Mathematical general 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.

Dynamics of Einstein–Aether scalar field cosmology

“…with constraint equation ρ m0 − 8B0 1−16B0 y 1 x 1 = 0. For solution (53), (54) for large values of time the scale factor has a power-law behaviour a (t) = a 0 t p , where p = p (B 0 ).…”

“…A very interesting theory of gravitational modification is the Einstein-aether theory [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54]. It corresponds to the class of Lorentz-violating theories of gravity, where one considers the existence of a unit vector, the aether, which is everywhere non-zero in any solution.…”

“…The action for Einstein-aether theory is the most general generally covariant functional of the spacetime metric g ab and aether field u a involving no more than two derivatives, excluding total derivatives [38,41,45]. Exact solutions and qualitative analysis of Einstein-aether were presented elsewhere, e.g., in [46][47][48][49][50][51][52][53][54].…”

“…For the analysis are considered spatially curvature and anisotropic perturbations. On the other hand, static anisotropic models for a mixture of a necessarily non-tilted perfect fluid with a barotropic equation of state (linear and/or polytropic equations of state) and a self-interacting scalar field were studied in [49,53,54]. In [55] it was presented the solution space of the field equations in the Einstein-aether theory for the case of a vacuum Bianchi Type V space-time.…”

Theoretical cosmology

“…We are primarily interested here in problems which we shall refer to as problems in theoretical cosmology, and particularly those that are susceptible to a rigorous treatment within mathematical cosmology. Problems in GR have been discussed elsewhere [8]. There are some problems in GR that are relevant in cosmology, and theorems can be extended into the cosmological regime by including models with matter.…”