 Okay, thank you. Let me again check that the microphone is working and Not too loud or too soft Yeah, actually this for the first lecture on the night after a party. This is I think a very good turn out GR 22 last week the the party went on till well past midnight. I don't think the turnout was Quite as good for the first morning lecture So I've been telling you about everything that I've Been talking about has been thus far has been in classical general relativity where I hope I've Displayed to you That there is really a remarkable Relationship between laws of black hole physics and laws of thermodynamics, but things get Really much much more remarkable when the quantum theory gets taken into account and Before I go to the first slide Let me remind you that in the laws of Thermodynamics and black hole mechanics. There's a correspondence between mass and energy and this even in classical general relativity is more than Just a mathematical correspondence of symbols and formulas mass really represents Energy in general relativity. There was also up to numerical factors Because the you know, we had a we have a D e equals t d s law of thermodynamics and in black hole mechanics We have a kappa over 8 pi D area these are corresponding quantities and these are certainly corresponding quantities, but how you split up the 8 pi numerical factor between these you know has arbitrariness in it, but some factor times kappa corresponds to temperature then and Some corresponding inverse factor times area Corresponds to entropy, but In classical physics This is where any kind of physical analogy ends because Although the mass is the physical energy of the black hole The physical temperature of a black hole is clearly by any reasonable definition absolute zero black holes Absorb anything you throw into it, but they don't by definition Amid anything in fact, you can even You know you get into this a step more deeply you can run Carnot cycles using a black hole You know doing thermodynamics lowering boxes near of radiation near a black hole in classical general relativity You can run a Carnot cycle with a hundred percent efficiency, which is another Statement that the physical temperature of a black hole is absolute zero That's not the situation though when you go to quantum theory and in fact quite amazingly black well there There are like three or three Absolutely amazing sub clauses of these sentences. First of all black holes Radiate in quantum theory now that makes no sense because I said Nothing can come out of a black hole this I'm drawing here a black hole event horizon and anything inside The event horizon By definition, I mean this is purely a definition of a black hole and the event horizon Being defined as the boundary, you know nothing Inside the black hole is going to influence an observer who I'm drawing here who's let's say far away from the black hole and Looking at at what's going on Well, you might say oh, that's Classical theory, but quantum theory all sorts of weird things happen. Well quantum theory Quantum field theory is just as causal as Classical field theory even though it's often not kind of presented in that way and of course what happens in quantum gravity, I don't know and What you even meet well what you even mean by space time in quantum gravity and the properties What a black hole would be and what properties it would have in quantum gravity? I don't know but the results. I'm talking about are in the context and The particle creation results are in the context of quantum field theory in curved space time Where the metric is treated classically, but the matter fields Are treated as quantum fields, so they're treated fully Quantum mechanically, I mean you can treat the metric also as a quantum object Paterbatively off of this classical Background so you can I mean you'll get into trouble if you try to go far in perturbation theory with that because of non renormalizability and other issues, but You know we can talk about gravitons Being produced and that's fine and they'll obey the same Sort of rules as the other and properties as the other quantum fields, but again in quantum field theory nothing Can propagate From the inside of the black hole to this outside observer, so how can the black hole be? emitting things Well, it's emitting things because in in essence, I mean one way of h plus over here describing it or thinking about it is using a notion of particles that would be a natural notion for stationary observers Outside the black hole to use now of course near the horizon Observers who were stationary near infinity With that notion of stationarity if the black hole is rotating you actually don't have stationary observers So what I'm really talking about here Well, we could restrict to the non rotating case, but I'm really talking about observers who were moving on orbits of time translation symmetry relative to the Killing horizon symmetry those orbits will be time like at least a little ways outside the black hole And with respect to that sort of notion of particles there are in fact, I mean this is the remarkable Hawking calculation there are I mean it's a little funny to draw it this way because Particles are really modes of the field and I'm using a particular definition of particles these things that I'm drawing would be vacuum fluctuations According to some freely falling observer across the horizon vacuum fluctuations not any any particularly different from the vacuum fluctuations in this room that we don't tend to pay a lot of attention to or notice but anyway the What happens? with a black hole that was let's say formed by gravitational collapse If you didn't form the black hole by gravitational collapse Then you would have to worry about what the proper initial conditions were because the extended space time would have this bifurcate Killing horizon and you'd need to have Conditions on your state on the white hole horizon one of Hawking's amazing Contributions in his original paper was to realize that you could get rid of that ambiguity by considering the physical case of Gravitational collapse to a black hole but anyway you have these Pairs of particles, which as I say alternatively could be viewed as vacuum fluctuations and would be viewed as vacuum fluctuations by some observer falling into the black hole, but I don't move faster than light, but the One of the particles one of the particles in the pair is inside the black hole and as always Was inside the black