# Frontiers Lecture: Demystifying Black Holes with Steven Gubser and Frans Pretorius

by AMNH on

Black holes may be among the most mysterious phenomena in the universe, but innovations in astrophysics are bringing scientists ever closer to unlocking their secrets.

Princeton University physics professors Steven Gubser and Frans Pretorius review Einstein’s theories of relativity and what they say about the existence of black holes. They explain some of the fantastical properties of black holes, and discuss how recent findings from LIGO, the Laser Interferometer Gravitational-Wave Observatory, may provide the first physical evidence of these extraordinary cosmic occurrences.

This Frontiers Lecture took place at the Museum’s Hayden Planetarium on December 11, 2017.

**Podcast: Download | RSS | iTunes (36 mins, 35 MB)**

**Frontiers Lectures: Demystifying Black Holes with Steven Gubser and Frans Pretorius — Podcast Transcript**

**Steven Gubser: **What I wanted to start with is a discussion of time and special relativity, because a large part of what goes on in general relativity can be understood in contrast to what happens in special relativity.

Okay, before we look at these animations, I want to demonstrate as well as I can physically what they're all about. I have here two identical, highly accurate stopwatches, purchased from Amazon, and I'm going to start them at exactly the same time and give one to Frans. He will sit still, while I trot off to the exit and then trot back, and when I recover this stopwatch from Frans and stop them, I notice, oh, gee, they read the same time, exactly 12 seconds.

Now, if in fact these stopwatches had been able to show their full accuracy of 15 decimal points, I would have noticed that in the femtosecond place, my stopwatch had less time elapsed on it than Frans' stopwatch.

That's an incredibly small effect because my feeble attempt to move quickly to the exit was still much less than the speed of light. So, that's where the animations come in. This is me moving in a flat space-time with a stopwatch, only I'm moving at an appreciable fraction of the speed of light. And what you see is quantitatively accurate. It is the slowing down of my time as compared to Frans' time on account of my motion.

Okay, so this is the twin paradox. I'm the traveling twin. Frans is the stationary twin. I go on a journey. He stays at home. I come back, I'm younger than he is. But if there's one thing you learned from relativity, it is that motion is relative, so you might suspect that I've analyzed this somehow wrongly or incompletely.

Let's take a look at this second animation to see another possible perspective where we think of the motion relative to me, and relative to me, of course, it's really Frans moving, so wouldn't I conclude from that second perspective that it should be the other way around? It should be Frans' clock which runs slower.

That's the paradox. Now, the way it's resolved is that there's a very straightforward distinction between my actual motion and Frans'. He was just sitting. He was not accelerating. In order for me to complete my motion, if I started with some initial velocity this way, I would have to accelerate back toward him when I got to the end point of my motion. I am, in short, the accelerating observable. That cleanly distinguishes me from him, and it really says, yes, it's my time that goes slower.

What I really want to leave you with here is the impression that special relativity is easy mathematically once you let go of the correct preconceptions. In other words, if you say to yourself, "Oh, yeah, Newton didn't quite have it right. Time is not absolute," once you've made that tremendous intuitive leap, the math is the easy part.

Now what we'll do is go on to general relativity, which is mathematically much more sophisticated, and I think I'll turn it over to Frans to tell you about it.

**Frans Pretorius: **Okay, thank you. Steve. Yeah, what we ultimately want to tell you about are black holes. What does all this time dilation have to do with black holes? Perhaps we can ask a slightly different question: How can we describe black holes? They're actually fascinating subjects.

But there are many interesting ways in characterizing black holes. The way we thought we'd do it today is by extending this analog of what Steve described as essentially time dilation to general relativity and gravitational fields and curved geometries.

As Steve mentioned, special relativity is actually a theory about space and time, or space-time. When you take that together with the speed of light being constant, you get all these fascinating conclusions that relative observers perceive time to flow differently. General relativity is also a theory about the nature of space and time, but what's different in general relativity is space and time can now have a curved geometry.

In special relativity, geometry is perfectly flat. In general relativity, it can now be curved. And what Einstein's theory why—okay, special relativity, the mathematics is easy. What makes general relativity so difficult is the mathematics that tells you how space-time is curved and what causes space-time to be curved.

