Topics in String Theory | Lecture 9
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Computer animation
Transcript: English(auto-generated)
00:05
Stanford University. OK, we have 10 minutes for schmoozing. Yes? You made a comment last time that horizons are always hot.
00:21
OK, right. I get it. So in a scenario where you've just got a- What do you mean? In a horizon of, what do you call it, the- Do you consider our universe to be hot? How is it hot? Do you mean how hot is it or how is it hot?
00:41
How is it hot? Like what the mechanism of it being hot? Exactly the same as in the black hole. Does that help you? All right, let's- Black hole has the energy of gravity and all this stuff,
01:00
right? There's some- All right. You know, that's pretty much exactly what I was going to start the lecture with tonight. So make sure I start on that. But I'm going to defer it until seven,
01:20
because that was about where I was going to begin. On the black hole, on the black hole temperature, we said you lowered the thermometer right near the horizon. If you did that experiment, we shot photons at the black hole to build it up. And you'd still measure the same entropy?
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Entropy, let's be precise. There's entropy, there's energy, there's temperature. What are you asking exactly? First, do we measure the same entropy because it would still be the same probability of the photon either being captured or missing? After the photon falls into the horizon or onto the horizon or whatever you want to call it,
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the entropy or the hidden information if you like, the information that's inaccessible to you because it's either inside the black hole or from your perspective, very, very close to the horizon and it would take forever for a photon to get out,
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the information that's hidden is one more bit. So every time you drop a single photon in, the entropy increases by one unit. So it's the energy that the ear parties is it like you've seen a higher frequency photon closer to the horizon? Yes. Yes.
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Yes. The question about the temperature of horizons, well, let me answer it now. The horizon is always hot. I'm going to come back to it in a minute. The horizon is always hot in the sense that if you lower the thermometer down, you would always detect a very, very high temperature
03:02
as you get very close to the horizon. So you can think of it as the horizon being a place or surface of extremely high temperature which is constantly emitting and reabsorbing photons.
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They're really quantum fluctuations and they're really nothing more than the high frequency quantum fluctuations that are taking place in a small volume of space in a small region of space. The horizon makes them real in a way which I'll come to, which we're going to talk about,
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makes them real thermal fluctuations in a rather interesting way. But every horizon always has the property that the temperature, the actual temperature of a real thermometer lowered down to the horizon would record a temperature which would be one divided by
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two pi times the distance to the horizon, the distance to the horizon in the denominator. Every horizon, whether it's a black hole horizon or any other kind, always has that property. You asked me a question now, I can't remember exactly what it was.
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When you hear the horizon, the time sort of freezes, does that mean that the photon that came in to build it? Yeah, yeah, yeah. Let's forget the photon that came in to build it up. Let's talk about the photons which are being emitted. Why, when you stand far from the horizon, does the horizon not, or does
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the radiation that hits you, why isn't it this very high frequency radiation that is emitted very close to the horizon? In other words, you drop a thermometer down, you discover that the region near the horizon, the so-called near horizon region, is very, very hot.
05:01
Why isn't it hot far away? I can assure you, if we created a blank wall over there, or that wall, and we heated it up to some very, very high temperature, and let's suppose it was a very, very big wall, and we heated it up to some high temperature, I don't think it would make much sense to say that as we moved away from the wall,
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that the temperature went down like one over the distance. It wouldn't. You would be blinded by high frequency photons pretty much no matter how far you got from that wall. The high frequency photons that came out of it, the gamma rays, the super gamma rays, everything else that came out
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of it, would blind you pretty much no matter how far you got from the very, very large wall. Why is it that that doesn't happen with the horizon of a black hole? Why, when you stand back from it, and get a good deal of distance between you and
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the horizon, why don't you get blinded by these very high frequency photons? And the answer, what's the answer? Redshift. Redshift. Redshift is another way of saying that as the photons propagate away from the horizon, they lose energy.
06:21
And they lose energy for exactly the same reason, kinetic energy, they lose energy for exactly the same reason that this piece of chalk does when you throw it up in the air. It is doing work, or in order to raise it, you have to do work against the gravitational field.
06:41
And if you don't do work to keep it moving with the same velocity, then the velocity will slow down. Well, with photons, don't think about velocity. Velocity is the wrong thing to think about. Think about energy. For a non-relativistic piece of chalk, there's a close relationship between its velocity and its energy. But what's really happening is
07:00
it's losing energy. For a photon, the relation between velocity and energy is different. Namely, the velocity is always the speed of light, and the energy is the energy. But it's still true that as the photon climbs out of the gravitational
07:20
influence of the black hole, it loses energy. So in the transit between the horizon, very near the horizon, where it's very hot, and far away, the photon loses an amount of energy such that a distant observer doesn't see the black hole as being
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very hot. He sees it having the Hawking temperature. And the Hawking temperature decreases with the mass of the black hole. Remember what it was. Hawking temperature of the black hole. And you can think of it as the temperature as seen by somebody looking at the photons
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or being exposed to the photons very far from the black hole. The temperature, T Hawking, is equal. I'll write the precise formula for no particular reason, but 1 over 8 pi times the mass of the black hole. There is an
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h-bar here, a Planck's constant, and some numbers of factors of c, which I can never remember. C cubed, I can't remember if it's upstairs or downstairs. I don't remember. I think it's h-bar. I think it's actually h-bar.
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8 pi times the mass of the black hole. So the bigger the black hole, the cooler it is. Why does it look cooler for a big black hole? Well, that photon that was created very near the horizon has a more difficult time getting away
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from the big black hole than a smaller black hole. In other words, it has to do more work. It loses more energy in escaping from a big black hole from the near horizon region than it does for a small black hole. So the effect is the temperature, the effective temperature seen by an observer far away
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decreases with the mass of the black hole. But if you take this formula, you interpret it in terms of the energy of typical photons. In fact, the temperature of black body radiation, the temperature of thermal radiation is approximately the
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same as the typical energy of any given photon. So this just says the photons typically have energy one over the mass of the black hole. And if you then trace that inward toward the horizon, and say how much energy did the photon have
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to have when it started out in order to escape and have this much energy when it got out, you'll find the energy is always the same. It's always about this one over two pi times the distance from the horizon. How close can you get to the horizon
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in a meaningful sense? Well, the Planck distance. The Planck distance sets the scale for all quantum mechanical distances. And so you might say then that these photons originated in a thin layer. This is the way it behaves. This is the way the system behaves. They have a thin layer
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perhaps a Planck length thick near the horizon or right at the horizon and it is continuously emitting photons and absorbing photons, emitting and absorbing them. The energy of the photons as high as you can imagine the Planck energy.
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Now, most of those photons don't get out. Most of those photons don't get out. What happens to them? They fall back. Here's the horizon of a large black hole and if the horizon is hot, it's continuously
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emitting photons but some of those photons go radially out but very very few of them. Most of them go off at some angle. The ones that go off at an angle are pretty much guaranteed to fall back into the horizon of the black hole. It's only a tiny
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cone of angles here that photons will escape. So in fact, the emission of photons is a very slow process. It's a slow process because most of them don't come out at an angle which is appropriate for them to escape and most
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of them simply just fall back to the horizon fall back because of gravitation. The ones that are coming out can bang into the ones falling in and guess what happens? They collide. They're very energetic. They make electron-positron pairs. They make everything that you can think about
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and they make this thin layer of very very hot material. It's like no material that we ordinarily encounter in nature. Much much hotter and the surface of it is continuously emitting photons which we see as having much much lower energy than their emission.
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That's the mathematical theory. That's the consequence of the mathematical theory of horizons. There's another way to think about it. Remember we've drawn pictures of black holes. The outside of a black hole was out here.
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The inside of a black hole was in here. Somebody a given distance from the black hole is moving along a trajectory which looks like this. Now there are quantum fluctuations that take place. Without trying to get into the depths of quantum field theory, let's just use
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the rough metaphor of quantum fluctuations being closed loops of virtual particles. So virtual particles are constantly being created and annihilated. An observer far away just sees those particles created. Doesn't
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see them actually. It all happens too fast. Here's here are the lines of constant time. Way out here is t equals infinity. Alright, somebody out here, a little
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fluctuation like that lasts a tiny amount of time. They can be detected. They can be detected by very high precision quantum electrodynamics and the properties of atoms and so forth. But there are also fluctuations that take place over here. Now, those
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fluctuations appear to be emitted from the horizon. The horizon is a place very close to this light cone here. They appear to be emitted and fall back into the horizon. That is exactly what these things are. They are
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the quantum fluctuations which take place half behind the horizon and half in front of the horizon and look to somebody on the outside as something emitted from the horizon and reabsorbed. Some tiny fraction of them have enough energy to get out and we call that Hawking radiation.
15:00
If you were on the outside, wouldn't you see it go from t minus infinity to t plus infinity? Sorry, again? When it crosses those 45 degree lines, you said there are t equals the lower ones. Yeah, well, okay, so let's not, yeah. There's a classical horizon which really looks like this.
