Roger Penrose just won Nobel Prize in Physics “for the discovery that black hole formation is a robust prediction of the general theory of relativity.” He shared it with Reinhard Genzel and Andrea Ghez, who won it “for the discovery of a supermassive compact object at the centre of our galaxy.”

This is great news! It’s a pity that Stephen Hawking is no longer alive, because if he were he would surely have shared in this prize. Hawking’s most impressive piece of work—his prediction of black hole evaporation—was too far from being experimentally confirmed to win a Nobel prize before his death. It still is today. The Nobel Prize is conservative in this way: it doesn’t go to theoretical developments that haven’t been experimentally confirmed. That makes a lot of sense. But sometimes they go overboard: Einstein never won a Nobel for general relativity or even special relativity. I consider that a scandal!

I’m glad that the Penrose–Hawking singularity theorems are considered Nobel-worthy. Let me just say a little about what Penrose and Hawking proved.

The most dramatic successful predictions of general relativity are black holes and the Big Bang. According to general relativity, as you follow a particle back in time toward the Big Bang or forward in time as it falls into a black hole, spacetime becomes more and more curved… and eventually it *stops!* This is roughly what we mean by a singularity. Penrose and Hawking made this idea mathematically precise, and proved that under reasonable assumptions singularities are inevitable in general relativity.

General relativity does not take quantum mechanics into account, so while Penrose and Hawking’s results are settled theorems, their applicability to *our universe* is not a settled fact. Many physicists hope that a theory of quantum gravity will save physics from singularities! Indeed this is one of the reasons physicist are fascinated by quantum gravity. But we know very little for sure about quantum gravity. So, it makes a lot of sense to work with general relativity as a mathematically precise theory and see what it says. That is what Hawking and Penrose did in their singularity theorems.

Let’s start with a quick introduction to general relativity, and then get an idea of why this theory predicts singularities are inevitable in certain situations.

General relativity says that spacetime is a 4-dimensional Lorentzian manifold. Thus, it can be covered by patches equipped with coordinates, so that in each patch we can describe points by lists of four numbers. Any curve going through a point then has a tangent vector whose components are Furthermore, given two tangent vectors at the same point we can take their inner product

where as usual we sum over repeated indices, and is a matrix called the metric, depending smoothly on the point. We require that at any point we can find some coordinate system where this matrix takes the usual Minkowski form:

However, as soon as we move away from our chosen point, the form of the matrix in these particular coordinates may change.

General relativity says how the metric is affected by matter. It does this in a single equation, Einstein’s equation, which relates the ‘curvature’ of the metric at any point to the flow of energy-momentum through that point. To define the curvature, we need some differential geometry. Indeed, Einstein had to learn this subject from his mathematician friend Marcel Grossman in order to write down his equation. Here I will take some shortcuts and try to explain Einstein’s equation with a bare minimum of differential geometry.

Consider a small round ball of test particles that are initially all at rest relative to each other. This requires a bit of explanation. First, because spacetime is curved, it only looks like Minkowski spacetime—the world of special relativity—in the limit of very small regions. The usual concepts of ’round’ and ‘at rest relative to each other’ only make sense in this limit. Thus, all our forthcoming statements are precise only in this limit, which of course relies on the fact that spacetime is a continuum.

Second, a test particle is a classical point particle with so little mass that while it is affected by gravity, its effects on the geometry of spacetime are negligible. We assume our test particles are affected only by gravity, no other forces. In general relativity this means that they move along timelike geodesics. Roughly speaking, these are paths that go slower than light and bend as little as possible. We can make this precise without much work.

For a path in *space* to be a geodesic means that if we slightly vary any small portion of it, it can only become longer. However, a path in *spacetime* traced out by particle moving slower than light must be ‘timelike’, meaning that its tangent vector satisfies We define the proper time along such a path from to to be

This is the time ticked out by a clock moving along that path. A timelike path is a geodesic if the proper time can only *decrease* when we slightly vary any small portion of it. Particle physicists prefer the opposite sign convention for the metric, and then we do not need the minus sign under the square root. But the fact remains the same: timelike geodesics locally maximize the proper time.