hole unless you trace it back to before the gravitational collapse occurred The other particle though well may well fall into the black hole most things Really close to the black hole do But such a particle may also make it out to infinity and be observed By this observer that I drew far from the black hole So this distant observer that I drew is in fact going to see a flux of particles which As far as he can tell seems to be coming from the black hole Well, of course that observer by definition can't see the black hole, but you can look in the Direction where the black hole is These particles are really seeming to come Perhaps from a white hole, you know in this bifurcate killing horizon Extended version of the asymptotic final state the the particles are would seem to emerge from the white hole Really close to the event horizon of the black hole at late times so that's a amazing thing number one that a Black hole Well, even a schwarz-dill black hole. Let me restrict to the Non-rotating uncharged black hole well Anyone would have that you would have asked before the Hawking calculation would have said okay in this dynamical period of collapse you've got a Strongly time-dependent space-time metric there's going to be some particle production arising from that and you know you'd see some observer would see some Flux of particles, but that should go away Quickly once the black hole is formed and settled down to a stationary final state Other people that you would have asked before the Hawking Calculation would have said That's what'll happen with a schwarz-dill black hole if you have a Kerr black hole a rotating Black hole then because of the super radiance phenomena you actually would get some spontaneous creation of particles I mean Zaldovich and Starbensky pointed that out in the paper where they First the first paper that noted and talked about the super radiance phenomena they pointed out you would get Particle creation, but that would just be Associated with the rotation of the black hole causing it to spin down. I mean analogous to if you had a Charged black hole you'd expect Schwinger pair production that would eventually discharge the black hole and then One would think you would get nothing in fact, I think the Hawking calculation arose because in Hawking's Visit to Moscow. I think Zaldovich told him about This work and Hawking was very skeptical about it and particularly skeptical about what? White hole boundary conditions you would Impose I mean unruh was working on this at the same time and actually realized that there was an issue of boundary conditions or I mean by boundary conditions. I really mean initial choice of initial quantum state but Hawking really resolved all that by going to the physical gravitational collapse case and when he did the calculation amazingly he found that Yes, you've got particle creation that would be some Messy thing that depends on the details of the collapse at very early times But instead of going to zero at late times you had this phenomena that I have been sketching here where the late time Observer would see a steady but non-zero flux of particles. So that's Already quite amazing, but what is even a lot more amazing and which is and also the reason why this fits in as in as the last lecture in black hole thermodynamics what this Observer sees well, I'm putting him as a making him a distant observer, but the Let's do the Schwarzschild black hole if There are modifications due to the rotation, you know due to the same effect that Zaltovich and Starvinsky had Pointed out and mentioned earlier that can all be taken into account, but let's stick to the Schwarzschild black hole where this Distant observer will see an exactly thermal spectrum of particles Well, and you doesn't really have to be a distant observer any Static observer will see a thermal flux of particles if you're in closer the The flux of particles will be blue shifted relative to infinity so you'll see a higher temperature here by the Standard tolman relation if you have a star the temperature of the The star in deep inside the star is actually if it's in equilibrium is hotter Then on the near the surface because by this same redshift law I mean that's not true in Newtonian gravity, but that is true in general relativity and that is true here for the black holes So That's a second so that you get a steady flow I said there were like three or four incredibly amazing things all packed into this one calculation, so I don't have a you know Pre-set list of these things, but just that you get a steady non-zero flux is amazing That that flux is exactly thermal is really amazing and That the temperature of this is the surface gravity is Maybe even more amazing because now this Dotted line that I drew here has completely disappeared and And The numerical factor has been calculated as well So kappa over 2 pi now. I'm using well The H bars G's and C's. I think I've set the Boltzmann constant No Boltzmann's constant is in there. Yeah, this is a formula if you stick this in for the Schwarzschild black hole that involves every constant of nature used in any Branch of physics basically well, maybe there's some extension exceptions, but Boltzmann's statistical physics constant H bar and then of course G and C are coming into the The picture so let me Leave the dotted line here, but there but Surface gravity over 2 pi really is the physical Temperature of a black hole now that's a very small temperature for a solar mass black hole if you put in the numbers You'd have to go to 10 to the fit well 10 to the 15th grams or so black holes You know would be radiating at a Well, you can do the numbers yourself choose whatever temperature you want and see what You know mass you would need but you'd need very small black holes to get a very big effect. In fact black holes would not be the Radiation from a solar mass black hole would in the present universe be totally swamped by the cosmic microwave background But if you wait long enough the cosmic what microwave background will redshift and this will be the dominant If there's nothing else falling into the black hole now one immediate consequence the Hawking also immediately Noticed as if you just do the back of the envelope calculation of the mass loss of the black hole Due to the Hawking radiation Well, it's thermal radiation. It's given By the Stefan-Boltzmann law you're really in a physical optics regime. So you can't Just copy the usual Numerical constants