Here's a schematic example that you've probably seen before. The one-word answer in words about what causes a curvation in geometry is matter and energy, and a lot of matter and energy. For example, there's Earth. Earth, especially compared to us, has a lot of matter and energy, and so it's curving space-time quite strongly about it.

One of the consequences of that, the thing that we probably experience most strongly, is that that accounts for the force of gravity entirely. In other words, from Einstein's perspective, you don't need a force of gravity; what we think of as gravity is really curvature in space and time.

The other thing that curvature does is it causes time dilation. Steve showed you an example where relative velocities are experienced by observers as time flowing differently. In a curved space-time like this, observers that are closer to the strong curvature, their time runs slower, as seen by distant observers.

Now, one thing to keep in mind in relativity, whether it's special or general, is that time from your own perspective, your own time is always flowing at the same rate. You never notice that you're slowing down. It's a relative effect. For instance, us, sitting on the surface of Earth, we don't notice time dilation. Compared to someone very far away, we probably also wouldn't notice because it's incredibly small for something.

Earth is quite massive, but in terms of how much time dilation it causes—for example, the GPS satellites that are a few hundred kilometers orbiting around Earth, they actually have to take into account time dilation to get a correct position. But the relative slowdown of clocks here on Earth compared to a GPS satellite is a parting 10 to the 10, so basically a nanosecond per second. It's an incredibly small effect.

Okay, but now we want to get to black holes, and we want to get to black holes through time dilation. Einstein says that space and time is curved, and it's matter and energy that causes curvature, and the more matter and energy that you have and the more density you can compact a certain amount of matter and energy together, the more curvature you have. Consequently, then, the more time dilation you'll have or the more that time will slow down when you're close to the strong amount of curvature.

Let's look at a sequence of objects, astronomical objects that we think are out in the universe, that start from a moderate curvature—the star, think of the sun—and go down to the most extreme case.

Think about a star like the sun. It's about a million times as massive as Earth, so that's a million times the energy. It's in quite a larger area than Earth, but the sun has a lot more gravity or curvature. If you could actually stand on the surface of the sun—well, you can't; you'll get burnt to cinders—but there would be a much larger time dilation relative to here on Earth.

Now, if we try to imagine compressing the star, and one way in which nature does that is at the end point of a star's life when it's burnt up all its nuclear fuel, then it's going to collapse down to a smaller object. And one object in the sequence of the life of a star is called a white dwarf star. A typical white dwarf is also about as massive as the sun, so about a million times the mass of Earth, but your typical white dwarf is about the same size of Earth. So, you have something that's about a million times as massive as Earth that's compacted into a size of roughly the radius of Earth. So, in a white dwarf, now time dilation is becoming quite strong.

We can go even further. The next stage in the evolution of massive stars is some of them won't collapse to white dwarfs. They've got too much mass, and they will collapse down to something called a neutron star. Now, a neutron star is an extremely dense object: If you take something again roughly the mass of the sun and you compress it down into something about the size of Manhattan, so 15, 20 kilometers across. It's a whole solar mass' worth of matter and energy compressed into about 20 kilometers.

There, the effect that this has on space-time curvature is so strong that the time dilation effects become easily measurable, 20 to 30 percent. For example, every second that passes 100 kilometers away from the neutron star, if you could be on the surface of a neutron star, only 0.6 or 0.7 of a second will pass.

Let's continue this thought experiment. Let's compress things even further. As this graphic illustrates, curvature is now the depth of the surface. You can imagine, let's compress this neutron star that was 20 kilometers, let's compress it all down to a centimeter. Now, that should have tremendous amount of time dilation, but a very interesting thing happens in relativity.

What general relativity says is that at some point, you can't compress matter further and keep it fixed in a stationary configuration. At some point, the matter will start to collapse on itself regardless of anything else that's going on, and space-time itself will start to collapse, and that's when a black hole is born.

So you might think that to get the most time dilation, you have to be at the center. But what Einstein's theory says is when a black hole starts to form, you actually get a surface that's called the event horizon where the time dilation goes to infinity.

