Introduction to Numerical Relativity

The universe we observe is governed by gravity, masterfully described by Einstein's Field Equations. However, unlike other forces in nature, gravity in General Relativity is self-interacting: the gravitational field itself carries energy, meaning that gravity creates more gravity. This behavior gives rise to a highly complex system of non-linear equations.

You might have studied famous exact solutions, such as the Schwarzschild or Kerr black holes. These theoretical solutions have opened up new ways of looking at reality, for sure, but they require imposing highly idealized geometric symmetries. When these symmetries break—as in the realistic case of two massive black holes orbiting each other—no exact analytical solution exists.

On the other hand, thanks to modern supercomputers, it is already possible to solve this problem by using the Numerical Relativity approach. This discipline overcame decades of stagnation thanks to the 2005 methodological breakthroughs, which finally allowed stable simulations of dynamic spacetime.

In this course, we will see how the simulation of merging black holes emerges quite naturally by applying the 3+1 split of spacetime, constructing valid initial data, and evolving it through time.

In particular, we will start from the theoretical limitations of Einstein's equations to understand why we need computational simulations, and we will analyze the complete anatomy of a binary merger (inspiral, merger, and ringdown).

From there, we will shift to observational milestones, seeing how Numerical Relativity was fundamental for the GW150914 detection and how matched filtering allows LIGO to "see" these waves amidst quantum and seismic noise.

After dealing with vacuum collisions, we switch to the realm of matter and exotic physics. We will explore the birth of multi-messenger astronomy with neutron star mergers, and see how these simulations allow us to distinguish black holes from hypothetical horizonless objects like boson stars. Finally, we will tackle the computational scale problem and the future of the field with Artificial Intelligence.

It's an introductory course, so no prior knowledge is needed. What are you waiting for?