yeah so okay so as I was just saying um yeah so the course title is uh black holes holography and Quantum information and this is the first time we are offering it my name is Chetan Krishnan and I'm a faculty here at uh CHP Center for hierarchy physics and um so you can contact me so generally speaking it's email is the best way to contact me so you can message me at dot AC dot in I think that's correct right so that's the extension yeah so and um and yeah and um so this course is basically about the let's say roughly about it's kind of like orienting about the quantum aspects of gravity with kind of black holes as the sort of the you know so sometimes black holes are called the hydrogen atom of quantum gravity because it teaches us many things even though we don't know anything about quantum gravity and hydrogen atom is this is the prototypical thing that we figured out when we first figured out quantum mechanics so quantum gravity we don't know anything the hope is that by figuring out black holes we will learn something about it so this is the broad idea but it's it's like in a much more primitive form we don't have a framework like quantum mechanics was for hydrogen atom at this stage for black holes okay so what we have is like a bunch of different half cooked ideas well we have some theories which are very good but which are so um you know we don't know how to solve them in a useful way so that we can say anything directly about uh let's say somebody who falls into a black hole okay so we don't really understand the is well enough so that we can make a clean statement about it but at the same time we do understand in some sense at least in some context we believe we understand the theory is underlying black holes okay so uh how are we going to solve these theories in a useful way so that we can understand what happens let's say at the Horizon of a black hole or of the singularity for Black Hole so these are the kind of questions that we would like to have answers to okay so the problem at least in one context we believe that we have successfully quantized quantum gravity and that context is what is called the area CFT correspondence okay so uh successful well I don't need to put codes I think it's a fair statement a successful quantization of gravity but not quite in our universe okay so so most of what I say today will very broadly be in the context of this area safety correspondence okay so today I'm today or in this lectures will be based in the context of ADA safety correspondence and today's lecture you know I'll say many words you don't have to get scared so we'll start more systematically starting next lecture but today's class is kind of like to throw some words around and scare you a little bit and see who sticks around you know so that's the real purpose of today's class but I'll not try to scare you too much so uh so so this is a theory of quantum gravity and this we believe is is the correct theory of quantum gravity but we don't know how to solve it you know I'm saying there are many reasons to think that this is the correct theory of quantum gravity in certain kinds of space times where the cosmological constant is negative okay we live in a universe where the cosmology constant is positive and the universe is you know accelerating so this is these are universes where the cosmological is negative but it contains so some you know so this is a this is a kind of space-time kind of universe where uh which is clearly different from ours but it is not so different that gravity functions differently you know what I'm saying it is not so different that it is just a completely different toy model where nothing interesting happens you can have real black holes collapsing forming evaporating so all of those things still happen in the area safety correspondence okay so uh nonetheless it is not the same universe as ours so but this is the reason why we believe that if we understand black holes if we understand all the black holes there that are present in the adsc Ft correspondence we would have understood black holes in the real world I think that's a fair statement okay so because there are classes of black holes that exist in Ada safety which are formed which can form due to gravitational collapse and then they evaporate and it has the conventional problems with black holes that we have for standard black holes and astrophysics like for instance the information Paradox how to explain entropy all of those problems are there so that's why we are fairly confident that we are not throwing the baby out of the bathwater by working with adsc FD correspondence and not a universe of the positive cosmosis constant okay so that's so this is the broad setting of my lectures but I won't you know occasionally I will talk about adcft but you may only see it kind of in the backdrop you may not always see it like coming up in the Forefront towards the second half of the lecture I think I'll say a bit more concretely about ad safety but this is the broad setting so this is where we are talking about this is I would say the modern setting for understanding black holes okay so a second point I want to make is that black hole as all of you know are formed by you know gravitational collapse and astronomy you know astrophysics and they're very interesting objects to astrophysically because they are you know compact objects and compact objects are of extreme interest purely from astrophysical point of view okay so when people have very recently they have looked at not very recently maybe in the last few years that have been like colliding black holes and so on so gravitational waves you've detected gravitational waves we have taken pictures of Horizons all of these things we have done okay Horizon is basically the surface of a black hole if you haven't if you're not that familiar with it so all of this we have done but all of those are interesting for the purpose of astrophysics my goal is not so much to emphasize that I will occasionally have reasons to connect with those things here and there because black holes are you know sort of on The Cutting Edge of physics so there are things that we think we may know which may not we may not know so in that sense the some aspects of astrophysics may be relevant for what I'm talking about but fundamentally my interest in uh black holes or the the focus of these lectures will be in trying to understand black holes are solutions of General activity or gravity okay so the the reason why we are interested in black holes is as a tool for understanding quantum gravity so these are we view them as Solutions of gravity which by understanding them in the full quantum mechanical theory of gravity we hope to learn about quantum gravity so this is this is a this is a theoretical Focus rather than a phenomenological astrophysical Focus okay and I'm not really an expert particularly on the astrophysics of life tackles but I think you know for the purposes of there are you know for instance there are there are very interesting things that happen around black holes and astrophysics for instance accretion disks okay so there's an interesting Dynamics there is magnetic fields which are very strong all of those things are very interesting but we will largely not talk about them but to unders the reason for that is that to understand them we mostly just need classical generativity together with electromagnetism or you know the standard physics that we are very familiar with we are trying to apply for physic you know these ideas we want to look at black holes as Solutions of General activity in such a way that we learn we we can try to understand things Beyond General activity so this is our goal okay so this is sort of the setting and um yeah so what did I what else did I want to say so this is sort of the basic orientation yeah so our goal is uh you know theoretical goals not astrophysical so this is one point so and uh so so in in some sense the there are you know so let me just say a couple of words about what black holes are as you know they are there there you know