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But it's always convenient and we all do this all the time. We draw a surface one Planck length away from that horizon. This is a surface one Planck length away from the horizon. Alright, it's one Planck length from here, one Planck length here, one Planck length here.
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It's a surface, if you like, if we wanted to draw this a different way, we would say here's the horizon and one Planck length away a thin layer and the point here, of course, is that the horizon is not really accurately defined to within the Planck length. Quantum fluctuations, not only
16:00
of ordinary particles, but of gravitation and the structure of the horizon, they all take place. So you really should think of the horizon as a sort of slightly thickened thing. If you think of it thickened like that, and the fluctuation takes place, then this fluctuation doesn't occur at time minus infinity.
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It occurs at some time in the remote past, or sometime in the past, it's emitted out and it falls back into the horizon over here. So somebody outside would say, look, I see some photons that look as though they were emitted from somewhere as very close to the horizon
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with very, very high energy. They pop out and they fall back in. That's what the mathematics gives. And it is the same no matter what the horizon is, whether the horizon is a cosmic horizon, whether it's a
17:01
now, for cosm... Oh, I guess... Are we ready to begin? We're happy. Ten after. Oh, we spent more time on that than I thought we would. Alright, well then. Good. Let's talk about cosmic horizons in the same vein.
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There are many ways to draw the sitter space. The sitter space is the space which is exponentially expanding. One way to draw it is just to draw a time axis which goes up to infinity. Draw the blackboard and keep track of the fact in our
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heads that the distance between any pair of neighboring coordinate lines here is exponentially increasing with time. That's one way to draw it. Another way to draw it is to
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squash all of future time all of future time down by a coordinate transformation. We did the coordinate transformation last time so that future infinity is a horizontal line like that.
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Corresponds to t equals infinity. I think I wrote down the metric for you in a form which maps t equals infinity to a finite place. I'll just rewrite it for you. The original metric was ds squared
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is equal to minus dt squared that's the usual thing plus e to the two h t dx squared where again dx squared stands for dx squared plus dy squared plus dz squared and this e to the two ht that's the exponential
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expansion of the space. Why does it come in with a two? Because it's really distance squared, right. Distance between points is increasing like e to the ht and so a given dx corresponds to a growing distance which grows as e to the ht
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and in the metric it's e to the two ht. Then we made a transformation just in order to we made a transformation of the time variable. We wrote the time variable t I think, what is h? Right, h is the Hubble constant
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in this space. Alright, it's also the expansion rate. We wrote t is equal I believe to one over h e minus e to the minus minus or plus minus ht
20:03
sorry t capital t capital t goes from minus infinity minus ht should be right capital t goes from minus infinity deep in the past
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what happens to this thing when t goes to minus infinity? It gets big, e to the minus ht gets big when t goes to minus infinity but it gets big and negative. Okay, so the remote past t equals minus infinity is also capital
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t equals minus infinity but the remote future when this t gets very big that happens when what? Do I have this right? I don't have this right excuse me
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still true when this t is large and negative this t is also large and negative but now what happens when this t is large and positive when this t is large and positive then
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this gets small and capital t goes to zero so t equals infinity is also capital t equals zero and this is just a trick it's a trick for getting the whole geometry or not the whole geometry but the part we're interested in the remote future in this case onto the blackboard now what happens if we rewrite this
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metric in terms of t let me just remind you it becomes minus sorry plus minus the t squared plus the x squared all times
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one over t squared one over t squared h squared to be exact notice that when t gets very small the distance between points gets very big just as it did on this side over here this is just a rewriting
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of this and t goes to zero is the remote future the nice thing about writing the metric this way is that in these coordinates light rays move on 45 degree angles that's very helpful in thinking about this geometry and asking who can communicate
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with whom let's get rid of this here why is it that light rays move on 45 degrees in this metric here well the rule for light rays is that the metric should be zero that the interval should be zero so the t squared
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minus the t squared plus the x squared equals zero it's the same as dt equals dx right and dt equals dx is just a 45 degree 45 degree line if the light ray was moving in the plane it would be 45 degrees in the plane if we're moving out of the plane it would be 45 degrees out of the plane but I
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there's only so much we can visualize with a two dimensional blackboard okay so that's the setup that's what the that's what space time in this kind of inflating um exponentially increasing world looks like here it is now let's imagine
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here's somebody you me I don't know and um in the real world of course people only get to live for a certain amount of time but right now we're gonna pretend we can live forever and so here's the world line of some observer and that world line does it really end
24:02
no it doesn't really end because there's an infinite amount of time between any finite point in here but on the blackboard we draw the observer as just ending at that point over here that's the remote future what can that observer see well he looks back and he sees
24:22
light rays coming at him from distant places and light rays move on 45 degrees I could I can add one more dimension to this if I uh I can fake a dimension by doing that
24:40
alright so we see light rays coming but I can't get I can't draw another dimension that's too many uh we would like x y and z but at least I can get x and y on the diagram here so this observer looks back and as time goes on gets to see more and more but never gets to see more of the space time
25:00
than is in the backward light cone here so in particular never gets to see anything out here or out here at least doesn't get to see it in any sort of usual sense so there's a region most of space and time almost everything
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is totally inaccessible to this observer now of course there may be another observer over here and that other observer sees something else there's some overlap okay some overlap but this person
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gets to see everything in here this person gets to see everything over here let's uh let's uh simplify it and just study one observer at a time for all practical purposes is universe space time universe is this backward light cone here now what does he see does he see
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does he see things a world which seems to depend on time you would think so you might think well everything seems to be exponentially expanding but let's just do a very simple exercise now let's consider this distance and compare it
26:21
with this distance and this distance and this distance and this distance basically this distance that I've drawn here is the distance from the observer to his horizon let's see how that changes with time it looks like it's shrinking but let's just check that
26:41
here's the metric here's the metric and let's put this point over here at t equals minus one let's put this point over here at t equals minus a half this one over here at t equals minus a quarter
27:01
incidentally that corresponds to uniform ticking off of proper time each unit of proper time would correspond to dividing this interval by half that's the way this transformation worked that's why there's an infinite amount of time in here and now let's calculate this distance
27:21
to calculate this distance we just look at the metric this is the metric and so these two points are separated by a spatial distance but no time distance so these two points are separated by zero
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time interval that means in this formula here there should be an equal sign there is no time separation at all between these two points so let's write it in a finite
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form not a differential form the square of the distance let's call it from a to b what is that no time separation what is the delta x between these two points can you tell
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this is a 45 degree angle this distance is capital t this distance is also capital t in fact let's not even bother writing this t equals minus one let it just be t let it just be t this distance here
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the horizontal distance is also the same as the vertical distance so we get zero from here but from here we get plus and delta x is the same as t squared but now divided by t squared h squared
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and what's the answer one over h squared in other words the distance from here to here is just one over h that's the actual real physical distance that you would measure
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with measuring devices but notice the answer doesn't depend on t the distance between here and here is exactly the same as the distance between here and here the delta x is smaller but the t is also smaller
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so when you write up to here the delta x gets shrunk by a factor of two but t also gets shrunk by a factor of two so the distance of the observer from the horizon is always the same the observer sees a horizon around him
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a sort of sphere and that sphere doesn't change with time in fact you might suspect that if you used coordinates that were appropriate to the observer inside one horizon
30:21
you might discover that the metric had no time dependence after all the distance to the horizon doesn't have any time dependence it's always the same and in fact many things don't change with time it just looks like they do you might expect that maybe there's
30:40
some way to rewrite this geometry and I'm simply going to tell you the answer I'm not going to work it out in detail but using coordinates like this, spatial coordinates which are appropriate to the absolute interior of this horizon region there is another way of writing the metric
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but it's only good for this region it doesn't capture what's going on on the outside but captures everything on the inside I'm going to show you what it is and you look at it for a minute and it may look somewhat familiar familiar but not
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familiar these are coordinates which are sort of built on the pattern that I've described here in other words they're coordinates in which the coordinate distance between here and here is the same as the coordinate distance between
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here and here this coordinate distance and the new coordinates will be the same as this will be the same as this and I'll show you what the metric looks like ds squared equals minus dt squared that's no surprise
32:01
it's actually proper time actual proper time and then the rest of it is plus oh no no sorry I did it wrong mistake I apologize for mistakes but
32:22
the mistakes happen when I haven't had my cookies yet minus what's it yeah right low blood sugar 1 minus h squared
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r squared dt squared plus 1 over 1 minus h squared r squared dr squared
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plus r squared d omega 2 squared remember what omega 2 is? right when you're sitting at the center and you look around you you see an angular sphere around you and this is just one way of
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not writing out the full spherical metric just simplifying it does this look at all familiar? yeah what's it look like? it does let me just remind you what Schwarzschild looks like exactly the same except h there
