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JIM ALEXANDER: Good evening, you're to the 46th Bethe Lecture. My name is Jim Alexander, and the next few minutes I'd like to tell you a little about the life and work of Hans Bethe, before I turn the microphone over to my colleague, Tom Hartman, who will introduce tonight's distinguished guest.
Hans Bethe was one of the titans of 20th century physics, who made profound contributions to almost every field of physics during a career that spanned 75 years of active research. He began his formal physics training at the age of 20. It was 1926, the year of the birth of biomechanics, and Bethe quickly mastered the new ideas and began to apply the principles of biomechanics, [INAUDIBLE] his equation to a wide range of phenomena and condensed metaphysics.
By the time he was 25, he had already published 10 papers, of which four stand even today as enduring classics. At the age of 27, he wrote two book-length reviews, one on the physics of one and two electron patterns behavior of electrons and [INAUDIBLE].
His approach to writing these reviews was to re-derive every result himself, extending it in new directions and to new depths that had not been previously probed. The reviews became textbooks for a whole generation of physicists.
That same year, 1933, saw the rise of the Nazi Party in Germany, and Bethe, because of his mother's Jewish heritage, lost not only his current position at the University of Munich, but also the possibility of any future academic position. So he left for England.
And in Manchester, he met Rudolf Peierls, who formed a friendship that lasted a lifetime, and a professional relationship that changed the direction of his research.
He launched into nuclear physics, and in 1934, he wrote [INAUDIBLE], a landmark paper on the structure of the deuteron and the short range nature of nuclear forces. Many other papers followed in rapid succession, and he soon became acknowledged as the world expert on all matters of the new field of nuclear physics.
Three years later, now at the ripe old age of 31, he wrote a 500 page, three volume review of everything known at that time about nuclear physics. Again, he re-derived every result, and extended every concept with new calculations so that, in fact, much of what he was reviewing had actually not been known until, again, he took to write the review.
The monumental work that resulted became known as Bethe's Bible, and stood for decades as the primary text on which another generation of physicists were made.
You need only peruse a few pages of this work to appreciate the clarity and precision of Bethe's language, and the unlabored manner in which he unfolds and systematically inspects each topic and subtopic in turn. And he wrote that work here, in this building.
After a sojourn in Manchester, he had come to Cornell in 1935, having been sought out specifically for his expertise in nuclear physics, a direction this department wanted to go. And the informal and collaborative style of doing physics that he had encountered first in the English and then the American physics communities, was firmly grounded for Bethe, and he later described the 1930s as his most productive decade.
In 1938, after a stimulating conference in Washington, he turned his attention to the problem of energy generation and stars, and quickly discovered the complex cycle of reactions that power not only our sun, but starts of all sizes, for which he later won the Nobel Prize in 1967.
At the outbreak of WWII, he plunged into work on military problems, and when the Manhattan Project was formed, its director, Robert Hoffenheimer, insisted that Bethe be head of the theoretical physics division at the Los Alamos Laboratory.
Unlike many of his colleagues in the Manhattan Project, Bethe never expressed regret or remorse about its outcome, but in the years that followed, he devoted himself tirelessly to arms control, ban on nuclear testing, and the goal of general disarmament.
He brought his scientific and calculation talents to bear on many critical technical problems that formed the hidden underpinnings of these urgent, political and social issues. He served frequently these years as advisor to Presidents Eisenhower, Kennedy, and Johnson. At the same time, he was shaping the physics department at Cornell, and he left the imprint we see still today in this [INAUDIBLE] collegiality.
He also left his imprint on the university as a whole. In the turbulent times of the late 1960s, particularly when armed students took over the student union and declared that Cornell had quote, "three hours to live," it was a nuclear physicist, of all people, who helped calm the turmoil. As then Cornell President, Dale Corson said, Bethe's was the voice of reason, the voice of clarity, the voice of responsibility, and the voice of conscience. And indeed, that's the same description could well apply to everything he did.
In later stages of his career, he took up new scientific issues including the supernova mechanisms. As an octogenarian, he solved a long-standing solar [INAUDIBLE] problem that stymied a generation of younger colleagues for 20 years, ever since Ray Davis first started counting neutrinos at the homestead [INAUDIBLE] in 1964.
It's hard to imagine any career of 75 years, let alone a career of such sustained productivity and immense breadth. As Kirt Godfrey said in his remarks at the Bethe Memorial in 2005, his intellectual output was on a scale that you would have considered impossible, if he hadn't actually existed.
The Bethe lecture series was instituted in 1977 to honor Hans Bethe and his many contributions to this department. Over the years, it has brought dozens up dozens of brilliant physicists together with technical and public lectures.
So on that note, I will turn this now over to my younger colleague, Tom Hartman, who will introduce this year's distinguished Bethe lecturer.
TOM HARTMAN: Thank you, Jim. So it is a pleasure and an honor to introduce tonight's speaker, Juan Maldacena. Professor Maldacena was born in Buenos Aires. After completing his undergraduate education and master's degree in Argentina he moved to Princeton, where he earned a PH.D in 1996, then to a post-doctoral position at Rucker's University, shortly followed by faculty position at Harvard.
Since 2001, he's been a professor at the Institute for Advanced Study in Princeton, New Jersey. During and shortly after his PH.D. Professor Maldacena uncovered a relationship between gravity and particle physics, which has literally redefined space and time.
This discovery sparked an entire new field of inquiry, and has led to dramatic progress in theoretical physics over the last 17 years. During that time, Professor Maldacena has continued to be a leader in the field, with important contributions to our understanding of gravity, quantum mechanics, black holes, string theory, particle physics, and the Big Bang.
He's been recognized with numerous awards, among them the McArthur Fellowship in 1999, Sloan Fellowship, Packer Fellowship the UNESCO [INAUDIBLE] Prize, the Sackler Prize, many others, including the [INAUDIBLE] in 2008 and the Miller Foundation Fundamental Physics Prize in 2012. He's a member of the Academy of Arts and Sciences and the National Academy of Science.
