 To save time, let me write a few things so that I can go over it quickly. Should I start? Don is late, but I hope he doesn't think it's at 2 o'clock. Okay, maybe we'll start. Okay, so let me now recapitulate everything that we did in the last three lectures because admittedly, I'm sure it's probably went a bit too fast. It was kind of a telegraphic introduction to some of these important concepts, but I hope I've managed to convey to you at least some important concepts which you can really sit down and do a calculation and when somebody says entropy, you should really understand what it means or somebody says, you know, ensemble, it should make sense to you. That was my goal, to illustrate each of these important concepts with an example. So we said that physical systems usually have some Hilbert space and they're described by some state, normalized to 1, so that is probably, it is an add up to 1. And a more general framework is this density matrix, rho, because very often we don't have exact information about the state of the system. So instead of talking about a state, we talk about a density matrix which satisfies these properties. And it is equivalent to a state, when it's a pure state, you can always write rho as a column vector times a rho vector, it's a matrix. So this notation may be unfamiliar to you, but it's kind of a standard in physics notation that it's a, this you should think of as a column vector and this as a rho vector. So therefore this product is a matrix, whereas this is an inner product, because this is now a column, rho vector times a column vector, which is a number. Whereas this is a matrix, because in general your state could be a sum over such, in some diagonal basis with some probabilities, p i's. And so you should think of these p i's, since the trace is equal to 1, sum over p i's is equal to 1. And you can think of p i's probabilities that the system is in the state i, and you don't really know exactly which state the system is in. And one can define a von Neumann entropy in this manner. And the point is that when we don't know exactly what is the state of the system, it's natural to talk about an ensemble of states, some ensemble of states. And by ensemble we simply mean a Hilbert subspace, some collection of states. In the micro canonical case, by ensemble we simply mean a Hilbert subspace. But so an ensemble, really what we want to do is we want to define a density matrix. So ensemble is just a collection of states. And we want to therefore define the probability that your physical system is in a given state with some probabilities, like this p i. If I specify a density matrix, I specify the ensemble, right? I specify the probabilities with which the elements of that ensemble occur. Is this clear? So the simplest ensemble is an isolated system of fixed energy. So only thing that we know about the system is that the total energy is fixed. You don't know anything about that system. So it's like a wine bottle which is completely hermitically sealed. So it cannot exchange any heat with any external world. So the energy of that wine bottle is fixed. And it can, the molecules of that wine can be in many different configurations. So there are really, because as you know the Avogadra number just to keep in mind is 10 to the 23. So if you take a bucket of wine, you have so many atoms and molecules in that. And so the number of possible states are humongously large. And so those are the kind of dimensions that we are interested in. So the dimension is that dimension of the Hilbert space. And in that case, the entropy so defined is simply the dimension of that Hilbert subspace, okay? And this is the famous Boltzmann relation which of course would be pointless to call it a famous relation unless you had a well-defined way of discussing what is the left-hand side because at this point is just a definition. But the important point is that S has an independently defined notion. And for that you need to go to the canonical ensemble. By that what we mean is, it's like a wine bottle at fixed temperature. So system at fixed temperature. Rather than having an isolated system of fixed energy, you have a system which is interacting with some heat bath. Like it's a wine bottle in a swimming pool. System with fixed temperature. And in that case, it depends on a parameter beta, which is just e to the minus beta h divided by some z of beta. It's normalized to 1. So z of beta simply trace e to the minus beta h. So given such an ensemble, so now what we are saying, we are saying that it's here, what this is saying is that this implies equal a priori probability. Sorry we don't, we started since, but I'm just reviewing. So here we have equal a priori probability because if the wine bottle is completely sealed, I cannot look inside. It is a reasonable assumption that okay, every state that can be there is equally probable. If I took infinite large number of copies of the wine bottle, sometimes the molecules will be going that way, sometimes the molecules will be going that way, but all possible macro states occur with equal probability. It's like the same thing that we do for a tossing coin, that we assume that equal a priori probability that it's head or tail. In canonical ensemble, we are assuming that the probability that a state with energy e will occur with lower and lower probability as the energy becomes higher and higher. And that scale is determined by temperature, so it's kind of exponentially cut off. Okay? In an ensemble, we can calculate the average energy, just the trace of the Hamiltonian with respect to rho. Because rho is giving us some probability and this is just the average energy. Okay, so very good question. So for infinite number of states, so I'm sort of assuming here everything is finite dimensional. We started with a two-state system, maybe an end-state system. There is a whole possibilities when the number of states becomes large, infinite when you take the large volume limit. There are possibilities of phase transition, which have to do, yeah, so it's a very good question, but something that I cannot answer in five minutes. Yeah, it's an excellent question and there is a... Yeah, so I'm implicitly assuming here that the Hilbert space is finite dimensional and phase transitions don't occur in that case. Okay, but it's easy to see that... See, this is what... This is basically typical energy of a state r times the probability to be in that... So I am just summing over all possible states r. Energy of that state is E r. E to the minus beta r is the probability. But I won't normalize probabilities, so I am dividing by E to the minus beta here. And what is this? This is nothing but 1 upon Z minus 1 upon Z del log del Z by del beta. E bar was minus del log Z del beta. And then it was natural to define log Z beta plus E bar beta is equal to S, because that beta is equal to del S by del E, since sometimes it depends on more variables. E is equal to T dS, delta is equal to 1 upon T, which is the heat. I mean, the amount of heat that you're adding. Q is the heat. And in fact, from here you can see where that fundamental definition came from. Actually, fundamental entropy is just a generalization of what Gibbs had defined for the canonical ensemble. Because you can see that this definition as a kind of a way to... It's a Legendre transform. If you think of log Z as a function of beta and its derivative gives you E bar, because we think of S as a function of energy, or average energy, and its derivative gives you beta as a function of E bar. This is the standard thing that one does in Legendre transforms, that if you want to invert some implicit relation, implicitly, if you want to invert a relation then this is the way to do it. But it's easy to check now because just take this row of beta. Row of beta is just minus beta H minus log Z, if I just take the logarithm. And if I multiply both sides by row and take a trace, I will get minus S is equal to... Let me put a minus sign everywhere, multiplied by row and take a trace. Then the left hand side is the entropy. The right hand side is beta times the average energy because trace of H, by definition, average energy is trace of H row. And trace of row since it's normalized to 1 is 1. So plus log Z. So that's the same relation as here. In fact, you can interpret this relation as a