that are in the geometric optics regime of a body, but it still scales as area times t to the fourth Area scales as m squared, but t scales as 1 over m so Smaller black holes radiate more and you can integrate this equation Basically in your head and if you put in the numerical factors you find the important thing is that a an isolated black hole It doesn't have anything falling into it Will evaporate completely as a result of this thermal emission in a finite time Again that finite time is pretty long for a solar mass black hole The 10 to the 15th grams that I mentioned is what you would need to get this time in Agreement with the age of the universe. So if you had 10 to the 15th gram Black holes produced in the very early universe somehow I mean no reason to think any were produced But they would be undergoing the final moments of evaporation right now okay, so this is now the new space-time diagram of a Black hole again taking into account the evaporation Of the black hole so an isolated black hole. So this Part of the diagram is what I drew for you fairly early in my first Lecture where you've got this is the one with the light cone straightened out and the angular Direction suppressed so each point here is a two sphere except for this line that represents an Origin of coordinates, so this is really a The points on this line are really points, but the points everywhere else are two spheres and this shows the Spherical outer surface of a collapsing body collapsing down to r equals zero and producing Singularity and a black hole is formed in that process and that you create this region of no escape So that's what I showed you before but now this black hole well, if it's a Solar mass black hole This is all you know 10 to the 73 seconds compressed into a little portion of the diagram so this is you know not You know not to scale so to speak if I'm using some time slice here that is Going up, you know in equal time steps in the equal proper time steps for observers outside the black hole let's say But the black hole is here slowly shrinking you can't See that in this diagram because I can't represent the size of the surface Having suppressed the angular directions and of course the horizon is always no but here it's shrinking down to zero radius and Presumably the black hole just disappears and again you'll just have some origin of coordinates some non singular r equals zero line and up here Presumably just empty flat space time out here. You'll have all the Hawking radiation that Was emitted by Not really the black hole, but the space time region around the black hole during the process so this is the picture of black hole evaporation and I've already put this up on the board, but now we're We only have one more dotted line in terms of are these quantities physically the same and We've fixed the numerical factor because we have the exact formula for the temperature so The kappa over 2 pi is the temperature that means one quarter the area Is what is supposed I can put the one quarter in now because I put the 2 pi in So does this represent? The physical entropy of a black hole Well, there's very good reason to think yes, but I'm going to pause on that for a minute to just mention that In some sense this phenomena has very little to do with the black hole It does have a lot to do with the black hole if you want a asymptotically flat space time and observers near infinity seeing radiation, but if you Just go to ordinary Flat space time and Kowski space time and Look at the Lorentz boost symmetry. I showed you the picture of the orbits of Lorentz boost symmetry in an earlier slide. It's probably as quick for me to redraw this as page back to that earlier slide, but This is what of This is the analog of a rotation So of course here in Minkowski space time, there's nothing special about this event. I just happened to choose that You know as my origin to define to pick out a particular Lorentz boost, but having done that the These two intersecting null planes In four dimensions these come out of the board as null planes Those comprise a bifurcate killing horizon associated with the Lorentz boost Symmetry and we can ask What does the ordinary? Minkowski vacuum state look like as far as these Accelerating observers are concerned. I mean the orbits of Lorentz boosts, of course are Accelerating world lines. They're uniformly accelerating the ones closer to the horizon are accelerating more than the ones Further from the horizon But we can take the Minkowski vacuum restrict it to the right wedge that I've drawn Where the Lorentz boost symmetries are have time-like orbits So the Lorentz boost killing field is time-like in this wedge if I take the Minkowski vacuum and restricted to the the The right wedge I can then analyze it From the point of view of using Lorentz boosts as my notion of time translation symmetry And if I say this is this right wedge is a stationary Spacetime in its own right, it's globally hyperbolic. It's a perfectly legitimate spacetime to apply quantum field ideas to If it weren't globally hyperbolic, there'd be determinism issues as there would classically But that's not an issue here in the right wedge If I have a notion if I have a time translation symmetry, there's a corresponding notion of particle associated with that and and that would correspond physically to the notion of Particles that would be observed by observers Following the orbits of the time translation symmetry so the Lorentz boost orbits or in other words uniformly accelerating observers and in Minkowski spacetime when I say It would correspond to what they would observe. I mean that you can you know make a model particle detector, which is some quantum mechanical system that interacts with the quantum field and You know it may have energy levels the and if your Particle if your particle detector makes a transition upward in energy These observers would interpret that as having absorbed a particle And I mean you and that Interpretation is completely consistent. It's the same interpretation as we use typically using inertial notion of time translation so the right in in this Minkowski spacetime we also have instead of Lorentz boosts a notion of time translation that Extends to the entire spacetime and you can again talk about what What you would mean by a particle detector that one of these observers would carry and How you would interpret the absorption of a particle, so I'm just using exactly the same Story here