Today we're fortunate to have him here to speak to us about black holes and the structure of space times. So please join me in welcoming our Bethe lecturer, Juan Maldacena.
JUAN MALDACENA: Well, thank you for the kind introduction. It's really a pleasure for me to be here to give this Bethe lectures. I've used many of Bethe's results in various branches of physics. It's really amazing how much he has done.
So I'll be telling you about black holes. And this is the outline of what I would like to tell you about. So I would like to tell you the history of black holes. And the history of black holes it's full of initial steps, and almost everyone who has talked about black holes was confused about some aspect of black holes. And I probably won't be an exception.
So black holes were discovered by Schwarzschild. I'll review how this happened. And even the creator of general relativity didn't believe that. So that's the first person who was wrong. And then it was understood that Einstein was wrong, and actually black holes could be formed from very massive stars.
And then it was realize-- Hawking realized that they're not black. So the name was even wrong. So there are black holes that could be red, there are black holes that could be white. So even the name is wrongly chosen.
And then he further said that they imply that quantum mechanics is incompatible with gravity. Then come some developments in string theory I'll try to summarize, which say that black holes actually seem to be compatible with quantum mechanics in the end, and this uses a certain relation between particle physics and gravity. So that's the outline of the talk.
But we'll start the talk by reviewing some important principles and some important facts about nature. And the first fact is this principle of relativity, s originally formulated by Galileo, saying that two observers traveling with constant velocity observe the same laws of physics. So if you are in the lab here at rest, or you are in a lab traveling, let's say, in an airplane, and you do some experiment, you would see the same results for the experiment. You wouldn't notice that you are moving if you don't look through the window.
Now that is fairly intuitive from the classical physics point. Well, it wasn't intuitive at the Galileo's time, but it's perhaps more intuitive in our time. We certainly travel in airplanes very fast, and we seem that we are stationary inside.
Now it becomes a little more surprising when you couple it with the one new fundamental law, which is the idea that there is a maximum velocity for the propagation of signals, which is the speed of light. And this is a second principle that Einstein incorporated into relativity. And this is surprising, because it is saying that the speed of light is the same for all observers.
So imagine you have some beam of light propagating. So we have some photons here propagating, some light waves. And now we have the same two observers that we discussed before. And how can it be that they see exactly the same velocity? And it turns out that in order for them to see this-- naively you would look at this and you say, well, this one on the top should see a lower velocity. That would be the intuitive answer.
And the first time you hear this about the two observers seeing the same velocity, you say, must be wrong. You must be saying something wrong. But actually, this is the way it works. And it works because this observer from the point of view of this one looks a little smaller, and time also for this observer travels a little slower. And in the end of the day, both actually measure the same speed of light.
Now this is counter-intuitive, but this is the way nature works, and this has been tested in numerous experiments. And that's how it works. Now this is sometimes expressed as the so-called twins paradox, where we match in two identical twins that are born. And then one is moved at high velocity relative to the other one, and then they come back many years later, and one is younger than the other. OK? That's how it works.
Now here, you might be a little surprised, because the situation was not symmetric between the two twins. So one suffered a little bit of acceleration at the beginning, and then acceleration at the bottom. And that broke the symmetry between the two twins. But anyway, so that's how it works. And this, again, has been tested in many experiments. Not with actual people, but with very precise clocks you can actually test this.
And the lesson I want to draw from this is that space and time form a single entity that be called space time. And our measurement of time, or our measurements of time on distances depend on how we are moving. So two observers moving relative to each other measure space and time differently. And we know exactly how to translate between the observations of these two observers.
The simplest way to think about it is that there is one thing which is space and time, and what we call time or space differs for each observer. So that's why in the rest of the talk we'll be talking about space times. So normally I'd say, well, you put together the word space and time. Time and space look very different. But it's this relativity and the fact that the moving observes mix them that forces us in physics to talk about space time as a single entity.
So that's why we talk about space time. Now we'll start discussing something slightly different, which is gravity. So we know that our intuitive idea is that heavier objects fall faster. We can certainly drop this and a piece of paper, and one will fall faster than the piece of paper. And then Galileo realized that once you remove the resistance of air, everything falls in the same way. So everything follows the same trajectory.
And in physics, we normally elevate this into a new physics law, sometimes called the Equivalence Principle, which says that everything falls in the same way. So if we are in a gravitational field under the influence of gravity, everything falls in the same way, independent of the composition.
This is not true for other forces. So for example, if you have an electric field and you have a charged particle, depending on the charge to mass ratio, the particles will move differently. But for gravity, this is not the case. Everything falls in the same way. And again, this had been tested with very high accuracy.
So when we have a space ship or the astronaut orbiting the Earth, both are, in some sense falling, around the Earth exactly at the same rate. Or if an airplane executes a free trajectory, Stephen Hawking falls in the predicted way, seems to float inside the airplane.
Now, Einstein developed the Theory of General Relativity to incorporate that principle. The idea is that gravity is due to the geometry of space time. So the idea is that a heavy object curves the space and time around it. So space and time are not flat, but they are curved by the presence of a massive object. And then a second particle that moves here will follow the trajectory of maximum time in that space time.
And the fact that different particles of different compositions are exploring exactly the same space time is the reason everything falls in the same way in a gravitational field. So all the particles are exploring the same nature of space time.
Now, it's important that space and time are both modified by gravity. And this is sometimes a little hard to visualize. So we'll discuss one particular aspect of it in some detail. So an important fact is that when we are in a gravitational field, the two observers who are at different positions in this gravitation field see time passing at different rates.
So let's say we have two observers, one at the bottom floor of a building and one at the top floor of the building. Then time for the observer at the bottom floor will go a little more slowly. So these little dots represent the clock. I didn't know how to do a rotating arrow in PowerPoint. Probably it's very easy to do, but sorry. At least I couldn't do it quickly.