In other words, if you could get to a point that's close to the event horizon—for example, if the sun were to collapse to a black hole, the event horizon would have a radius of roughly three kilometers. So, if you could somehow hover out there, a distant observer will see a clock having come to a complete halt. Time would have frozen completely from the distant observer's perspective.

**Steven Gubser: **Okay, let's take a look at another animation or two that illustrate the points that Frans was making.

I have here first of all just a representation of the Schwarzschild solution, where I chose to be a little further from the horizon than Frans was. We're both hovering in place at fixed elevations, and we're going to do this thing with the stopwatches again.

What you're seeing here is exactly the kind of gravitational redshift that Frans was talking about, and maybe you can think about it in H.G. Wells' terms that if you want to travel into the far future and remain more or less young while you do it, one of the ways to do it is to hover near a black hole horizon or very, very close to a black hole horizon, and then the universe's time will just pass you by. And then when you come out again, it's the future, but you're only a few days older.

Another way to do this of course is to travel to a very far distant galaxy and then come back. That's the ordinary twin paradox way to do it. You might say in order to stay young, you can jog, if you can jog really fast. You can live deep in the gravitational well. You can commute to New York City. Any kind of motion will help you out in staying young. But I'm afraid it's only by a few picoseconds at the levels of motion that we can usually achieve.

Okay, so now let's see what happens if we do something a little different. I'm going to start at the same elevation as Frans, and I'm going to jump like this, and then when I come back down, we'll see where my clock stands relative to his.

Here I am going up, and I'm in freefall, just as you saw me as I really jumped in the room, and now I just landed and went back to hovering. But we stopped our clocks. Now you might say, Wait a minute, something really new has happened. This time, my clock went further forward than his.

That's very puzzling when you first see it because you say, wait a minute, I'm the traveling twin, right? Frans stays in place. I travel. I go up and then down, and yet, it's my clock which is further forward, so something is totally backward from everything we've seen before.

Why? It's because of what Einstein thought of as the principle of equivalence. Frans, in order to hover, in order to stay where he is like he's staying right now, he has to constantly push against the force of gravity. He's doing it with the chair right now. Whereas when I'm actually jumping, before I land, while I'm actually mid-air, I'm in freefall. In the point of view of Einstein's principle of equivalence, I'm feeling absolute weightlessness, and that is equivalent to essentially not feeling the gravity that Frans has to push against.

So, really, from the point of view of the principle of equivalence, it's Frans who is the accelerating observer. It doesn't look like it, but I'm the one who's in freefall while I'm doing this jumping thing.

But let's see one more demonstration here. What would happen if I continued on down toward the black hole horizon? I go up, my clock is running a bit faster than Frans', and then I start going down, and then, wow, my clock just completely comes to a halt while ordinary time far from the black hole, as well as to an extent Frans' time, continues on.

So, this is a case where as I'm heading toward the horizon closer and closer, my time just comes to a screeching halt, really. And what does this mean? It means that if an external observer were to watch me falling into the black hole, it would look like I never did. But from my point of view, it's not that way at all.

At Second No. 140.1, I would be inside the horizon, headed inexorably for the singularity.

**Frans Pretorius: **Okay, so now let us ask a question, then. We described the event horizon as being a place where this time dilation goes to infinity. An observer outside never sees you cross the event horizon, so what happens when you do cross it? Like Steve said, from his watch's perspective, he did cross it at Time 140.1. So, has he gone beyond infinity, in some sense? The answer, kind of astonishingly, is yes, in terms of this notion of time dilation.

Let me explain what it means going beyond infinity. It gets back to what I was saying that at some point, what relativity says is you can't take a given amount of energy or matter and compress it into a finite-sized blob and just let it sit there. At some point, it will have to stop collapsing. But it's not just the matter that collapses; it's space-time itself which is collapsing in on itself.

And what happens in a black hole when you do that, when you take this neutron star example and compress it beyond this three-kilometer event horizon, is space and time completely flip character crossing the horizon.