well the answer is I mean the the the kind of mathematics involved in Giving the description of the quantum state as a particle state with respect to Lorentz boosts Here is essentially the same problem as Hawking was doing in the particle creation Calculation, I mean there isn't particle creation here. It's two different descriptions You have the vacuum state with respect to ordinary time translations and you're asking what is How was that state described in the right wedge by these Accelerating observers and the answer is that it's a thermal exactly a thermal state. In fact, there's a rigorous so the this What is an accelerating observer see was first analyzed by unruh in an in an attempt to get more insight into what was going on in the Hawking calculations and that's so The fact that these observers would see a thermal distribution of particles at a temperature equal to the acceleration over two pi That's referred to as the unruh effect, but in fact at exactly the same time You know algebraic quantum field theory tight You know rigorous mathematician tight people for completely different reasons having nothing to do with physics were interested in looking at how the Vacuum state really punker a invariant vacuum of any quantum field theory would look in the right wedge when Using the notion of Lorentz boosts for time translations and the statement is that it for anything, it's Exactly a thermal state a KMS state You know in exact accord with the unruh effect that I checked in the business in yano this agnano this agnano victim on theorem was Submitted for publication within about a month of when unruh's paper was submitted for publications of these were really done quite Simultaneously and of course it completely independently. It was nearly a decade before anyone noticed that there was any relationship between these two so The bottom line on this is that a uniformly accelerating observer Feels himself or herself to be in a thermal bath of radiation at This unruh temperature And in fact in some sense the you know the Hawking effect that I described Well, if you take observers that go very close to the horizon As I already said their temperature is going to get blue shifted so they're going to see thermal radiation of particles at the blue shifted Hawking temperature, but the surface graph as I mentioned in the first lecture the surface gravity is the limit as you go to the horizon of The redshift factor times the acceleration so in the limit as you go to the horizon this formula becomes the unruh formula so In some sense for stationary observers just outside the black hole the black hole has nothing particularly to do with the radiation they see which However as you go out to infinity now the observer the stationary observers are nearly inertial And they will still see a non-zero radiation. It's not given by this acceleration formula anymore Okay, so now let's go to this bottom line, so is the area of the black hole the Physical entropy of the black hole Well, this has already kind of come up in various this question has sort of been asked in various Forms I mean in the question periods and so on of you know Well, how would you calculate the entropy of a black hole? Well, I don't know how you would Calculate the entropy of a black hole until you had a quantum theory of gravity and Not just the name of a quantum theory of gravity But an actual theory that from which you could actually do actual pose questions and do and calculate actual answers or whatever and you know I Think everyone would agree that we're not We're not really there yet. I mean there is the this exception of the Calculations done in the context of string theory that I commented on in response to a question a couple of days ago But again, that's not really Taking you know the true description of a black hole and identifying what is Going on with the black hole and I think it's really questionable as to whether even the ordinary Thermodynamic ideas that I spent a lot of time sketching yesterday in terms of phase space volumes and so on that was in the classical case, but you know densities of states or numbers of quantum states with given Macroscopic type properties would be the corresponding thing in the quantum theory. It's really not obvious to me that those ideas will even Carry over to the black hole case. So I don't know how to make progress on Erasing this line We're convincing progress certainly You know without physics You know taking some major further leap so that we understand the quantum nature of a black hole fully and we understand also What the entropy will mean in that context? But there's very good reason nevertheless by obviously more indirect arguments To believe that this really is the physical Entropy of the black hole in the same sense as you know, whatever formula you have for an ordinary matter system is the physical entropy of that matter system and that The argument for that really Is well this so-called generalized second law of thermodynamics because in fact Although I've Written down the second law of thermodynamics in any process the entropy Never decreases and I wrote down the area theorem as The area never decreases Both of these laws Separately have serious problems So what's the problem with the second law that? has survived centuries and No, but people have always tried to find problems with it, but nobody's found a problem. Well, there's a sort of trivial problem in some sense with the ordinary second law which is I can have some you know bucket of Entropy here. I mean this is a space-time diagram So I'll have to draw it in time too, but I can have some Box of gas that has a lot of entropy or whatever and I can just throw that Into the black hole now. We don't really know what happens when it goes into the black hole, but presumably it goes into the singularity I mean, maybe we don't really have singularities in quantum gravity But anyway as far as I'm concerned out here this box of gas is gone and You know, I mean, I suppose part of its world line is still You know world tube is still in my past, but this is so incredibly redshifted You know, there aren't going to be any photons or anything else after a rather short while coming from here so This box of gas is gone into the black hole and if I look around me At how much entropy there is in the universe? I'll find the entropy has decreased from what it was at this early time You might say, okay, well, why don't you go into the black hole