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or h squared is replaced by twice mg and then it's divided by r everything is exactly the same if you take the parentheses and put this in here likewise over here has the same kind of form
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and in fact it looks awfully different to have h squared r squared here but the important thing what was important? where is the horizon? Schwarzschild right but the horizon is where this thing is equal to zero
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that's the place where clocks slow down it's the places where distances are infinitely contracted and it's also the place which separates the inside of the black hole from the outside of the black hole you can also write this in the form r minus 2 mg over r doesn't matter how you write it the real
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importance of this term is the place where it vanishes where it equals zero that is where the horizon is likewise here same idea the place where this thing vanishes which is r where is that? that's r equals 1 over h
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where is it? r equals 1 over h that's the distance to the horizon so this is another way of writing exactly the same
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geometry except something really breaks down if you try to make r bigger than 1 over h something goes berserk here you really don't get out past here the whole metric gets a little
35:42
bit silly space turns into time, time turns into space we don't need to worry about that right now you let r go from zero to 1 over h and this geometry covers the entire cone there, everything that's inside that cone and it is
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everything that can be seen by an observer by a single observer well because you weren't here to begin with because we're talking about an exponentially expanding universe e to the, where the scale factor is
36:21
e to the ht alright so your, um, series of lines up there they're all length 1 over h these? no oh here, the horizontal ones yes they're all length 1 over h
36:41
and therefore they're like the distance to the horizon here from r equals 0 to 1 over h alright it takes a little bit of mathematical trickery to rewrite this metric a few coordinate changes, there's a few transformations of coordinates that you have to do
37:00
and when you've done those two coordinate transformations you'll find that this patch it's called sometimes the causal patch or the static patch and one thing you notice about this metric, the most interesting thing about or the many interesting things about it first of all it has a horizon but second of all, none of the coefficients here
37:20
depend on time they do depend on r, but they don't depend on time and in that sense, in this form the metric is completely time independent, and it's another way of saying that what an observer sees around him just doesn't change with time it's static
37:41
it always looks the same, distance to the horizon is always the same, whatever the observer does at one time if he does the same thing at a later time he'll see the same thing now, it is true that if he takes two particles let's ignore any forces between them let's suppose
38:00
he takes two particles and lets them go what will happen? we're ignoring any attractive forces space between them expands and so they go flying off so you say, well that doesn't sound time independent, but the point is if he does the same experiment at a later time, he will see exactly the same
38:21
thing. Take the two particles let them go, and they will slowly accelerate apart from each other take two particles at a later time, do exactly the same experiment they will slowly accelerate away from each other
38:42
now, once we get to this point here we can kind of see that a black hole horizon and a de Sitter space horizon are very very close relatives the fact that everything that
39:02
you can say about the near horizon region of the Schwarzschild black hole is pretty much the same for the for the de Sitter space I'm not going to try to prove that but the similarity of the metrics are pretty clear
39:20
and they in fact, near the horizon they are almost identical very close to the horizon but the whole global picture of things is quite different whereas a black hole is something that we stand on the outside of and look at and emits Hawking radiation
39:41
the whole picture here is quite the opposite we are on the inside the black stuff is on the outside, the stuff we can't see and yes, there is Hawking radiation being emitted from the horizon for the same reason it's emitted and absorbed emitted and absorbed, sometimes a photon
40:02
gets to us, but we can also ask the question what is the energy of a photon by the time it gets from near the horizon to us, and there, there is a big redshift factor this metric just like the Schwarzschild metric has associated with it a gravitational
40:22
potential, this metric also has a gravitational potential I'll draw it for you as a function of r as a function of r, there is a gravitational potential, let's see if we can draw it
40:42
it's upside down, it looks like this the meaning of that is it takes work to pull something from the horizon to the center equivalently, an object emitted from the center as it makes its way toward the center will lose energy, will lose potential energy, it has to
41:02
climb up out of this potential and will lose energy another way of talking about redshift you can work out the redshift easily enough from the metric, I'm not going to do it now but a photon emitted near the horizon will lose energy on the way out here such that the Hawking temperature
41:20
the temperature seen over here is of order let's see, what is it well it's of order one over the distance from the horizon of order one over the distance from the horizon in Planck units, how big is
41:42
our horizon, how big is the horizon our our horizon our horizon meaning to say the horizon that we think we know about on the basis of known expansion properties of the universe this distance one over h
42:00
one over h is about twenty times tenth of a ninth light years twenty billion light years the redshift of a photon from the horizon to where we are
42:23
is by this factor of one over twenty times ten to the ninth, that's a lot so the photons that we see coming from the horizon are of incredible, basically the answer is that the photons that we see have a wavelength
42:40
which is about ten billion light years so those are not very energetic photons, we are not likely to see them what kind of experiment would we do to see them? well, we would do the same kind of experiment we do with a black hole except we do it from the inside
43:02
Alice takes her fishing pole and at the end of her fishing pole, instead of a worm she has a thermometer and she casts it out, they've got a very long line of fishing casts it out the expansion of the universe pulls on the thermometer or tends to be
43:20
accelerated out away from her and she waits ten billion years or so there's also a it's not just a fishing line it happens to be a coaxial cable or something that she can read off the temperature and she waits ten billion years and then
43:40
puts a thumb on the spinning reel to stop it she doesn't want it to go through the horizon so it stops over here and she waits until the signal comes back that's another ten billion years and she reads off the temperature and the temperature she'll read off is high on the other hand if she just sits here with her thermometer
44:03
the temperature that she sees is red shifted and is very, very, very low far lower than anything that can be produced in the laboratory it's not in particular the microwave background temperature has nothing to do with that it's the de Sitter temperature
44:21
that's the properties of the world that we live in as it will be a couple of hundred billion years from now empty, apart from us sitting at the center everything else having passed out through the horizon some amount of stuff gravitationally
44:42
bound together might include our galaxy and a couple of other galaxies that happen to be not being dragged away with the ambient expansion and everything else will be essentially empty except for this very, very, very feeble
45:00
radiation radiation from the horizon it's a sort of grim prospect as various people have noted astronomers at that time will be confused they won't understand why they're alone in a very big universe in fact they'll have a very hard time measuring how big the
45:21
universe is, there won't be anything out there to look at and the possibility of measuring this radiation, well that's not going to happen either yes, Michael well
45:41
yeah exactly the same as the black hole here's the black hole, from our perspective, in a classical analysis we would see things getting closer and closer and closer and closer to the horizon you could say exactly the same thing over here let's draw it, here's our observer
46:04
somebody falls through the horizon, our observer over here never gets to see them fall through the horizon as time goes on for our observer here, he looks back and as far as he can tell as time goes on longer
46:21
and longer Bob says Alice is getting flattened against the horizon yeah, so it's exactly the same as the black hole basically a red shift and then basically disappear fade away so if somebody else is able to fall through the exact same
46:41
spot okay, classically classically, just without worrying about any quantum mechanics okay, here's what he sees sees may be the wrong term since the wavelength
47:00
of radiation that they send out gets so long that they'll never see it but here's the mathematical description, let's see, alright here's the horizon and here is who was it, Alice? Alice, and what's the other one? Jane here you are, Jane
47:24
Alice falls towards the horizon and gets squashed Jane gets closer and is starting to get squashed Alice gets closer Jane also gets closer in other words, a kind of sedimentary
47:42
structure forms near the horizon that just gradually gets closer and closer and skinnier and skinnier now that's a classical description what actually happens quantum mechanically is more complicated and involves interaction with the emitted radiation and so forth, which
48:01
spreads them all out over the place and ionizes them, cooks them, does terrible things to them but from a classical perspective you would just lose track of them as they get squashed closer and closer in there and you'd lose track of them because the radiation from them would get more and more and more rate shifted as they get closer and closer
48:26
okay
48:46
Alice has no problem with the horizon she sails perfectly happily through. Bob just loses track of her, but if he tries to do an experiment which sees what happens to her
49:01
we come back to this very strange situation, this Heisenberg microscope situation where the only way to do an experiment is to shine some high frequency light on Alice and as Alice gets closer and closer to the horizon the wavelength of that light
49:21
has to get smaller and smaller so the answer, I'm sure by now you know is that the attempt to look at her will cook her yes, Kevin simplistically
49:40
all those effects of the horizon of the black hole were caused by the gravitational well, if you will, of the black hole what's the equivalent here? well, there is a gravitational, alright good alright, let's do a little exercise for any kind of gravitating material
50:01
which we usually call mass but we could use the word energy for any kind of distribution of energy or mass there is around it or in its neighborhood or somewhere there's a gravitational potential alright, the gravitational potential is usually called phi
50:21
phi is such that its derivative its gradient is the gravitational force on an object, on a unit mass on a test mass, on a unit mass now, phi satisfies an equation on the left hand side of the equation is something involving phi and on the right hand side of the equation is the mass density
50:41
this is Newtonian physics, let's do Newtonian physics what's on the right hand side let's just say the mass density which I'll call rho, and it's a function of position in general, it might just be a point source at the center what's the left hand side? I've just written phi but of course that's not quite right anybody know the right
51:02
not the divergence but close, you're close something called del squared which is basically the second derivative with respect to the distance, r d second by dr squared
51:20
of phi for a radially symmetric distribution of mass is equal to rho this would give us, for example for a point mass, it would give us the usual coulomb the usual Newtonian potential energy now, vacuum
51:40
energy is an interesting possible candidate to put on the right hand side and let's do so is my sine right is del squared phi minus rho or plus rho I can't remember, I'm not going to try to figure it out it's one or the other
52:04
it's not important alright, what would happen if there were vacuum energy in addition to the pockets of ordinary energy that sit there well, you would put a constant on the right hand side what would the constant be? the constant would be the constant
52:20
vacuum energy, sometimes called cosmological constant other times it's just called h squared it's basically proportional to h squared in fact, that's where h comes from so here's a formula for the gravitational potential of a space
52:40
which is filled with a uniform distribution of mass it's the vacuum energy absolutely uniform, not only doesn't it change with position it doesn't even change with time alright, what's the solution of this equation? nope
53:00
easier than that it's phi equals plus or minus we'll figure out the sine later h squared r squared let's differentiate well, maybe there's a factor of two what's the first derivative of this?