And so this is what happens. And this has been tested experimentally. And works this way. And for a case of the person at the bottom floor of the building and one at the top floor of the building, you really need a very sensitive clock, because the difference is one part in 10 to the 15. So a very, very tiny difference. But there are such very accurate clocks where it can be measured indirectly. But you can actually see this effect.
So this gives rise to another kind of twins paradox where, imagine you have a massive body here, a very heavy object. And you have the same two identical twins. And then one spends some time near this very massive object, and then we come back out. This one will be a little younger. So this fact that time goes more slowly, it works for all clocks, even the biological clock.
So Einstein wrote down his equations that determine the shape of space time. And very soon afterwards, Karl Schwarzschild found the geometry outside a massive spherical body. So he solved equations for a spherically symmetric configuration, so the simplest configuration you can imagine, and he found that solution.
And that solution is a particular curved geometry. And in particular, that geometry tells us how time flows at different radial positions. So imagine you are sitting at some fixed distance from the star, and you have a clock, and you can see how the clock moves, how time passes at that position. And this solution that Schwarzschild found tells us exactly how time passes. So there is a rate of time far away, and this is almost constant far away, because we are far away from the body, we have flat space far away.
But then, as we get closer, time slows down a little bit. So this one travels a little more slowly than this other one. And suppose we have a star of the same mass and we make it smaller. Then we are exploring more of this solution that Schwarzschild found. And we find that time slows down even more. Again, this is an effect that has been tested. If you put a clock at the surface of the sun, it runs smaller by one part in one part in 10 to the 7, compared to a clock far away.
Also, the clocks that are in orbit, for example, that are in the GPS satellites run at the different rate than clocks here on the surface of the Earth. And you have to take those effects into account to really measure your position. So these are effects that are important and are measurable.
Now, there is something very strange that happens, which is that if you make the star smaller than a certain size, it looks as if time comes to a complete standstill. So time doesn't flow anymore. So if you go to a certain radius, which we now call the black hole radius, but you have a star smaller than this radius, then at this position, time seems to have stopped completely.
And this is what Einstein found very strange and thought that these objects could not possibly exist. How can there be an object where time is not passing? It looks like completely wrong.
It was later understood that actually the geometry continues beyond that surface so there is some other region of space time beyond that surface. And that even though an observer from outside thinks that time has come to a standstill, someone who is fallen in the will fall into this black hole and will go through that region, shoot through that region, and according to his own time, it will take a finite amount of time for him to go through this region. He will not see anything special when he crosses this region.
However, after crossing the horizon, he will end up crashing into a singularity. So there is a point in his future where space time completely crashes, and he will eventually die at the singularity.
A very nice analogy for understanding this is to view space time as a river, with apologies to Heraclitus. So imagine the following configuration. Imagine we have a river where the velocity of the water starts increasing more and more until you get to a certain waterfall. Now on this river, we'll have some fish. And these fish can travel with a maximum velocity, which we'll call c. That's the velocity of light. So that's analogous to the velocity of light.
And so we have a fish here who's in a region where the velocity of the water is smaller than the maximum velocity at which the fish can swim. That fish can avoid falling into the waterfall. So he can swim upstream and not fall in the waterfall.
On the other hand, if you have a fish-- there is a point in the river where the velocity of the water is equal to the maximum velocity of the fish and a fish that is sitting at this position will just stand still. And so that's analogous to time coming to a standstill at the horizon.
On the other hand, if you have a fish in this region where the velocity of the water is bigger than the maximum velocity at which it can swim, well then inevitably, no matter how fast the fish tries to swim , it will fall into a singularity.
Now something important is that if this is, let's say, a very muddy river, and the fish cannot see the shore, this fish will not see anything special when it crosses this point where the velocity of their water is equal to the maximum velocity at which it can swim. This a perfectly reasonable point in the river. Nothing special is happening to it, and it will not notice the fact that once it crosses this region, it will inevitably fall into the singularity. So that can only be noticed if it can compare the position to infinity or to the shore.
OK, so there are a bunch of ideas I wanted to transmit, which is that the horizon is a smooth surface, and nothing special happens there. It's an imaginary surface. Here, this position of the horizon is a place where the velocities are equal. But a fish that is swimming here doesn't see a flag, a sign saying this is the horizon. Doesn't see anything like this.
Similarly, when you are falling into a black hole, you don't see a big sign saying, you are falling into the horizon. You are crossing the horizon. You feel like regularly, just moving in space, and nothing special happening to you. There was one point.
The second it that a star, if it is very massive-- I didn't explain why-- but a very massive star can actually collapse into a black hole. So there are objects in the sky that seem to be black holes. So let me talk for a few moments about those objects.
So there are basically two types that have been seen. So some are produced by the collapse of very massive stars, so masses which are a few times the mass of the sun. So those stars are [? believing ?] the course of their evolution to eventually form-- so they're going to explode into supernovas, and then they could form black holes.
There are such stars, and there are also black holes that are the center of galaxies, which are very, very massive. They are of order a million or a billion solar masses, and they are very big. They have a size of four [INAUDIBLE] the solar system. While this one's settled to a size, that horizon size I was talking about, it's about 10 kilometers in this case.
So for example, if you're falling into these black holes, you can fall through them for about eight minutes, or a few minutes, and nothing special would happen to you. Except when you fall to the singularity, you really feel something very bad.
Now how do we see these black holes if they are so black, and so on? So in principle, we can see them by noticing how they bend the light rays coming from stars that are behind them. That same principle, in practice, it typically-- you see them if you are a little more lucky, and you have, for example, a black hole that is near a big star, and then some material from the star falls into the black hole, and then it gets heated up to high temperatures and emits x-rays or some other radiation that you can then see.