Actual particles are not test particles! First, the concept of test particle does not take quantum theory into account. Second, all known particles are affected by forces other than gravity. Third, any actual particle affects the geometry of the spacetime it inhabits. Test particles are just a mathematical trick for studying the geometry of spacetime. Still, a sufficiently light particle that is affected very little by forces other than gravity can be approximated by a test particle. For example, an artificial satellite moving through the Solar System behaves like a test particle if we ignore the solar wind, the radiation pressure of the Sun, and so on.

If we start with a small round ball consisting of many test particles that are initially all at rest relative to each other, to first order in time it will not change shape or size. However, to second order in time it can expand or shrink, due to the curvature of spacetime. It may also be stretched or squashed, becoming an ellipsoid. This should not be too surprising, because any linear transformation applied to a ball gives an ellipsoid.

Let be the volume of the ball after a time has elapsed, where time is measured by a clock attached to the particle at the center of the ball. Then in units where Einstein’s equation says:

These flows here are measured at the center of the ball at time zero, and the coordinates used here take advantage of the fact that to first order, at any one point, spacetime looks like Minkowski spacetime.

The flows in Einstein’s equation are the diagonal components of a matrix called the ‘stress-energy tensor’. The components of this matrix say how much momentum in the direction is flowing in the direction through a given point of spacetime. Here and range from to corresponding to the and coordinates.

For example, is the flow of -momentum in the -direction. This is just the energy density, usually denoted The flow of -momentum in the -direction is the pressure in the direction, denoted and similarly for and You may be more familiar with direction-independent pressures, but it is easy to manufacture a situation where the pressure depends on the direction: just squeeze a book between your hands!

Thus, Einstein’s equation says

It follows that positive energy density and positive pressure both curve spacetime in a way that makes a freely falling ball of point particles tend to shrink. Since and we are working in units where ordinary mass density counts as a form of energy density. Thus a massive object will make a swarm of freely falling particles at rest around it start to shrink. In short, *gravity attracts*.

Already from this, gravity seems dangerously inclined to create singularities. Suppose that instead of test particles we start with a stationary cloud of ‘dust’: a fluid of particles having nonzero energy density but no pressure, moving under the influence of gravity alone. The dust particles will still follow geodesics, but they will affect the geometry of spacetime. Their energy density will make the ball start to shrink. As it does, the energy density will increase, so the ball will tend to shrink ever faster, approaching infinite density in a finite amount of time. This in turn makes the curvature of spacetime become infinite in a finite amount of time. The result is a ‘singularity’.

In reality, matter is affected by forces other than gravity. Repulsive forces may prevent gravitational collapse. However, this repulsion creates pressure, and Einstein’s equation says that pressure also creates gravitational attraction! In some circumstances this can overwhelm whatever repulsive forces are present. Then the matter collapses, leading to a singularity—at least according to general relativity.

When a star more than 8 times the mass of our Sun runs out of fuel, its core suddenly collapses. The surface is thrown off explosively in an event called a supernova. Most of the energy—the equivalent of thousands of Earth masses—is released in a ten-second burst of neutrinos, formed as a byproduct when protons and electrons combine to form neutrons. If the star’s mass is below 20 times that of our the Sun, its core crushes down to a large ball of neutrons with a crust of iron and other elements: a neutron star.

However, this ball is unstable if its mass exceeds the Tolman–Oppenheimer–Volkoff limit, somewhere between 1.5 and 3 times that of our Sun. Above this limit, gravity overwhelms the repulsive forces that hold up the neutron star. And indeed, no neutron stars heavier than 3 solar masses have been observed. Thus, for very heavy stars, the endpoint of collapse is not a neutron star, but something else: a *black hole*, an object that bends spacetime so much even light cannot escape.

If general relativity is correct, a black hole contains a singularity. Many physicists expect that general relativity breaks down inside a black hole, perhaps because of quantum effects that become important at strong gravitational fields. The singularity is considered a strong hint that this breakdown occurs. If so, the singularity may be a purely theoretical entity, not a real-world phenomenon. Nonetheless, everything we have observed about black holes matches what general relativity predicts.