53:24
with respect to r h squared r, what's the second derivative? h squared that's this so look at that, a uniform absolutely uniform distribution of energy, and there should be a minus sign
53:42
here yeah, there should be a minus sign here if we do it right so we see that there's a potential a gravitational potential you can think of it as being due to this uniform distribution of energy distributed all over space, and what does it look like?
54:00
it looks like this minus h squared r squared over two what does this mean? this means ok, let's follow it a little bit that's the potential energy that I
54:20
at the center of my geometry I'm special, I'm at the center of my geometry, you may think you're special and you're at the center of your geometry, and that's fine with me, but I know that I'm special and I'm at the center of my geometry I make measurements, I make measurements on test masses, and I find they move in various ways and I find they move as if
54:41
they were in a potential which went like this, incidentally what does the force do to this potential? no? the force is the derivative of the potential
55:00
alright, so the derivative of the potential is just r so this is a force which increases it's repulsive that's the minus, it pushes you down, and the force increases linearly with distance so any two objects if you just start them out
55:22
at rest they will start to accelerate away from each other and they'll eventually start picking up some speed, how long does it take them to pick up any appreciable speed? that depends on h basically 10 billion years for it to pick up any significant speed, but if you wait 10 billion years it will be
55:41
moving like a rocket ok, so no not like a rocket, like a photon or close to a photon alright, so what does all this mean? supposing you start an object right up at the top, literally at the top what happens to it? it stays
56:00
there, right? it's at a point of unstable equilibrium well that's another way of saying that me, in my own frame of reference if I take an object which is located right where I am it'll stay there you don't have to be too sophisticated about it
56:20
I am my own object and I don't move away from myself well, not sometimes, but not usually ok, now what happens if I take an object a little bit away from me and again we are ignoring all other forces, we can take into account the other forces we're ignoring
56:40
so I start somebody out at rest 100 meters away from me, that's over here what happens to them? they start moving away and in fact, in this kind of potential they will move with an accelerated motion that will be if we take into account relativity and everything, it'll be this exponential
57:02
expansion so that's just a statement, that the space in between things is expanding exponentially so this kind of picture is a kind of Newtonian description of the same physics as the exponential expansion of the universe it is as though there was, number one, a uniform distribution
57:23
of matter around me and number two, a gravitational potential energy which is pushing everything away yeah we do have to consider the other forces we certainly do the other forces, even a person yeah, right so if I took, let's take the potential energy
57:43
between you and me only gravity for simplicity alright, there's another contribution I'm standing at the center and I'm thinking about you and I see that you have an ordinary gravitational Newtonian potential energy relative to me in addition to this
58:02
alright, first of all what would the ordinary gravitational potential energy look like? it would look like that pulling me toward, pulling you toward my center it would look like that, one over r
58:20
one over r potential alright, if I add that to this, it would look something like that so if I'm close enough if I'm close enough then I get pulled you get pulled toward me if I'm right at this distance
58:40
or you're right at this distance then you're in a kind of unstable equilibrium but if you're slightly farther away than the top of this then you get pushed away so this is exactly the situation that the Andromeda galaxy appears to be in here somewhere it's falling toward us a few galaxies
59:00
away, they're accelerating out away from us so this is a model just based on the expansion of space and it acts as if it has this gravitational effect then we did talk and in describing it we also talked about vacuum energy which is it's a quantum effect, it has nothing to do with the
59:22
and it behaves the same and so they both actually are there which are both no, no, no, no, no there's one thing you can call it vacuum energy you can call it cosmological constant there's only one thing and
59:41
if you want to mimic it or model it in Newtonian physics you model it by a uniform mass distribution but it's one thing it's not two vacuum energy, cosmological constant dark energy, all the same thing
01:00:00
all correspond, as I said, if you want to model it in the Newtonian context, you simply imagine a uniform right. But on the other hand, we're saying this is also a consequence of the simple expansion of space and the relativistic. Well, you can either say it's a consequence of it or a cause of it.
01:00:23
I mean, they go together. I think in this case, it's a little bit of expansion. Say it again. Pardon the time. Say it again. Pardon me. I'll do it. It seems that the same phenomenon has a completely classical description and a completely
01:00:43
quantum description. Well, the origin of the dark energy, the cosmological constant, the vacuum energy, is most likely quantum mechanical in origin. But once it's there, it's just energy. In fact, I mean, you know, the mass of the electron
01:01:05
has a contribution to it, which is quantum mechanical. But once we know what the mass is, we simply use it in classical physics as if it were just an ordinary mass. So while it's quite true that the dark energy
01:01:21
cosmological constant may ultimately be due to quantum fluctuations, nevertheless, by the time we average over those quantum fluctuations, what we see is a uniform distribution of energy, very small to be sure. If it wasn't small, then we'd be rocketing away
01:01:41
from each other. It's very, very small. In natural units, in Planck units, it's something like 10 to the minus 123. It's a very, very small number compared to any other natural scale in physics. And it sets the scale for our, and it's the reason,
01:02:02
its smallness is the reason that our horizon is so big. What would the world be like if that cosmological constant was much larger? In other words, if H was much larger. Here's the distance to the horizon, sorry, 1 over H. What if H was much, much bigger?
01:02:22
Then we'd be surrounded by a smaller sphere. We would be living in what effectively looked like a much smaller universe, even though it was, even though, what do I say, H being big, right? If H is big, that of course means a more rapid expansion,
01:02:40
but it also means a smaller horizon. So we'd be sitting in the smaller horizon, but with a much larger repulsion between things. If H were larger, the repulsion would be larger, and the size of the horizon would be much smaller. So we'd be much more tightly restricted,
01:03:02
but we'd also be subject to a force which would be, you know, causing everything to explode. So there is a limit to how big the Hubble, the cosmological constant of the dark energy can be, and we'd still be able to measure it.
01:03:21
We'd be able to talk about it. Would that mean that there's a relationship between Planck's constant, which is related to quantum fluctuations, and the cosmological constant? Yes, in a sense. There's a relationship.
01:03:41
The language to use is not Planck's constant. Planck's constant is a thing which carries dimensions. It's always good to talk about dimensionless things. If you work in Planck units, you're automatically talking about dimensionless things. And, of course, knowing what the Planck unit is
01:04:03
does involve, in ordinary measurement units, does involve knowing what H bar is. But, yeah, the answer is, in Planck units, and that means everything is dimensionless,
01:04:21
it means that the Newton constant is one, it means the Planck constant is one, it means the speed of light is one. In those units, where all the fundamental constants of physics, the most fundamental constants of physics are set equal to one, in those units, the cosmological constant energy,
01:04:41
the dark energy, is 123 orders of magnitude smaller. So that's one of the great puzzles of physics. What's the origin of that very, very small number? Along those lines, I've heard that not all theoretical share the idea that vacuum energy should be identified with the cosmological constant
01:05:02
because of that 143 order. If you go out in the world and you start asking people, is the Earth flat or is it round, you'll find 40% of Americans don't know that it's... And I wouldn't say... The situation is...
01:05:24
Well, let's be a little bit charitable. It's not quite as bad. Yeah. Now, there are some serious physicists who question it, and ten years ago, you might have reasonably questioned it. Ten or fifteen years ago,
01:05:41
you might have reasonably questioned it. The bounds from experiment get tighter and tighter and tighter. What they mean by saying that the cosmological constant isn't a constant is the hypothesis that basically the age varies with time and decreases, decreases, decreases,
01:06:02
and will eventually settle down to zero. The idea that the vacuum energy, whatever it is, is a temporary, transient thing which will settle down and eventually not be there. Just one quick question. When they keep saying on these TV shows
01:06:22
about how everything's being lonely and dark and we'll be all by ourselves and all that kind of stuff, we wouldn't even know significant truth because you can't see galaxies other than maybe a drone with the unaided eye. No, with a telescope, you can. Yeah, by saying we're not a telescope, Oh, that's probably true.
01:06:43
Yeah, without a telescope... Yeah. Yep. Yeah, who cares? So, you know, it's a very interesting question.