And then you notice that all this radiation is coming from a very small region. So with this emission has some variability over time scales which are very, very tiny. So some time scales which are comparable to the time it takes for light to cross a very small region, a region of this order of the size of the black hole.
And this way, people have argued that you have black holes. This is some of the evidence we have for black holes. For black holes at the center of the galaxy, people find that-- this is a galaxy, for example-- and they find some very powerful source in the middle of jets off of gas and so on, which are huge. Notice that these are the size of galaxies. And they come up to the conclusion that the only thing that is powerful enough to give rise to these jets could be a very massive, very big black hole in the center.
They came to this conclusion before they went in and looked with the Hubble Space Telescope, and they found some disk of dust and gas orbiting some very compact objects. So this is 400 light years, the black hole that would be, let's say, light minutes there in the center. It's not seen in the picture.
And it's believed that many galaxies similar to ours have these black holes at the center. Even ours has a black hole of this kind at the center. Now one question that is often asked is whether black holes are dangerous. Now, they're dangerous only if you fall in. The gravitational force outside the black hole is the same as the gravitation force of a star of the same mass. And that we saw when we were discussing the Schwarzschild solution.
The nearest one is very far away. And if you're worried about the dangers of gravity, you should worry more about falling from a ladder. But black holes are really good for theoretical studies. And I will discuss from now on a few theoretical properties of black holes, which make them interesting, and post very interesting theoretical questions.
So the first interesting property of black holes is that they are very universal. So the final form of a black hole is independent of how it is made. So it's only characterized by its mass and its angular velocity, and in some cases it's charge. In actual nature, the charge is not very important.
In other words, there are stars which can be-- there all kinds of stars. There are red stars, yellow stars, blue stars. But if they collapse they all form the same type of black hole. So for the same mass and same rotation angular momentum, or rotational velocity, we have the same black hole.
So in order to specify a star, you need to give its chemical composition, its history, its stage during the nuclear evolution, and so on that Bethe worked so much to find. But the black hole is much simpler, just a single solution of the equations with only one parameter, which is the mass or perhaps also the angular momentum-- two parameters, very simple.
That's one aspect that is simple. There are other sort of funny regularities or simplicities about black holes. And the next simplicity is this one that was found by Stephen Hawking. And it's the following. So imagine you have two black holes that collide. So when you have two black holes that collide, that's a complicated process, and you have to really solve the equations and it's fairly complicated to solve the equations. Only in the last few years computers were powerful enough to actually be able to solve the equations and be able to calculate what happens when you collide black holes. You emit gravitational waves, and you emit gravitational waves.
And in particular, this is telling you that the mass of the final black hole will be smaller than the sum of the masses of the initial black holes. Because some of the mass has gone, some of the energy has gone into producing these gravitational waves. But nevertheless, the area of the horizon always increases. So the area of this horizon has to be a little bigger than the sum of the areas of these two horizons.
And this is a very simple inequality that comes out of somehow complicated equations. So if you write down Einstein's equations, it's suddenly not obvious that this will be true. But there are some arguments you can make even without solving the equations that this will be true. That's what Hawking has done. And so that's an interesting property.
Now these two properties that we discussed are properties that hold according to the standard Einstein theory, so the theory of general relativity, which is what we call the classical theory. So classical theories are those where if we give the initial conditions, we can then evolve to the future and find the evolution in a deterministic way.
Now once we include quantum mechanics, we find a new surprise. And there's no surprises there can be black white black holes. So what Hawking found was that the laws of quantum mechanics implies that black hole emit thermal radiation. This is a theoretical discovery. This effect has not been seen in an actual black hole. The reason is that for black holes, which have masses comparable to the mass of the sun, which are the types of black holes that are produced naturally in nature, this temperature is very, very tiny. It's so tiny that it cannot be measured, unfortunately.
But the temperature increases as the size decreases. So a smaller black hole would have a higher temperature. So in particular, a black hole which has the mass of a continent-- take all of America, you collapse it into a black hole. This is this would be very difficult to do, but theoretically we can do it. We can calculate what the equations tell us.
And in that case, the temperature would be such that the radiation that comes out will look like white light. So this black hole would be light. So a smaller black hole would be red, and an even smaller black hole would look like a tiny, white dot.
Now what is the origin of this radiation? So I just will now tell you a little cartoon derivation this radiation, of why this is the case. This is essentially what Hawking arguments was. And the reason is related to a property of relativistic quantum mechanics. A relativistic quantum mechanics is the theory we use, the theory that results of joining quantum mechanics and special relativity.
I won't be able to tell you much about it relativistic quantum mechanics, but the only thing that will be necessary for me to tell you is that in that theory, particles can be created and destroyed, and in that theory, the vacuum is-- even empty space can be viewed as a rather complex object where particles are constantly being created and annihilated. So at every point in time you create a particle with positive energy and one with negative energy, and they annihilate each other very quickly, so quickly that you cannot notice that there was a particle with negative energy.
So in flat space, the energy of an isolated particle always has to be positive. And there is no net particle creation. But in the presence of a black hole, due to a different notion of time in the interior, this particle which, from the point of view of the outside, from the point of view of the energy notion from the outside, has negative energy, has no problem existing in the interior of a black hole. Because the notion of time is really different in the interior.
And in this case, at the horizon of a black hole-- so this represents the horizon of the black holee-- this pair creation can occur. And one of the particles can escape all the way to infinity. That's the one that we see far away as Hawking radiation, as radiation. And the other falls into the singularity. So this is something that comes from considering the quantum mechanics of particles in the neighborhood of a black hole horizon.
So what this implies is that a black hole can emit radiation. And as it emits radiation, it loses some mass. And therefore, these black holes have a finite lifetime. Now let me mention the lifetime of a few black holes. So if you had a black hole of the mass of the sun or the Earth, it will live much longer than the age of the universe. So this is a very slow process.