and then you can count that well even if you go Into the black hole at late times. You're not gonna be able to To see in any meaningful sense that entropy so that I mean at least if the singularity is anything and the interior of the black hole is anything like it's Like it is in classical general relativity so Now I mean Well when these ideas were first Being discussed as they were at Princeton when I was a graduate student I mean, I I didn't particularly Find the personally find this troubling at all I mean, you can do the same story if you take a bucket of baryons or whatever and you know late 60s early 70s Baryon conservation was believed to be an exact law of nature and throw these baryons into the black hole and you have Baryon, you know a failure of baryon conservation if I count up the baryons in the universe It's now smaller than then when I started so Wheeler was not at all Somehow perturbed by that and had the word he was very good at making up Words for things and he had the word Transcending the law of baryon conservation, so it wasn't violated. It was transcended and I Think he was happy with that. I was happy with that certainly that was fine I was equally happy with the second law of thermodynamics, which I Would have viewed as a much less fundamental law than the law of baryon conservation, which I did Why not transcend that but Somehow Wheeler was very bothered by that and got Beckenstein to think about that problem. I thought that was a really Not very fruitful Problems to work on so this is in you know 1970 71. So this is several years before the Hawking calculation, which was 1974 I mean announced I guess in January 74 but Anyway, that that's the problem with the ordinary second law But there's a more serious problem with the area theorem once you know about the Hawking effect and so on so the area theorem is a theorem that is based on Hypotheses including the weak cosmic censorship, but that's not the troublesome hypothesis the the troublesome hypothesis goes back to the Rachiduri equation which critically uses this assumption in the argument and With Einstein's equation Well, there's a trace term, but these are null vectors So I can replace this by TAB the stress energy tensor And this is just a positive energy condition. So That shouldn't be problematic to assume that or that, you know Wouldn't necessarily sound I mean in certainly in classical general relativity that would not be something That one would worry about but in quantum field theory all local energy conditions are Violated it's very easy to make up quantum states that have locally negative energy density And indeed that's What happens here in the black hole and that better happen because you're getting in this Hawking effect a Flux of positive energy to infinity And if you're going to get energy balance, which you are going to get in this I mean at least in this You know small perturbation of a stationary black hole. You definitely will automatically get energy conservation in quantum field theory There has to be a flux of negative energy into the black hole and there is but that violates the argument and indeed from what I've said, it's already obvious that it's Violated because I said a black hole will in fact completely evaporate In a finite time its area thereby will go to zero and it started out large so The area theorem is Genuinely problematic But what Beckenstein proposed is Why not add up? all the entropy of matter outside black holes to the black hole area where I've thrown in Appropriate constants of nature into this formula, but this is just the one-quarter area that's supposed to represent in this analog in the analogy the Entropy of a black hole Because then When I throw something into Throw some ordinary matter into a black hole I'm gonna decrease the entropy of matter outside black holes, but this ordinary matter with you know Nice chunk of entropy or whatever that I'm throwing in will satisfy energy conditions And that will tend to increase the area of the black hole and you don't have to increase it a lot to get I mean this is a in ordinary units a very very large number So you don't have to increase the area very much in square centimeters to get a lot of entropy On the other hand in the Hawking effect You're you are going to decrease the area, but you're spewing out this thermal distribution of particles From the black hole and that's going to have a lot of entropy for you know the amount of energy that you're Spewing out and again, that's enough to keep this quantity satisfied This quantity non decreasing at least in a quick back of the envelope calculation, so It's interesting to probe Whether in a little more careful detail Whether if you try to do things optimally you can in fact violate this generalized second law and a very promising way to do that which Bill Unruh and I analyzed By now 35 years ago, I suppose would be well instead of carelessly throwing some box containing Entropy s and of course it will have some energy e associated with it instead of just careful instead of just Carelessly tossing that into the black hole Slowly lower it into the black hole to the black hole and then Drop it in or let its contents fall in when you're very extremely close to the Horizon so the idea is the entropy that you lose is going to be the same Wherever you dropped this box from But the change in black hole area well you can calculate that from the first law of black hole mechanics that we have up here and so that's going to depend on how much energy is delivered to the to the black hole and the idea is if you lower it to The horizon the amount of energy that actually gets delivered to the black hole Well, classically at least is just the redshift factor times the Locally measured or rest energy, you know locally measured rest energy of what's in this box so if you lower it slowly and carefully that rest energy should stay constant but One way you can calculate this calculate the work done in your laboratory as you lower it and that will give you this conclusion that You know well if you get it arbitrarily close to the horizon you'll extract the entire rest energy of this box Back in your laboratory, and you'll deliver no energy to the black hole so it looks like you can lose Entropy s but keep your area change again using the first law because you've Made the mass change arbitrarily small you can make the area change of the black hole arbitrarily small and Violate the generalized second law