Lectures on black holes and linear waves

By Mihalis Dafermos, Igor Rodnianski. 2008

346 citations recorded by the source corpus.

These lecture notes, based on a course given at the Zurich Clay Summer School (June 23-July 18, 2008), review our current mathematical understanding of the global behaviour of waves on black hole exterior backgrounds. Interest in this problem stems from its relationship to the non-linear stability of the black hole spacetimes themselves as solutions to the Einstein equations, one of the central open problems of general relativity. After an introductory discussion of the Schwarzschild geometry and the black hole concept, the classical theorem of Kay and Wald on the boundedness of scalar waves on the exterior region of Schwarzschild is reviewed. The original proof is presented, followed by a new more robust proof of a stronger boundedness statement. The problem of decay of scalar waves on Schwarzschild is then addressed, and a theorem proving quantitative decay is stated and its proof sketched. This decay statement is carefully contrasted with the type of statements derived heuristically in the physics literature for the asymptotic tails of individual spherical harmonics. Following this, our recent proof of the boundedness of solutions to the wave equation on axisymmetric stationary backgrounds (including slowly-rotating Kerr and Kerr-Newman) is reviewed and a new decay result for slowly-rotating Kerr spacetimes is stated and proved. This last result was announced at the summer school and appears in print here for the first time. A discussion of the analogue of these problems for spacetimes with a positive cosmological constant follows. Finally, a general framework is given for capturing the red-shift effect for non-extremal black holes. This unifies and extends some of the analysis of the previous sections. The notes end with a collection of open problems.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/784c9e13c525df3cc65aaba30835315762da9f53

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In Astronomy and Astrophysics

1. Paolo Pani
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In Nuclear and High Energy Physics

1. V. Cardoso
2. Paolo Pani
3. Jorge E Santos
4. Ó. J. Dias
5. Benson Way
6. S. Aretakis
7. R. Brito
8. P. Blue

Citations in the corpus

• Local decay of waves on asymptotically flat stationary space-times (2009, methodology): “ the spherical modes; they establish a t−2 local decay for each spherical mode. Switching to Kerr, the first decay results there were obtained by Finster-KamranSmoller-Yau[17]. LaterDafermos-Rodnianski[12]wereable toextend their Schwarzschild results to Kerr, obtaining almost a t−1 decay. Finally, we”
• A new physical-space approach to decay for the wave equation with applications to black hole spacetimes (2009, neutral): “For background, see Section 4.4.1 of [8].”
• Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations I (2011, methodology): “The first boundedness result for solutions of the wave equation on slowly rotating Kerr (|a| � M) spacetimes was proved in [23] and decay results were derived in [2,24, 49 ].”
• Price's Law on Nonstationary Space-Times (2011, neutral): “r the trapped set as well as Strichartz estimates are contained in the paper of Tohaneanu [67]. For related work 6 JASON METCALFE, DANIEL TATARU, AND MIHAI TOHANEANU we also refer the reader to [67], [20] and [2]. For large angular momentum |a| < M a similar estimate was proved in [19] for axisym”
• Tests for the existence of black holes through gravitational wave echoes (2017, background): “There are strong indications that the exterior Kerr spacetime is stable, although a rigorous proof is still missing [140].”
• Numerical methods for finding stationary gravitational solutions (2015, methodology): “A proof of the existence of exponentially growing massive scalar wave superradiant solutions in Kerr spacetimes was recently established in [452–456].”
• Slowly decaying waves on spherically symmetric spacetimes and ultracompact neutron stars (2014, background): “In this case, we can actually avoid the time frequency localisation altogether, as it is easy to construct a current (modelled, say, on the Schwarzschild current of [1]) for the spherically symmetric mode.”
• Horizon Instability of Extremal Black Holes (2012, neutral): “For an exhaustive list of references see [15].”
• Quasi-Normal Modes and Exponential Energy Decay for the Kerr-de Sitter Black Hole (2010, methodology): “r certain constants Cand 0independent of u. For the Kerr metric, the local energy decay is polynomial as shown by Tataru and Tohaneanu [34, 33], see also the lecture notes by Dafermos and Rodnianski [13] and the references below. Outline of the proof. The starting point of the construction of R g(!)”
• Hidden symmetries and decay for the wave equation on the Kerr spacetime (2009, methodology): “) The crucial positivity and negativity properties arising from conditions (i)-(v) can be found in the work of Dafermos-Rodnianski [17, 20].”