definition of the von Neumann entropy. That's how you motivate the von Neumann entropy. It gives us a very fundamental relation to the first law of thermodynamics. Energy is conserved. If you add heat to this room, its total average energy goes up and the amount of energy, the change in energy is equal to the amount of heat that you add. Otherwise, energy would be lost. And this, but what Karno discovered was through his Karno cycle and so on, how to define various steam engines and so on, he realized that heat actually is related to entropy. I mean, he defined the notion of entropy through this relation. And why did he define such a weird thing? Why did he do this? He was trying to understand the most efficient way of designing a steam engine. And he was able to formulate a second law. Now, if you define the entropy in this way, then second law of thermodynamics is that delta H is always greater than or equal to zero. Entropy always increases. Or it can be zero. The increase can be zero if it's really slow irreversible process. Entropy never decreases. That was his motivation for defining this entropy. And the brilliant insight of Boltzmann was to relate that thermodynamic concept to something having to do with the dimensions of a Hilbert subspace through this somewhat complicated chain of reasoning. And how does this help you? So for example, what is second law of thermodynamics? We know that you can break an egg easily, but you cannot put back an egg so easily. Whereas the fundamental laws of physics are time irreversible. So why does that happen? Or if I have all the molecules half of this room and I remove the partition and I wait long enough, you'll find that they're all spread over. But you will never see a situation that you start with this and suddenly all the molecules will go to the half of the room. So what is happening is that disorder increases. And a simple way to understand that the entropy here as initial S1 is much less than S2y because here the Hilbert subspace that is available to you of fixed energy is much bigger. The molecules can be all over the place. And so the dimension of this Hilbert space is much, much bigger, exponentially large compared to the dimension of this subspace where all the molecules have to be confined. Therefore since the dimension in this natural tendency is just a probabilistic, probabilistically it's like if you have a jigsaw puzzle and you just shake it and open it, it's much more likely, just probabilistically that you will find it broken and not that it is assembled. There's nothing fundamentally forbidding that possibility that suddenly the jigsaw puzzle reorganizes itself. But it's highly unlikely if you shook millions of boxes maybe once in a while you'll find that the jigsaw puzzle reassembles itself. But most of the time you'll find that jigsaw puzzle breaks itself. So using this idea of Boltzmann you can explain the second law as a consequence of just the fact that in a spontaneous process generically the dimension increases. The dimension of the Hilbert space available to you is increasing and so the entropy is increasing. Actually in this case, in fact we take the energy to be a function of entropy because I have, there is some Legendre transform involves actually I should probably write it E by ds times ds. Yeah, so you should think of just like temperature and volume are some properties of the system. You should think of energy and entropy as properties of the system in a given state. Now we are trying to find relations between the two. So E can be a function of s and v. Okay, now if you think about it like that this is nothing but the temperature times ds and this is nothing but the minus the pressure times dv. So if you want to make it kind of axiomatic then you can just define the pressure to be this and define the temperature to be this. These relations are valid in what is called a reversible process where you are slowly varying things and the second law of thermodynamics is really telling you when you are far away from equilibrium. So let me actually take an example suppose you want to understand why heat flows from hot to cold and not the other way around from cold to hot. Well it follows from the second law of thermodynamics because if I have a heat bar here one at temperature T1 and the other at temperature T2 and suppose it therefore this one which is bigger will give out some heat to this bar and heat it up so that eventually they both reach the same temperature. Then what is the change in entropy? Change in entropy for the first one is negative as you are right for the first one is negative because it has lost heat divided by T1 but for the second one the change in entropy is positive but it is receiving that same amount of heat so heat is conserved energy is conserved so whatever amount of heat this bar has given out must be the same as the heat that this bar has absorbed the energy has to be conserved so the energy given out by this one bar is the same as the energy absorbed by this bar so delta Q is the same but it is absorbing at a lower temperature so the total change in entropy of the total system is what it is delta Q which I am taking delta Q to be a positive quantity so delta Q times 1 upon T2 minus 1 upon T1 according to the second law this has to be positive or equal to 0 this means that T2 has to be less than so what we have shown is that here is an example where energy is conserved manifestly because the swimming pool is giving heat to the wine bottle and so some energy is being given out so that energy of this bar decreases the energy of this bar increases the energy of this bar increases total energy is conserved the entropy of this decreases by an amount which is less than this as long as T1 is bigger than T2 and therefore we can explain all these phenomena like the heat flows from hot to cold or the gas goes from one end to the other by a single law namely the second law that the total change in entropy has to be positive but notice that this is an irreversible process reversible process would be if the temperatures are really very very close to each other and it's a very slowly happening process then it's called a reversible process in that case you can really think of various thermodynamic quantities as just functions and these derivatives are simply these ordinary differentials on some space so you should think of as far as this law is concerned here think of E as just a function of two variables and then you are trying to express S as a function of T and V or you can express T as a function of E and V sort of so on you can play by taking Legendre transforms you can define various quantities in terms of each other so what we want to take away from this is that there is a very fundamental notion called entropy there is a notion called temperature let me say this is a fundamental notion called entropy S as a function of E dS by dE allows me to define one over the temperature there is a notion of temperature and S is equal to just the number of microstates log of sorry the first law follows and the second law follows sorry first law doesn't follow first law is a statement about the energy conservation but you can identify the entropy that occurs in the first law with this quantity so what does it have to do with black holes so I was trying to review what we did in the last three lectures very quickly just the relevant pieces that we needed today so this part we just did yesterday but perhaps now you fully understand what is going on so admittedly it's a bit telegraphic so for example I have not explained to you how to go from micro canonical to canonical and so on so these I will be happy to discuss in a tutorial or privately if there is whoever is interested in introduce a notion of a quantum field first we introduce a notion of a classical field and the simplest example was some map phi of x mu scalar map from some manifold real numbers and you could do a Fourier transform of this and there was some normalization so I could write it as e dot k dot x a k of t let me see minus i omega t this is just a Fourier expansion and it satisfies some wave equation more generally I will apply some on some manifold mj phi equal to 0 which you can think of as a d star star d if you like differential forms more if you are in flat space it is just a usual wave equation it is like