01:07:07
I've thought about it myself, and I've known other people who've thought about it. How would astronomers of that distant time in the future... Earth is still here. They're still here. You know, there's some questions and problems,
01:07:22
but nevertheless, we're all still here, and we have our telescopes, and we've lost our history books. The Internet has failed. We've lost our history books, and we don't know very much. How might we become aware of the fact that there was a finite horizon out there,
01:07:42
that there was a dark energy, that things are accelerating away from each other, and it's very, very hard to imagine any... I can't think of any way that they'd be able to tell. The cosmic microwave background, what would happen to it? It redshifts to the point where it's completely undetectable.
01:08:02
What about the galaxies that we count in order to measure the Hubble constant and so forth? They're not there anymore, and it just becomes very, very difficult to imagine how they could reconstruct, uh... But without galaxies being born within the...
01:08:23
New galaxies are not born from empty space. No, but they would still... But we're adding to it with the dark energy. No, the dark energy doesn't... It won't create galaxies. They would... Actually, before we knew any of this astronomy, there was something called Olber's paradox,
01:08:40
and so they would have to have some explanation for that. No, there wouldn't. There wouldn't be any stars out there. No, there would. There would be our galaxy, and... No, Olber's paradox, let me remind you, Olber's paradox was about the idea of an infinite uniform density of stars out to infinity. If you just take the stars from within our galaxy,
01:09:01
how much light do they give? We know how much light they give. Olber's paradox was a paradox of what happens if the universe is filled with stars out to infinity and they're not... They didn't know about, uh... About expansion of the universe. They're just sitting there. No, but they wouldn't know about galaxies, so they would just look out and see stars.
01:09:21
Yeah, but they would... Right, but they would see it into the stars. They would see the galaxy... They would see the island universe that astronomers before Hubble believed in. Yeah, including Einstein. Including Einstein. But he stuck the cosmological term in there just in case, you know, to make it static.
01:09:41
So they make that same mistake, so to speak. Uh... No, they would just... They would... Yeah, right. The stars in our galaxy would be smearing against the horizon, right? So they would go out in the... No, no, no, no. They'll just be just where they are now. They'll be, uh... Where are they? How far... How big is the galaxy?
01:10:02
100,000 light years, right? Small. Tiny microscopic, uh, size. No, I'm saying that the horizon will be shrinking until... Horizon doesn't shrink. Horizon doesn't shrink. It's constant. No, no, it's not constant yet. Uh... It'll be another 23 billion years. Okay.
01:10:21
Right, but we're talking about 100... Yes, yes. And how big will it be in 20 or 30 billion years? Few galactic radii. No, no, no, no, no, no, no. It will not be a few galactic radii. It will be about 20 billion light years. From the dark energy that we know is there, from the measurements of everything we know about,
01:10:42
the cosmological constant is a certain number. How big? Big enough to put the horizon about 20 billion light years away. Now, tomorrow, basically forever. So they'll be sitting in this big container,
01:11:01
uh, 20 billion light years across, with 100,000 light year, uh, galaxy, uh, uh, that they can see, everything else having been swept clean, and how do they detect the fact that there is a outer, uh, boundary?
01:11:22
Let's call it, uh... Isn't there some rate at which stars are ejected from the galaxy? No, when... Yeah, stars are... Stars are ejected from galaxy. That's true. You'll have some small number of objects you can track for a while. You can track them for a while,
01:11:41
but you won't be able to track them out to cosmolo... You know, uh, well... How do we know that that doesn't already happen? Which? Stars evaporate from the galaxy? That, you know, that edge of the universe, that 20 billion light years away... Is? Is... You know, we're not able to see the rest of it.
01:12:02
We can see it. We can see it, uh, we can, we can see... We can measure the, um, the expansion rate and reconstruct, reconstruct it. Right? Where there's enough of stuff around to be able to reconstruct it at the present time. Suppose the, uh, the event horizon would be just outside
01:12:24
the distance to, uh, the Andromeda Galaxy. Wouldn't the, the... That would be quite noticeable. Yeah, then, then you'd be able, because the Andromeda Galaxy wouldn't behave the way it behaves now. Right. Because the component with the expansion is so large. Yes.
01:12:41
Yes. But it's not... The, the horizon is not going to shrink in. The horizon is going to be at the place where... Yeah, it's going to be at... With the numbers that we have today, with the numbers that we have today, well-measured, checked, many different ways,
01:13:02
the prediction for the future is that the horizon will be at about 20 billion light years. That's where it, uh, that's, that's where it will be. Yeah. In other words, h is constant during, uh, during this kind of expansion. And, um, the distance to where you have to go,
01:13:21
to where the speed, where you see the speed of light, that's going to be constant. Uh, what would be the expanse happening out there? Not that I know of. I saw the, the, the...
01:13:40
Yeah, but we can't see that part. That's out beyond the horizon. That's out beyond the horizon. That will not be visible to us or to our future, uh, descendants. Now, what is true, there is something which is true.
01:14:00
Uh, they're called Poincare recurrences. I'll tell you what a Poincare recurrence is. First of all, your picture of... That's the inside. Your picture should be that there's some temperature here, a very low temperature from our point of view, a very high temperature out near the boundary. But you should think of it as a thermal system.
01:14:22
It, in fact, has an entropy. The entropy is just the area of the horizon in Planck units. It has an entropy, a certain finite entropy. And so you can think of it, to a large extent, as a cavity with a finite amount of entropy in it and some thermal gas. The thermal gas is mainly out near the horizon
01:14:43
because of this effect. It all tends to congregate out here. And it's constantly fluctuating. Now, a thermal system, let's talk about, uh, thermal systems, a box, a box of gas with lots of molecules in it.
01:15:04
After it comes to thermal equilibrium, and in fact, what will happen is everything will come to thermal equilibrium. When everything passes out through the horizon, it will be in thermal equilibrium. What is a gas, a box of gas like
01:15:21
when it's in thermal equilibrium? Well, it's like nothing. I mean, it's a big, dull box of uniform density. Nothing much happens. It just sits there and, uh, sits there forever and ever, right? Yeah, if it's radiation in there, it's a black body.
01:15:41
If it's molecules in there, the molecules, and they're fairly, let's say, they're fairly dense, then apart from tiny little fluctuations, nothing ever happens. Well, that depends on time scales. There are time scales. Basically, statistically, anything can happen.
01:16:01
There are no rules. There are no rules. There's only statistics, only probabilities. That's all there are are probabilities. You can really ask, what is the probability that all the molecules suddenly come together in such a way that they all wind up in the corner of the room? There's an answer to that.
01:16:20
There's an answer to that, the probability. It is basically E to the minus the entropy of the gas. The entropy is roughly the number of molecules of the gas. So if we have a box with 10 to the 25th molecules, this is E to the minus 10 to the 25th. You wait a time of order E to the 10 to the 25th.
01:16:44
It's called the Poincare recurrence time. And basically, the Poincare recurrence time is the time scale for extremely rare and unusual fluctuations to take place. An extremely rare and unusual fluctuation might be all the molecules go off into the center.
01:17:02
Or it might be they all assemble themselves into an elephant or whatever. That time scale is of order what's called the Poincare recurrence time. And it's normally extremely huge. On that time scale, there's constantly things going on,
01:17:22
fluctuations taking place which take you far out of thermal equilibrium. So far out of thermal equilibrium that in the return to thermal equilibrium, all sorts of complicated vortices and other things might form in this before it returns to thermal equilibrium. This is also thought to be true for the sitter space.
01:17:44
That if you really wait long enough, how long? E to the 10 to the 120. What units? It doesn't matter what units.
01:18:01
Believe me, it doesn't matter what units. In any units you can think of, it will take E to the 10 to the 120 of those units in, let's say, years for recurrence to occur. And on that time scale, all kinds of things will take place. Among other things.
01:18:20
Right. But short of such impossibly long time scales, nothing will happen. It'll just be heat death. The same kind of heat death that takes place in here when everything comes to equilibrium and there's no more interesting things happening.
01:18:44
Well, more mild fluctuations could happen. Fluctuations could happen. I'll give you an example of a fluctuation that can happen. We can even give a rough estimate of how long between such fluctuations. Instead of the whole world returning to the original Big Bang,
01:19:02
which might happen, that would be this time scale here, we could ask for a more modest kind of fluctuation. A more modest kind of fluctuation might be a fluctuation happens in which a solar system accidentally forms. I told you an elephant could form. Okay, yes, an elephant can form. And the time scale for elephants to form is not
01:19:22
e to the minus the entropy of the whole gas. It's e to the minus the entropy of an elephant. What's the entropy of an elephant? Roughly the number of molecules in it. In the same way, if you wait long enough, by accident, by pure thermal fluctuation, a solar system might form here.
01:19:43
That might be the cheapest way of making an observer. The cheapest way, probabilistically speaking, of making an observer might be for a fluctuation to happen and to create a solar system. But I can think of something a little cheaper. I can think of just the earth forming.
01:20:02
That would be even cheaper. The entropy of the earth, well, you might worry a little bit if there was no sun, then you wouldn't have any warmth. But let's not worry about that. You might form an already warm earth that keeps itself warm for some period of time. And you say, well, maybe it's not enough time for people to evolve.