so Beckenstein was aware of this issue from the beginning and made argument tried to make arguments about You couldn't you'd have to have some minimum temperature here first and then The box would have to be a finite size. You couldn't look Lower it all the way to the horizon that eventually evolved to Bound entropy to energy ratio bounds on the size of the box those arguments and You know I think don't work actually for this purpose But there's a much simpler reason why this process doesn't work and that has to do with Well the effects that I've just been telling you about with well, particularly the unruh effect here of the accelerating observer near the in flat space time, but what's relevant is the Accelerating observer here near the horizon because if you're going to lower this very slowly then of course this Box is going to be nearly stationary. It's going to be on a very highly accelerating trajectory and It's going to from the point of view of the from the stationary point of view it's going to be surrounded by a thermal bath of particles okay, but That thermal bath is not at a uniform temperature because the temperature is Varying as the redshift factor and as a result of the temperature gradient There is automatically for thermal radiation going to be a pressure gradient Which means there's going to be a buoyancy force just as though you were located you were Lowering this into a fluid So in fact, one of the things I had to do in this paper is look up the original Archimedes pre-print on this to give proper credit for Where the optimal place to drop the boxes because you don't want to push it past its floating point I mean and but you have to figure out where the floating point is and Archimedes Figured that out You know when it displaces its own mass is where the floating point is and that was part of the calculation so we Think said in the paper that that this was when we Finished the calculation and got that I think we had a footnote saying that this was also done independently by Archimedes and Gave the citation But if you take this buoyancy force into account, then you don't Recover all the energy at infinity That was in the rest mass of the boxes a little residual left over and if you Take into account that residual. It's exactly what you need to keep this generalized second law satisfied, so what am I Saying I'm saying that if you take the entropy of matter outside black holes and add a quarter black hole area you get a Quantity that as far as one can tell never decreases Well, what would be the interpretation of this? law well, this is the entropy of Everything in the universe besides the black hole you're looking at That must be the entropy of the black hole this must be the second law of thermodynamics, I mean I just don't see How could it be otherwise or whatever so in any case that is enough for me to erase this line and conclude that There's not only a perfect mathematical analogy, but the laws of black hole mechanics I think are the laws of thermodynamics applied to black holes and of course understanding Why that should be black hole entropy remains you know one of the most interesting questions in Physics or at least in any of the areas of physics that I Know about or think about Yeah, so I've just said the same thing here that the apparent validity suggests of this generalized second law suggests that a quarter area really is the physical entropy of a black hole Okay, so I have 20 minutes a little less than 20 minutes left that that actually Complets this basic story on black hole thermodynamics, but there is An issue. I mean which is usually referred to as the black hole information paradox. I Don't have any Belief that the word paradox should be used there So I would tend to call it the black hole information issue that is Related to this. I mean it ties. I mean depending on People's thoughts of what entropy means and so on it may tie into this and I thought I should at least Mention that now I expected to have more like half hour or 40 minutes left to talk about this So I'll go through this a little bit more Quickly than I otherwise might have But this issue well it has to do with quantum entanglement and That's probably I mean if you've taken quantum mechanics courses you're you probably have some familiarity with with entanglement, but if If you have a system that consists of two subsystems each of which would have Hilbert space h1 and h2 Then the rules of quantum mechanics say that the Hilbert space of the joint system is the tensor product of these Hilbert spaces Now what's weird about that and what gives quantum mechanics? quite a bit of the weird properties that It has Is that I mean if all states in the tensor product were product states Then that would be easy to kind of understand you have this state Representing the first system in that state representing the second system when they're together but you have linear combinations of things of this sort and Those linear combinations well they might be re-expressible as just a product of states, but generically they can't be expressed as a as a Product of states in which case even if these systems are not interacting so they're not talking to each other nevertheless, what? You know what you measure in the second system is correlated with what is measured in the first system Right, this is you know the Einstein-Rosen-Podolski paradox is based exactly on the final state of your Whatever you did your decay of a particle or whatever being an entangled state So if you have an entangled state, and you just want to describe One of the systems let's say h1 you can't describe that as a vector in your Hilbert space You can describe it as a density matrix, but there is no pure state There is no vector in the Hilbert space that You know will describe will give you the correct probabilities of outcomes of all measurements and Entanglement is I mean just occurs all over the place all interactions basically will result in Entanglement, I guess I'll possibly say a bit more about that in In Quantum field theory entanglement is even more of an essential feature of quantum fields And it plays a big role in the Hawking effect actually as well, and that's really quite easily seen just if you take a Scalar quantum field this is the standard expression for the two-point the vacuum two-point function of a scalar field in flat spacetime and The expectation of the product Goes as one over geodesic distance squared between the events