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Stability and Instability in the Mathematical Analysis of the Einstein equations

Final Report Summary - STABMAEINSTEIN (Stability and Instability in the Mathematical Analysis of the Einstein equations)

The first main objective of the grant was to make fundamental progress on the black hole stability problem. This has been achieved in two papers (joint with M. Dafermos and I. Rodnianski), which prove the linear stability of the Schwarzschild solution against gravitational perturbations (arXiv:1601.06467) and decay estimates for the Teukolsky equation on slowly rotating Kerr backgrounds (arXiv:1711.07944).

The above papers unravel an important structure in the Einstein equations near the Kerr family of solutions and open the door to a full resolution of the non-linear stability conjecture of the Kerr family of black holes, something that seemed out of reach at the beginning of the grant. Further work on the full non-linear problem is currently in progress. One of the key new insights was a physical transformation theory for solutions of the Teukolsky equation, which leads to a coupled system of wave- and transport equations, which could be exploited to prove decay estimates. Another main insight was the complete understanding of residual pure gauge solutions in the context of a double null gauge, which we used to linearize the equations.

A very interesting by-product of the research was the discovery of novel conservation laws (``Conservation laws and flux bounds for gravitational perturbations of the Schwarzschild metric”, Class.Quant.Grav. 30 (22) 2016) and their relevance for the stability problem, which have later been connected to the canonical energy by Prabhu and Wald.

Finally, during the period of the grant a different approach to the stability problem was taken by my PhD student Thomas Johnson who proved the linear stability of the Schwarzschild solution in a generalised wave gauge (as opposed to the double null gauge employed in the papers above). See arXiv:1810.01337.

The second main objective was to understand the non-linear instability of anti-de Sitter space. Here fundamental progress was made independently of the ERC grant proposal by G. Moschidis (Princeton University) who proved a non-linear instability result for the spherically symmetric Einstein-Vlasov system. The analysis of anti-de Sitter spacetimes carried out in the context of the grant proposal therefore took a different route than expected. Together with my ERC postdoc Arick Shao we developed a program establishing a classical version of the holographic principle of theoretical physics through semi-global Carleman estimates on asymptotically anti-de Sitter (aAdS) spacetimes. A major achievement here was to connect the geometry of aAdS spacetimes, more precisely the trapping of null geodesics near the conformal boundary, to the behaviour of solutions to PDE both from the point of view of unique continuation and global dynamics (decay estimates)

Finally, in connection with these two objectives a research group was established at Imperial College, which at the peak time of the grant (2015-2016) consisted of 6 researchers (some hired independently by the department) working on objectives related to the grant and attracting visitors from all over the world. The two postdocs hired by the grant have now moved to permanent positions at Queen Mary University London and a postdoctoral position at Princeton University.