Numerical relativity beyond astrophysics.

Though the main applications of computer simulations in relativity are to astrophysical systems such as black holes and neutron stars, nonetheless there are important applications of numerical methods to the investigation of general relativity as a fundamental theory of the nature of space and time. This paper gives an overview of some of these applications. In particular we cover (i) investigations of the properties of spacetime singularities such as those that occur in the interior of black holes and in big bang cosmology.

A positive-energy theorem for Einstein-aether and Hořava gravity.

Energy positivity is established for a class of solutions to Einstein-aether theory and the IR limit of Hořava gravity within a certain range of coupling parameters. The class consists of solutions where the aether 4-vector is divergence-free on a spacelike surface to which it is orthogonal (which implies that the surface is maximal). In particular, this result holds for spherically symmetric solutions at a moment of time symmetry.

Spherical topology in cardiac simulations.

Computational simulations of the electrodynamics of cardiac fibrillation yield a great deal of useful data and provide a framework for theoretical explanations of heart behavior. Extending the application of these mathematical models to defibrillation studies requires that a simulation should sustain fibrillation without defibrillation intervention. In accordance with the critical mass hypothesis, the simulated tissue should be of a large enough size.

Numerical simulations of generic singularities.

Numerical simulations of the approach to the singularity in vacuum spacetimes are presented here. The spacetimes examined have no symmetries and can be regarded as representing the general behavior of singularities. It is found that the singularity is spacelike and that, as it is approached, the spacetime dynamics becomes local and oscillatory.