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Mathematical general relativity

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Abstract

We present a number of open problems within general relativity. After a brief introduction to some technical mathematical issues and the famous singularity theorems, we discuss the cosmic censorship hypothesis and the Penrose inequality, the uniqueness of black hole solutions and the stability of Kerr spacetime and the final state conjecture, critical phenomena and the Einstein–Yang–Mills equations, and a number of other problems in classical general relativity. We then broaden the scope and discuss some mathematical problems motivated by quantum gravity, including AdS/CFT correspondence and problems in higher dimensions and, in particular, the instability of anti-de Sitter spacetime, and in cosmology, including the cosmological constant problem and dark energy, the stability of de Sitter spacetime and cosmological singularities and spikes. Finally, we briefly discuss some problems in numerical relativity and relativistic astrophysics.

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Acknowledgements

I would like to thank Piotr Bizon, Mihales Dafermos, Luis Lehner and Frans Pretorius for detailed comments on an earlier version of the manuscript, and Eric Poisson, Claes Uggla and Clifford Will for useful remarks.

Funding

Financial support was provided by the Natural sciences and Engineering Research Council of Canada (Grant No. 4-4046).

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    Alan A. Coley

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Coley, A.A. Mathematical general relativity. Gen Relativ Gravit 51, 78 (2019). https://doi.org/10.1007/s10714-019-2559-5

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