The Tolman–Oppenheimer–Volkoff limit is not precisely known, because it depends on properties of nuclear matter that are not well understood. However, there are theorems that say singularities *must* occur in general relativity under certain conditions.

One of the first was proved by Raychauduri and Komar in the mid-1950’s. It applies only to ‘dust’, and indeed it is a precise version of our verbal argument above. It introduced the Raychauduri’s equation, which is the geometrical way of thinking about spacetime curvature as affecting the motion of a small ball of test particles. It shows that under suitable conditions, the energy density must approach infinity in a finite amount of time along the path traced out out by a dust particle.

The first required condition is that the flow of dust be initally converging, not expanding. The second condition, not mentioned in our verbal argument, is that the dust be ‘irrotational’, not swirling around. The third condition is that the dust particles be affected only by gravity, so that they move along geodesics. Due to the last two conditions, the Raychauduri–Komar theorem does not apply to collapsing stars.

The more modern singularity theorems eliminate these conditions. But they do so at a price: they require a more subtle concept of singularity! There are various possible ways to define this concept. They’re all a bit tricky, because a singularity is not a point or region in spacetime.

For our present purposes, we can define a singularity to be an ‘incomplete timelike or null geodesic’. As already explained, a timelike geodesic is the kind of path traced out by a test particle moving slower than light. Similarly, a null geodesic is the kind of path traced out by a test particle moving at the speed of light. We say a geodesic is ‘incomplete’ if it ceases to be well-defined after a finite amount of time. For example, general relativity says a test particle falling into a black hole follows an incomplete geodesic. In a rough-and-ready way, people say the particle ‘hits the singularity’. But the singularity is not a place in spacetime. What we really mean is that the particle’s path becomes undefined after a finite amount of time.

The first modern singularity theorem was proved by Penrose in 1965. It says that if space is infinite in extent, and light becomes trapped inside some bounded region, and no exotic matter is present to save the day, either a singularity or something even more bizarre must occur. This theorem applies to collapsing stars. When a star of sufficient mass collapses, general relativity says that its gravity becomes so strong that light becomes trapped inside some bounded region. We can then use Penrose’s theorem to analyze the possibilities.

Here is Penrose’s story of how he discovered this:

At that time I was at Birkbeck College, and a friend of mine, Ivor Robinson, whose an Englishman but he was working in Dallas, Texas at the time, and he was talking to me … I forget what it was … he was a very … he had a wonderful way with words and so he was talking to me, and we got to this crossroad and as we crossed the road he stopped talking as we were watching out for traffic. We got to the other side and then he started talking again. And then when he left I had this strange feeling of elation and I couldn’t quite work out why I was feeling like that. So I went through all the things that had happened to me during the day—you know, what I had for breakfast and goodness knows what—and finally it came to this point when I was crossing the street, and I realised that I had a certain idea, and this idea what the crucial characterisation of when a collapse had reached a point of no return, without assuming any symmetry or anything like that. So this is what I called a trapped surface. And this was the key thing, so I went back to my office and I sketched out a proof of the collapse theorem. The paper I wrote was not that long afterwards, which went to Physical Review Letters, and it was published in 1965 I think.

Shortly thereafter Hawking proved a second singularity theorem, which applies to the Big Bang. It says that if space is finite in extent, and no exotic matter is present, generically either a singularity or something even more bizarre must occur. The singularity here could be either a Big Bang in the past, a Big Crunch in the future, both—or possibly something else. Hawking also proved a version of his theorem that applies to certain Lorentzian manifolds where space is infinite in extent, as seems to be the case in our Universe. This version requires extra conditions.

There are some undefined phrases in my summary of the Penrose–Hawking singularity theorems, most notably these:

• ‘exotic matter’

• ‘something even more bizarre’.

In each case I mean something precise.

These singularity theorems precisely specify what is meant by ‘exotic matter’. All known forms of matter obey the ‘dominant energy condition’, which says that

at all points and in all locally Minkowskian coordinates. Exotic matter is anything that violates this condition.