A black hole with the mass of an ordinary object-- so let's say 100 kilograms-- so if you took 100 kilograms and you formed a tiny black hole with it, it would emit radiation and it would evaporate completely in less than a millisecond. So that would be worse than a nuclear bomb. So this can be a very rapid process for very tiny black holes.
Now fortunately, those black holes are not ordinarily forming in our neighborhood. In fact, the only black holes that are known to form are black holes of the mass of the sun or even bigger. A black hole of a mass of the order of a mountain produced at the beginning of the Big Bang would be evaporating right now. So that might be something we might see if they were produced at the beginning of the Big Bang.
Now, we don't know whether such black holes were produced at the beginning of the Big Bang. We have no evidence that they were produced at the beginning of the Big Bang. But this would be perhaps the only hope to see this radiation directly at the level of black holes.
Now people discuss sometimes very tiny black holes that could be produced in particle accelerators, in some strange theories of physics where there are large extra dimensions and so on. Such black holes would decay very quickly by, well-- if a black hole of a mass of 100 kilograms evaporates operates in less than a millisecond, a very tiny, tiny, little black hole will evaporate much more quickly.
Now I should say that even though this effect of Hawking radiation has not been seen directly in black holes, There is a very similar effect when you have an expanding universe. And that effect is thought to produce the initial fluctuations for cosmology. And so it's an effect that we indirectly have already seen.
Now these thermal properties of black holes lead to interesting theoretical puzzles. So they seem to lead to contradictory properties. Already, we have that classically, black holes can only-- their mass can only increase. But on the other hand, due to these quantum effects, their mass can decrease. So there seems to be two opposite conclusions.
But there are very concrete puzzles that we have due to the black hole temperature. And the first question is, let's say roughly speaking, why are black holes hot? And the second one is one that is called information law. So I'll briefly talk about these two problems.
So first I need to remind you of the concept of temperature. So Boltzmann explained temperature by saying that it's due to the microscope motion of the constituents of the system. So if we have air, it could be colder or it could be hotter. So the only difference between cold air and hot air is that in hot air, the air molecules are moving faster than the ones in cold air. That's the only difference.
And so what this implies is that whenever you have something that is hot, there is something moving. Any hot object has some little moving parts that are having some random jitter, or some random motion. So whether it's air, or it's a liquid, or it's a solid that is vibrating, everything has some moving components in it. And that's the origin of temperature.
So in other words, heat is due to the motion of the microscopic constituents of matter. And you can somehow quantify how many moving parts the system has, and that's done by a concept which is called entropy. And there is a first law of thermodynamics which tells us what this entropy is, the number of these constituents, if we know the energy and the temperature.
And if you calculate it for the case of black holes-- well, Hawking gave us a formula for the temperature of the black holes-- and then we can calculate the entropy, which ends up being the area of the black hole, the area in units of the Planck length, which is very, very tiny length. So this is a Newton constant, and when you express this combination of [INAUDIBLE] in terms of length, it ends up being a very tiny length.
And what this means, this denominator being so tiny, means that the entropy for black hole is a very, very big number. So if you have any microscopic black hole, its entropy is going to be huge. It's an entropy which is much, much bigger than the entropy of any material with the same size.
But this formula then gives us an interesting interpretation of this area law that Hawking had found using Einstein's equations. So he found using these deterministic Einstein's equations, the area of black holes always increases. And if you make the scientification between the area and the entropy, then the area increase implies the second law of thermodynamics, which says that the entropy should always increase. This order should always increase.
And so in some mysterious way, Einstein's equations, which are deterministic classical equations, know something about thermodynamics, the laws of thermodynamics. So the puzzle is then, what are these microscopic constituents of a black hole? So a black hole is hot, has some entropy. What is really moving inside the black hole?
Given that black holes are so universal, so these microscopic constituents should be the microscopic constituents of space time in some way. So our understanding this will help us understand the structure and nature of space time itself. So it doesn't matter where you made the black hole-- out of protons or neutrons or dark matter. So all these black holes look the same, and their entropy, we think, is going to be explained by the motion of some microscopic constituents of space time.
So that was one of the problems. The second problem is the problem of information loss. So we said we can form a black hole in many different ways. So we can throw various things inside black holes. But it looks like black holes always evaporate in the same way. If we calculate that evaporation using the loss of gravity and quantum mechanics, treated with the accuracy that we know how to use, and with the laws of physics that we know, it looks like the radiation that comes out of black holes is completely thermal, and has no information of the stuff that went into the black hole itself.
Now if the black hole had a quantum mechanical description according to standard laws of quantum mechanics, then we ought to be able to, in principle, perhaps not in practice but the least in principle, be able to have equations that tell us what comes out of the black hole. And what comes out should depend on what goes in. So in the laws of physics, all the laws of physics that we know, are such that if you have different initial conditions, you have final-- well, you have two different final configurations.
Sometimes the final configurations could be fairly similar to each other, but there has to always be some very tiny differences between them. In other words, if you have a sheet of paper and you write a letter on the sheet of paper and then you cut it in lots of little pieces, and you throw them away, you throw them in the floor, at first approximation you have a lot of scattered little pieces of paper. And the information of the letter you wrote seems to be lost.
But if you patiently gather all the little pieces and you put them back together, you can recover the initial information. So the information of the letter that you wrote in the original sheet of paper is really there. And the question is whether the loss black holes are such that the same happens in that case, where you can, in principle, take the radiation and reconstruct what fell into the black hole.
This is a question, not because we ever will want to necessarily do this in practice, but it's a question because we like to understand the questions that govern the quantum mechanics of space time. And we want those equations to satisfy those physical principles that we know all other equations in nature satisfy.
On the other hand, if information was lost, so if it was really true that no information comes out the black holes, then we are a little lost, because we don't really know how to modify quantum mechanics. And all of the attempts that have been made of modifying quantum mechanics have run into various problems. But in order to answer these questions, we need to have a theory that puts together quantum mechanics and gravity. So we need a theory that can treat space time in a quantum mechanical way.
