Computational physics is one branch of physics that solves problems in physics by implementing computational algorithms. Especially in theoretical physics, the study of numerical physics is significantly useful and often used for simulations.

One of the most exiting branches of computational physics is numerical relativity. It is the astrophysical study to simulate objects in space-time universe like stars, black holes, and gravitational waves, based on Einstein’s Theory of General Relativity.

The physical law in space-time can basically be represented by one equation: the Einstein Equation. Therefore, numerical relativity is primarily the study of numerical solution of the Einstein equations:

Gij = 8πTij.

These are16 coupled hyperbolic-elliptic nonlinear partial differential equations (which reduce to 10 by symmetry). They equate the curvature of space-time (Einstein tensor G) with the energy and momentum in the space-time (stress-energy tensor T). As the result, the differential equations evolve time and space derivatives. Simulating the movements of objects in space-time on computer requires the methods to take spatial derivatives and time derivative. I introduce some of the practical methods we learned in class to deal with these problems.

Special derivatives
Einstein equations by themselves cannot be differentiated exactly. Therefore, we need to change them to the forms so that they can be differentiated exactly. Fourier series expansion, expansion in terms of Chebyshev polynomials are used for this purpose. These methods are effective because they have high convergence rates.

Time derivatives
Runge-Kutta equations of fourth-order are often used in this step. As we increase the order of Runge-Kutta equations, we can acquire arbitrary convergence rate.

Numerical Relativity group at the Albert Einstein Institute is expert in these simulations, and I site their resulted images.

General information of Numerical Relativity

Image and Movie Archive of the AEI Numerical Relativity Group