that they are uh their systems which have which start from Fairly mundane initial conditions that's what is interesting about them because you don't have to have like a you know you started a sufficient amount of matter you're guaranteed to produce a black hole it is not just that you know you have to arrange things or something like that if you start a sufficient amount of matter we expect that our theories predict and we see that you'll always produce black holes okay black holes are a generic initial conditions can lead to the formation of black holes in general motivity so they are generic uh generic endpoints of gravitational evolution and by gravitational Evolution we mean generativity okay so how many of you have taken some course on General activity pretty much all of you but some of you not so uh I think you will need some general activity so who have not taken a course on generativity so there are some people who have neither taken nor okay you have not taken so um so I don't think you need a full course on General activity but I think you may you know it will I think you'll be it'll be it will be a little bit difficult if you don't study some things a little bit Advanced I think generativity you will need okay Quantum field Theory my demands are less intense you just need to do uh so so let me write the prerequisite here so prerequisites first is General activity so I don't think you really need a full understanding of Relativity or a full course even necessarily but you should have seen Einstein's equations what I mean by this you should have seen Einstein's equations and you should you should have seen the schwa Shield solution of the Einstein equations so if you have seen these two things you should be okay I don't think you need much more than that then I still teach you okay so uh but these two things I think it is we can't really go back all the way you know what I'm saying so these two things if you have not seen it it's not very hard to kind of pick it up I mean according to many people it's like generativity is a subject that you can learn over a long weekend so it's not uh and I think it is not an exaggeration so I would not say the same thing about bottom field Theory which is also a prerequisite it will take you probably a lifetime to learn but um we don't need all of quantum field Theory fortunately we just need to know a little bit of free scalar field Theory which is basically the first chapter of any standard book in minkowski's Baseline in minkowski space so yeah so the prerequisites are actually you know for the for the kind of terrain that we are planning to cover it's not that much okay so um yeah so this is I would say this is uh this you need more than this but this is also easier you know what I'm saying and this is just you this is really just the first chapter of any textbook on Quantum field Theory that's really all you need you won't need much more than that okay so um right so those are the prerequisites and uh if I have to recommend one book for this maybe I think it'll be threateniki's first chapter second chapter or something where he talks about quantizing scale of fields that you can take a look and uh General activity I mean there are many books um maybe let me just write cattle or D inverno which is a book that I used last time I taught relativity and it is a surprisingly good book so but also you know so the point is that but that's a slightly bigger amount of material but you don't have to know all of it if you can sort of get a hang of Einstein's equations and you understand structural solution partial solution is the name for the black holes the simplest black hole solution in general activity okay so these if you can if you have some understanding of these two things I think you should be fine so so that's as far as prerequisites go and oh yeah so I should also probably mention something about how the course evaluation or whatever so since I you know as I've said this before nobody likes making courses making exams nobody like giving exams probably like taking exams nobody like grading exams I particularly don't like grading exams so what we are going to do is we'll have like some I'm going to base it on attendance and class participation mostly and maybe like there is really a presentation at the end so this is my standard routine for all courses so that's what we'll do and the split will roughly be I think uh the presentation will probably be about 50 and the rest will be 50 okay so that's what we are planning to do so uh uh right so that's what that is and there is no real uh um textbook that we will follow very intently but I'll I gave you some references so one of them who is this is a series of lectures by Daniel Harlow on some Jerusalem lectures on Quantum black holes something like that so and this is so this number if you are not seen it you should familiarize yourself with that sort of numbering you know how to say so if this number seems magical to you that's actually a you know red flag so so it means that you you can just Google that number you might it'll answer your questions okay so that means it's basically uh it's an archive number it's archive you know has these it's a repository for preprints on on the web and this is the archive number for the particular review that I'm talking about so if you just Google it you don't even have to know what archive is if you Google this number I'm pretty sure the paper will come up okay so uh okay so that's one thing I'll give you but I may not strictly follow it but uh but you know these are standard so it gives you a context of the kind of things that I'm going to talk about that's the most important thing about it so okay so and the reason uh are black so let me finish this here so the reason why we are interested in we are trying this the courses oriented in this way is because in the last uh well last 40 years we have been making steady progress on understanding Quantum aspects of gravity and in the last decade there has been an extraordinary effort in some sense to make more progress okay to especially to understand this issue called the information Paradox I'll tell you what the information Paradox is in a bit bit uh but maybe towards the end of this lecture um and so because of this the sum of these topics are very timely and relevant and they are kind of in the active research in some sense at least by the end of this lecture we will be talking about things which are fairly close to The Cutting you know edge of research let's say so that's one reason why this course is let's say the the relevance of the course for the current uh Zeitgeist so to speak and uh so that's that's uh that so and this will give you a general orientation of the kind of material that I'm hoping to cover even though it is slightly outdated um so yeah so so this is what we want to talk about so the the reason why so okay so with uh what's the question no so um so yeah so what I wanted to say yeah so so what are the what are the questions what are the orienting questions that we want to answer okay so the orienting questions are one is black hole entropy and the second is the so-called information paradox so um so yeah so these are the two main questions that one has for black holes when you're talking about uh black hole physics the two main uh targets for once people understood so the main two Targets in the in the in the for in the formulation of a quantum theory of gravity for instance would be the ability to explain this as well as this from our Theory okay so string theory is a theory which is able to do the first part for certain black holes okay so what a certain non-generic black holes or non-generating in a suitable sense black holes so and this is a problem which at the moment I mean there are various uh claims of success of having resolved it and that we understand various bits and pieces of it but our understanding of this is nowhere near as good as our understanding of this and even our understanding of this is not that great as I said it is for some classes of black holes and string