The Penrose–Hawking singularity theorems also say what counts as ‘something even more bizarre’. An example would be a closed timelike curve. A particle following such a path would move slower than light yet eventually reach the same point where it started—and not just the same point in space, but the same point in *spacetime!* If you could do this, perhaps you could wait, see if it would rain tomorrow, and then go back and decide whether to buy an umbrella today. There are certainly solutions of Einstein’s equation with closed timelike curves. The first interesting one was found by Einstein’s friend Gödel in 1949, as part of an attempt to probe the nature of time. However, closed timelike curves are generally considered less plausible than singularities.

In the Penrose–Hawking singularity theorems, ‘something even more bizarre’ means precisely this: spacetime is not ‘globally hyperbolic’. To understand this, we need to think about when we can predict the future or past given initial data. When studying field equations like Maxwell’s theory of electromagnetism or Einstein’s theory of gravity, physicists like to specify initial data on space at a given moment of time. However, in general relativity there is considerable freedom in how we choose a slice of spacetime and call it ‘space’. What should we require? For starters, we want a 3-dimensional submanifold of spacetime that is ‘spacelike’: every vector tangent to should have However, we also want any timelike or null curve to hit exactly once. A spacelike surface with this property is called a Cauchy surface, and a Lorentzian manifold containing a Cauchy surface is said to be globally hyperbolic. There are many theorems justifying the importance of this concept. Globally hyperbolicity excludes closed timelike curves, but also other bizarre behavior.

By now the original singularity theorems have been greatly generalized and clarified. Hawking and Penrose gave a unified treatment of both theorems in 1970, which you can read here:

• Stephen William Hawking and Roger Penrose, The singularities of gravitational collapse and cosmology, *Proc. Royal Soc. London A* **314** (1970), 529–548.

The 1973 textbook by Hawking and Ellis gives a systematic introduction to this subject. A paper by Garfinkle and Senovilla reviews the subject and its history up to 2015. Also try the first two chapters of this wonderful book:

• Stephen Hawking and Roger Penrose, *The Nature of Space and Time*, Princeton U. Press, 1996.

You can find the first chapter, by Hawking, here: it describes the singularity theorems. The second, by Penrose, discusses the nature of singlarities in general relativity.

I’m sure Penrose’s Nobel Lecture will also be worth watching. Three cheers to Roger Penrose!

Theoretical work is rarely eligible for a Nobel prize because of a combination of two rules. One is the rule that you mentioned, that only work that has been experimentally confirmed is eligible. The other is the reason why Hawking could not share the prize posthumously with Penrose, which is that the award can only go to living scientists. Either rule is reasonable by itself, but in combination, they mean that there could be no prize for this work of Penrose and Hawking if Hawking had done it alone or if Penrose had not lived such a long time (he is 89).

There is also the rule that the nobel prize can only go to a maximum of three scientists at the same time, so Hawking could not have shared this years prize even if he had lived. Of course other combinations of scientists would have been possible, say Penrose and Hawking alone. But I guess the nobel prize committee also liked to combine theoretical and experimental work this time.

Interesting, I was not familiar with that rule! So maybe, morbid as it seems, they had to wait for Hawking to die before they could award a prize for this topic. It's strange that they would have this rule and still feel able to divide the prize in such a way that half is for the theoretical work (to go to Penrose alone) and half is for the experimental work (split between two people). That made it seem like they really wanted to split it evenly between all four people and Penrose only got Hawking's quarter because Hawking had died.

I don’t think so. The theoretical side of Hawking and Penrose has been around for a long time, so they could have received a joint prize earlier. I think that the experimentally confirmed aspect is important, and the observers who got half the prize played a major role in that. So, the time wasn’t really ripe until after Hawking had died.

It so happens I have seen both Hawking and Penrose in person, Hawking many times during my undergraduate time at Caius, Cambridge and Penrose once at a public lecture in Portsmouth. Respectively they were and are great men, but IMHO Nobels should not be given for theory.

