How The Penrose Singularity Theorem Predicts The End of Space Time (Script) (Patreon)

The Nobel prize in physics this year went to black holes. Generally speaking. Specifically it was shared by the astronomers who revealed to us the Milky Way’s central black hole and by Roger Penrose, who proved that in general relativity, every black hole contains a place of infinite gravity - a singularity. But the true impact of Penrose’s singularity theorem would is much deeper  - it leads us to the limits Einstein’s great theory and to the origin of the universe.

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Black holes have haunted our theories of gravity since the 1700s. When John Mitchell and Pierre-Simon Laplace explored Newton’s law of universal gravitation, they realized the possibility of a star so massive that it would prevent even light from escaping its surface. Few people took these “dark stars” seriously - especially when we learned that light didn’t really behave as Mitchell, Laplace, and even Newton assumed. And then of course we figured out that not even gravity worked as Newton had told us. In 1915,  Newtonian gravity was superseded by Einstein’s general theory of relativity. But that wasn’t the end of black holes or dark stars; it was their resurrection.

Karl Schwarzschild solved the equations of general relativity soon after Einstein published them, revealing that a sufficiently dense ball of matter would be surrounded by a surface where time froze. Beneath that “event horizon” all matter, light, space itself was doomed to fall towards a central point. At that so-called singularity, the gravitational field becomes infinite. But physicists tend to be dubious about infinities - more often than not they turn out to be a glitch in the theory. Einstein himself doubted that the black holes could form in the real universe, and even if they could, they certainly shouldn’t harbor singularities.

After all, Schwarzschild’s solution didn’t say anything about HOW a clump of matter could reach the densities high enough to produce an event horizon - nor even whether such dense matter would really contract into a single point. It only showed that, once so-contracted, the resulting black hole was stable.

But the intriguing and terrifying possibility that black holes might really exist inspired some of our greatest minds to find ways they could exist. In 1939 Robert Oppenheimer and Hartland Snyder showed that a perfectly spherical, perfectly smooth ball of dust could collapse into a Schwarzschild black hole, singularity and all. Makes sense - if all particles are falling directly towards each other on a perfect collision course, the math has to land them all in the center.

But this didn’t convince anyone - what in our universe is perfectly spherical or smooth? It was assumed that normal, messy objects could never collapse into a perfect point. Surely any tiny deviation from spherical symmetry would cause particles to miss each other at the last instant. That might, for example, result in the infalling matter flying back outwards again. So, for example, a star that collapses at the end of its life might entirely rebound as a giant explosion.

And now we come to the mid 1960s. Roy Kerr has just figured out how to describe a rotating black hole in Einstein’s theory. In it, the central point of infinite gravity is spun out into a ring - but it was a singularity nonetheless.  The Kerr solution was still highly symmetrical, so the same old arguments held against these things forming in reality.

It took a young Camridge physicist named Roger Penrose to prove that black hole singularities were utterly unavoidable in Einstein’s theory.  Penrose’s singularity paper is deceptively short - just a couple of pages in Physical Review Letters However that brevity is deceptive. But some say this 1965 paper produced the first truly new advances in general relativity, half a century after the theory was published.

So what did Penrose discover? How did he manage to peer into the mathematical heart of the black hole? And why did this deserve a Nobel prize? Penrose set out to show that an event horizon and a singularity would form for any distribution of matter, no matter how messy, as long as that matter was compacted into a small enough volume of space. At least according to general relativity. And his singularity theorem succeeded in that - but it succeeded in so much more, as we’ll see.

Today I’m going to give you a sense of Penrose’s breakthrough, and why it revolutionized general relativity. For a more complete picture I can refer you to a good grad program in physics - but let’s see how much we can do in a handful of minutes. Basically, Penrose showed that according to Einstein’s theory plus a couple of assumptions,  black holes must contain singularities. And this is true regardless of how the black hole formed.

He did it in a clever way - by showing that the grid of spacetime literally comes to an end inside a black hole. In general relativity we map the fabric of spacetime with  gridlines that we call geodesics. These are the paths traveled by an object in freefall in a gravitational field. The path traveled by a ray of light is called a null geodesic. They are the gold-standard for gridding up spacetime.

