A note on the Klein–Gordon equation in the background of a rotating black hole

Abstract

This short paper should serve as a basis for further analysis of a previously found new symmetry of the solutions of the wave equation in the gravitational field of a Kerr black hole. Its main new result is the proof of essential self-adjointness of the spatial part of a reduced normalized wave operator of the Kerr metric in a weighted L 2 -space. As a consequence, it leads to a purely operator theoretic proof of the well posedness of the initial value problem of the reduced Klein-Gordon equation in that field in that L 2 -space and in this way generalizes a corresponding result of Kay ͓"The double-wedge algebra for quantum fields on Schwarzschild and Minkowski spacetimes," Commun. Math. Phys. 100, 57 ͑1985͔͒ in the case of the Schwarzschild black hole. It is believed that the employed methods are applicable to other separable wave equations. © 2009 American Institute of Physics. ͓DOI: 10.1063/1.3037327͔ I. GENERAL INTRODUCTIONThe linearized stability of the Schwarzschild black hole follows by combining the ReggeWheeler-Zerilli-Moncrief decomposition of gravitational perturbations of the Schwarzschild metric 16,19,13 with a result by Kay and Wald 12 that proves the boundedness of all solutions of the wave equation corresponding to C ϱ -data of compact support. The proof of the last rests on the positivity of the conserved energy.The question of the linearized stability of the Kerr black hole is still an open problem whose outcome is of major importance to general relativity. In comparison to the case of the Schwarzschild black hole, the solution to this problem is considerably more complicated. Mainly, this is due to two facts. First, a decomposition comparable to that of Regge-Wheeler-Zerilli-Moncrief does not yet exist in this case, although the recent finding in Ref. 6 gives hope that such a decomposition might exist. In contrast, a partial decomposition based on the Newman-Penrose formalism depends on the choice of a tetrad field, i.e., is gauge dependent even under "small" coordinate transformations. 2 Second, a conserved energy for the solutions of the wave equation exists, but the energy density is negative inside the ergosphere. This fact excludes at least a direct application of the so-called "energy methods" to a proof of stability of the solutions. The total energy could be finite while the field still might grow exponentially in parts of the space-time. But recently a local stability result has been proved that the restrictions of the solutions of the wave equation to compact subsets K in space are elements of L C ϱ ͑K͒ with a norm converging to zero for t → ϱ. 9 Because of the absence of a decomposition of the Regge-Wheeler-Zerilli-Moncrief type, the question of applicability of the last and other similar results to the question of linearized stability of the Kerr metric is still open.As mentioned above, Ref. 6 contains the surprising find of a new symmetry operator that a͒ Electronic

On the stability of the massive scalar field in Kerr space-time

“…An analogous operator has been found for the operator governing linearized gravitational perturbations of the Kerr geometry [20]. A recent study finds another such 'symmetry operator' which only contains a first order time derivative and commutes with a rescaled wave operator [7]. Differently to Carter's operator, this operator is analogous to symmetry operators induced by one-parameter group of isometries of the metric, in that it induces a mapping in the data space that is compatible with time evolution, and therefore describes a true symmetry of the solutions.…”

“…Proof. For this, we use the notation of [7]. According to the proof of Theorem 4 of [7], the underlying sets of X and X := L 2 (Ω s , (r 4 /∆) sin θ)) are equal; and the norms induced on the common set are equivalent, the maximal multiplication operator T r 4 /(∆Σ) by the function r 4 /(∆Σ) is a bijective bounded linear operator on X that has a bounded linear inverse; the operator H, related to A 0 by…”

Wavy way to the Kerr metric and the quantum nature of its ring singularity

“…This form of choice for the solution of the Klein-Gordon equation is also considered in [32], for exploring the new symmetries of the solution of the wave equation. For the metric (93), the Klein-Gordon equation with the assumed solution can be written as…”

“…Since, complete separation in the spatial and time derivatives is not possible, one may define the right hand side of Eq. (105) as the spatial part of the reduced normalized wave operator [32], without imposing k = 0 (s-wave). However, even if we consider this case, the result would not change, because, the constant number k do not contribute near r → 0 and r → ∞.…”