Background
First, some background. In xxx, Einstein published xxx which outlined his general theory of relativity. This theory explained how space and time should be thought of as a single 4 dimensional entity known as “spacetime”. General Relativity is theory of gravity. It explains how matter creates gravitational fields and how this affects the movement of other matter. In principle it explains everything for which quantum effects can be ignored (i.e. we are not working on subatomic length scales).
I say “in principle” because the theory is very complicated. In fact, if you write it out in full it involves xxx equations. It was so hard in fact that very few solutions were know. The first known solution is “Minkowski spacetime”. This is so simple as to become boring. It describes a universe where there is nothing - no matter, no gravity, no anything. Nothing happens. It is easy and, consequently, boring.
We want to describe interesting things. So suppose we insert a particle into our universe. To keep things as simple as possible, suppose this particle has mass but no charge and suppose it isn’t moving. This about as straightforward as can be without being boring Minkowski spacetime. What happens now? The resulting spacetime is known as “Schwarzschild spacetime” and already things get interesting. In fact, it took people years to fully understand this spacetime. In particular, it contains one of the key features of general relativity which continues to puzzle scientists to this day - black holes. A black hole is a region of spacetime which cannot communicate xxxx.
Oppenheimer
I was at the cinema this week and saw a trailer for Christopher Nolan’s newest film “Oppenheimer”, due be released next year. The film will focus on J. Robert Oppenheimer, “father of the nuclear bomb”, in his capacity as director of the Manhattan Project. Arguably no scientist has had a greater effect on the world. In this post I’d like to focus on some of his other work which is closer to my own area of research. “On Continued Gravitational Collapse” by Oppenheimer and his student Hartland Snyder was published on 1 September 1939, the same day Germany began the invasion of Poland which triggered the start of World War II. This paper is significant for being the first explicit example outlining how a black hole could, in principle, be formed in nature.
If you’ve read xxx then you’ll know about the Schwarzschild spacetime. This is the simplest spacetime which solves Einstein’s equations of general relativity [1] (excluding the boring Minkowski spacetime which contains no matter or gravity). The important feature of this spacetime is that it contains a black hole - a region of spacetime from which one cannot send signals to anyone far away. If you are in a black hole you are stuck. People on the outside can send you messages but you cannot reply.
It is important to stress that just because this spacetime solves the equations does not mean we should expect it to exist in nature. There are a number of reasons for this which I won’t go into (for now), but in this case the most important is that the spacetime is “stationary”. This means that it doesn’t change in time. It will always look the same and has always looked the same. This leads to the following question: how did the black hole form? It was not clear from Schwarzschild’s example whether black holes could exist in nature because nobody had come up with a detailed description of how they could be created.
Enter Oppenheimer and Snyder. They studied a simplified model of stellar collapse, arguing that once a sufficiently large starxxx has burned all of its fuel it will collapse under its own gravity to form a black hole. The particular model they studied involved a number of idealisations. They assumed that the set up was spherically symmetric, meaning that the star is not rotating. This greatly simplifies the Einstein equations, making it possible to solve them using pen and paper (hence the relative brevity of the paper which comes in at only 5 pages). They also assumed the star was made of pressureless matter known as “dust”. This also simplifies the equations, however they argued that pressure would act to slow down the collapse but would not affect the end result. Under these assumptions, Oppenheimer and Snyder showed that an observer collapsing inwards with the star would eventually be unable to send signals to their far away friends - they have entered a black hole! Moreover, if the observer carries a clock, they will measure a finite time before they enter the black hole (for a star with mass around 1 thousand times that of our sun and with the density of water, this would take around 1 day).
Now that we have a mechanism under which black holes can be formed, should we now expect to find them in nature? Well, not exactly. There are still a few issues. The most pressing is the symmetry assumption used by Oppenheimer and Snyder. Their model described a star collapsing towards a single point. Eventually, enough of the star became concentrated in a single region that a black hole is formed. But in nature, nothing is ever perfect. In particular, stars 1There’s a slight subtlety here. What I mean by this is that this spacetime solves the vacuum Einstein equations. It only makes sense to talk about solutions of Einstein’s equations once you have specified either the left or right hand side - otherwise anything can be a solution! It is also confusing at first that this black hole spacetime is also a vacuum, but it is, and I don’t want to make a big deal of it here, are not perfectly spherically symmetric. They wiblle and wobble. They have lump and bumps. Could it be that the black hole resulting from the collapse of a star described by Oppenheimer and Snyder is a result of the perfect symmetry of the initial set-up? It is entirely plausible that if we give this set-up a little poke, and in doing so break its spherical symmetry, then something entirely different will happen. The star might start to spin as it collapses. Or bits of it might fly past each other and off into the distance. It may never concentrate enough of its mass into a single region to form a black hole. Perfect spherical symmetry is in some sense infinitely unlikely. You need the star to have zero spin. Not 0.001 or 10−100. Exactly zero. That is never going to happen. A good analogy is this. Suppose you have a needle with a weight at the top and you want to balance it on its point on a flat table. In principle this is possible, it satisfies all the laws of physics. In practice however, its virtually impossible. Even if you hold your breath and close all the windows to prevent a draft, you need to set your initial conditions up more or less perfectly. As the needle gets thinner and thinner, the set up needs to be more and more perfect. If the needle became infinitely thin you’d need to be infinitely perfect. You can’t be.
So do black holes form in nature? The next breakthrough on this question came from Roger Penrose and Stephen Hawking in the form of their incompleteness theorem2 . This dropped the spherical symmetry assumption, meaning the result applies to nature, but it was more vague about the outcome. The theorem essentially says that “something bad happens”. This “bad” thing might be a black holes, but the theorem doesn’t guarantee it. But that’s a story for another day.
[1]I am of course addressing the question of black hole existence from a purely theoretical viewpoint. The answer is almost certainly yes. Not least because we’ve got photos of them!