generalization of the Laplace equation it is like del x I mean gradient square and if you going from classical field to quantum field simply what it does regard the Fourier coefficients as oscillator as this creation annihilation operators the oscillator algebra so basically as oscillators operators corresponding to oscillators with the commutation relation a k dagger notice that because of this equation the sum is only over the special k because omega is determinant in terms of k this implies if you just omega k square is equal to k square k a k dagger is equal to 1 or we can write it more generally a k a k dagger prime is equal to chronicle delta in this way of looking at it a quantum field is really no more complicated than just a collection of oscillators in the oscillator frequencies the oscillators are labeled by the wave vector k it is a special vector it is a vector in rd ok I am now here I am doing r1,d but more generally it can be some general manifold m1,d so k is a special vector in rd and omega is determinant by this condition now there are various generalizations possible of this this is what is called a massless vector field massless scalar field mass simply means you put still a linear equation but you write this as delta phi is equal to phi and in this case your relation changes omega k square becomes k square minus m square sorry k square plus m square and this tells you that I can identify omega k with the energy because if I square sorry omega k omega k is plus minus omega k square is this if I identify omega with the energy it really saying e square minus momentum square is equal to m square and this is the basically the version of the equal to m square relation see when k is 0 it says that e is equal to m we have put the speed of light to be 1 there are factors of c there is a measuring time and this is really the relation e is equal to m square but if the particle has momentum then the relation is this is the relation between the energy momentum and mass by quantize I mean by looking at a field we ended up with the notion of a particle so that is the only thing I would like you to take away from this so once again I will be happy to explain quantum field in more detail this point of view I can explain in more detail I hope you get the general idea so some of these things I have been I have to say I have not been able to do it more but it really requires time so I cannot do it in 15 minutes also consider nonlinear wave equations so nonlinear field equations for example I could consider this plus m square phi and this is a linear equation but I can also add to it some nonlinear term and notice that the same equation has kind of we are interpreting in two different ways at one level we are thinking of it just as a classical field equation where there is no mystery where classical field equation we completely understand phi is just a map and this is some nonlinear partial differential equation satisfied by that function there is no mystery about it but then by interpreting its Fourier coefficients as oscillators we are thinking of it as an operator-valued function or operator-valued distribution in more general situations and then defining this equation really becomes very very tricky and as we saw you can run into infinities very quickly because you are dealing with infinite number of oscillators collection of infinite number of oscillators once again this I did very telegraphically but renormalization theory way of defining this regularize to consider only finite number of oscillators effectively because there are some cut-off epsilon which is going to take to 0 in the end so as long as epsilon is finite you are dealing with just finite number of oscillators so there is no problem but you have to take the epsilon going to 0 limit in a very particular way and the particular way has to do which again I have to admit I didn't do basically take all these coefficients so there was a question also you could have had also overall normalization here which is z epsilon which you can remove and put it here as long as z epsilon is non-zero you can have all kinds of infinite number of terms 5 to the 5 lambda 5 epsilon the claim is of the renormalization theory is that by adjusting various coefficients lambda I epsilon you can take epsilon going to 0 limit at the same time lambda 1 epsilon for some finite number of principle you could have had an infinite number of terms here the main result of renormalization theory is that okay we are adjusting these various lambdas for such terms which are local as you can see local terms so I am not allowing terms like phi x some green function x y phi y basically all these divergences can be removed okay so this is a statement which requires a two semester course and I am not going to try to prove it but again I will be very happy to discuss this but there is a in perturbation theory this is a different way to state renormalization theory that by adding terms which are local you can remove most of the all the divergences and you need to add only finite number of local terms see if you needed to add infinite number of local terms then you would not be able to do any calculation but just by adjusting that there is a algorithm for doing it which depends on what symmetries you have and so on and so on this infinite sum of oscillators can be made sense the collection of infinite oscillators can be made sensible and we saw just one simple example of that in this ground state energy calculation but okay let me come to that later on let me just move on but at the level of classical fields there are many other things we could have considered a connection one form or it could be even a Lie algebra valued for some group G Mu dx Mu so this is Lie algebra valued for if you are thinking of some principle wonder of group G and then the let us consider G is equal to U1 for simplicity then the equation of motion in this case is dd star is the Laplacian on one forms so it is exact generalization of this similarly you can consider a metric tensor so it is now a tensor valued map instead of considering the scalar valued map that is a metric tensor field and once again if you want to think of it the metric tensor field as some kind of a terms of oscillators it also satisfies a Laplacian equation which is called the Lichnorovitz Laplacian the interesting thing is that in this case it admits a very beautiful and essentially unique to second orders in derivatives non-linear completion and those are the Einstein equations you can write them as R mu nu minus half G mu nu R equal to 0 or if you have some matter present okay we can take it to be 0 for now okay so I hope the logic is clear so far that we have classical fields which satisfy classical field equations like a simple Laplace equation those are linear equations similarly if you have one form field it satisfies another simple one form equation which is again the generalization of the classical Laplace but such equations admit a non-linear generalization so in general you should consider non-linear equations the transition to go from a classical field to quantum field in the linearized case is very simple you just do a Fourier expansion you regard the Fourier coefficients as oscillators and you know everything about oscillators you can calculate this partition function you can calculate Hilbert space dimensions of Hilbert space and so on but the non-trivial thing is the R yes so for example for the one form the equation looks like something like this again it's minus del square by del t square plus grad square A mu plus there is an additional term which is del by del x mu of divergence of E is equal to 0 del lambda A l another way to say that is this is the just the usual Laplacian there is an additional term possible now which was not there for the scalar field there is a whole story about gauge fixing and so on you have to but okay but if you choose this gauge what is called the Lorentz gauge then it looks very much like wave equation I am now considering U1 gauge field U1 gauge field the covariant derivative because the Lie algebra of U1 is a yeah in that case the covariant derivative acting on A is the same as D sorry have I yes so yeah let me remind you so given a metric on the manifold G mu nu Riemann told you that okay there are two or two second derivatives of metric