theory we have explanations for it and that has taught us quite a bit okay as I said in fact it is by studying this black hole entropy questions in string theory that we stumbled upon the adhcft correspondence okay so uh so this has been a very fruitful line of research and it still is uh but this is more of a kinematic problem so what I mean by that is black hole entropy the question of black hole entropy is the question of what are the degrees of freedom of black holes okay so that's roughly speaking the knowledge of the Hilbert space of black holes but in order to fully understand Dynamics in understand physics we not only just need an understanding of the degrees of freedom but we also need to know how those degrees of freedom evolve and the second question about the information Paradox which I will tell you in Broad outline today is basically a question about Dynamics okay and the question of uh this is that's why it's a much harder problem and that's why our understanding if it is still very uh you know flimsy it's a it's a sharp contradiction in what we believe is our understanding of evolution of systems that contain black holes so that's what information Paradox is so you could even think of that as very basically as the level of what happens to things that you throw into a black hole we don't fully understand okay so that's another way to look at the uh the black hole information Paradox question so let me start by saying um what um so so the so so one of the so I think that's you you I haven't said what black holes like not explain escaping Etc I'm assuming that all of you have heard all those things right so you black holes are things from which light can't escape so which means that we believe that almost nothing can escape so these are the kinematic basic features of black holes and theories of gravity that we have they're they're generically they form so that's the broad context so uh yeah so the the primary question so in a sense what we want to do in the context of our questions is uh treating a black hole is an ordinary Quantum system with entropy this is this seems to be a very productive idea okay so uh so let me just write that treating black holes as ordinary Quantum systems with entropy has turned out to be a very productive idea and the entropy of black holes is so this is sort of the um so the entropy of black holes is area divided by 4G Newton okay our G Neutron this is this is all of these are you know as I said many of the things that I say today you will kind of have to take it at face value we will gradually develop it okay so this was a this the idea that black holes of entropy was first suggested by beckenstein so in some sense beckenstein is the person who sort of inaugurated the kind of ideas that we will be interested in this course okay so he observed that s has to be proportional to area because you know so so he he so he what happens to the entropy of a thing that you throw into the black holes nothing can come out of it somehow the entropy seems like it is lost and black holes at the time people believe that they have zero temperature so it seemed like the first law of thermodynamics would be broken the second law of terminal depending on how you look at it there is various versions of phrasing it but you will run into trouble with thermodynamics okay so and the way he addressed that problem is by attributing an entropy to the black hole and so he wanted to understand how should we but if you're looking at from outside how are we going to attribute entropy to a black hole and so initially the one possibility is that maybe the mass of the black hole captures entropy but the point is that mass of the black hole you know you know it's it's there are like so it is not so in for instance in black hole collisions it is not very clear that mass this total mass in some suitably defined way is a constantly increasing quantity okay so for instance gravitational radiation can take away Mass which is why we see ligo for example okay so but it is one can show at the time there was an old theorem that in fact Hawking had proved for other reasons um before which showed that the entropy sorry the area of a black hole can never decrease in any process that involves multiple black holes for instance he had shown that as a theorem so using that fact beckenstein conjected that the entropy of a black hole should be proportional to the area of the black hole okay area of the Horizon of a black hole Horizon means the surface of no return okay so that surface has an area which we can compute using standard techniques in uh you know in basic relativity or you don't even really need relativity for that so and if you compute that area he conjected that the S should be proportional to the area so this was sort of the starting point where black holes we understood as somehow some they have some thermodynamic significance okay so um yeah so that's the sort of the starting point but this was people who are suspicious of it you know so and uh as often happens so and uh but what what did what it what what was uh shown by Hawking using a different calculation was that this entropy is in fact precisely equal to in uh so this is what he showed so Hawking showed this he showed it in a slightly indirect way he computed a certain temperature which is now called Hawking temperature and he fixed this proportionality constant okay so this is what he did so this is called the beckenstein Hawking entropy and what is surprising about this entropy and so okay let me tell you how um so yeah what is surprising about this entropy is that unlike for instance the Solutions in other theories of our favorite theories the solutions of uh General activity when you fix let's say the mass and the angular momentum let's say we are just working with pure Einstein gravity with no electromagnetism or charge or anything so if you just have Einstein gravity then if you fix the mass and angular momentum of an of a space type of an object of a black hole of a thing at Infinity then that uniquely fixes the you know solution okay if you fix it at Infinity then the solution is completely fixed so that seems that in general activity there is only one solution corresponding to a given Mass black hole okay but this is problematic if you want to attribute an entropy to it because typically we think of entropy as the number of solutions or number of states in a system which have certain macroscopic quantities right so in general activity if you come fix the mass and angular momentum which seems to be in most physics this would be like if you precisely know the energy and angular momentum you have fixed the state so that doesn't but it still leads to some form of an entropy okay so the fact that black holes in general activity do not have free parameters so one way in which a black hole could have uh you know if you fix the mass of a black hole at infinity and the angle of infinity it could be that you know the maybe this metric should have had some profile as it was reaching Infinity do you understand what I'm saying like if you have a scalar filter an electromagnetic field you know let's say some field you know so usually all our theories are field theories like generativity is also a field Theory so if you have a field Theory what happens to the field like just the fact that you're fixed let's say some quantum numbers Associated to the field at Infinity does not fix the solution uniquely okay you can have profiles you can have some functional form for the function which dies over infinity various things it can have all kinds of things just some you know total quantum numbers and infinity will not uniquely fix a solution because it's just a general feature of you know this is something that your if you solve a scalar field equation I mean depending on boundary conditions Etc you can find many solutions okay so that's not the case here you hear what happens is if you fix this total quantum number at Infinity the solution gets completely fixed and that is the reason why this entropy in classical General activities