EPSRC Centre for Doctoral Training in Analysis

Originally from

I have an undergraduate degree in Math and Physics as well as a masters in Computer Science (focusing on Scientific Computing), both from the University of British Columbia. For my Master's thesis I studied the multigrid algorithm, borrowing techniques from complex analysis to analyze its convergence.

Areas of interest

My interests are quite broad (maybe even slightly scattered) and include topics in pure mathematics/physics, computing, and computer graphics. In addition to my official research I like to amuse myself with a steady stream of informal projects, ideally including components from at least two of these fields. For example, I am currently applying photon mapping techniques from computer graphics to study gravitational lensing.

Outside of mathematics I am a runner and enjoy casual hiking. I am also a huge fan of Thai and Indian food, which I derive great enjoyment from attempting to cook at home.

Work

I did my long miniproject with Ulrich Sperhake on numerical relativity. I wrote computer code simulating the gravitational collapse of a massless scalar field - a toy model of black hole formation that was studied extensively in the eighties and nineties (both from an analytic and numerical perspective).

I wanted to go beyond just a mathematical understanding of the problem and address questions like "what does gravitational collapse look like?" and "what if I was watching a black hole form in my living room?". To that end I borrowed techniques from computer graphics and built a raytracer on top of my simulation, resulting in the following two animations.

Both movies feature a spherically symmetric imploding scalar field - in one case imploding in the CCA common room, in the other above the great wall of China. As the field implodes spacetime curves in response, to the point that a singularity is almost formed - but in these cases the concentration of energy is insufficient, the field disperses to infinity and spacetime flattens out.

The case where a black hole actually forms is harder as the spacetime coordinates become badly conditioned. This kind of problem is ubiquitous in numerical relativity - for example, the original work of Choptuik in the early nineties was done using a coordinate system that breaks down the moment an event horizon forms. To handle this I needed to implement an adaptive mesh refinement algorithm, which took me a couple of months beyond the end of my miniproject.

One of the things I really wanted to get out of this portion of the project is an understanding of how the black hole shadow - an optical effect consisting of an apparent dark disk visible in the direction of a black hole - forms (look for the black circle in the picture of a Swartzschild black hole in front of St. John's college below).

What is the reason for this effect? In the Swartzschild case of an eternal black hole that has existed since the beginning of time, it is a property of the space of the solutions to the geodesic equations - namely, the property that not all past directed null-geodesics starting at a fixed point in spacetime outside the event horizon make it to infinity. On the other hand, black holes which have existed for any finite amount of time do not have this property. This seemingly contradicts the expectation that a black hole that has been around for a long time should look more and more like one that has existed forever.

What I have found is that for black holes created a finite amount of time in the past, instead of a sharp black circle one gets a fuzzy dim circle that gets sharper and blacker over time, converging as expected to the Swartzschild limit. In this case it is the gravitational redshift, rather than a property of the geodesic equations, that is responsible for the effect. This is illustrated in the following two videos:

(Special thanks to Marco Reinhardt for providing the spherical panorama of the Berlin Holocaust monument used in the second of these videos)

Publications (other)

What does gravity look like? preprint, to be published in the 2014 Edition of Eureka Math magazine.

Email

R.Hocking [at] maths.cam.ac.uk