Null geodesics  traveling into or past any gravitational field tend to be drawn together - to converge, or be focused. This is gravitational lensing. Gravitational fields produced by regular matter always produce this convergence. It would take negative mass or negative pressure to cause light rays to diverge. This focusing property of gravitational fields is called the weak energy condition of general relativity, and it almost certainly holds in black holes.

So null geodesics traveling down into a black hole are going to converge - no big surprise there. But the crazy thing about black holes is that null geodesics  beneath the event horizon that are trying to travel outwards also converge. Actually, the whole idea of the event horizon is tricky here - there are multiple ways to define an event horizon depending on where you sit compared to the black hole. Penrose came up with a more precise idea - that of the trapped surface. That’s any closed surface that has this property that null geodesics - and so any light  - pointed outwards from the surface actually move downwards. Any closed surface “inside” the black hole is considered a trapped surface.

Penrose showed that at least some of the parallel null geodesics leaving any trapped surface, either up or down, had to converge. Had to cross each other and come to a focus. And he also showed that, for rays departing a trapped surface, it’s meaningless to continue to track the progression of space and time beyond one of these focal points. In other words, space and time end at the focus.

I can only give you a vague sense of why this is the case. Penrose proves it by showing that impossible contradictions arise otherwise. Imagine a pair of light rays emerging from the same point and then focused back towards each other. Both are equally the shortest path between those points. Now imagine extending one of those paths just a little. The distance from the starting point to the end of that extension is still the same whether we take the left or right arc. But now we can find an even shorter path - one obtained by smoothing out this kink. And that has to be shorter than both original paths. So there’s the contradiction - we found a shorter path than the supposedly shortest paths.

This tells us that the null geodesic does not continue past the focal points AS a null geodesic. Null geodesics terminate at these focal points within a black hole. We call this geodesic incompleteness. Because they are the grid we use to map space and time, geodesic incompleteness means space and/or time end at these termination points. It doesn’t just freeze, they literally cease.

Let’s try an analogy - the geodesics we use to map the surface of the earth are longitude and latitude. Lines of longitude come to a focus at the north and south poles. A line of longitude is the shortest distance to the north pole - the quickest way for you to increase your “northness”. But if you travel past the north pole you’re no longer on the shortest path to any new point that you reach, and at the same time you reached the end of north - maxed your northness - and started traveling south again. Well, in a black hole you don’t reach the end of north, you reach the end of time or space. They are dead-ends to reality.

So yeah, geodesic incompleteness is pretty freaky. Prior to Penrose it was generally held that geodesics could be traced indefinitely into the past and future. All of spacetime should be a smooth, if curved structure - a manifold - cleanly defined everywhere. Penrose showed that space and time could have holes in it. These holes tend to be associated with infinite spacetime curvature - infinite gravity. In other words, singularities. Penrose didn’t actually show what type of singularity would form in a black hole - just that some type was inevitable.  In a Schwarzschild black hole, time ceases at the point-like central singularity, while in Kerr black holes space ends at the ring singularity.

With Roger Penrose’s discovery, black holes and the singularities within had to be taken more seriously.  But the true utility of his singularity theorem went well beyond black holes. Just after Penrose published his paper, a young graduate student was inspired to apply the singularity theorem in a very different way. That student was Stephen Hawking. In his PhD thesis Hawking showed that you could use the same arguments to investigate the behavior of geodesics traced backwards through our entire universe towards the Big Bang. Now at this point we’d known for 40 years that the universe is expanding, and probably started in a much denser state. In fact the latter had been verified by the detection of the cosmic microwave background just the prior year in 1964.

We knew that geodesics converge towards each other looking backwards in time, but doesn’t mean they had to all meet. They could, for example, miss and weave past each other in an insanely dense but not infinitely dense knot. This might be the case if the universe underwent cyclic big bounces.

Hawking showed that Penrose’s arguments about black holes also applied to the universe - that geodesics traced backwards in an expanding universe had to truly meet - to form a singularity, which meant they had to terminate. Time itself could not be traced beyond this point - which suggested that time really started at the Big Bang. Hawking and Penrose further developed these ideas together, publishing a clean proof in 1970. We now call the combined proofs of geodesic incompleteness the Penrose-Hawking Singularity Theorems.