are very puzzling quantity okay classical General activity cannot immediately explain what is the origin of the centropy this is coming from the interpretation of you know the necessity of uh the fact that we don't want to violate the loss of thermodynamics if you if you have a black hole space time that's where it is coming from and the way in which we first noted this fact so this this fact that you know these the numbers mass and angular momentum can completely fix a solution is called a no hair theorem okay so no hair theorems are a feature where quite broadly there are ways in which slightly you can evade these things in some not very interesting ways but broadly speaking no hair theorems are a feature of General activity okay so they are a feature of the relativity and this is the reason why the entropy of a black hole is a serious puzzle because classical General activity alone cannot do that for you okay and that's one reason also to think that classic is relativity may be some sort of a coarse grain thermodynamics like Theory and the microscopic theory is something else altogether and which has ultimately that has that is what we have found for instance in the adcft correspondence okay in our one functioning theory of quantum gravity it seems to be the case that general relativity is just some emergent approximate thing that shows up which is not there in the full UV complete description UV complete means the full you know the full microscopic description okay so that's the so no hair theorems are a feature of classical Journal activity and they are in tension with the existence of entropy so yeah so the so this is this is sort of the place where um black hole physics you know in the in the 1960s black holes you know black hole physics kind of went through sort of a generalized you know kind of a revolution I think in the 1960s when curve is a new cylinder mathematician who uh he discovered while he was in Austin that um these these guys he discovered the most General solution at the time when he discovered it he discovered the rotating black hole solution the short Shield solution was discovered you know rational solution I mean uh you know let me just write down the Shoshan solution so this is this is what is called this is the standard form in which the spherically symmetric black hole of generativity I'm writing some of these things because you know you can decide whether this course is at a suitable level for you know what I'm saying so I think that so if you have seen this is the minkowski metric in polar coordinates okay and the thing that is sitting above it if you set m equal to zero you find that you get them income schematric right so that solution is what is called the short Shield solution even though that's not the form and it's partial wrote it down trochl discovered a solution I think within months after Einstein wrote down his theory you know he wasn't uh he was somewhere and he was in battle and he died pretty much months after he discovered it so uh but he did not write it in this form it was written down by another guy named roster so uh and and that was also very quickly after Einstein's theory came up and Isis Theory came out in full form in 1915 and I think it was within months if I understand correctly or maybe yeah that's what I may not be slightly lying but uh at least morally it is true okay so he uh so this was the first solution of General activity that was not minkowski which was known as an exact solution and this solution stayed like that for almost 50 years you know and nobody managed to find a solution which had rotation uh on top of this so this is a solution which is spherically symmetric but most astrophysical objects for instance are not quite spherically symmetric they are actually symmetric they have rotation so you don't really always this solution is sometimes a reasonable approximation but it's not really a it's not really it cannot definitely be an exact solution okay so what Kerr did he managed to change that situation I don't remember exactly when but probably in the 1960s early 1960s I think so he wrote down a solution of General activity which contains two parameters instead of one so this solution contains only one parameter which is the mass of the black hole okay he wrote down solution which contains a mass and an angular momentum and that sort of kick-started what is called the Golden Age of black hole physics I would say golden age of classical black hole physics and then in the wake of that Discovery people came across various uniqueness theorems and this no hair theorem is an example of a uniqueness theorem okay so they came to this conclusion that if you fixed you know two of In classical in black holes in four-dimensional three plus one dimensional Einstein gravity they understood that if you fix the mass and the angular momentum of a solution you completely fix the solution at least you know very minor assumptions with very minor assumptions okay so this is so this was observed by uh so this was this was you know this nowhere theorem was shown by Israel okay this was various versions of No Hair Theory so no hair theorems are pretty technically difficult theorems and a pretty technically difficult uh theorems In classical generativity and we will not say anything about them we'll just use them as facts okay so there are uh yeah so the and that it does not use the kind of there are two kinds of technical complications or technical difficulties that classical generality usually has one of them has to do with proving these things called Singularity theorems okay so I will talk about that in the next you know how much time do I have so I think I'm going farther than I expected yeah so we have time so I'll talk about um so the next thing I want to kind of give you an overview of the the general lore of black holes today so that's the goal of today's lecture so and uh so the next thing I want to talk about other than no hair theorems is singularity theorems okay so and Singularity theorems are one kind of so both require fairly sophisticated mathematics okay so uh and uh the kind of mathematics that usually one studies for in you know so-called Global aspects of General activity which some of you may have studied is the methods required for proving The Singularity theorems but these theorems do not they are not they are not proven using those techniques and they're very different techniques and they're still very hard techniques so I don't think most people who study Advanced relativity usually study no you know the singularity theorem techniques but these methods are generally speaking also equally difficult but I have not met that many people who are experts on them that's Singularity theorems are you know and neither am I neither am I so uh so yeah so I just wanted to say that I wanted to give a hats you know tip to Israel because it's a very important very crucial theorem for our purposes uh but I think that pretty much everybody treats it as a black box I mean it's a reliable black box there are very good reason for believing it but the full proof is not easy anyway so um so yeah so so that's the first ingredient first thing that I'll say and uh or first or second thing that I'll say and the next thing that I'll say is um a bit of classical black hole physics which is related which is uh theorem first proved by Penrose okay so this is so Singularity theorems are basically the statement that if an object has sufficiently collapsed uh then it basically always contain a singularity inside it and a singularity is a place where um uh you know your there's the the differential equations the the solutions of the differential equation basically blow up so the statement of Singularity theorems are that very fairly generic initial value data uh in general at classical relativity will evolve and form singularities so this is basically the statement of penrose's Singularity theorem and this is uh yeah it's strange in my opinion by the way that Penrose got the Nobel Prize and not curved it's a bit surprising because this