What is your take on the fact that a mathematical physicist has won the Nobel Prize for physics? Is that the first time that has happened? Actually, Penrose is a mathematician by training. (Fair enough I suppose since physicist Edward Witten, who also has a degree with a major in history and minor in linguistics and worked as a journalist and studied economics and maths for a while before coming to physics—not that it took long since he was a full professor at the IAS at Princeton at just barely 26—has won the Fields Medal.)

I don’t know if Roger Penrose is the first mathematical physicist to win the Nobel. But Maxwell got a degree in mathematics from Trinity, and Newton and Hawking were Lucasian Professors in mathematics at Cambridge, so there’s certainly no obstacle to mathematically minded people doing good physics.

Then there’s Einstein, some guy, did papers on Brownian motion and relativity. But he got the Nobel for his paper on the photo-electric effect.

As a youth I was outraged that Einstein won the Nobel for the photoelectric effect instead of special or general relativity. But reading Pais’ biography

Subtle Is the LordI changed my mind. Pais points out that of all known fundamental particles, the photon was the most controversial for the longest time. Einstein basically proved that light comes in particles, and set the stage for the quantum revolution. Pais calls this Einstein’s only truly revolutionary discovery—in the sense that it did not follow naturally from previous concepts, as special relativity or general relativity do. It was a radical break from existing physics, and we’re still struggling with it.Thanks for your reply John. In the context of 1905, Einstein was following up on Max Planck’s 1900 work on quantisation.

A theoretical paper can’t “prove” anything, it can only suggest an experiment. This is the famous ‘disprovability’ of Karl Popper

Einstein’s work on the photoelectric effect together with the experimental data proved the existence of photons. I know about Popper; I’m just using “prove” in the everyday sense like: “if you sit down on your chair and it breaks, that proves it wasn’t strong enough to support you”. (Yes, maybe there was some other cause… )

” But reading Pais’ biography Subtle Is the Lord I changed my mind.”Indeed. Einstein himself said that his idea of the photon was the only time he had been really radical.

Special relativity was “in the air” and someone would have stumbled on it sooner or later. General relativity is rather different. (I like Pais’s comparison of the former with Mozart and the latter with Beethoven. Now, physics is waiting for Bach.) However, at the time there were doubts and little experimental verification. The golden era of GR didn’t start until after Einstein’s death, and in the last 40 years of his life there was little experimental evidence for it. Interestingly, much of that later came about because of technology based on quantum mechanics, which Einstein also had a hand in developing. The now basic issue of the difference between coordinate and actual singularities (mathematical; not necessarily physical) wasn’t even understood until Finkelstein’s work on that, which was after Einstein’s death.

Pais’s book is one of the greatest, if not the greatest, scientific biographies. Walter Moore’s biography of Schrödinger is also very good.

I was there! Here is a video of some of the greats from the golden era of GR reminiscing on the good old days. The whole video is worth watching. Dramatis personae:

Rindler died about a year and a half ago. Cecile DeWitt is also no longer with us. Again, get out the popcorn, sit back, relax, and watch the whole thing. Again, it is all worth watching, but Ted Newman’s contribution is particularly good. (I have a connection to the protagonist in one of his stories.) You can see my balding head from time to time; I’m sitting behind Joe Taylor.

Forgot the link: https://nsm.utdallas.edu/texas2013/events/

Or https://www.youtube.com/watch?v=iH8btReqv4c&feature=emb_logo

pwmiles:

You should read Kuhn’s

Black-Body Theory and the Quantum Discontinuity, 1894–1912. He argues that Planck did not believe in the quantized emission or absorbtion of light-energy untilafterEinstein’s work. Of course, not everyone agrees with Kuhn’s conclusions, but the issue isn’t as cut-and-dried as your comment suggests.For a more recent evaluation, see Revisiting the Quantum Discontinuity.

I also recommend this, which is short and free:

• Helge Kragh, Max Planck: the reluctant revolutionary,

Physics World, 1 December 2000.A quote about Planck’s 1900 paper:

Another:

Apparently, one reason Einstein did not get the Nobel prize for general relativity was because Gullstrand was on the Nobel committee. Gullstrand thought Einstein was wrong. Ironically, the Gullstrand Painleve coordinates that were intended to prove this, are now known to be an equivalent formulation of the Schwarzschild solution.