And such a theory is string theory. So let me say a few words about string theory. So string theory was a theory that started to be developed in the late '60s. It's a theory under construction. So we don't know all aspects of the theory, but we know some aspects of the theory. And it's a theory of quantum gravity. So what this means is it's a theory that describes space time in a quantum mechanical way.
And it reduces Einstein theory under ordinary circumstances, so that these low energies or long distances. And it's a theory which we don't know whether it's the correct theory to describe nature yet. It's just a theory. But it's really the only consistent theory that we have to describe space time. There are other approaches but they are nowhere near the consistency and internal consistency that string theory has.
And one thing that will be particularly important for us in this discussion is that it can describe in a complete way certain universes with negative curvature. So there are certain universes that are quantum mechanical, where they contain gravity and so on. And we can describe them in a complete way.
And so that's what I plan to discuss now. So I won't discuss all the details of string theory. I only will discuss one aspect of it. And the suspect is sometimes called holography. So it's the fact that in certain space times, so there are space times with negative curvature. So they are, in a way, the simplest kind of space time, which is not flat. So we can have spaces which are flat, we can have spaces which have positive curvature, and spaces with negative curvature. So this is the simplest space with negative curvature.
And here Escher, a Dutch painter, has painted for us a nice little representation of a space with negative curvature. So this is a projection of the space with negative curvature to the two-dimensional surface. So here, each of these fish has the same proper size, but due to the effects of the projection, this looks smaller. So the inverse effect happens when you try to project the positively curved of surface of the sphere of the Earth on a two dimensional piece of paper.
Then there are some regions near the poles in the standard projection that look very big. And if you do that, then it might seem that Greenland is much bigger than Argentina, but we know that that's not the case. So here, these fish here near the boundaries are being shrunk. But in a certain sense, we can describe, according to this principle of holography and according to these theories in string theory, we can describe all the physics that happens in the interior of these space times, in terms of a theory that lives only on the boundary, that lives on space that has one less dimension, the boundary of space.
So if we were interested in describing a four-dimensional space with a negative curvature, this boundary we are talking about would be like a sphere that sits very far away. So it's a two-dimensional sphere. So let me describe this with a picture, or with a little animation.
So we have some object here moving in the interior. And this object is described by a bunch of interacting particles on the boundary theory that are, again, themselves also moving. And this is a relationship that's been explored in great detail during the last 15 years or so. And more and more aspects are being understood, and more aspects of this mapping between the physics in the interior and the physics in the boundary have been understood.
So that's how we describe objects in the interior. And we can describe any object in the interior, even a black hole in the interior. And if we have a black hole in this space, then it can be described in terms of the theory that lives purely in the boundary. So we have a gas of particles or a fluid of very strong interacting particles on the boundary that describe the black hole. And not only the black hole, but the whole space time inside. So they describe the whole space time inside, together with a black hole.
And in this case, the temperature entropy of the black hole can be understood as arising from the motion of these particles. So we had this question of, where does the entropy of black holes come from, and so on. And the according to this relationship, the answer is that-- once you map it to this problem on the boundary-- it comes from the actual motion of these particles on the boundary. So then the temperature and entropy have an explanation, which is the same as the explanation for any other system that has a temperature in terms of the motion of microscopic constituents.
But the weird thing is that in this case, the microscopic constituents are on the boundary of space time. They are not localized in the interior of space time. We don't have little pieces of space time in the interior, but we have these objects, which move in the boundary.
The theory on the boundary obeys the standard rules of quantum mechanics. So therefore, the black hole in the interior also base their roots on quantum mechanics. And if you believe in this relationship, if this is a contractual relationship between the fixes in the interior and the boundary, there is a lot of evidence for this conjecture. But if you believe that it's absolutely true, then black holes are consistent with the laws of quantum mechanics.
And this is a definite prediction of string theory. But one that is very difficult to test, because it's difficult to produce black holes. Now this relationship between the theories in the interior and the theories of the boundary can also be used to learn about strongly interacting theories or strongly interacting systems.
So in that relationship, what I've discussed so far, we've trying to discuss our universe, or some universe that we're interested in with gravity, in terms of particles that live on the boundary. But you could also do the opposite. So you can think that boundary theory, so this theory of particles, are the particles that we have in our own space time, the particles that we know. They could be the particles of a condensed matter system, or the particles that describe nuclear physics.
And the interior would be some imaginary, curved five-dimensional universe. And this relationship says that these two are related. And in this case, you can use black holes as sources of information.
So for example, an application of what I was saying before, so an application of these ideas is that you can model the heavy ion collisions in RHIC, or the Large Hadron Collider. And one point of view is you can view them as creating a new face of matter-- so some new face of matter where the protons lose their identity and they form a soup of quarks and gluons. So this is a state of matter that is believed to have existed in the beginning of the universe, and which is being recreated in these high energy collisions.
So you view it in this way, or alternatively, you view is as creating a black hole in a weird, five-dimensional space. And this point of view can be useful for doing some calculations. And usually, we can do these calculations if we change a bit the theory.
So we treat not exactly the particles of our own universe, but particles which are kind of similar to the ones in our own universe. And in those cases, we can really do the calculation this way. So these ideas can be used to think about these problems, to provide toy models for these problems.
So in conclusions, black holes are fascinating objects where the geometry of space time is deformed in dramatic ways. And black holes give rise to interesting theoretical challenges once you put them in with the laws of quantum mechanics. And string theory can describe black holes in a consistent way, a list as viewed from the outside. And one thing we are learning is that space time seems to be an approximate immersion concept which arises from the interactions of morally [? mentary ?] particles that live on its boundary.
And there are many interesting open questions. So for example, we would like to understand how to describe the interior of black holes. And also we would like to understand how to use this idea of holography to describe expanding universes like ours. And these are questions that are now being explored. They're being explored in many places, in particular the second question here-- Tom Harmon, one of the professors, he has been having very interesting ideas. Well, thank you.
