Mathematical General Relativity

TL;DR

This paper surveys fundamental open problems in classical and quantum general relativity, including singularities, black hole uniqueness, spacetime stability, and issues related to quantum gravity and cosmology, highlighting ongoing challenges and research directions.

Contribution

It provides a comprehensive overview of key unresolved problems in general relativity, connecting classical, quantum, and cosmological issues in a unified framework.

Findings

Discussion of the cosmic censorship hypothesis and Penrose inequality.

Analysis of the stability of Kerr and de Sitter spacetimes.

Identification of open problems in quantum gravity and numerical relativity.

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.