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Dr Rita Teixeira da Costa: Einstein and partial differential equations

Mathematician Dr Rita Teixeira da Costa studies partial differential equations, with a special focus on Einstein’s equations in the theory of General Relativity. The JRF shares her interests in Mathematics and Physics. 

Rita Teixeira da Costa

Where did you study before you came to Cambridge?

Before Cambridge, I was an undergraduate student in Portugal, where I am from, at Instituto Superior Técnico in Lisbon. But my degree is not in math: I studied Engineering Physics.

What led you towards mathematics?

As a student in Técnico, I was fortunate to get to try out research in both mathematics and physics: I worked for some time with Professors Vitor Cardoso from physics and José Natário from math, and I received a grant from Calouste Gulbenkian Foundation to do some undergraduate research in math with Professor João Pimentel Nunes. These projects turned out to be complementary: through the former, I learned general relativity from the physics perspective; from the latter I learned a lot of the analysis and geometry that goes into the mathematical theory.

What are partial differential equations and what are Einstein’s equations?

Partial differential equations are the mathematical language in which we describe the evolution and motion of physical systems around us. Einstein’s equations are a system of partial differential equations which encode the theory of General Relativity. This is a physical theory explaining gravity, i.e. how massive objects attract each other. It may seem esoteric, but General Relativity is something we all use every day, when we open Google Maps on our phones!

Before General Relativity, our understanding of gravity came from a former master of Trinity: Isaac Newton. The big new idea that Einstein introduced in the early 1900s is that time and space should not be considered separate entities: they are part of a flexible membrane we call spacetime and which obeys Einstein’s equations. The equations are best summed up by the words of my academic great-grandfather, John Wheeler: “matter tells spacetime how to curve, spacetime tells matter how to move”.

General Relativity is a theory in Physics; what role can a mathematician have?

Physics, as all sciences, is based on the scientific method, which works roughly as follows. Scientists make observations, and then build a model which explains them. But that model must be tested; how? First, we use the model to extrapolate, predict what happens in a different setting from that of our original experiments. Second, we run an experiment in this new setting to check if our predictions hold up to reality.

The test is only valid if the two steps are completely independent: a scientist cannot be biased by the empirical evidence when trying to understand what their model predicts would happen, or they risk a false positive. The only way we make sure this does not happen is to have, in the first step, predictions be rigorously derived from the model. That is exactly what mathematicians are trained to do!

Rita presenting her work at the program ‘General Relativity, Geometry and Analysis: beyond the first 100 years after Einstein’ at the Mittag-Lefler Institute in Sweden. (Credit: Sami Kozłowski)

And what predictions are you trying to make from Einstein’s equations?

During my PhD under the supervision of Professor Mihalis Dafermos, I have been studying black holes. These are spacetime membranes with a region (the “hole”) from which not even light can travel fast enough to be able to come out once inside. Scientists believe that all supermassive stars end up as black holes, which are, therefore, seen as an equilibrium state. I am interested in what happens if you disturb the equilibrium: if we throw in something at the black hole, does it absorb, wiggle around a bit and settle back to being a normal black hole? Or does it turn into something new?

This question requires us to look very closely as black holes. But if we zoom out from a black hole in the universe, we see plenty of other objects around it. And if we zoom out even more, they all blur together and look like a homogenous lump. In fact, scientists know that, at very large scales, a homogeneous lump is a great model of the universe! Unfortunately, it is not entirely clear if this large-scale model can be rigorously derived from our description of the universe at smaller scales, i.e. close to stars and planets, etc. Understanding how to mathematically bridge together the small and large scales is another thread of research I am eager to start as a JRF at Trinity.

Black holes solutions to the Einstein equations were discovered as early as in 1915, but our first picture was only obtained in 2019. (Credit: Event Horizon Telescope Collaboration)

What are you looking forward to most as a JRF?

Cambridge is an amazing place to do research in General Relativity, but Trinity even more so: so many fellows, past and present, physicists and mathematicians, dedicate themselves to the study of this beautiful theory! I am honoured to have been invited to be part of this community.

I am also very much looking forward to chatting with scholars in other disciplines, both in Sciences and Arts: I believe ultimately we should all strive to be ‘men of the Renaissance’, learning a bit of everything, though personally I find that math keeps me too busy to be very diligent about this! Nevertheless, I have always been especially interested in Biology: I took a few courses in it as an undergrad and, more recently, interned for a summer at the Sainsbury Laboratory. I am also an avid reader of novels (I am currently ‘collecting’ books from female Nobel prize laureates in literature), and recently, some politics, economics, and history books too. I am eager to learn more from the experts at Trinity!

Rita Teixeira da Costa

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