This is probably the right time to tell you how the singularity theorems must be wrong - or at least point to wrongness. They give us the predictions of pure general relativity, and assume the various energy conditions. We know that general relativity is not the entire picture - and this might be the most powerful result of the singularity theorems. Remember, we should be dubious when we see infinities and singularities in our theories. Penrose showed us that singularities are unavoidable in general relativity, and so GR must break down at those points. The resolution must be the union of general relativity and quantum mechanics - a theory of quantum gravity, from which both quantum mechanics and relativity are just approximations. Such a theory may tell us what really happens to geodesics when they approach and merge. And even, hopefully, what geodesics - and space and time - really are. Penrose’s singularity theorem is a big part of what set us on the path to this yet-undiscovered greater theory.

Roger Penrose won this year’s Nobel prize in physics for his contributions to our theoretical understanding of black holes. He shared it with Andrea Ghez and Reinhard Genzel, who proved to us the existence of the Milky Way’s central supermassive black hole by monitoring the crazy orbits of stars in the galactic core. The work of Ghez and Genzel and other astronomers have guaranteed the existence of black holes, which means their are places in the universe where general relativity must break down or produce singularities. Nobel-worthy insights, and all from some bright ideas about how light rays travel and terminate at the singular dead ends of spacetime.

632 COMMENTS

https://www.youtube.com/watch?v=EagNUvNfsUI&lc=UgzeetroAm3H89yFcqF4AaABAg

Anonymous makes some important points. I’ll summarize - in Einstein’s relativity the definition of simultaneous events is relative. It depends on relative velocities. Does this mean that the idea of a “now” slice through the block universe is meaningless? Well, maybe - but two observers can certainly construct a map of events across the universe that they would define as happening simultaneously at a given instant in time: their slice of now. Even though they have to wait until all that light reaches them to do so. So the “now” slice has a precise meaning. What it does not have is the property of clearly and currently existing to the exclusion of other slices of the block universe. And that’s where the relativity of simultaneity connects to eternalism. If you can ascribe actual existence to one slice of the block universe - to one observer’s definition of the present, then you should ascribe existence to the slice of the present as defined by any potential observer on that first slice. And that fills the block universe, past and future.  The alternative seems to be to NOT ascribe reality to any observer’s present. But then what exists? Just the one observer? At any rate, the relativity of simultaneity doesn’t “prove” eternalism - rather it shows that presentism or the “growing block” interpretations are incoherent.

https://www.youtube.com/watch?v=EagNUvNfsUI&lc=UgyRdfOw17vqZdlM3-N4AaABAg

Speaking of presentism, Carl Stanland raises a point that highlights another objection I have to the concept. If only the instant exists - what is that instant? Is it a very short stretch of time, like one Planck unit of time? Or is it a truly instantaneous stretch of zero seconds? As much as physicists hate to talk about consciousness - I think it’s relevant to bring it up here. Our conscious experience “exists” - in fact it’s the only thing we can observe directly. But that experience almost certainly is emergent from the action of our neurons, and those neurons take time to fire and communicate across the brain. So something that “currently exists” is smeared out over at least 100 milliseconds of time. At least that length of present should be thought of as real. You might argue that our conscious experience rides the leading edge of the brain’s action - emerging in a true instant, synthesizing prior brain activity. But for that to be a true zero-time instant, then it would need to be the instantaneous state of the brain - the momentary spatial configuration of its particles - at that instant that gives rise to conscious experience. So then does a frozen brain just experience the current instant eternally? Most probably not - rather, consciousness surely requires dynamical processes - a flow of information. Which means at least a short chunk of the clock universe has a real “current” existence - much large than a Planck time. And if that much time can “exist”, why not all of it?

https://www.youtube.com/watch?v=EagNUvNfsUI&lc=UgxjsOZJ_FACdktiTrt4AaABAg

Guilhereme Marquesani takes issue with my characterization of Einstein as the smartest man in the block universe. Dude, I said block universe - not universES. Rick is the smartest man in the multiverse. Also, I’m pretty sure Einstein was another renegade from the Citadel of Ricks. The resemblance can’t be a coincidence.

https://www.youtube.com/watch?v=EagNUvNfsUI&lc=UgwbFOaGiQdBkaO-8jx4AaABAg

https://www.youtube.com/watch?v=EagNUvNfsUI&lc=UgyrLe1ABW6wbIxnVZF4AaABAg

And for those uninitiated into Rick and Morty, there are other funny pop-culture references in the comments. Drake gave a nice Douglas Adams quite: “Time is an illusion, lunchtime doubly so”. Grandpa’s place reminds us of Dr Who’s experience of time as “big ball of wibbily wobbly timey wimey…stuff”. But really, you should just go watch all of Rick and Morty. Think of it as a co-requisite study course to PBS Space Time.