to figure out the geometry the first derivative allows you to define the Christopher symbols some combination of the first derivatives of the metric and that allows you to define a Riemannian connection and the second derivatives a combination of those second derivatives allows you to define a Riemann tensor okay and all the information about the curvature then can be obtained in terms of the Riemann curvature and various derivatives obviously but you can recast everything in terms of Riemann tensor because everything can be of course understood in terms of arbitrary number of derivatives of the metric but the Riemann's formulation tells you that given a Riemann tensor and its covariant derivatives you can formulate everything in terms of the Riemann tensor so you should think of Einstein's equation as simply a generalization of this equation so G00 actually is a Newtonian potential which as you know satisfies Laplace equation if you now add time to it you will get a wave equation for G mu nu and if you add non-linearities to it then you will get the there are only a unique way to add non-linearities maintaining diffeomorphism in variance and that basically is uniquely determines this equation that's why Einstein was able to guess it it's kind of a unique way generalize Laplace equation La Poisson equation maintaining diffeomorphism in variance if you identify the metric with the so this is going to bring us into black hole story now of course now you should also quantize the metric so again metric also has just like light so here what happened was light has also various oscillations so this is why light is like a wave because this equation tells you that it electromagnetic wave equation this is the electromagnetic wave equation and the electromagnetic vector potential satisfies a wave equation and the light of different frequencies is simply a solution of this equation going in different directions similarly a gravity wave is simply an equation of this Lechnorovitz type of Laplacian again a Laplacian equation for small perturbations but the full non-linear generalization unlike here for the scalar field the non-linear generalization was kind of bit arbitrary could have added 5 to the 4 whatever here diffeomorphism in variance more fixes it to second order is in derivatives that brings us to black holes should I remind you is this all Riemann tensor you remember for example Riemann tensor is antisymmetric two indices symmetric in this two indices you can think of it as a curvature two form for the spin connection and so on there are various ways to think about the Riemann tensor think of it as a curvature two form and the Ricci is basically obtained by sigma mu sigma nu it's a contraction of the Riemann tensor R mu nu here the Ricci tensor is sigma lambda R sigma mu lambda nu the repeated indices are summed over it's like a trace the Ricci tensor is a trace of some kind of trace of the Riemann tensor and Einstein's equation expresses and Ricci scalar is a further trace so R is g mu nu R if you look at this equation for g mu nu is equal to minkowski metric eta mu nu plus small h mu nu small fluctuations and ignore the nonlinear terms you'll get a wave equation for h that's the link between what we have been talking about and Einstein's equation Einstein's equation is just a nonlinear version of a wave equation it's just a second order partial differential equation which in some linearized approximation looks like a wave equation but it's a nonlinear partial differential equation and whose nonlinearities are fixed by the tensor structure of the Riemann tensor so that everything is manifestly diffeomorphism invariant you could apply the same story I mean there are differences but okay in spirit it is similar so you have metric and you have connection one forms in the case of gravity but the connection one forms are expressed in terms of the metric by the Riemann compatibility condition and then the differential equation is only in terms of the metric so what are black holes now I am going to speed up a bit so it's a manifold m1,3 with a metric g mu nu the simplest example of the metric you thought was g mu nu was eta mu nu which is a diagonal analog of a Euclidean metric minkowski metric this is clearly a solution of Einstein's equation because for a eta mu nu Riemann tensor vanishes so Ricci tensor vanishes are vanishes the equation is trivially satisfied so minkowski space as it should be is certainly a solution of Einstein's equations otherwise we would be in trouble otherwise we would be living in a space time or approximately a space time which is not a solution of Einstein's equation so Einstein's claim is that our space time manifold admits a metric which must satisfy this equation so what are the possible solutions and one of the very simple very elegant and beautiful solution which was found by Schwarzschild it's this form no, think of single black hole existing independent of us so it's a manifold by itself so I am thinking of an isolated we have been thinking about isolated systems so an isolated black hole is just a the entire manifold is this black hole we actually think about black holes is that you have to patch together so if you have let me first describe a single black hole isolated black hole we could be living in the presence of a single black hole suppose the sun was a black hole then no other stars are present then we would be living in the black hole space time and the remarkable statement is that which is called the no hair theorem that if you want a spherically symmetric so this is spherically symmetric because there is omega 2 here this is a spherically symmetric black hole with no spin sorry r is like the radial coordinate omega 2 is the solid angle on s2 the d omega 2 is like the solid angle basically this is d theta squared d theta d phi squared g is Newton's constant m is the mass of the black hole let me come to the no hair theorem a bit later sorry I will describe it later I will come to it later let me first explain to you what is a black hole and then I will come to the no hair theorem so far this is a metric this is a solution solution of Einstein's equation notice that as r goes to infinity it goes to flat space time it becomes a minkowski metric r goes to infinity this is just the minkowski line element but something terrible seems to be happening when r becomes equal to gm the metric is some components of the metric are going to 0 some components are diverging so we need to analyse this geometry little bit carefully near the r equal to 2gm which is what I will do and this is just a simple change of coordinate it will turn out so incidentally this singularity led Einstein to declare that this solution doesn't make sense he actually wrote a wrong paper a published paper saying that why such singularity cannot occur in nature his argument was not correct and basically you couldn't accept this fact that there is a singularity it turns out that singularity is actually completely innocuous it's a coordinate singularity it's just a bad choice of coordinates but it hides an important notion which is called the event horizon which is what we will discuss now so let's just analyse it near horizon geometry r near this is called the horizon but okay it's equal to 2gm so this is just a change of coordinates let me call this is equal to psi we can see that ds square is equal to minus psi 2gm dt square plus d psi square over 2gm plus this r horizon square times d omega 2 square this is just a change of coordinate from here to there I write r minus 2gm I basically write r is equal to 2gm plus psi and then I want to expand I can take 2gm out and I want to expand imagine that psi divided by 2gm is small here I expand and so on so therefore I drop terms of order plus terms of order 1 over 2gm now I am further going to make another coordinate change rho square is equal to 8gm times psi what it means is that it's of order psi over so plus terms which are order psi over 2gm square yeah that makes sense so that's what I wrote here I write r is equal to 2gm plus psi I take out 2gm and I regard psi upon 2gm small compared to 1 and I just expand so if I do this the metric takes the following form so this is actually the reason is that rho is like a geodesic coordinate if I move one unit of rho for fixed T if I move one unit of rho I am