so it blows up as the points Get close to each other, but it's easy to show if there were no entanglement then the expectation of the Entanglement now of the field at these two different points the field is now the observable if there were no entanglement Then the expectation value of the product would be the product of the expectation values Which is zero and this is very non-zero when the points are close to each other so The information loss issue I mean has to do with the fact that when you Form a black hole there will be entanglement between the state of the quantum field outside the black hole and inside the black hole and indeed this entanglement that I these Particles that I was talking about or drawing early on are tightly entangled the Hawking Therefore what comes out as Hawking radiation is entangled with things that are inside the black hole But you don't even need Hawking radiation to have a lot of entanglement between the inside and the outside of the black hole You know you could do EPR experiments. I already have Some stuff here going into the black hole. I could have some other stuff Here that doesn't go into the black hole and I could have these Systems be highly entangled and Now I would automatically without even any Hawking radiation have a lot of entanglement of the stuff out here with stuff in the black hole at late times so the Situation then I'm redrawn the evaporating black hole space-time diagram If we start with some initial pure state at some early time Prior to the collapse We're gonna at some later time where time is some appropriate choice of space like hypersurface, but if I Choose this kind of space like hypersurface the evolution from here to here You know will of course give us a pure state up here But there'll be a lot of correlations and entanglement between the State outside the black hole and the state inside the black hole So if I'm just looking at what's outside the black hole That will be have to be described by a mixed State so what I've said so far is I think not controversial or well, I guess it would be contested by people who Want to try to distort this picture enormously to avoid the conclusion that I'm that I'm about to draw but So let me rephrase that as saying that this doesn't the the fact that there are correlations here doesn't buy itself in and of itself bother anyone that I know but what Causes what results in controversy and people talking about paradoxes is what happens if the black hole Evaporates completely because if the black hole evaporates completely in this semi classical picture I mean what you get up here after the black hole has evaporated is the evolution of What's here outside the black hole? this quantum stuff here is entangled With stuff that used to be inside the black hole But now the black hole has completely disappeared and you end up with a mixed state So in this process If you treat it semi-classically you get evolution from a initial pure state to a mixed state So if you believe that's impossible Then that then you can use the word paradox because Here is a clear argument that that's what the final state Should look like so and This is what people mean when they talk about information loss into the black hole I mean in fact Knowing the state up here. You wouldn't be able to Reconstruct the initial pure state there'd be Different pure states that would have different properties inside the black hole that could Result in the same final mix state so that is for sure a striking conclusion and It is definitely worth asking what you know What could go wrong? I mean things definitely can go wrong because this argument was based on a on a semi-classical picture where I was treating the black hole space time Classically, I mean it was evaporating that was taking the quantum Effects into account, but I was still treating the metric classically to get this conclusion. So if So if you want to ask the question what could have gone wrong You know could something have gone wrong to have given us the wrong conclusion that we get a mixed state up here Well, it seems to me that if something is going to go wrong, it would have to go wrong in one of these I mean it's Useful to classify what could have gone wrong in one of these four regions I mean, of course, it could have gone wrong in more than one of these four but but For reasons, I don't understand if and this has been going on for well more than 40 years People are extreme many people are extremely unhappy with the idea that you'd end up with a mixed state here and are trying to Find a way not to end up with a mixed state So one way you could avoid ending up with a mixed state is you don't actually form a black hole and that's the Fuzzball idea so instead so you in this picture you gravitationally collapse to some incredibly tiny radius Outside the Schwarzschild radius and then all of a sudden you tunnel in to some very quantum Configuration, I mean the fuzzball idea is the prime Example of that and now you don't actually make a black hole Although somehow or other this fuzzball emits Hawking radiation like things but you end up With a pure state. I mean this Seems to me one of the most radical proposals. I mean we're talking about a completely Low curvature regime here Why classical physics should just catastrophically break down when it's not great? I mean the curvatures here could be less the curvature in this room And why we should have some catastrophic breakdown. I Definitely don't understand but this this viewpoint has a Number of advocates Well, I've said a few things if you don't form it just at the right time. It'll be too late and well You've got a lot of causality issues as well, but I don't think I need to dwell on this So another place where things could be wrong with the picture that I show is in this Evaporation regime so you go ahead and form the black hole much as in classical general relativity, but then the Black hole of apparition doesn't I mean people still want Hawking radiation to be emitted, but if you're going to start building up these Correlations and entanglements you're going to likely be sunk so one way to avoid building up entanglements and And so on is to convert the the horizon into a singularity So again, this is a really in my view radical Proposal because again, you're in a low curvature regime here the horizon is you know some globally defined Object that has no particular local significance. We wouldn't know it if we were if there was