singularities are inside black hole Horizons so essentially a singularity is something that you will never see so and considering the fact that you know Nobel Prize is about actual ability to see the black hole black hole people are actually when they say when you take a break a picture of a event horizon or something like that you are basically looking outside the Horizon you know you're not seeing inside so and the current all the things that we are really testing are solutions of you know trajectories of particles or photons or whatever right around the curb black hole so that's what we are really checking so it's a bit surprising that Karen did not get the Nobel Prize but Penrose did but okay Penrose is okay too so you know it's fine so uh but anyway so the so this is so this is the second sort of major ingredient that we will not really talk much about so the our goal is always to go to the quantum side of things but I want you to know the classical lore that existed before and this is one of the second piece of things so this entropy and uh you know I will talk about black hole temperature in a second those things will be very important for us but these no hair theorems Singularity theorems Etc are more classical things which you will not really have much to say about okay but we will believe them yeah and uh by the way any questions by any point please feel free to stop ask etc etc so uh uh right so so the so the so this observation so I you know I told you that beckenstein determined or proposed that black holes should have an entropy based on pure thermodynamic arguments like if you throw something into a black hole what happens to it it has to have an entropy otherwise if the thermodynamics is broken so but I also told you that the coefficient of this proportionality was fixed by Hawking in this way okay and a very important thing about it is to notice this is the boltzmann constant this is G Newton but the most important theory is that this is one by H cross okay so this is not proportional to H cross this is proportional to 1 by H cross and this will play an important role for various things so which means yeah I mean there are yeah it's not I mean you will be you will be able to find expressions in your quantum mechanics text which contain one by H cross but uh there is a sense in which getting an 1 by H cross on the denominator is much harder than getting it on the numerator okay so and that's this is this basically comes from indirect arguments but any quantum theory of gravity should be able to reproduce it and as I told you like in certain black holes in string theory you know people have been able to reproduce them okay so so that gives you strong reason to believe that both this expression is correct as well as the string theoretic constructions that lead to them are also probably correct okay so and this is a fundamental test We Believe of all theories of quantum gravity so all gravity theories basically contain black holes okay except some very low dimensional ones etcetera so all interesting uh black gravitational theories contain black holes and essentially it is because of you know the fact that if you have there is no mechanism for repelling things in gravity if you add matter things just keep collapsing so there's no mechanism for stopping the collapse and if you put enough matter you can overcome any other Force that's the primary this is basically the I give you a proof of the existence of let's say black holes okay so gravity is always attractive and uh you know there is uh if you put enough matter the gravitational collapse is a stronger phenomenon than any repulsive mechanism that you can put in place so it'll always Crush things to nothing so and that's what happens with the black holes okay so uh um right so uh so yeah so this one by H cross is important um and uh so how did he fix that okay so how did he fix how did Hawking fix this and the way he fixes in a completely different calculation and this is some a calculation that we will do in detail Hawkings calculation of black hole entropy sorry Buckle temperature so he showed that by doing Quantum field Theory like for of a simple scalar field like the one that you study in the first chapters in minkowski space is you can use the quantum filter of a free scalar field in the background of a black hole and he showed that these black holes you know this uh the there is a there is a there is a spectrum of scalar radiation at Infinity which is at a certain temperature so this is something that we will do okay so this calculation basically helped them to establish that black holes have a temperature which is H cross times Kappa okay where Kappa is called the surface gravity I mean we will fix it for various black holes in various contexts so um so so this it's I only want to emphasize the H cross here this Factor depends on various you know geometric aspects of black holes this is not important for our purposes right now but the thing I want to emphasize is that it is proportional to H cross okay this one was inversely proportional to H cross and this one was proportional Trace cross Okay so so and one fact we knew just from the solutions of General activity that we have for black holes okay the solution to General activity that we have for black holes we always knew that Etc okay so the reason I'm writing dots because if you have electrically charged black holes where you have electromagnetic field as well and not just relativity then you will have an expression of this form okay so this is the area and this is the what is called the surface gravity and in most of my discussions I will set you know H cross and c and you know various other things to one okay so it will sometimes be important to keep G and H cross and this particular example I'm keeping also H cross because here so this expression note that this is a completely classical generativity expression very important fact okay so this is some expression which just contains the parameters of the black hole so you get some expression like this and this equation if you assume that this equation is uh you know is giving you the temperature then using this guy here I can write this as TDs is DM minus Omega DJ whatever stuff okay so this is Steve Hawking if you like I'm setting all the the constants Etc you know we can fix so if you fix a all the constants correctly you will I'm not being here this is the only place I'm being very careful about all the uh constants in the problem okay so here I'm just writing so by using this relation I am able to rewrite this equation Hawking showed that Hawking temperature is proportional to H cross times K and from the first law you know from this equation that is valid for black holes and generativity we can show that t Horizon that if you come in late don't ask just walk in okay please go in so uh th times d s is uh yes so dth times d s is DM so um so this equation is the first law of thermodynamics okay so this is uh so this you can think of in this way and you see that this looks like the first law of thermodynamics right and that's how we know that this A and S are related okay so use so Hawking Computing the temperature this formula was a formula in general activity that you already know this is sometimes called the first law of black hole thermodynamics okay but at this point it is not a law of thermodynamics it is just some formula relating parameters of a black hole okay but then by using this formula and this a you know then we can interpret this entropy as being related to S and that's how Hawking fix this coefficient okay so this was the proportionality was established or suggested by beckenstein and by Computing the temperature of the black hole and knowing the first law of black hole thermodynamics and putting it in quotes because this is a purely classical relativity statement just from the equations of General activity you can show this from there you can truly interpret this equation as a thermodynamic equation for black holes okay so that's that's sort of what gets kicked off the sort of the sort