01:20:24
Don't worry about it. The fluctuation involves the creation of people, too. But you could even do cheaper. If all you want is a single observer, you could imagine a single observer, or just his brain, just his brain being nucleated by random fluctuation.
01:20:40
Such things are called Boltzmann brains. And serious physicists worry about computing the relative probability of ordinary observers with a reasonable history like us versus Boltzmann brains. They don't want a universe in which the typical observer
01:21:06
is a Boltzmann brain. That's not a good thing if when you calculate the properties of the universe that you find out that the overwhelming majority of observers are Boltzmann brains that appear spontaneously out of nowhere.
01:21:22
And serious physicists worry about whether their cosmology will be dominated by Boltzmann brains instead of Darwinian brains that evolved. I'm not kidding you. It's a serious business.
01:21:44
Quicker way to do what?
01:22:02
No, you're right. I should qualify my statement that it's impossible to do it except on time scales which are big enough to explore such large regions. Right, that's exactly right. These forces are such that they only become appreciable
01:22:23
at distant scales of, let's say, some fraction of a billion light years. So, yes, if you could go out and explore... Yes, of course you're right. That's a much shorter time scale. You could go out and send out your explorers, map out the geometry with light signals,
01:22:44
measure the properties of space on large scale, space and time, and if you had yourself a few billion years to do it in and rockets that could get out there and explore that large size, yes, then you could definitely tell. But if you're restricted to, you know,
01:23:02
the usual lifetime of an experiment, well, just any kind of ordinary time scales, you would have a much harder time telling, and I'm not sure how you would do it. Yeah, but you're absolutely right. You could imagine sending out explorers to map out geometry, and they would find this.
01:23:24
So the time scale of the experiments would be red-shifted. Well, no, the time scale of the experiments would be of order a few billion light years. I'm just saying that it takes longer the more you wait. It takes longer the more you wait.
01:23:40
Longer to do the experiments to find out what we know now. Yes, it would take much longer to do the experiments. Yes, that's right. That's right. Is the observer sending... No, no, I think you're right. It is a matter of redshift in a sense, yes. The observer sending information back sends radiation that is modulated,
01:24:01
and that's redshift. That's okay. Don't even do that. Don't even think about it. Just send the observers out. Let them make measurements with light rays. The light rays take a billion years to get from one place to another. That's okay. We have a billion years. And then just bring the observers back and... And you're right. There is a redshift factor, but we face exactly the same redshift factor
01:24:22
when we measure it today. We measure today the cosmological constant by monitoring the cosmic mackerel background. That's been redshifted by a factor of 1,000 or so. So... So what is the information content of a highly redshifted signal?
01:24:43
You don't have to redshift that much. If you want to explore the universe on a scale of, let's say, 4 billion light years on the side, and that's enough. Probably a billion light years would be more than enough to detect the expansion.
01:25:05
In a billion light years, the redshift is not a big deal. It's a small fraction of one, a tenth or something like that. So you're not limited by redshift in making these measurements.
01:25:20
If you were, we wouldn't be able to do them today. The planets and so forth... Not the planets. The stars and galaxies that we see out there that we count and which form our measuring rods and so forth, those are out pretty far, a few billion light years. But we're not limited in detecting them
01:25:41
by their redshift from us. Since we're talking about Poincare recurrence probabilities, and, of course, all this stuff is based on, you know, quantum mechanics, quantum field... No, that does... Yeah, Poincare recurrences are not quantum mechanics. They're classical... I'm not saying that's true,
01:26:01
but all this is based on predictions, on the kind of laws of nature that we are aware of today. What's the probability that in the future humans will discover laws about nature that will make all this wrong? Probably one. But, you know, yes,
01:26:20
your point is well taken, but it's also true that the most single robust set of rules that has never been shaken at all are the rules of thermodynamics and statistics. So if I had to bet on which laws would be most likely to survive
01:26:42
for a very long time, arbitrarily long time, I would guess those which were based on statistics, probability theory, namely statistical mechanics. But do we know? No. I would also make the point that statistics and probability, what they ultimately mean is also something that is quite mysterious
01:27:01
and we might have a different view of it. Yes, I think we will have a different view of it. I was going to talk about... Let's see, what time is it? It's an hour what? I know, I know, I know. What time is it really? All right. I thought I would amuse you with some other things
01:27:22
that physicists worry about today that sound very bizarre. Can I ask if this is a good time to bring in the question that you brought up about the ultraviolet red shift connection? I mean infrared, the ultraviolet infrared connection? Yeah, I never quite get to it, do I?
01:27:42
No. It is the right time, but I'm not going to do it anyway. Another ten years? Yeah. Someday we'll get to it. Someday we'll get to it. I was going to tell you
01:28:02
about a puzzle that has to do with probability theory that is really sort of obsessing a lot of physicists and I'll tell you where it comes from. The first thing which we're not going to talk about tonight
01:28:21
is the idea of, except I'm just going to mention it, is the idea of eternal inflation. I'll draw a picture of it for what it's worth. And then I'll tell you what the nature of the puzzle is. First we want to get a picture on the blackboard. A picture of the sitter space or an expanding, inflating universe
01:28:42
looked like that. Everything below here is the world, everything above there doesn't mean anything and this is the end of time. One of the things that was discovered sometime in the 1970s by a physicist by the name of Sidney Coleman was that the sitter space like this is probably unstable.
01:29:01
But it's unstable in a very specific way. It's unstable with respect to nucleating little bubbles in which the cosmological constant might be smaller. I'm just going to draw you what the picture looks like. If a bubble nucleated
01:29:21
over here in which the cosmological constant was a little bit smaller or maybe a lot smaller what would happen is a region would grow would grow and eventually fill up a region of space like that.
01:29:42
This region of space in here this region of space time in here is itself an entire universe. Don't ask me why it's a bubble but it's an expanding bubble and that expanding bubble expands out to something infinitely big up here and it corresponds in some sense
01:30:00
to a universe. With time many of these bubbles will form. In fact the population of them will exponentially increase. Why? It has to do with the exponential expansion. The population will exponentially increase.
01:30:21
This is a picture that cosmologists love to draw. These different bubbles could be different from each other. It's not really terribly important. They might be different from each other. Things go on in them and this process is imagined to keep going and going and going. Each one of these tiny little bubbles
01:30:40
is as big as the original one I drew here for the same reason that this line here is as big as that line. It's just that they occur later. And they fill up all the future infinity here
01:31:01
with a kind of fractal structure of exponentially increasing population. There could be people in there. There could be whatever you like. The populations exponentially increase. There are some very funny puzzles about probability theory having to do with
01:31:21
exponentially increasing populations. And they have really gotten in our way. I'll give you some examples just for fun. I'll tell you some examples and then I'll tell you how it applies to this. Imagine the following Godanken world. It's not an experiment. It's just a world.
01:31:42
In fact, it's not a Godanken world. It is our world. It's a world in which the population is exponentially increasing. Unfortunately, this is approximately true. Number two, it's a world with finite resources. I think it's true.
01:32:01
But let's take it to be the case that for one reason or another that nobody really has the vaguest idea what the total resources are. This I think is not true. Unless you're in Congress. Unless you're in Congress. One half of Congress anyway.
01:32:22
Yeah. So we will take it. These citizens who live in this exponentially growing population, they really don't know, have no idea at all what the resources are. But they are interested in the probability
01:32:40
that they want to know what the probabilities are that their resources will last for another generation or so. How can they give an answer to this if they don't know what the resources are? But we'll add one more postulate. And the one more postulate is one that we always make in science. I'll give you some examples of it.
01:33:00
And it is that we who are doing the experiment are typical. Typical means the following. It means that if there are a great many possible outcomes, but more, but by some large majority the observers who are doing this
01:33:20
have a large probably, well let's see how to say this. Yeah, okay. Let's talk about coin flippers. Let's talk about coin flippers. I hate to digress. But let's get into coin flippers. We got zillions and zillions of these people who are obsessed with coin flipping. And what they love to do is they love to flip a coin a million times.
01:33:41
All of these people do this. And they count how many heads they get and they count how many tails. What they do with it, I don't know. Alright, but a typical one of them, or one of them asks the question, what bet should I make for how many heads and tails do I get? Not exactly, but you know, within some reasonable bounds.