“so there’s certainly no obstacle to mathematically minded people doing good physics”I agree. But there is an obstacle to them getting the Nobel, in that it needs to be an “invention or discovery” or at least confirmed by observation, as opposed to “pure theory”. So what I meant was that it is interesting that it went to Penrose even though he is usually thought of as a “pure theory” guy.

Not that it matters in the grand scheme of things, but I have no objection to Penrose getting the prize for this work.

Interesting is that there have been several prizes for astrophysics in the last few years; until not that long ago, there had only been (depending on the definition) four or five.

“space is infinite in extent, as seems to be the case in our Universe”.Assuming a trivial topology, the universe is finite if . Observationally, we know that it is very close to 1, but the sign is unknown, so the Universe is either infinite or very big, but I would say that the jury is still out on whether it is infinite.

Hmm, I hadn’t known the jury was still out! It’s a pretty big question.

Here is something I never understood, maybe you have an answer. When you write “what we really mean is that the particle’s path becomes undefined after a finite amount of time”, whose time are you talking about? For example, for a Schwarzschild black hole, the Schwarzschild time to cross the horizon is infinite. Do you mean that the integral along a geodesic from any spacetime point to the singularity of the interval is finite? Do you know where I can find this calculation?

Tomate wrote:

Yes. This is the time that would be ticked out by your watch as you fell into the black hole—the ‘proper time’.

It’s in any decent textbook on general relativity. (We can use this as a definition of “decent textbook”, making the statement tautologously true.) I’m having trouble finding the calculation online, but the result is here:

• Geraint F. Lewis and Juliana Kwan, No way back: maximizing survival time below the Schwarzschild event horizon.

This studies how to maximize the time it takes to hit the singularity by firing your rocket as it falls into a black hole. Once you cross the event horizon there’s a finite upper bound!

Thanks!

John wrote:

John, Plato started this line of thought. Paraphrasing a bit >>Surely in the words of our own language (Classical Greek in his case) can be found all the ideas that are necessary<<

It doesn’t work. My hero Popper devoted the entire first volume of ‘The Open Society and its Enemies’ to rubbishing Plato. Another great book on this topic is “What is Mathematics Really” by Reuben Hirsch, sometime emeritus professor at the U of Albuquerque NM.

Well, one might argue that Quine did a pretty good job “rubbishing” Popper. And when it comes to the history of science, Popper’s “one counterexample disproves a theory” has turned out to be a rather useless viewpoint.

Hirsch’s philosophy of mathematics is definitely off-the-beaten track. Intriguing, but hardly Holy Writ.

I know, Popper’s star has faded. He’s generally thought to have come second in the long-running debate with Kuhn. But Popper, to my mind, better captures the process of discovery. He offers the metaphor of a net: your hypothesis is a net which you cast into the future, to see what it will catch.

I love this and I think it’s a model of problem-solving that is followed by people in many practical fields: plumbers, electricians, car mechanics. The vital point is to distinguish the hypothesis, or theory, from the discovery (or observation), which is a fact.

My original proposition, perhaps crudely advanced, was that “proof” needn’t have the same meaning in science as in mathematics (or, say, law or history). The methods and criteria are different. I think it’s fair to trace this error to Plato; call it over-reliance on verbal categories.

I don’t fully subscribe to Hersh’s attempt at a ‘humanist’ compromise between Platonism and anti-Platonism. But his book at least has the merit of pointing up the contrast in an accessible way.

Quine (to oversimplify no doubt) took on board the 19th century elaboration of the real number line, and held that such a conception of number was indispensable to physics. Again, this seems to me a step too far.

“I’m sure Penrose’s Nobel Lecture will also be worth watching.”

And here it is:

BTW, I’m quite sure you also posted this at the Café, but I couldn’t find it there. The October 2020 archive only shows three entries, none of which are this post.

Thanks!

No, I don’t think I posted it there.

Okay, I stand corrected. Thanks.

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