The main theme of the 2020 Nobel Prize in Physics is black holes, and it has been awarded to Roger Penrose, Andrea Ghez, and Reinhard Genzel. The Nobel committee awarded half of it to Ghez and Genzel for their pioneering work in showing that there is a supermassive black hole at the centre of our galaxy, the Milky Way; the other half has been awarded to Roger Penrose for showing that ‘black hole formation is a robust prediction of the general theory of relativity.’ This three-part series of articles focuses on Penrose’s contribution.

Roger Penrose is currently 89 years old and is an emeritus professor of mathematics at the University of Oxford. Before retiring, he held the prestigious Rouse Ball Professorship of Mathematics chair at the University of Oxford. Penrose has collected many accolades to his name, including a knighthood, and is a legendary and maverick figure among physicists. He is known for his incredibly creative and somewhat unique geometric approach to solving problems in physics which has led to some creative innovations such as Penrose tiling and clever geometric illusions like the Penrose triangle and Penrose stairs. The Penrose stairs have made their way into popular culture; they have been referred to in movies like Inception and have served as an inspiration behind Michael Lacanilao’s viral Youtube video The Escherian Stairwell.

Another quirky aspect of Penrose that is perhaps worth a mention is his books on popular science. Penrose wrote the now-famous The Road to Reality: A Complete Guide to the Laws of the Universe which is over 1,000 pages long and is aimed at a popular audience — the layman. The irony is that what Penrose deems a layman book actually covers a decent deal of undergraduate physics literature plus some advanced concepts. He starts with basics like Pythagoras’s theorem, ventures into general relativity and towards the end of the book he is discussing concepts like quantum gravity, which many starting physics students would struggle with.

I mention the book because this article is celebrating Penrose’s work and the article is written in the spirit of The Road To Reality. However, the treatment here is nowhere near as exhaustive for the discerning layman. It is relatively easy for me to say that Penrose got the Nobel Prize for showing that black holes are consequences of the theory of general relativity or saying something along the lines of ‘He got the Nobel Prize for his work on the singularity theorems’ etc., but I feel like that both cheapens and underscores the significance of his work. So, instead, my aim is to take you on a pseudo-historical journey which builds on the evolution of ideas from Einstein to Penrose and describes both what Penrose did and why his work is essential in the later development of general relativity by putting it in a somewhat historical context.

With that said, allow me to guide you through this journey as we tug on some of the most profound concepts regarding space and time while pushing even those concepts to their limit.

#### Einstein and General Relativity

In 1915, Albert Einstein wrote down the equations for his theory of general relativity (GR) which describe how space-time behaves in the presence (and absence) of matter. Below is the compact form of this equation where the speed of light c is set to 1.

$G_{\mu\nu} = 8\pi GT_{\mu\nu}$

These are the Einstein field equations, and they describe the dynamics of space-time. The mathematical details are not essential to follow the article, but if you want a sense of intuition for what is going on, the left-hand side of the equations describes the geometry while the right-hand side describe energy and matter and its contribution to gravitational dynamics. The physicist John Wheeler describes the equations in an interesting manner: ‘Space-time tells matter how to move; matter tells space-time how to curve.’

It’s the Einstein field equations that allow us to learn about the exotic phenomena in the universe such as black holes (we’ll get to them soon!), gravitational waves, and even the dynamics of the universe itself. Since these equations are the bedrock of modern cosmology, their importance cannot be overstated.

Before I move onto the next part of our journey, you may have noticed that, in an attempt to describe general relativity above, I wrote down one equation, but I have used the plural ‘equations’ to describe it. This is not a grammatical error! The mathematics allows us to express the equation neatly, but this is, in fact, not one but a system of ten simultaneous equations that must all be solved together to get a solution. Such a system is difficult to solve because it comprises ten coupled non-linear partial differential equations. In simpler words, the equations are immensely difficult to solve for an exact solution. As a matter of fact, after writing down his theory of general relativity, Einstein himself doubted that one could give an exact solution for his equations because he thought it would be impossible to solve them analytically.

Fortunately, he was wrong about that.

#### Schwarzschild Geometry, Static Black Hole, and Singularities

Just a few months after Einstein wrote the Einstein field equations (EFE), Karl Schwarzschild, a German physicist and astronomer, provided the first exact solution to the Einstein field equations (many more followed in the coming years and decades). What makes this achievement more impressive is the fact that Schwarzschild served as a lieutenant during the First World War and he worked on the solution while he was in trenches on the Russian front.

Schwarzschild went on solving the equations by making assumptions and exploiting some symmetry relations like any good physicist. He started by assuming a spherical massive object (by massive I mean it has a mass; it does not imply that the object is big) with mass M and wanted to model the geometry around this object. I’ll break the assumptions down into two steps; note that these assumptions and descriptions are about the space-time geometry around the massive spherical object (e.g. a star) and not about the massive object itself.

1-  He modelled the gravitational field around a massive static object as spherically symmetric (non-rotating).
2- He looked at a specific case of Einstein field equations known as the Einstein vacuum equations. These are the form the Einstein field equations take in an empty space-time i.e. space-time devoid of matter. In such a space-time, you still have a gravitational field because space-time is still curved (for the more avid reader, in empty space-time the Riemann curvature tensor ≠ 0 which is what needs to be ascertained for there to be no gravitational field).

Taken together, the spherically symmetric vacuum space-time geometry external to a massive object came to be known as Schwarzschild geometry, and this would later define what a black hole (at least one kind of black hole) is. The equation below is the final form of the Schwarzschild (the first) solution to Einstein field equations (where, once again, c=1).

$ds^2 = -\bigg(1-\frac{2GM}{r}\bigg) dt^2+\bigg(1-\frac{2GM}{r}\bigg)^{-1}dr^2+r^2d\Omega^2$