01:34:01
I'm going to do an experiment. I'm an experimental physicist. All experiments involve statistics especially in quantum mechanics. And when we make predictions, we're making predictions analogous to the prediction of coin flippers who flip coins. Well, what's the answer? The answer is there are some coin
01:34:21
flippers who flip all heads. There are some coin flippers who flip all tails. According to the standard theory of probability, the overwhelming majority flip about a million flips, a million plus or minus a thousand square root of
01:34:41
n. So, if you're at all smart and you're a good experimental physicist and you know the rules of statistics, you will put down your prediction. Your prediction will be, what did I say, half a million plus or minus a thousand. I think I probably said it wrong. Half a million plus or minus
01:35:01
a thousand, that'll be your best guess. And if you're very far away from it, you'll probably say there was something, you probably won't believe that you are an outlier. You're more likely to believe that your theory was just wrong. But there are going to be those people who are outliers and they of course will believe their
01:35:21
theory is wrong and they'll be wrong. We always make the assumption that we are typical. So, in this expanding population, let's make the assumption that we are typical. We are like the majority of all observers who ever lived. Then
01:35:40
what bet do we make for how long the resources will last after the present time now? Well, not necessarily the last one, but there won't be more than two or three generations after us. And the reason is, in an exponentially expanding population,
01:36:01
like now, about half the people who ever existed, exist now. About three quarters of the people who ever existed, existed within two generations. About seven-eighths within three generations. So, if we're typical in an exponentially expanding population, we are
01:36:21
near the end of that expansion. We're near the end of that expansion. Now, that's a very puzzling conclusion. It doesn't, it sounds like there's something wrong with it. How can you possibly know that if you don't know what the resources are? On the other hand, it is true that no matter what the resources are,
01:36:42
the most people will be near the end of the resources. Right? I give you another example of this kind of exponential expansion. You've all seen, I'm sure you've seen, it's my favorite drawing.
01:37:02
I come back to it over and over again because it has some deep mathematical significance. It's Escher's drawing called limit circle number four. You all see it, angels and devils, and I'm not going to try to draw angels and devils, but it has a big, let's just restrict ourselves to devils. I prefer devils.
01:37:20
It has a big devil in the center, it has smaller devils out here, it has smaller devils out here, and so forth. Now, in fact, the geometry
01:37:41
is such that they're really all identical to each other. There are symmetries of this diagram which move points, which can move this point into this point, it'll move this point into something small over here. It's just the way that it's drawn, makes some things look smaller and some things look bigger. It's completely symmetric, but
01:38:00
of course when Escher actually drew it, he did have to draw the small things small and the big things big. Now, here's the question. Let's define an angel or devil, excuse me, a devil which has been drawn with a body
01:38:20
part missing. Let's call him an amputee, okay? Definition of an amputee. If the devil was drawn with some body part missing, we'll call him an amputee. Now, these devils go to sleep and they wake up in the morning and they wonder, am I one of the unfortunate amputees or am I one of the
01:38:40
one of the whole devils? If you look at Escher's drawing and you extrapolate Escher's drawing past the last visible devil and you believe that you can extrapolate it, you'll look, you'll see, Escher was awfully good at drawing. He didn't draw any devils with missing body
01:39:00
parts. So your conclusion would be, with very high probability you're not an amputee. On the other hand, Escher had finite resources. Finite patience, finite time, finite amount of ink. And that means, as he got tired out near the very, very edges, surely
01:39:22
very, very close to the edges, you'll find that almost all of the angels and devils will be slightly miss-drawn, right? Of course they will be. They won't have heads or they won't have tails or whatever it is, the ones out near the boundary. Now, what is the ratio of the ones which are adjacent to the boundary
01:39:41
to all of them? You know the answer? No. There are about as many out near the boundary as there are in the whole thing. Obviously there are fewer at the boundary. A factor of a half. And that's because the number of them
01:40:02
grows exponentially as you make bigger and bigger circles here. They grow exponentially. And so if you actually count, if you cut it off let's say, by putting a circle around here, now count all the ones that are adjacent to the circle and all the ones on the inside
01:40:22
you'll find are about equal. It's the same mathematics as saying about half the population, no matter how long into the past the population had been growing for, today about half the population, one generation, constitutes about half the
01:40:40
population of anything that ever existed. In the same way, about half of the devils are right on the edge, wherever the edge is, we don't have to know how much resources Escher had. It doesn't matter how much he had. No matter how much
01:41:00
he had, as he draws more and more and more of them, the ones near the edge will always be about half of all of them. And so, if you wake up in the morning and you don't know whether you're a whole devil or half a devil, the chances are about 50% that you're half a devil, that you're an amputee. This is very puzzling because
01:41:20
you could say, well, how can I say that when I don't know how much resources there were? Well, the answer is it doesn't matter how much resources there were. Okay, now we come back to here. We are again facing a problem
01:41:41
of exponential population growth. In this case, population of little bubble universes. Oh, let me come back, let me come back to, let me come back to angels and devils for a minute. Come back to angels and devils
01:42:00
for a minute before we go on to universes. Escher did have angels and devils. Let's ask a different question. The different question being when you wake up in the morning, are you an angel or a devil? 50% probability, right? There are as many
01:42:21
angels as devils. And the mathematics of Escher's drawing if it went on for infinity, in some sense that would be true. But we know that Escher had finite resources which means we have to draw an edge to this. The answer for the ratio of devils to angels is extremely
01:42:42
sensitive to the way we draw that line. It's extremely sensitive to the way we could draw the line. For example, every time we come to an angel, let's go around it that way. Every time we come to a devil, we go around it that way. And now we push this edge out further and further. By a
01:43:00
factor of about three quarters, we will have more devils than angels. It's all in the edge here. It's all in the edge. And so the assignment of probabilities to angelness or devilness is just extremely sensitive to what Escher did just
01:43:20
as he was running out of resources. It's ridiculous because you simply don't know anything about his resources. You would think the answer was 50-50. But if at the end he had a little bit less pink paint than white paint, the answer would be many, many more white than pink.
01:43:42
Same exact thing here. You want to know the relative probability of different kinds of bubble universes forming. Blue ones, pink ones, and so forth. The population is exponentially growing. How do you decide? You try
01:44:02
to decide the same way. You say, let's chop this off at some time and count the number of blue ones versus pink ones. But the problem is the answer is super hypersensitive to the exact way that you draw this line. And that has led to a problem that
01:44:21
cosmologists, physicists like myself call the measure problem. Measure as in probability measure. How do you assign probabilities in a world of exponentially increasing populations? The answer always is it is extremely sensitive to the way that you cut things off, to the way you
01:44:41
and your imagination in the universe. This is a very, very big problem for cosmologists because they want to be able to use their theories in a predictive way and make predictive predictions. What kind of prediction is there? Or
01:45:01
if not predictions, at least explanations of things. And so in this kind of world, if it's really true that there's an exponentially increasing population, we run into new kinds of questions about probability theory and about how the use of statistics which really
01:45:21
just have baffled, really baffled us. My guess is of course that we're asking the wrong questions. We're probably asking the wrong questions. But that doesn't help to say that we're probably asking the wrong questions. It doesn't help unless you can say what the right
01:45:41
questions are. Is there any connection to the renormalization or to the odd and even strains? I remember it's odd and even strains, but both the fermionic strains where the
01:46:02
number of loss letters varies with the spin of the strain. Well, there are relations with renormalization theory. There are relations with renormalization theory. Renormalization theory is also a situation where again
01:46:21
you start your picture of the world as having a finite number of degrees of freedom per unit volume. And then you double the number of degrees of freedom in each volume. And then you double it again and you double it again. You don't double it really. You double it in your imagination and ask how things converge. What we're talking
01:46:41
about of course is you start picturing the world as a lattice. You can't handle a continuous infinity of degrees of freedom. So you start the world as a lattice and you make your calculations based on one degree of freedom
01:47:01
per unit wavelength of an electron or something in quantum electrodynamics. Then you say I want more accuracy so I divide the world in every cell in half again. And I recalculate. And in each case I ask the same kind of questions. What's some relative population
01:47:20
of regions of space with this property or that property. And again not with respect to real honest time, but with respect to your mathematical sequence of descriptions the population is exponentially increasing. Again you get infinite answers. You have to deal
01:47:41
with those infinities. And in quantum field theory there are techniques for asking questions which are not sensitive to the let's call them the edge effects. There are techniques for extracting out things which are not sensitive to the details of how you chop
01:48:01
this up. So yeah, there are correspondences and similarities between questions in quantum field theory and these kind of questions in cosmology. And people are trying to exploit those similarities, but not I must say with any great success here.
01:48:25
Yeah. Well, what's called eternal inflation creating yeah, it's deeply connected with inflationary theory.
01:48:41
Can two nucleations be close enough that they interact? And if they did we should see it on the sky. There's a little funny patch on the sky. There are funny patches on the sky and there are again very serious physicists, astrophysicists, cosmologists trying to sort out the
01:49:01
details of what you would see on the sky if two of these bubbles collided. We more or less know. You'd see a small patch on the sky which had a little colder or warmer temperature. And of course you do see patches on the sky with a little bit of warmer and colder temperature. But to be more detailed
01:49:22
you'll have to get really detailed to see that that's what they are. One of the details would be the polarization of the microwave background around them. It's kind of interesting. If you saw one of these patches in the sky, and there is one, it's called a cold spot. A cold spot, and nobody knows what it's due to. It's a cold spot on the sky
01:49:40
and a cold spot in the sense of microwave background. Alright, cold spot. It's a candidate for a spot where another bubble collided with our bubble. It's a candidate. What more can you say about it, other than it's a candidate? Well, you can try to work out in more detail what such a
01:50:00
patch would look like. And so, for example, you discover that light is polarized in particular. This is the direction of polarization around this in a particular characteristic way. And people are doing this. They're working out the consequences of such a bubble collision. Of course,
01:50:21
should this be confirmed, this, that it does look like a bubble collision and in detail it does, this of course would be a major, major revolution in cosmology. The discovery and detection of other bubbles
01:50:40
which collided with our bubble would be very, very major. The difference is that the inflation factor, the age in these regions, distinguishes the bottom from the tops. Is the bottom region expanding faster? Bottom region typically is expanding faster.