actually just increasing the distance ds is proportional to d rho square if I keep all of the coordinates constant right so it's like a geodesic distance and the surprising thing is that you can now ignore this factor this is some two sphere so there is a two sphere so I have some fixed radius and I am looking at the the two dimensional plane r T plane r is equal to 2gm I always have to cross it with a two sphere s2 whose radius is 2gm m13 is a two dimensional plane m13 near r is equal to 2gm as r approaches 2gm is a two dimensional plane times the two sphere I have just done a very simple coordinate change the surprisingly this two dimensional plane whose metric is this let me write it as rho square a square T square d rho square this is m11 this whole thing I am calling m11 so the black hole space time is really looks like m11 cross s2 near the horizon this one so d rho square is d rho square I just integrated this equation did I miss I must have missed you are absolutely right there is a xi here I am absolutely right sorry the other way around thank you for catching that mistake this was an important mistake notice that therefore rho is basically like a square root of xi then it will come out sorry about that essentially that is happening because here I have 1 minus 2gm and here I have 1 minus 2gm inwards but now comes a kind of a very punch line in a way that m11 is essentially r11 m11 is actually just flat it looks weird so let me prove it to you I think I have time sorry a is equal to just to find notation and also is 4gm rho square a square will be rho square divided by 16gm whole square so let us look at r11 is a minkowski space right I can define u and v coordinates and then it is easy to check ds square is equal to minus du dv right it is the minkowski version of going to complex u you can think of as z and this is z bar except that because you have minkowski you do not need an i right so you know that euclidean metric is dz dz bar and now let me define new coordinates u is equal to 1 upon a e to the a small u make a minus sign mistake here confuse everybody I think it is correct what I have written here and v is equal to minus 1 upon e to the minus e v so ds square is equal to minus e to the a u minus v du dv this is capital U and this is small u no no I am defining a new coordinate u small u by this coordinate transformation and I am defining another coordinate small v sorry sorry v it is du dv so basically I am going from capital U capital V to small u small v coordinates so far so good now let me write ok it might look to you like a long now define small t small x it is easy to check that ds square is equal to therefore e to the 2 ax minus dt square plus dx square now you define rho is equal to 1 upon a e to the ax is that ds square is equal to minus rho square a square dt square plus du square actually this might look to be a very trivial change of coordinates and indeed it is but it really took physicists I don't know 40 years to really figure this out and if Einstein had understood this he would have not made written wrong paper because this is if it is a coordinate change then no geometry quantity is diverging then you man tensor in some orthonormal frame is perfectly finite there is nothing there are no curvature there is no funny business happening there but this also leads us to notion of an event horizon and the reason is that so let's look at the minkowski space see minkowski space normally we draw it as a two plane so time is going this way x is going that way so constant u capital u that is my capital u was t plus x increases u is going this way and v is going that but notice the u coordinates the small u coordinates a quarter of the space only the first quadrant in other words as u goes from minus infinity to plus infinity capital u goes from 0 to infinity so capital u goes from 0 to infinity this is your u is equal to minus infinity and u is equal to plus infinity somewhere here v is equal to minus infinity is here and v is equal to plus infinity is here that corresponds to only v is equal to infinity and v is equal to 0 so let me check if it is correct v is equal to minus infinity capital v is also minus infinity sorry it goes from minus infinity to 0 so you cover only this part of the space time so that was the weird thing about so as you can see I mean mathematicians like Lorenz sorry they like Euclidean space manifolds and so on because Lorenzian things it makes hyperbolic everything is complicated wave equation is more complicated than the plastic question and so on but in fact a lot of the interesting physics also depends on that complication in particular this kind of a phenomena of part of the space is missing okay it can happen also in Euclidean space but here it will happen in a more dramatic way but in this case so suppose by mistake you are using U and V coordinates to describe Minkowski space-time and you discovered the Minkowski metric in these coordinates right then you would be missing part of the space-time and then suddenly you will find that you know if you throw a bar is actually leave you know you're not able to describe it you will exhaust the entire range of your coordinates your coordinate patch doesn't cover the whole space but there is a simple solution to it we just change coordinates to capital U capital V and extend the capital U capital V coordinates to their full range and then you recover the entire space-time and what we have discovered is that the black hole looks very similar to this space but in this funny coordinates and in this funny coordinates so constant time slice capital T slices are going like that so this is the time of this observer but small t is actually going from here to there like that small t is going at minus infinity is going like that small time is going from minus infinity to plus infinity but it's not able to cover the entire space-time here here of course we have in the Minkowski space we know that T is going from minus infinity to plus infinity Minkowski space-time is defined by this range right just like a Euclidean space is defined but yeah all of R11 so if I start with all of R11 so if I start with an R11 what I'm trying to show is that let me start with the entire of R11 in this case I'm going to run the argument in reverse okay let me start with an entire R11 so there is no argument for you that the whole space-time exists and let me make this funny change of coordinates the thing is that you could take that the point of you that okay I declare that to be my manifold but then what will happen is that certain geodesics I mean certain geodesics will you want to make your manifold to be geodesically complete so certain geodesic will end suddenly so you throw a ball and even though you have not really used in infinite amount of time your geodesic will seem to end so the technical term is you want to manifold to be geodesically complete no so let's look at let's look at Minkowski space right you could always take half of Minkowski space one quarter then what will happen is that some observer who's sitting here at rest their lifeline will suddenly end right so that doesn't seem to be a reasonable physical situation where our life just comes to everybody's life here will suddenly come to an end okay you could you may want to live in such a world but a more reasonable space-time manifold would be one which in which the geodesics continue but light doesn't stop there so no the light continues so that's what I'm going to tell you it will not come out but it will but the geodesic will continue so so this surface that's exactly the point now which will come to now so this suggests that if I want to extend my so the general principle is the following that okay this is also true in Euclidean manifolds right if I Euclidean manifold if you chose if you got just half of space then you'll get your geodesics will end right and you want to know how far how much can I extend the geodesic then that will be the completion of that manifold then I may choose to chop it up and just decide that okay I want to only talk about this part of the manifold but which is not what one normally does right so you the same philosophy want to apply the minkowski signature that you want to extend your geodesics as far as possible okay without running into problem and actually this surface is there is but