Let me explain it in this way. Taking a step back, one thing that we also know that happens in black holes is when black holes form, they have these things called singularities in the inside, which are bad places. They rip everything apart. The tidal forces get infinitely strong. If you fell into a black hole and you reached the singularity, in one direction you'll get stretched out into an infinitely thin, long line, and squashed in the other direction. You'll be spaghetti-fied.

Let's try to understand what this going beyond infinity is.

I've read our book, so I know that I should not cross the horizon of a black hole, but I come across this black hole in my journeys through the universe, and I see, okay, the black hole's over there. The thing is, I know the black hole's there, the horizon is over there. The singularity, where is the singularity? Well, it's inside the horizon, so I shouldn't cross it.

But let's suppose I kind of lose track of where I am, and suddenly I cross the horizon. It's like, oh, damn, I've crossed the horizon, what now? Where's the singularity? I've got to get away from the singularity. And that's the wrong question.

As soon as you've crossed the horizon, because of that space-and-time flip character, the question is not, Where is the singularity? It's, When is the singularity? And the answer: It's in your future.

So, once you've crossed the horizon, you can—and if it's a large black hole, you won't feel any tidal forces yet. You won't even necessarily notice where the horizon is. You can start, in a panic, shooting your rockets off in any direction. You can go as far as you might be able to go, but the problem is you're still going forward in time, and so you will reach the singularity.

And that's also one way of thinking or understanding why the event horizon is this one-way boundary. It's because the only way—once you've crossed that boundary, the only way you can get out again is to go backwards in time. And as far as we know, there's no way in which the laws of physics allow you to go backwards in time.

Okay, so we've explained a little bit about black holes in terms of time dilation. I think what you might appreciate is that black holes are completely crazy predictions of Einstein's theory in the sense if anyone asks, "Does something as crazy as that actually exist in the universe?" perhaps paraphrasing Carl Sagan, this is a really extraordinary prediction of Einstein's theory of relativity. And if we have such an extraordinary prediction, we need extraordinary evidence.

And so, now we'll switch to black holes in the universe. Is there evidence that such bizarre properties of space-time actually exist where space-time can undergo gravitational collapse?

The two pictures that we're showing over here is the evidence before LIGO's historic discovery of the merger of two black holes a few years ago, which we'll spend a bit of time talking about later. This was the best evidence that black holes existed before, but I phrase it as being very strong evidence that black holes actually do exist, but it's completely circumstantial.

Let me explain these two plots. The plot there on the left, what that's showing is over 20 years of observations of the orbits of stars around the dynamical center of our galaxy. It's in the constellation Sagittarius A. At the very center, there's a weak radio source that was labeled Sagittarius A*, and what this one snapshot is showing, those bright images of stars at one particular time, and then the trajectories, the ellipses and the circles, are the orbits of the various stars over the past 20 years.

What you can just see is that they're orbiting nothing. Each of those stars are perhaps the mass of the sun, and they're orbiting something which is not visible on this image. If you just apply Kepler's laws, like how much mass must there be in the center for those starts to be able to go around in roughly 20 years, you get a number that's about 400 million times the mass of the sun.

So, what on earth—not what on earth. What in our galaxy could there possibly be there that's four million times the mass of the sun but it's invisible? And so, well, let's say a black hole is one answer. It's consistent with it being a black hole. But as I said, I'd say it's very strong evidence but it's circumstantial. We don't see the black hole; we see the stars.

Okay, that all changed in terms of evidence for these weird things a few years ago when the LIGO, Laser Interferometer Gravitational-Wave Observatory, in the United States detected the collision of two black holes. These are the two LIGO interferometers, one in Hanford, Washington, the other in Livingston, Louisiana. They're these four-kilometer-long Michelson interferometers, essentially, that measure distances very, very accurately.

We'll explain in a second what gravitational waves are and why they want to measure those distances so accurately. This is the first confirmed gravitational wave signal that they measured in 2015. And since then, there have been many other detections—actually, five other black hole detections—and more recently the detection of binary neutron star collision.

What this is showing is that this is the key result that they published in the paper in *Physical Review Letters*. This is really the scientific data, and what the wiggles are showing you, there are a lot of curves and a lot of tracks, but the main thing to focus on is this oscillatory wave that increases in frequency and amplitude. It reaches a maximum, and then it dies down.