of the sort of the quantum uh side of black hole physics so this Singularity theorems and no hair theorems and all that they were all classical relativity so that was sort of the wake of let's say you know um uh in the wake of curse discovery of the curse solution so that kind of you know that is called the Golden Age of black hole physics and it culminated in this calculation that Hawking did in 1973 I think which is uh so and that calculation basically showed that black holes actually have the temperature and should I do something is it still audible yeah okay so uh and that's what that's what basically this uh this that's sort of the thing that started off so so now that comes us to the so this is sort of the you know I told you there are two main things that will be kind of our orienting stars in the course and one of the black hole entropy and one of them will be information Paradox okay so and of course we'll talk about many many other things related to adcft this that Etc but all of them will be guided in some sense by these two things so I already explained at a very superficial level the black hole entropy question so now I'm going to say uh this uh what is it that what is the information Paradox because now we have enough context okay and the context is basically that if you have a black hole that is sitting somewhere and it has a temperature and it's radiating right it's radiating means it's losing energy and mass so asymptote far away from the black hole it's steadily very slowly but it's losing mass so which means that at some point it will evaporate completely and you will get you know just the thermal radiation right so the black holes are not the final state of gravitational collapse and in other words so Hawking's calculation basically suggests that the temperature that is you know the radiation that is left over from the evaporation of a black hole at the end of its evolution is the most entropic thing in the universe okay so that object is what finally the end state of every collapsing thing would be this is what you know at least if you take Hawking's calculation it primer facing that's what it means but it also gives you a problem it's just that at least Hawking's calculation the way originally it was calculated it looks like the spectrum is exactly thermal it's a perfect black body Spectrum okay and that means that the initial State you know the way you form a black hole is that you started in some initial State and then you let it evolve this is the time Direction and at some point you form a black hole Etc so this is you know why this I'm calling this a black hole Etc you'll see later so and uh when this happens and later on it basically is radiating this Hawking quanta okay so it radiates these things and at the end you have you're left with you know all of the the initial the information about the initial State and if you think of it you know we believe that quantum mechanics is the real description of nature so this was some initial state which was a pure state right it's a pure uh a cat in the Hilbert space but finally when it's once it's evaporated what your n what you end up with is basically a thermal density Matrix a thermal density Matrix basically means you know a row which is basically of this form because that's what thermal radiation means thermal radiation means that the the the density operator if you are familiar with that is of the form of the canonical ensemble that's what it means okay so or you know like this if that was a little too highfalutin the or you know initial state which was which had information in it which depends on the details of the initial configuration ended up looking like just purely thermal Spectrum right so that is that is something that you do not expect in a unitary theory of quantum gravity because unitarity basically says that pure states have to evolve to Pure States okay so and that is the sort of the first iteration of the information Paradox it's not really a paradox even though Hawking thought it was but it's close enough to a paradox that it can be sharpened into a true contradiction okay and we will talk about that and that's one of the ways you know so this question even though the way I am talking about it is not obvious the this question can be phrased as a statement about the smoothness of the Horizon of a black hole so what I mean by smoothness of the Horizon is uh yeah so that's sort of the next big ingredient for our discussion by the way any questions at all feel free to ask I mean especially if there are a lot of students you know sometimes there is kind of like a stage aspect to it so you may not feel comfortable asking so if one of you are brave enough to ask the question then others will follow so feel free at any point you know especially in today's lecture it's like we are just talking we are among friends you know what I think so you're just talking about General philosophy of black holes so anyway so uh so the thing that I wanted to say was that this particular question that Hawking asked and Hawking you know Hawking argued that if you if this is he no he did his calculation and the Spectrum looks thermal so Hawking argued that this means that information is truly lost okay so information is truly lost means that quantum mechanics breaks down that's what it means so uh in other words unitarity unitarity you know Schrodinger's evolution is what every all the everything evolves by schroding your equation we believe that you know we are we are we will be entirely conservative in this entire course in the sense that we always believe that quantum mechanics is correct you know so uh yeah I mean if that goes wrong I don't know what to do so uh so assuming that quantum mechanics is correct we expect so this cannot happen but Hawking you know Hawking is a brave man so what he did was that he came to the control he uh declared or he thought that the information is lost and unitarity breaks down in uh gravitational Evolution but this is this doesn't seem to be what is happening at least in many toy models and now we have sufficient understanding of various situations where we believe that this may not be the case okay so in fact one of the proponents of the counter idea is tooth dolphed so Toft was the first guy a first guy or maybe one of the first people to basically say that um to basically suggest that you know um information may not be lost in black hole evolution so uh and it was taken over you know it was further pushed forward by Susquehanna various others and ads EFT correspondence is actually a realization is you know if the area CFT correspondence is correct right I would not told you what adsc FD is but I did mention what it is it's a theory of quantum gravity but we believe that if the ads CFT corresponds is correct there is no way that information can be lost the statement that a perfectly unitary Quantum field theory is can be used to describe quantum gravity in space time so the negative cosmological constant okay so let me repeat that statement so area safety correspondence is one of the implications of the idea safety correspondence is the statement that quantum gravity in your space times the negative cos logical constant can be described using a perfectly unitary standard Quantum field Theory uh um you know Quantum field Theory so that means that if any process that happens in the bulk of the space time in the space time like formation of black holes evaporation all of them should have a unitary description so information possibly cannot be lost okay we still don't understand the mechanism through which that information is information Paradox is restored but we know that it should exist you know I'm saying that's what Ada CFT correspondence gives gives us so and so in that sense talk to us uh more correct than Hawking in this particular case and uh so that's so that's the so that's one so that's you know so we believe that the universe is you know so information is in fact comes out so I also want to say a couple of different ways of looking at the information Paradox which may