there's nothing wrong with it because no curvature is diverging nothing is happening all curvature quantities are finite so there is no reason to regard this point as any special then here and why should something just which falls in I mean why should I just declare that the space time ends there so let me try to continue the space time as much as I can continue and that is known as the Kruskal extension of the black one and it's sort of clear so it should be heuristically clear to you but you can actually make it it's there in my notes I will okay eventually when I write them I will distribute them actually eventually when they're done completely but just as here the idea of the extension is clear that instead of using small u and small v coordinates I try to use capital U capital v coordinates and extend the range of you and we as far as I can that's the idea of an extension right and then I will get a manifold where the geodesics can continue as far as they can and essentially you discover space time like minkowski with one problem is r equal to 0 you remember was a real singularity you remember I totally agree here yeah this I agree so this I totally agree this I agree with you okay so I have been so you agree with me that near here I can follow the procedure that I have described and that already tells me that there is life beyond r is equal to 2gm inside there is some other region which is not accessed by my rt coordinates okay you can actually there is a another coordinate which is an r star coordinate which which is related to the r coordinate by and your small u and v coordinates are really like t minus r star or how was it t plus r star and v is equal to t minus r star how I can do the analysis do the analysis on a complete manifold without doing this near horizon approximation but the near horizon approximation makes it clear that we are really dealing with a something like very similar to minkowski space right there's nothing very special about this r is equal to 2gm surface and then you can extend that analysis more generally at other values of r in terms of these coordinates these are coordinates and you can write the whole metric yes yes the product structure is valid only near r is equal to 2gm and the metric therefore takes the form minus 2gm upon r e to the minus r divided by 2gm du dv r square du omega square okay says that when r goes to 2gm is basically du dv okay there is some overall scaling methods so the point is that I did that analysis because it's easier to illustrate but otherwise you can do this I mean after all Pruskal got his name attached to it so it was it's a work of several decades okay of at least two or three decades for people to discover that the black hole space-time had a full extension this r is a small r no so if you just take that solution that I told you and if you rewrite it it's actually the same notice that r equal to 0 is the real singularity because the Riemann curvature actually diverges there in orthonormal coordinates the causal diagram looks very different from what you thought see what did you think if you look at the solution from far away so let's try to look at the solution from far away okay this actually this solution in fact describes the gravitational field around the Sun very well okay so if you take the Sun this is the Sun and if you are at a distance r from the Sun it looks exactly like this now for the Sun for the solar mass 2gm is of the order of 3 kilometers okay whereas the Sun has a huge radius you know million kilometers so 3gm is somewhere inside here okay so for the Sun you have to change the solution once you go inside but the solution is valid outside of the Sun but imagine that the Sun really collapsed into a black hole then what will happen is that Sun will get keep shrinking and it will really go into inside this this is your 2gm surface so when you're standing far away from the Sun constant r is like a time like direction right you're standing at fixed radius and time is flowing upwards but look what is happening as you come closer this is r is equal to 2gm this is r is equal to infinity so when r is equal to infinity the world line of that observer if I if there is a person standing here he just sitting at radius fixed r and the time is flowing up so this is time coordinate right but as you come closer and closer time actually flows like that and r is equal to 2gm is no longer surface is no longer a time like line but it becomes like a null is like a light like that's the main conclusion and that's called the event horizon it's really null line cross s2 okay and that is called an event horizon so is this clear and black hole now I can now give you a formal definition of a black hole so you know of course know the definition of a black hole from popular thing that the gravitational field of the black hole is so strong that even light cannot escape it right because if you throw a ball up it falls down but if you throw it with escape velocity like a rocket it can escape the gravitational field of the earth but if the earth becomes more and more dense then eventually the escape velocity will be the speed of light and then nothing will be able to escape that's the heuristic picture of a black hole but now we can be more specific a black hole is a space time with an event horizon and event horizon is a stationary null surface I have explained to you what is a null surface it's clearly a null surface because it's like a speed of it's going at the speed of light it's an s2 which is going at the speed of light right so so that's why it's a black hole you have an s2 which is like a star so if you have a star there is an s2 and it's clearly stationary right the Sun is there and it's not changing so the boundary of the Sun is a stationary surface for sure but but it's stationary and time-like but null I also mean the same way as another way to say this is light like right if I go closer and closer to the Sun my time is still flowing and I'm just standing away from the Sun and the time is flowing I don't need to go at the speed of light but if I want to stay at the surface r is equal to 2gm I have to go at the speed of light otherwise I cannot stay at r is equal to 2gm so another way to look on this diagram for minkowski space where is everything is clear v is equal to 0 this line is a null line right look at the minkowski space right minkowski space this is a time-like line so if some observer sitting here he will keep going straight up some observer sitting here keep going straight up whereas this is a this is a trajectory of a light ray this is the light ray going in this direction if I want to stay fixed on this if I want to fix my coordinate to be v is equal to 0 I will have to move at the speed of light otherwise I cannot keep v constant but we discovered by this coordinate change that r is equal to 2gm is exactly like capital V is equal to 0 therefore if I want to stay at the black hole horizon stationary I have to move at the speed of light that's the meaning of a stationary null surface okay there are a few subtleties but I think this is pretty good I mean this is I think an accurate definition of a black hole a black hole is a space time with a stationary null surface which is called the event horizon and now it's clear to you why you cannot escape see in this case however see look at this here this surface is null but it is not stationary because it is changing with time light is light rays moving is changing with time so the funny thing about this event horizon is it is both stationary and null at the same time it looks like a surface of the star which is out there which is not moving but if I want to stay at that point I have to move at the speed of light and I cannot just go I cannot sit at rest there yeah so that's the main difference you see at the surface of the star I can sit at rest it's a stationary surface in Minkowski space here this is a null surface but I'm changing with time right my I'm changing with time the black hole horizon is the funny geometry is is just a question of how the metric is behaving the metric is so funny that R is equal to 2gm is both a stationary and a null surface and such space time which admits the stationary null surface is called a black hole and now you can make it more precise now if something crosses R is equal to 2gm right the