01:51:01
And our region is somewhere in here. And it had a history of different expansion rates. So it's complicated. It's a complicated story. But it does include this idea of exponential population growth.
01:51:21
So one could foresee a situation in which we truly confirmed that there were bubble collisions with our world. We would have confirmed something like this picture, at least a piece of it. Then it becomes a really serious matter to try to get predictive and to ask what the relative probabilities of different kinds of bubbles
01:51:42
are. And that's where we run into this trouble of exponential population. The answers would be very dependent on the question of whether they were finite resources or not. What is a finite resource? I don't know what a finite resource is. It comes to an end somehow. And the details of the way that it came to an end
01:52:02
would be would determine what relative probabilities were. No, it's more complicated than that. The place where they collide, depending on details, there's something called the
01:52:21
domain wall, which separates them. And the domain wall on one side has one behavior. The domain wall on the other side has another behavior. But the point is the domain wall would be visible to us, or could be visible to us.
01:52:41
We would need a bit of luck. Here we are. Here's our observer. If the domain, if the bubble collides over here, some sort of bubble collision over here, this observer doesn't get to see it. But if it collides over here,
01:53:01
then he does get to see it. So there's different scenarios, different situations. But it does, it doesn't seem totally out of the question or the numbers that we know that there could be bubble collisions, that there could be visible bubble collisions. And if so,
01:53:20
they have a particular signature, a particular behavior. They're fairly characteristic. And if the cold spot is such a bubble collision, we will know. We will know in time. What do I bet? Well, I don't know what to bet, honestly. Do the bubbles have four dimensions?
01:53:40
Yes. What about the curvature of expansion? Or the curvature of expansion whether it's convex or not? These bubbles are usually negatively curved, Friedman-Robertson-Walker universes.
01:54:01
They have negative spatial curvature. The bubble collision would certainly modify the curvature in some characteristic way. But it wouldn't turn the world from a world with a negative curvature to a positive curvature. That or that it couldn't do. But then,
01:54:20
yeah. Kind of a related question, but is it possible to assign a probability to the statement that string theory is correct or incorrect given that we have almost no empirical evidence one way or another? I don't think so.
01:54:40
50-50 would probably be. There would be one spot. That's a matter of individual opinion for which I don't think there's any value in trying to assign a possibility. I'm not even sure I know what it means.
01:55:02
I'll give you an answer that I've given before. There is a mathematical structure, a very precise mathematical structure which exists. It's mathematically consistent. Mathematicians have won big prizes
01:55:21
for proving theorems that were first suggested by theorists on the basis of string theory, which the mathematicians didn't know. So the consistency of the internal consistency of it has been extremely well checked. There is a structure.
01:55:40
The structure has gravity and it has quantum mechanics in it. So in that sense, string theory is a well-defined exact... Now of course there's a very remote possibility that somebody will discover an inconsistency in it, but it seems extremely unlikely
01:56:00
a mathematical inconsistency. On the other hand, that part of it which is mathematically precise, let's call it string theory with a capital S, with a capital letter, string theory, capitalized. It is
01:56:20
supersymmetric. It is, um... It's supersymmetric. The world is not supersymmetric. The world, as we see it, is not supersymmetric. The mathematical structure which is well-defined and for which we have a lot of confidence that it really exists.
01:56:41
You can call it quantum gravity with supersymmetry. It is not the real world. So in that sense, we know now, no question, string theory is not the right theory. 100%. We know it for sure.
01:57:06
Breaking the symmetry, nobody really knows how to do. It is not part of the mathematically precise part of the theory. We all expect that this thing that we call
01:57:21
capital string theory, there it is, string theory, it has many, many, many different versions. A huge number of different possibilities lie within this. We all expect that it's contained within something bigger, something bigger that we don't know
01:57:40
how to analyze. It's more or less like the situation where, maybe it's like the situation, where physicists in the time preceding Newton might have discovered the formula F equals ma, and the formula
01:58:01
the force is equal to m1 m2 times g over r squared. But they didn't have the mathematical tools to analyze anything but the simplest orbits. So they said, alright, we will use symmetry, the symmetry of circular orbits, we need one little bit of extra
01:58:20
information, we need to know the acceleration of a circular orbit, that's not too hard. You can do it on the basis of some simple symmetries, and you can therefore figure out Kepler's which one of the laws is it? One of the Kepler laws which tells you the period is a function of radius.
01:58:43
You know that the theory that you're working with, the theory of circular orbits is not correct. But you just don't have the mathematical power to be able to go beyond it. I don't know if that's what the real situation is, but it could be like that. So we know that the theory of circular orbits is
01:59:02
contained in something bigger, don't know how to analyze it. We know that the perfectly idealized super- symmetric string theory, the one that we really know how to analyze, is not the real world. It looks a little bit like the real world, but it's not the real world. And the question is whether it's contained in something bigger
01:59:22
which we could call string theory with a small s. But the answer is we don't know very much about this. We know a great deal about this. This has been enough to tell us all sorts of things
01:59:40
about the combination of gravity and quantum mechanics, things about black holes, things about all kinds of things. But to say one way or another, what's the probability that string theory contains a or that there is an extension of it
02:00:00
a bigger version of it that contains our world right over here, I don't know. Of course, in some sense, well, of course that's true. Our world is someplace. Here's the theory of someplace. If you add this to this, you get something bigger.
02:00:20
But that's not helpful. We want to know whether in some interesting way the theory of the real world is some extrapolation from what we already know. And there's no way to know. Does it contradict the string theory? Contradict the possibility? Only capital string. Does it contradict the possibility that what?
02:00:42
Does the capital string theory rule out the four? Capital string theory doesn't rule out the presence or doesn't rule out the existence of small string theory. But what it does say is that nature is not, as we know it, is not described by capital string theory. All right, so we know with certainty
02:01:03
that a bigger structure has to be found. And whether that bigger structure that contains our world will sort of be contiguous with, connected to the smaller structure of capital string theory, this, we don't know. We can all make our bets.
02:01:21
You're saying that string theory does have, has no symmetries of breaking like the H particles and things like that? People write loads and loads of papers about the symmetry breakings of string theory. The symmetry breakings in particular are the symmetry breakings of supersymmetry.
02:01:40
And we know a lot about the mathematics of supersymmetry, but we don't know a precise version of how to combine the breaking of supersymmetry with string theory. That is outside the scope of what we know. That doesn't stop people from, you know, you make an approximation here and there,
02:02:02
and then you start rolling, and you can do some things. The problem is that you're making approximations, but you don't know what those approximations are approximations to. You make some standard approximations that we use all the time, and you start calculating, but there's no backbone to the thing
02:02:21
that you're approximating. So at the present time, I think the safest thing to say is string theory, as we know it and as we can... The rigorous version of it is certainly not the real world. There's, in my view, lots of reason to think
02:02:41
that the part of it that we know about is not the whole thing, that it could be bigger. It probably is almost certainly bigger. But how to say whether the real world is part of it that's very much a prejudice that different physicists share in different proportions.
02:03:01
The string theory, it contains gravity. Does it contain special relativity or a special relativity apostolate for string theory? No, it really does require special relativity. Now, special relativity can be violated by the environment.
02:03:23
We're sitting here in a room where everybody's at rest, right? And the world does not look Lorentz-invariant to me. Your presence here violates my sense of Lorentz-invariance. It also violates... We talked about this last time, didn't we? Yeah. So you can have what physicists call a background,
02:03:41
which means stuff in it, which picks out a particular frame of reference. But the underlying rules for string theory do have Lorentz-invariance. That's a prediction. It's not a postulate.
02:04:01
Yeah, I would say it's a prediction. But the most robust... Yeah, I think so. But the most robust prediction is that it has gravity. There's no way for it to not have gravity. It has gravity, but is it consistent with Einstein's theory of general relativity? It is. I mean, the background independence, the general covariance, is in the string theory also.
02:04:26
Yeah, this question about background independence is a confusing one that... Yeah, it is background independent up to a point.
02:04:41
Up to a point, yes. It's hard to change the asymptotic boundary conditions continuously, but it should be hard to change the asymptotic boundary conditions continuously. How do you change asymptotic boundary conditions continuously? You can't do any operation that would change them.
02:05:01
But short of asymptotic boundary conditions, changing the background locally in space-time, yeah, the string theory does have that. And you don't have to take my word for it. This is something that's been long studied since the 1980s,
02:05:24
works of people like Curtis Callan and other people have studied that. No, we're finished tonight.
02:05:41
Okay, so... For more, please visit us at stanford.edu.
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