And what we think this is—and again, science, how certain are we that black holes exist? Well, science can never say 100 percent about anything like this, especially not something so far out there, but we can give good evidence. And in the previous examples, I said they were circumstantial pieces of evidence. Here, this evidence is the first direct evidence that black holes exist because what these waves are representing is as black holes are moving around and they're about to collide, they're churning up space-time, producing gravitational waves, and this signal is really telling us how the space-time about event horizons is exciting these gravitational waves.

This is really a signature that's coming from strongly curved space-time in the vicinity of event horizons of black holes. Unfortunately, if relativity is right—or perhaps fortunately—we'll never be able to see even with this direct measure what's happening across the event horizon because even gravitational waves can't escape, but this is telling us that space-time is as bizarrely curved as Einstein predicted in the vicinity approaching the horizons.

I'll just show an animation. This is from a group of researchers that made a simulation of this event just to sort of illustrate what it looks like. If you can start playing it, you can see those two little black dots. Those are the two black holes in orbit, and this event that LIGO saw, they were each about 30 times as massive as the sun.

Incidentally, this animation is being played at about a factor of a thousand slower than what the event really happened, so it's really being slowed down. These black holes are orbiting each other at a sizable fraction of the speed of light. These green waves, that's the gravitational waves that are being emitted. There, they coalesced into a single black hole, and then that burst of radiation left the system and it started propagating throughout the universe.

In fact, this event, based on how loud it was in LIGO, the amplitude of the waves, we can infer that it occurred about a billion light-years away from Earth. So, a billion years into the past, those two black holes collided. They produced those gravitational waves that traveled through the universe and eventually hit the LIGO detectors a couple of years ago.

Now we can explain a bit how gravitational waves work.

**Steven Gubser: **Right. Frans gave a good explanation of what we want to understand, which is what are gravity waves really? What is this thing that LIGO observed and we're so excited about? Well, I did my best to draw it. It's a successive compression and expansion of space-time in different directions.

So, in this illustration, gravity waves are traveling from left to right, and what we've pictured, just to guide the eye, is slices of space-time across the gravitational wave. This is a three-dimensional wave that's traveling from some distant source way off to the left, most likely some black hole merger, and as it passes any given plane, first as you see in green, that plane is expanded horizontally and then it's compressed horizontally, as the red arrows show, while being expanded vertically.

That expansion and contraction cycles over and over, and the cycles last—what is it?—about a hundredth of a second for the typical gravitational radiation from these sources. And that's no accident because it's a scale of the rotational period of those objects just before they merge.

Now, this picture of a gravitational wave can be compared perhaps to something you've seen before, which is a similar picture of a light wave.

What's especially exciting about the most recent LIGO discovery, which was the merger of two neutron stars, is that they were able to compare the rate of travel, the speed of gravity waves, versus light because the merger of neutron stars produces both gravity waves and light. And the merger event that they saw happened approximately 100 million years ago, and yet, the gravity waves and the light arrived on Earth only about 1.7 seconds apart.

So, that means that gravity waves and light travel at the same speed to approximately 15 decimal places or better. It is immediately one of the most accurate measurements in all of physics.

**Frans Pretorius: **I guess to connect to what Steve was saying, I can also just—perhaps if you can play the merger animation again as I'm explaining it.

The two black holes are in orbit. It's this orbital motion which is producing the gravitational waves, the stretching and squeezing that's sort of going outwards. The green there is telling you the amount of stretching and squeezing and it's propagating outwards. But gravitational waves carry energy, and so the consequence of that energy emission is that the black holes spiral in.

But then the closer they become, the more tightly bound they are. The faster they whirl around, and that's why the amplitude reaches this crescendo and then dies down, leaving behind a quiescent, stationary black hole.

As Steve was explaining, how do we measure gravitational waves? Well, they're disturbances in the geometry of space-time—again, this is Einstein's theory of relativity. It says that gravity is a geometric phenomenon, and so, how do we measure distances? Well, we have to come up with very, very accurate rulers, and that's essentially what this interferometer is.

