A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds

By Mihalis Dafermos, Igor Rodnianski. 2008

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We consider Kerr spacetimes with parameters a and M such that |a|≪M, Kerr-Newman spacetimes with parameters |Q|≪M, |a|≪M, and more generally, stationary axisymmetric black hole exterior spacetimes $(\mathcal{M},g)$ which are sufficiently close to a Schwarzschild metric with parameter M>0 and whose Killing fields span the null generator of the event horizon. We show uniform boundedness on the exterior for solutions to the wave equation □gψ=0. The most fundamental statement is at the level of energy: We show that given a suitable foliation Στ, then there exists a constant C depending only on the parameter M and the choice of the foliation such that for all solutions ψ, a suitable energy flux through Στ is bounded by C times the initial energy flux through Σ0. This energy flux is positive definite and does not degenerate at the horizon, i.e. it agrees with the energy as measured by a local observer. It is shown that a similar boundedness statement holds for all higher order energies, again without degeneration at the horizon. This leads in particular to the pointwise uniform boundedness of ψ, in terms of a higher order initial energy on Σ0. Note that in view of the very general assumptions, the separability properties of the wave equation or geodesic flow on the Kerr background are not used. In fact, the physical mechanism for boundedness uncovered in this paper is independent of the dispersive properties of waves in the high-frequency geometric optics regime.

Subfields: Astronomy and Astrophysics, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/1904e1b31cbf2de81066625c71be1f6f81c8aefb

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Citations in the corpus

A new physical-space approach to decay for the wave equation with applications to black hole spacetimes (2009, unknown): “…, say. t ∗ (The coordinate vector ﬁeld ∂ t ∗ in ( r, ) coordinates will do for instance.) by Let us note that (EB-Mink) is replaced This non-degenerate energy boundedness statement was originally proven in [5] but can in fact be shown independently of decay results, as in [7] and Section 3 of [8].”
Local decay of waves on asymptotically flat stationary space-times (2009, unknown): “obtained by Finster-Smoller [18], and Finster-Kamran-Smoller-Yau [17]. Full uniform energy bounds were proved in the the case of the Kerr space-time with small angular momentum in Dafermos-Rodnianski [14]; their result applies in eﬀect to a larger class of small perturbations of the Schwarzschild sp”
Hidden symmetries and decay for the wave equation on the Kerr spacetime (2009, methodology): “The key idea of [19, 49] is to start with such a function F , to replace the phase space coordinates for null geodesics by their conjugate Fourier or spectral variables, and to use the the resulting F in A.”
Stability and Instability of Extreme Reissner–Nordström Black Hole Spacetimes for Linear Scalar Perturbations II (2011, methodology): “ates (t;r;;˚) in 1916 [46] and 1918 [42]. In these coordinates, g= g M;e= Ddt 2+ 1 D dr 2 +rg S2, where D= D(r) = 1 2M r + e2 r2 1The redshift vector eld was used as a commutator for the rst time in [24]. LINEAR STABILITY AND INSTABILITY OF EXTREME REISSNER-NORDSTROM I 5 and g S2 is the standard met”