TOM HARTMAN: So we have time for a few questions. If I can get the microphone [INAUDIBLE]. Just shout it out, I guess.
SPEAKER 1: OK. So you mentioned [INAUDIBLE] the black hole [INAUDIBLE] there is an anti-particle like, fall into a black hole. But why only anti-particle fall into black hole? [INAUDIBLE].
JUAN MALDACENA: Yeah, so it's not what we normally call an antiparticle. It's a particle with negative energy.
SPEAKER 1: Then why [INAUDIBLE] particle with negative energy fall in--
JUAN MALDACENA: Because the particle with negative energy cannot exist outside. So we can only help force particles of positive energy outside. The thing is that from the inside point of view, it has positive locally measured energy. But from the outside point of view, it would have a negative energy. Yeah.
SPEAKER 2: Don't you this positive energy and negative energy particle to [INAUDIBLE] inside [INAUDIBLE] the boundary, which is--
JUAN MALDACENA: Yeah, they're created in some neighborhood of-- yeah, the question was, don't you need these two particles with both positive and negative energy are created exactly at the horizon? And the answer is no. They can be created within some small region near the horizon. And the smallness of the region depends on the mass of the particle. So for very light particles, this region could be quite big, of the order of the size of a black hole, and for smaller particles it will be near the horizon. Yes?
SPEAKER 3: Can objects have imaginary mass?
JUAN MALDACENA: Can objects have imaginary mass? Yes and no. Depends on what you call the mass. Ah, sorry. I forgot I had my own microphone. I think the most honest answer is, not really. So there is something which you can call imaginary mass, which is when you have a certain field that this unstable, and it's at the maximum of its potential. And then in those situations, the field will go to the minimal of the potential and then particles around that state would have positive mass. So let's say good, stable states of matter, you can't. Yes.
SPEAKER 4: So if I understood you correctly, [INAUDIBLE] all black holes, these very large ones, will evaporate. [INAUDIBLE]. Also, if I understood you correctly, [INAUDIBLE] observer someone [INAUDIBLE] black holes [INAUDIBLE], infinitely. How does anyone [INAUDIBLE]? How does anyone ever actually--
JUAN MALDACENA: Yeah, that's right. So from the point of view of the outside, you never fall into the black hole. From the point of view of the person who's falling in, he or she will fall into the black hole. So even though-- yeah. So that--
SPEAKER 4: Those seem like very [INAUDIBLE] of observations. From the outside, I watched--
JUAN MALDACENA: Well, yeah, I should say a little more. So let me be a little more precise. What do you really see from the outside? So from the outside, you see the object falling into the black hole, but time, we said, was going more and more slowly. So any signal that the subject is sending is going to be slowed down. So the object will look redder, and all the radiation coming out will have a longer and longer wavelength, and so on.
And at some point, anything that comes from this object, the whole energy of this object will be less than the Hawking temperature. And at this stage, it will be completely masked by the Hawking temperature. And to all intents and purposes, it will have completely disappeared. So that's what the observer outside sees.
And this happens relatively quickly compared to the lifetime of these black holes. This happens in a time which-- so all signals of the object that is falling in are decreasing exponentially, with a half life, let's say, of the order of the light crossing time, so the time it takes for light to cross the horizon. Yes.
SPEAKER 5: So when a photon is emitted from these [INAUDIBLE], it probably has to be [INAUDIBLE], right?
JUAN MALDACENA: Yes. Sorry, say it again? The photon--
SPEAKER 5: So when the photon is emitted [INAUDIBLE] horizon space, it has to have [INAUDIBLE]?
JUAN MALDACENA: No. Well, according to the calculation that Hawking did, or a calculation you can using quantum fields around the black hole background, the radiation that the particles that are emitted from the near horizon region do not carry any information.
SPEAKER 5: How's that possible [INAUDIBLE]?
JUAN MALDACENA: Well, the basic idea is that this radiation is coming from this randomness in the vacuum. So they have as much information as they vacuum has. And so there is essentially no information. The vacuum is a unique state. And that's why you don't have any information of the matter at fell in.
Let me say this in a slightly different way. So the matter falls in, and all signals of the matter are being red shifted, right? They are being sort of dumped down, and they disappear after a relatively short time. And you revert back to the vacuum in the near horizon region. And then the emitted particles, the [INAUDIBLE] are reflecting properties of the vacuum, and not of the matter that fell in.
So you are basically ripping the vacuum apart into two pieces-- one going out and one falling into the black hole. And there is some entanglement of the quantum fields in the vacuum. And this is what gives rise to the thermal nature of the black hole-- the black hole radiation. Yeah.
SPEAKER 6: You have said that the string theory can describe black holes in a consistent way, if it's from the outside. Now, earlier in the lecture, you talked about these outstanding issues like why black holes [INAUDIBLE]. Can you clarify exactly how string theory helps [INAUDIBLE] those issues?
JUAN MALDACENA: Yeah. So what string theory gives you is a microscopic accounting of the entropy. And it does through these dualities, so through this holographic relationship. So you have the black hole in some space, in this negatively curved space time. And string theory replaces the whole space time by a theory that lives on the boundary. And in this theory on the boundary, you can calculate the entropy of the black hole. And it turns out that the entropy you calculate is the same as the one you calculate geometrically from Hawking's formula. Yeah.
SPEAKER 7: So this notion of space time having a boundary-- is it simply a mathematical assumption trying to figure out [INAUDIBLE] to--
JUAN MALDACENA: Yeah, so the question I think was, what is this notion of space time having a boundary? Yeah, I went a little quickly through this. So in these negatively curved universes, they look basically like a universe with a gravitational potential well. So there is a gravitational potential always pushing you inside.
So if you send in a massive particle from the center, this massive particle will come out and then it will bounce off the gravitational potential and then come back into you. So a massive particle can never make it out to really infinitely far away.