superficially look different but they are intimately connected okay so one way to look at this is you know sort of clear which is that what happens to the things that you throw into the black hole you know like somehow the story is that uh uh you know black holes have a region called a singularity so I'm drawing these pictures we'll explain these pictures actually the first thing I will do is explain these pictures in some detail in the first couple of lectures okay so this is the picture uh can you rephrase the question I mean I from the way I said it it is exactly that right so you have information means what was what do you mean by information is the same you know I the the coherence of the state is another way of characterizing this in describing information so coherence means like you know if if a state If a pure State a cat evolves into a density Matrix we say that information is lost in some sort of continuity equation of probabilities violated right so that's another way to so I mean it depends a little bit on what you mean by information and this is the specific sense in which I'm using it okay so the information here means that probability is conserved so probability will not be conserved if uh uh this guy have all stood to this guy okay so there is we don't first of all we don't even know how it happens we don't have a mechanism for you know well I mean we can write some limb blood like equations and all that but the point is that pure State evolving into a thermal State like this we necessarily lose information probability is not conserved so that's what I mean by information balance I mean you can also think of it as the breakdown of quantum mechanics because that's what it is in quantum mechanics information is always preserved right so so yeah so so uh any other questions sorry to get it repeat sorry what is interacting with support sorry yeah well I mean if if you're if you're thinking about the universe as being described by a state it can you can think of it as a closed system right so I'm talking about the wave function of the universe here or any any system like you whatever closed system that contains a black hole okay so here what I'm saying is that you start with some initial State you evolve it you form a black hole and once the black hole evaporates some information has to be lost according to Hawking's calculation and if you take it at face value okay so uh right yeah so uh I mean you can couple it to and make it an open Quantum system and all that but this particular aspect of it will still remain intact so you can look at some region of space-time and phrase it as an open Quantum problem but it's still this is the fundamental issue okay so uh yeah so yes so so yeah so so far you know I have not really emphasized singularities much and I generally will not you know in the in the course of this course so to speak so but the um so yeah maybe we'll talk about it here and there but off most of the time our attention will be on the horizon okay a horizon is the defining property of a black hole the existence of an event horizon is what defines a black hole in the sense that you know if you have an event horizon is basically a region of space-time from which classically information cannot reach infinity that's what an event horizon is okay so for instance somebody you know yeah so something that happens here cannot this is the asymptotic region or the boundary of space-time in this picture so these pictures are sometimes called Penrose diagrams so we will talk about them later in the next maybe lecture so let's say something happens inside this is the region that is called the inside of a black hole so if something happens here it goes and hits the singularity of the black hole so the statement that the black hole nothing can come out of a black hole is a statement that whatever happens anywhere in here cannot get to here because you know at best you can go along 45 degree lines because speed of light is limited okay so these statements if you are not super familiar with them uh you know just ask me or if you're completely if you're completely baffled by them then maybe it is you know this course is a little too advanced for you okay so uh so this is uh so if you're sitting here and this is the light cone of this region in special relativity you may have seen it so this light cone is the region the future Litecoin is a region that is accessible to a person who's sitting here so but if you are sitting here on the other hand you will only have access to this region and that does not reach the asymptotic boundary so this is what is called the asymptotic boundary this is the region at Infinity of Any Given space time okay so that's what this whole picture is for and um yeah so this is this so this so yeah so this is what is called The Singularity and this region here which is called this is our so-called future Event Horizon so a person who is sitting here can fall inside a black hole but once he's inside the black hole is not going to be able to get out is it sort of clear to all of you is there if somebody is not clear I think you know the level at which it is not clear you should explain it to me you should mention it to me so that I can kind of readjust and recalibrate how I proceed okay so um so anyways so yeah so this this object The Horizon is going to be and and one of the things about Hawking's calculation is that he assumes that the principle of equivalence so let me erase this because this is kind of a important point and right it in little more detail so um so one of the crucial things that Hawking needs to assume as we will see in our calculation is that he has to assume that this region of space-time respects your relativity okay what that means that's that assumption is called the smoothness of the horizon okay smoothness of the Horizon is basically the statement that a freely falling guy into a black hole free-falling geodesic into the black hole will will just experience uh you know principle of equivalence at the Horizon okay so that's what is kind of the defining property of a generic point in space time so the Horizon as far as a local uh local involving um you know server is considered has a is uh is exactly like minkowski's space-time okay so this is this is the this is sort of the starting principle of the principle you know of the general activity which basically says that anywhere in space even if there is a gravitation if you are in free fall you will you know you will local experiments will look like minkowski experiments okay so Maxwell's equations will work the same way all of that stuff okay so locally according to a freely falling Observer the space time around him looks like there is no difference he cannot do a local experiment and detect the fact that he is you know detect anything you know what I'm saying so you cannot detect Gravity by doing a local experiment while in free fall so that's the elevator experiment in Einstein's you know general relativity and all that so a key assumption that Hawking makes is that at the Horizon of a black hole okay so the surface of no return for a free-falling person there is nothing strange that happens so nothing the free-falling guy he will not recognize it when if he is falling into the black hole in fact for a sufficiently large black hole and this is not a crazy assumption you might think that okay black holes are collapsed objects maybe they're dense you know the horizons you know maybe something violent happens but the point about a black hole is that if you take sufficient amount of mass and form you know form it to collapse it and form a black hole what it does is that the whole Horizon can be arbitrarily weakly curved you know what I think so for instance I wrote down the schwart shield metric so if you have a black hole that forms from a huge amount of mass let's say you know a Galaxy sized black hole it's you know galaxies is maybe not that good but it's still pretty good so and if you form a black hole it's sufficiently huge amount of mass you can make the Horizon to be arbitrarily