geodesic that don't want you to take wanted to forbid that person cannot come out without traveling fish faster than the speed of light so it's just it's a stationary null surface so once you cross that you cannot come back because you'll have to move at the speed faster so look what happens basically R is equal to at infinity your R is time like as before as you normally think as we're used to thinking if you're far away from the star R is a time like surface I mean I'm sitting at R equal to some fixed R and my time is flowing so I just plot the time and that's the trajectory of the observer there as you come close to the horizon it becomes light like so here it is time like here it becomes light like and here it becomes space like you can go you can go inside to go inside you don't need to go faster than the speed of light right even a trajectory which is at rest it will go inside but you cannot come out that's the point not this is not a geodesic this is actually not a geodesic at all geodesics are just straight lines let's say roughly speaking the geodesics a straight line they will keep so any observer if you throw a ball if you throw a ball towards the star the geodesic will be that it will go on to the star similarly it will go on to the black hole but it go on to a black hole meaning what it will cross this surface R is equal to 2gm and then it will happily continue you thought that there was a surface there there is no surface there it will happily continue and then it will meet a singularity in its future because now R is equal to 0 has become a space like slice it is a it's in your future so this is sometimes stated in popular literature as in this paradoxical way that space and time change their role what it really means is the coordinates that you thought were describing space and coordinates that you just thought were describing time far away from the black hole are actually coordinates of a null surface near the black hole and are reverse their roles inside the black hole mass of the black hole is the parameter m and that actually will okay I have okay no for this black hole eternal this black hole it doesn't change for a classical black hole it doesn't change it's a solution for a fixed m parameter it's a solution some answers equation and nothing changes it it's called an eternal black hole because it's no in the case of charge black holes you get ideas yeah no it's not in so there are situations now okay is this clear didn't get very far but okay I think I was being too ambitious okay so now the final thing I want to explain to you is the entropy of the black hole and then we are at least done with the title of my lectures so why does such a space time why do we associate an entropy with such a space time okay so but is this clear that the space time of a black hole looks like this r is equal to 2gm r is equal to 0 is like a is it space like surface it's like in your future in fact there is a two copies of this it turns out and r is equal to infinity somewhere here the full fiscal extension looks like this and this part is sometimes called a white hole and this is called a black hole so let's not go into that the main point is that you have this weird surface which is both stationary and none and that is responsible for the entropy and temperature of the black hole you see suddenly this is a very beautiful connection between geometries space time geometry and quantum mechanics and thermodynamics somehow this was why Hawking's discovery was such a shock to many people and it can be motivated in the following in my number of ways so I told you that entropy never decreases so Beckenstein said okay if I take a hot water of bucket and throw inside of black hole I can reduce the entropy because nothing comes out of the black hole you just told me there is an event horizon so the entropy that is I can just get rid of all the entropy in the universe outside of the black hole by just pumping it inside the black hole and the therefore the entropy outside of the black hole will decrease how can that be he said the black hole must have entropy proportionality area and he argued and there are theorems in this classical general relativity that show that area of a black hole is always increases in any such process you know if I throw a truck inside of a or if I coalesce to black holes the initial areas a1 plus a2 is always smaller than the area of the final dark hole yeah so area yeah if you remember that area it just 4 pi times the r is equal to 2gm which is 2gm square so for example if you take two black holes and coalesce them they will form a bigger black hole energy is conserved you can check that so this is actually a non trivial theorem which was proven by several general relativists over the years he said that if I associate entropy with the black hole then I can save the second law of thermodynamics because then the entropy of the black hole plus the entropy of the hot bucket of water together they will increase see when I say that entropy always increases as we saw in the case of these two metal bars entropy of this can decrease an entropy of this can increase there's no you can decrease the entropy of part of the subsystem that's not a problem but what you cannot change is the total entropy what you cannot decrease is the total entropy so he said that okay but this leads to second paradox the first paradox is that second law of thermodynamics why is violated but this leads to second paradox because if it is entropy then it must have temperature because as we saw del s by del e energy is equal to m energy is the same as the mass of the black hole and del s by del m was 1 upon t has been proved in our thermodynamic course therefore if I change the mass of the black hole slowly then the entropy given by this formula this depends on the mass by this relation I can calculate what it is and you get that it must therefore have temperature which is 8 by gm so if you want to avoid paradoxes of the geometry of space-time with the laws of thermodynamics you're forced to associate an entropy with a black hole and the temperature with a black hole there is an h bar did I put h bar I have been putting h bar equal to 1 c h bar will appear here which is here let me sorry maybe I made a mistake let me say yeah my h bar is 1 so I will just put 1 now why do we get this 4 here all that you required is just it should be proportional to area that requires a calculation this 8 pi and that was Hawking's calculation okay but let me recapitulate so we saw that space-time has to be is a manifold remain in manifold with a metric the metric must satisfy Einstein's equations one of the solutions of Einstein's equations is the Schwarzschild solution if you now try to make the Schwarzschild solution as geodesic is complete as possible you learn that that solution so in fact this whole space-time is a solution of Einstein's equation if I choose to work directly in the UV coordinates they will satisfy Einstein's equations everywhere and here there is a singularity so actually here there is a geodesic in completeness which is really genuine and nobody knows what to do with it so this is the famous problem of singularities of classical general relativity but here there is no singularity here everything when we could be right now on a horizon of a black hole and we wouldn't know not a steady but we could be just falling through a horizon of a black hole and we wouldn't know any difference because curvature is weak everything is small if you had a huge black hole there would be no difference locally between the horizon of a black hole and flat space-time as we saw it's almost Minkowski space-time with a sphere which is very large so if you so that was one point and now if you have such a surface which is a one-way surface that things can fall in and they cannot come out then the second law of thermodynamics would be violated unless you associate entropy with it and since second law of thermodynamics is some very sacrosanct principle of physics we let's associate an entropy with it proportional to the area which goes very well with the area theorem of black holes so together everything will increase and it looks all very nice but then your force to conclude that it has temperature and that competition was done by Hawking to actually show that the black hole does have a temperature