actually a massive shell Elapsing on us at the speed of light and we were actually already inside of black hole horizon Of course, that would be this early stage So maybe you don't need quite the firewall then but anyway That is the firewall idea is another one that has Possibly the largest number of advocates So Let me not dwell on that since I'm running out of time a third possibility is that all of the information escapes out in some sort of oh well Sorry, the third possibility is that the black hole Does what? I'm saying it does but when you get down to around the Planck scale it just stops evaporating and Then so then you have the black hole existing forever And you still have a mixed state out here, but if you include what's inside this remnant Then the state is pure so again. I don't really see what good that does you I mean it's sort of The issue The key issue would be can you interact with this remnant or can you not interact with this remnant? If you cannot interact with the remnant, then I'm not sure what good it did you to have the remnant and say oh All the information it's still there. It's in that remnant. We just can't access it and if you if you can Access You know interact with the remnant then these remnants have arbitrarily high entropy at Maybe Planck mass energy, so why wouldn't you you know? Thermodynamically, they're enormously favored and why don't you spontaneously create those? I mean so remnants don't have many advocates that I'm aware of and then finally maybe you the black hole does evaporate but everything that was in the Classically described in the singularity comes out in a final burst and that restores the information This is actually an idea that I've been looking at Carefully myself. I mean there's a nice Model with moving mirrors of Hoda Schutzhold and unruh In a paper several years ago Where you would get Hawking radiation, but end up with a pure state But in fact one can show I I can show that Although you're effectively getting entanglement with vacuum fluctuations You have to admit real inertial particles in this in this process and again the usual arguments against bursts as Having too much needing too much energy to carry off the information I believe really do apply even with this modified idea Arising from the Hoda Schutzhold and unruh work So why do people not believe in? Information loss I mean given that the semi classical theory predicts it and You know it's really hard. I think if I haven't shown you that it's impossible I think I have shown you that it's not going to be easy to modify the semi classical Picture in a reasonable way to get it. Well, the reasons I've heard over the last 40 years have to do well three things that the first statement is that this Information loss Violates unitarity. Let's see. I think I'm going to run about three minutes over I'll finish this up quickly, but I realize I'm at the legal end of my time But I think you might want to hear some of this So the trouble is that unitarity is used in two different senses It's used to mean conservation of probability and if we had a failure of conservation of probability That's bad, but that's not what's being proposed. I hear we're just talking about Pure states evolving to mixed states and if your final State is described on something. That's not a Cauchy surface You'd expect it to be mixed and that's not a violation of quantum mechanics. That's a prediction of quantum mechanics so if you evolve in flat space time a massless scalar field from a usual T equals zero slice But now to a hyperboloidal slice that runs off to null infinity The evolution is perfectly well-defined, but the state on this hyperboloidal slice Will be a mixed state and that's the kind of thing that's going on in the black hole of apparition And I'm not sure now you're entangled with radiation that went off to infinity Prior to the time of this slice, but I don't see the problem. There have been arguments of that if you had pure states evolving to mixed states you'd have Failure of energy and momentum conservation. I mean a paper of banks peskin and seskin is Widely quoted as showing that and I mean their paper is fine, but their model is a kind of Markovian process where there's no memory of previous Behavior in the evolution. I mean it's the Lindblad equation is what they're Using you can make up other model Evolutions and in fact unruh has given a nice model with a hidden spin system Where you'll get for your quantum system? Evolution from a pure state to a mixed state with no violation of you know with exact energy conservation holding so I don't really see that as an issue and Finally, I mean in the last 20 years the main argument that there has to be information loss That there can't be information loss there has to be something going on that Gives you a pure state at the end is the this ADS CFT correspondence and I think since I'm Gone my extra three minutes that I've already said I'll just say on this that You know there are a lot of implicit assumptions in these ADS CFT arguments like ones that I've Listed here, and if you took various of these assumptions literally and applied them to classical General relativity you would get results that are blatantly wrong So I mean I think a lot of work really needs to be done to make the ADS CFT arguments into a genuine argument that I mean Something in the nature of a proof or whatever that that if the Well first you've got to really formulate the ADS CFT idea, but then after formulating it You know under the assumption that that's true a proof that Black hole of apparition can't take you from pure state to mixed state. I think is really very much called for rather than just saying oh Quantum gravity has to be equivalent to some conformal field theory and conformal field theories are unitary So gravity has to be unitary so a black hole Can't end up in a mixed state After it event I mean the final state after black hole of apparition can't be a mixed state So I'm for the time being until this Some explanation is given as to how information is regained. I'm definitely sticking with information loss but the main conclusion of the whole four lectures is that I think I've shown as I hope I've shown you with Four lectures worth of pretty packed-in material. There's really a amazing connection between gravity quantum theory and thermodynamics and We're certainly not at the end of the story and I expect more Major insights to come in the future. Okay, so thank you