What LIGO really is is a very clever way of measuring distances extremely accurately. It turns out that even if a tremendous amount of energy is released in gravitational waves, the changes in distances are incredibly small. That's why we've never measured—we don't perceive gravitational waves. The effects are so very, very small.

For example, in that event, the two 30 solar mass black holes that collided, they released about three solar mass' worth of energy in that last 20th of a second. If you think about it, three suns' worth, E = mc^{2}, liberating gravitational waves in a fraction of a second. The luminosity, it had such an extreme luminosity, if you were to compare it to, say, the luminosity in starlight, in that instant where the black holes merged, they outshone all the visible light in the universe combined. So, it's almost unimaginable.

But then the thing that's perhaps so astonishing as well is despite this tremendous energy release, the amount of warping of space-time is almost infinitesimal so that by the time it reached these interferometers on Earth, the change in distance over the four kilometers of the LIGO arms was less than the diameter of a proton.

That's the experimental challenge of LIGO. How can you measure changes in distances less than the diameter of a proton over a four-kilometer period? Interferometry is what they chose to do, and Steve will now explain how an interferometer works.

**Steven Gubser: **Okay, so let's start with the image on the lower right. That's a very simplified version of the LIGO detector

Here, this is a laser. This is the beam splitter. These are the two mirrors that reflect the light back to the beam splitter. The light comes from the laser to the beam splitter. It goes back and forth to the two mirrors. The light then recombines at the beam splitter and goes to the readout device.

That's the general construction of it. Now let's understand what it's trying to do.

In an ideal world or a simple world, this length here between the beam splitter and the one mirror would be exactly the same as this distance. If they were exactly equal, then there would be the same number of wavelengths of light going to and from each of the two mirrors so that when the light comes back to the beam splitter, it recombines constructively.

If instead a gravitational wave passes by, it will change the relative sizes of those two arms, it will change the relative length of those arms, so that the light does not interfere constructively but, rather, destructively. And you can see the difference from the interference pattern.

That's what the LIGO people are looking for. They're looking for their interference pattern to shake just a little bit, and they better be able to control for vibrational noise—literally noise—so as to isolate what is really due to gravitational waves.

**Frans Pretorius: **To conclude, then, let's just mention briefly this latest discovery that LIGO announced a few months ago is that they discovered the merger of two neutron stars. Again, neutron stars are each about 1.4 times the mass of the sun. In the early parts, a black hole and a neutron star merger are very much the same. They both have what's called the chirp waveform. It's the motion of the two compact objects about each other that produces the almost constant pitch, but as they spiral in, the frequency increases and the amplitude increases, and eventually it's this runaway and it ends in a burst.

The astonishing thing with this neutron star is it was in LIGO for the black holes for a fraction of a second. LIGO, what happens is it hears the sounds of the universe. It doesn't see it because the frequencies are in the audio range. And so, it heard this neutron star in spiral for several minutes.

And then as Steve also mentioned, at T = 0 it merged, and then 1.7 seconds later these gamma-ray observing telescopes saw a gamma ray from that same area. And then over the next few days and months, literally dozens of telescopes and satellites pointed at that part in the sky, and they were able to find exactly where the electromagnetic counterpart was. They saw the events first in gamma rays, just a second or so afterwards, and then they saw optical, radio, X-ray, infrared.

And perhaps one of the fun little facts about that is the characteristics of the optical light that they saw is very consistent with something called the rapid nuclear capture process, or the r-process, which is what we think is where heavy elements in the universe form, such as gold and platinum and lead.

It was always thought that possibly supernova could account for some of them, but the last 20 or 30 years was a real problem. Simulations couldn't produce the observed abundances of these heavy elements, and it was theorized that perhaps neutron stars can do this. And here was this evidence now that actually the heavy elements like gold are produced in neutron star collisions.

**Steven Gubser: **This is why I like to say that we can now answer a question that LIGO scientists could not when they were asked, "What is the music of the cosmos that you are hearing?” They said, "We're not sure.” Baroque? Jazz? Now we know: heavy metal.

All right, I think we've gone through most of what we had to say. I hope you are intrigued to check out what more we have in our book.

Thank you for your attention.

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