of this kind and I was hoping that I could do that calculation today but maybe not for the purposes of our connection with modular forms the question becomes that raises a third paradox what are the microstates in other words what Hilbert space are we talking about whose dimension is exponential of this entropy D as a function of m let's say such that s is equal to log dm yeah what is the what how do we associate and this is the problem that string theory has answered partially I mean I would not say that it has fully answered this question but it has in the partial answer is already quite spectacular therefore actually first of all you need a notion of a Hilbert space right in not even to ask this question you need a notion of a Hilbert space so what you require is a quantum gravity so you need some definition of Hilbert space of quantum gravity the leading I mean the basically the only candidate really which is seems to work case the string theory that Hilbert space of quantum gravity I should be able to identify states which account for this entropy and this is a completely universal formula depends on the area of the black hole now this has been generalized in other dimensions with charge with spin and it's always area upon four it's a completely universal formula but that area depends on other parameters q and g and so on but it's always the area of the sphere of the horizon sometimes just yeah if it's a spinning black hole it's more complicated but there's a notion of a horizon defined for charged black holes and spinning black holes which is very similar to this notion of horizon this reversible is just a process so given any system no no so in order to define entropy I need to talk about slow motions and as I meaning if I want to entropy is not tied or energy is not tied to reversible or irreversible refers to a physical process okay whereas entropy is a property of the system at any given time just like at any given time this room has energy and this room has volume and room has entropy I mean any given system you can associate with it and energy energy and temperature and volume thermodynamic quantities and reversibility is refers to if I change the parameters or if I change the volume of this room abruptly or slowly that's a different question so I can always associate an entropy so that does not require us to discuss at a given time I can associate with an entropy which is not independent of whether I'm discussing a reversible or irreversible process the black hole has entropy in a thermal equilibrium let's say here so if you're in a thermal equilibrium with some heat bath the black hole you will discover has an entropy and has a temperature and it can stay in equilibrium it's exactly like a hot body hot which can be in equilibrium if you surround it with a heat bath at the same temperature the I say partially because we don't really know how to associate any microstates in this picture but if you take Newton's constant to zero for a class of black holes with charges let's say charge q the single charge q such that the mass is basically determinant is proportional to q these are called extremal black holes that depends on the charge q if you look at the same system as g goes to zero this is a g much bigger than one let's say and if you take g much less than one the same system is described by some brain system the same state we start with a so you have a Hilbert space of string theory suppose what do I mean by Hilbert space of string theory I should be able to associate a state so suppose I have string theory in m1 3 cross k and let's say this k has some winding s1 cross 85 a string a brain one brain wrapping this s1 once let's say carrying some momentum along with it in the remaining four m1 3 when g is much less than one it looks like a point particle and here there is a one-dimensional field theory living on this brain one plus one-dimensional right so you have m1 3 and cross k across s1 let's say here at each point there is an s1 but only at this s1 sitting at the origin there is a string wrapping a brain localized there which really means there is a localized field theory or a bundle it's like a it's would have been a fiber bundle if there was a bundle everywhere but this is more like a shift that you have a field theory living only it's like a skyscraper only living at that point and nowhere else and therefore I can associate with a state in this Hilbert space of this theory this is a state that I'm calculating it has some charge which depends on the number of windings let's say n and the momentum shift or whatever that theory is living on that brain in the simplest example is 24 bosons that we discussed to calculate and the energy and I just have to now calculate the number of states for a given n okay I'm not describing this very well okay let me make one real small comment and then I will stop because I think the main point is that instinct theory by varying the Newton's constant because you saw that the metrically depended on this combination gm if I took Newton's constant to zero this term will drop out and it will look like flat spacetime so I can ignore gravitational effects completely when g is comparable when this effect is noticeable so there is a way to take the limits in such a way that you can effectively regard that as a state in minkowski spacetime state meaning an object particle localized here but it's not really particle because it has all kinds of excitations inside it is a particle from the four-dimensional point of view but it's really a one-dimensional object which has wiggles on it but in some internal space which I cannot see from the four-dimensional point of view and those wiggles are described by a quantum field theory of 24 bosons and the number of these wiggles is the number of microstates in which that brain can be and that I can calculate and that in that case you you'll find that log dn this is the famous partition problem of ramanujan is 4 pi square root of n and now you crank up g to be much bigger than 1 and what you will get is that this term that we were ignoring 1 minus 2 gm upon r of course this is a more complicated black hole it's not a schwarzschild black hole but it's a generalization of schwarzschild black hole which depends on some charges and so on but the factor that appeared there in the metric gets modified by terms like that okay so as g becomes large this terms become important and the same system I can view as a black hole now and I calculate this entropy by using the area formula and what I find is that this terms are to be equal to find a black hole solution very good so the particle has some mass right because particle is also particle has this 24 bosons living on it but because it has tension this string has some tension it has some mass and now the question is gm the combination gm it has some charge also from the four dimensional point of view because of this winding number so the question is if I can ignore the gravitational reaction back reaction if I can treat the metric to be almost flat then I can do the brain computation but if I increase g then I should really look at regard this as a star you know because it has mass and the gravitational field around it will not be negligible in fact it can start growing is area will start growing as we saw because the horizon area is gm yeah so the real parameter is that the radius of the horizon is roughly like gm or in this case it could be gq gq question is that you have to compare that with some other parameter which is the string scale there is another scale in the problem and whether this is big or small you can ignore this or not ignore it if you can ignore it then it looks like flat space you're doing a enumerative geometric calculation you're just doing a simple calculation of computing modulate spaces of this or that or some Fourier coefficient of modular form whereas if this quantity becomes big then you can no longer treat the metric to be a flat metric but you have to use the metric corresponding to the black hole and it turns out to be a black hole metric and the area of that black hole metric you can calculate and the entropy you can calculate remarkably it agrees precisely with all these coefficients in a rather non-trivial way okay i think i'll stop here i think i've perhaps lost you but okay i could i did my best