A light rain principle could go infinitely far away and then come back. So we need some boundary conditions [? at the ?] infinity. But it gets very red shifted when it goes far away. So in some sense, it's also bouncing back to the interior.
So in some sense, physically, it is as if you were living in a box of some fixed size because of this confining gravitational potential. And when we talk about the boundaries, just some region that is very far away. So you take a very large size and that's the region we're talking about. Yeah.
SPEAKER 8: Can you tell us about singularity?
JUAN MALDACENA: The singularity-- yeah. So that's a very good question, and it's related to the first problem that we don't know-- how to describe very well the interior of black holes, and exactly what happens at the singularity. So we don't know how to give a description of singularity that is better than what we have in general relativity.
SPEAKER 8: So why [INAUDIBLE]?
JUAN MALDACENA: Why?
SPEAKER 8: Why [INAUDIBLE] statistics of singularity?
JUAN MALDACENA: Well, the singularity is predicted by the laws of general relativity. The question is-- and there the questions fail. Now the question is whether there are other equations that do not fail, that are better. And we lack those equations. And we like them so much that even there are people who question even the fact whether black holes really have a smooth interior, and so on. There is some debate going on on this subject. Yeah.
SPEAKER 9: So with the holographic principle [INAUDIBLE] if stuff gets absorbed in a black hole, then information still [INAUDIBLE]. But if the information-- if the black hole evaporates, then [INAUDIBLE] with all the photons that come out of that hole, and have no information, that is still information on the surface.
JUAN MALDACENA: I'm not sure if I understood the question. So the idea is that the Einstein equations and the formalism of treating the quantum fields around the fixed background are an approximation, is an approximation to the physics, which is similar to the thermodynamic approximation. So in the same way that we apparently lose information when we do thermodynamics-- for example we have entropy increase and so on-- if we did the calculations in the exact theory, we wouldn't be losing information.
So in this holographic descriptions, on the boundary theory you don't lose information, because you can always do the calculation in the microscopic theory.
SPEAKER 10: So with the [INAUDIBLE]?
JUAN MALDACENA: So the question was whether with a theory without boundary information is lost, right? That was the question. So the only cases we can treat completely precisely are these cases where we have a boundary. In cases of space times without a boundary or space times that are undergoing going cosmological expansion, for example, we don't have an alternative exact description. But we think that since black holes, in the end of the day, are localized subjects, eventually there will be a theory that describes them in a unitary way, to the extent allowed by the theories in this compact space times, or a cosmological space times.
We view this as a lack of our understanding of the theory, rather than [INAUDIBLE] problem [INAUDIBLE] physics. Yeah.
SPEAKER 11: So the radiation [INAUDIBLE] from the black hole is coming from this emptiness. Why are you saying that the black hole evaporates? Seems like it's giving out radiation but from the inside.
JUAN MALDACENA: Yeah, so the black hole, so it's giving out-- yeah, I'm sorry. The question is, so this radiation is coming from the neighborhood of the black hole, especially there is this particle that comes from the outside of the black hole. So why is the black hole evaporating at all?
We can calculate this radiation. And we see that these negative energy particles are decreasing the mass of the black hole. So total mass of the black hole is becoming smaller and smaller. So that's something we can definitely calculate.
And then as the black hole becomes very, very small, then we can't calculate anymore. But we think that when you have a very tiny black hole, probably what would happen, but probably it will not have much entropy. And so it would probably explode. And so even though we don't know what happens in the final stages, we think that it would be over soon.
TOM HARTMAN: Speaking of which, let's take two more.
SPEAKER 12: So ultimately, how does a better understanding of a black hole contribute to our understanding of nature in general?
JUAN MALDACENA: Yeah. So the question is, how is the understanding of black holes contribute to understanding of nature? Well, just trying to understand more black holes forces us to understand better space time, space time and quantum mechanics. And there is a question that we don't understand, which is the beginning of the Big Bang. So there was a singularity in the beginning of the universe. And that's one we would like to understand.
And we hope that by understanding better black holes, we'll eventually understand the beginning of the Big Bang. So that's the big motivational question. Now I can answer to you more in practice it is that by thinking about black holes, you discover this duality, this, for example, this holographic duality, which you can then apply it to other areas of physics. So that's in practice also how it has contributed to other areas of physics. Yeah.
SPEAKER 13: So with the beginning of the universe, the energy [INAUDIBLE]. How come the universe didn't end up being just a single black hole [INAUDIBLE]?
JUAN MALDACENA: Yeah, so the question is, in the beginning of the universe, the energy density was enormous. And so why didn't the universe start being a black hole? And in some sense, the universe is-- there is a singularity. So this large amount of energy causes a singularity. And it's the initial singularity.
So there was an initial singularity due to this, let's say, large amount of energy or expansion. So this amount of energy implies in the same-- let me say differently. Essentially, the same argument that says when a star is small it should collapse into a block hole, so into something that contains a singularity in the future, says that when we go to the past, we should find the singularity.
TOM HARTMAN: Before we conclude, Jim has an announcement.
JIM ALEXANDER: Just briefly, I wanted to let you all know there will be a reception held in 401 or the physical sciences building starting immediately after, now. And you're all welcome to come. And if you don't know where it is, just follow the crowd.
Einstein has taught us that space and time should be combined into a dynamical entity called "space-time". Spacetime is deformed, or curved, by the presence of matter. Black holes are objects where this deformation is very drastic, leading to a region in the interior from which nothing can escape. Quantum mechanics implies that black holes emit thermal radiation, or Hawking radiation. The theoretical study of quantum aspects of black holes has led to interesting connections between spacetime and more ordinary quantum systems.
Juan Maldacena of the Institute for Advanced Study explores the deep connection between black holes and quantum field theories, September 24, 2014, as part of the Department of Physics Bethe Lecture Series.