Wubba Luv a Dub Dub

633 COMMENTS

Last week we continued our discussion of determinism in relativity and the block universe, but this time weaving in what quantum mechanics had to say  about the whole mess. Quantum mechanics had a few things to say - you guys had more. Let’s get to the comments.

https://www.youtube.com/watch?v=1JCRDaa3ehk&lc=UgwkUJ2_t3ez_8QO7Mt4AaABAg

Zoltan asks if there’s any possibility to observe or interact with any of the other world, and if no how is the idea falsifiable? Actually yeah, we can interact with the other worlds. - AKA branches of the wavefunction. That’s exactly what a quantum interference experiment is seeing - multiple possible “realities” overlapping and either stacking or canceling out to either strengthen or cancel out certain ongoing branches of the wavefunction. Many Worlds says we find ourselves in one of those branches. The problem isn’t that we can’t observe the other branches at all - just that we can’t observe them on a macroscopic scale. There is one proposal by Steven Weinberg called the Everett-Wheeler telephone that allows a type of communication between worlds, but only works if quantum mechanics works in a particular non-linear way We should probably do an episode on it.

As for falsifiability - I’ll try to give Sean Carroll’s argument for this one. Many Worlds can be considered  the “cleanest” interpretation of quantum mechanics. It’s what you get when you assume nothing but the evolution of the wavefunction under the Schrodinger equation. Other interpretations - Copenhagen, pilot wave theory, etc. actually add things to the pure evolution of the wavefunction under the Schrodinger equation. So we can look for those extra things - the moment of wavefunction collapse, the guiding function or the empty wavefunctions of pilot wave theory - if we continue to try and fail to detect these required additions, then “pure quantum mechanics” becomes more likely, and some would argue that means many worlds.

https://www.youtube.com/watch?v=1JCRDaa3ehk&lc=Ugwx70i6RIyUn7gJgmJ4AaABAg

Michal Grno points out that we forgot to mention Carlo Rovelli's relational QM interpretation. Actually, I didn’t forget - I had a whole section on it but realized there was no way to do it justice especially because we haven’t done an episode on that before. So how about we just do that episode. Because I agree - relational quantum mechanics is a really profound perspective, in which the “real” quantum states aren’t the entities but the relationships between the entities - pretty relevant for addressing the reality of other parts of the block universe.

https://www.youtube.com/watch?v=1JCRDaa3ehk&lc=UgzXUjfLnzHpD43jZRJ4AaABAg

Alan Foxman struggles with the concept that there’s no objective “now”. Even though we can only see the universe of the past - e.g. a star 100 light years away being a century in the past. Surely if we could teleport there we’d see the modern version. That’s true, but here’s the issue: if you and your friend do the same teleport, you’ll end up at different times depending on your relative velocities. Your velocities determine your definition of the present.

https://www.youtube.com/watch?v=1JCRDaa3ehk&lc=UgwaUXNdv_wtrXbD5H14AaABAg

This is all very hard and Coffea gives us a way out. This could be “That moment when a boltzmann brain is the simplest explanation.” You’re just random particles accidentally assembled from the void with your current memories and experience of the world and are about to evaporate back into it. Prove I’m wrong! There, science, finished! Maybe I’ll teach next semester this way and walk out 15 seconds into the first lecture. Nice.

This week we want to extend our heartfelt thanks to … …, while our singularly grateful for your support, our appreciation is by no means geodesically incomplete - rather it extends infinitely into the future AND the past. With your help Space Time might do the same. Well, not into the past - hard to make new past episodes before the first one. Although if anyone could Gabe could - hmmm. Well, for now we’ll keep pushing for the future, thanks to the support of all of you.