Mathematical methods for numerical relativity

Physics context

Numerical relativity is the art of solving the Einstein equations, and so constructing a spacetime, by numerical methods on a computer. The numerical approach becomes necessary when a solution in closed form or a suitable perturbation approach are not available for the problem one is interested in.

Two such problems are supernova explosions and the final merger phase of a binary system of two black holes or two neutron stars. These are the most energetic events in the universe, and are believed to power gamma ray bursts from gravitational potential energy. They are also the most likely source of gravitational wave signals at frequencies of a few hundred Hertz. LIGO detected the first binary black hole merger in 2015 (published in 2016), and in 2017 LIGO and VIRGO jointly detected the first binary neutron star merger. The after-effects of that merger were subsequently seen in all wavelengths of electromagnetic waves. Modelling gravitational wave sources is essential to interpreting the data. Electromagnetic and neutrino observations see the fireball that surrounds these events, but gravitational waves show us the motion of mass in the centre, and should help us to understand matter under extreme conditions (higher than nuclear densities).

Mathematical underpinning

The Einstein equations are partial differential equations. The most common numerical approach to solving them is to write them as a time evolution problem for the metric (shape) of three-dimensional space. These evolution equations are turned into algebraic equations by sampling all functions on a 3-dimensional grid of points, and approximating partial derivatives by finite differences.

General relativity does not have a preferred coordinate system given a priori; a coordinate system must be generated on the fly as the spacetime is calculated. This gauge freedom also has the effect that the initial data mentioned above cannot be specified freely but are subject to constraints , elliptic equations not containing time derivatives that the initial data must obey. Finally, boundary conditions must be implemented that reflect the physical situation one is trying to simulate.

Writing the Einstein equations in such a form that numerical simulations are stable and accurate is a central problem that has kept numerical relativity back throughout the 1990s, but two independent breakthroughs were made in May and November of 2006. The first simulation of a binary inspiral for more than a single orbit was carried out in F. Pretorius, Evolution of binary black hole spacetimes, Phys. Rev. Lett. 95 (2006) 121101. I have made a key contribution to the formulation of the Einstein equations used in this paper, see C. Gundlach et al., Constraint damping in the Z4 formulation and harmonic gauge, Class. Quant. Grav. 22 (2005) 3767. See Caltech Engineering and Science for a journalistic account.