 below.…”

The $\Lambda$ to zero limit of spacetimes and its physical interpretation

“…This condition is obviously violated for a spacelike slice in a homogeneous, spatially flat FLRW universe. 2 Interestingly, the problem of global non-linear stability for black holes has recently been solved for slow-rotating black holes in a de Sitter universe [16], 3 while the corresponding problem for asymptotically flat spacetimes remains one of the big challenges in the field of mathematical relativity, see [20,19,2,3,13,14] for recent progress on the linearised problem and [18] for the full non-linear problem under strong constraints. In this paper we will use the qualitative properties of how the Λ → 0 limit is approached to give a heuristic argument that the Einstein equations are a legitimate approximation to the fundamental Einstein-de Sitter equations, for calculations in the short-range regime.…”

“…That is, (2 + 1)-dimensional anti-de Sitter space 2. This was pointed out to the authors by Beatrice Bonga in private communication 3 This is arguably the physically relevant case if the cosmological constant is, in fact, positive.…”