weakly curved okay so here for instance you notice that this one you know this is I wrote this equation here so this was my uh you know equation for the Schwarz field metric so this radius this you know the r is equal to 2 GM by C square is this is what is called The Horizon of the black hole for the short Shield metric so there this this piece of the metric looks weird right so 2gm by C square is equal to R is what is called the short Shield radius so I let me write it like this so which means that if the mass of the black hole is large the radius becomes large so radius is inversely related to curvature right so which means that the bigger the black hole or the more massive the black hole the black hole is becoming less and less uh curved you know and that means for instance so in principle it is possible that so so it means that it's also very not dense you know because if you divide the mass by some suitable volume so in fact you know volume needs to be defined but if you do that you will find that the black hole is actually very you can make it arbitrarily less dense so most astrophysical black holes are pretty intense places but if you if you could construct a very very very large black hole it will be arbitrarily you know well behaved locally you know what I'm saying and this is the reason why in one way of understanding why we expect that the principle of equivalence May hold here okay so for instance we could be passing through the Horizon of a black hole right now and our Fates in the future may be sealed but there is locally we may not be able to tell the difference so this is a reasonable assumption from many perspectives okay from the perspective of their relativity it is not a crazy assumption at least you know we will question it in various ways as we go along but it is not a crazy assumption to think that the Horizon of a black hole is a reasonable place okay so uh yeah sure yeah true yeah see equals principle is always broken up to Tidal forces right even now on the earth we experience tidal forces so but these Idol forces are tiny so the question is how you know the equivalence principle is basically a statement that the metric and the you know Christopher symbol Etc can be expanded around the flat space value so the question is if it breaks down I mean the corrections are always there but the corrections cannot be measured in uh you know without doing an experiment that spreads over some space and time that's the sort of the operational statement so and that certainly so to the extent that the black hole breaks that you know you it also breaks it in any geometry where there is a horizon sorry any geometry where there is gravity right even here on Earth we have tidal forces it's controlled by the mass of the mass of the Earth in exactly the same way there will be a similar amount of tidal force in the black hole also but it is not more than what is expected from an object of that Mass okay so uh so this yeah so the principle of equivalence is uh you know so the the reason why it is a bit concerning to think that maybe the equivalency principle holds at the Horizon is because of the fact that you know you know that you can see from this metric and which we will discuss in the next lecture is that this seems to be something strange happening at the metric here okay it looks like the one of the one of the components of the metric has collapsed has become trivial and the other one blows up so this the next piece is Dr Square right so this is your next component and then there is the angular part so this is your metric and this guy also blows up and this guy becomes trivial so it might look like at least one representation of the metric it looks like the space time is getting going crazy but on the other hand we one of the things we know about generativity is that we can do coordinate Transmissions so there exists coordinate Transformations which you can do to rewrite this metric in a form where it is smooth at the Horizon okay so and the principle of equivalence is the statement that we have the freedom to do those coordinate transformations so if you believe that we have the freedom to do those coordinate Transformations then at the Horizon you're breaking sorry you're uh you are you know your the black hole has a smooth Horizon okay and that's the reason so if you believe in general activity black holes seem to have a smooth Horizon and that's how uh Hawking computed that assumption went in in a very crucial way in fact you can show that the Assumption of the smoothness of the Horizon directly translates to this thermality of the Hawking Spectrum okay so the fact that it is thermal is directly related to the smoothness of the um Horizon so if you so if you so so there are many ways of resolving the information Paradox you know so the question is whether we can retain all these features that so yeah so I didn't say why it is so right so one of the ways one of one you know I told you that the information Paradox in the original way that Hawking framed it is basically the statement that a state forms and you know some pure State forms and evolves into a mixed state so that's a violation of unitarity so but you can actually argue at least in some versions of the information Paradox that it is equivalent to the um you know the it can be phrased as a as a tension with the smoothness of the Horizon so you can you know there is a there is a there is there is a sense in which you can phrase the maybe I should say that a bit differently a little bit later not maybe today so uh yeah so so the but let me just say that in very uh crude words that the smoothness of the Horizon is uh while respecting black hole thermodynamics the the ability to respect black hole thermodynamics while at the same time having a smooth Horizon that is unitary is really at the Crux of the questions about black hole information Paradox so the way I stated that the only reason I wanted to emphasize it is The Horizon is going to play a crucial role in all of these discussions essentially because it directly translates to the thermality of the Hawking spectrum and that is responsible for making the black hole evaporate okay so smooth Horizon is an important idea which we will also talk about in a few different ways uh and uh what is the time now yeah so uh so I think that that brings me how tired are you so the for the first class day I think some of some of your vagrants I expect to leave by the next couple of lectures so I think that maybe uh you know instead of uh you know instead of like uh yeah I think this is kind of this is kind of like the semi-classical discussion of black holes is kind of an okay overview so we will uh say a few more words about the the way where the course is headed Etc maybe in the next lecture and then also uh start with uh you know maybe the causal structure of black holes and so on and so forth okay so that's what we'll do okay so any questions so in fact we are taking it at face value so when we say that you know the entropy of a black hole is given by this right so what we are doing when we say that we want to have like a count of the microstates of the theory then we are taking it at face value we are saying that our entropy is a real boltzmann entropy and by doing that is indeed how we are you know coming to this idea that we need to be able to count it so count the number of degrees of freedom and that's why I'm saying that the you know the entropy of a black hole is kind of a kinematic question so it's a question about the num the number of degrees of freedom of the of the black hole and this um you know evaporation this evaporation and information loss information loss a more dynamical problem because it really is talking about how things evolve you know we need to have a early time and a late time in order to talk about information Paradox but in order to talk about this we need to look at the black hole State and count the degrees of freedom so it doesn't talk about time Evolution there's no there is no real well I mean there is some indirect way in which the hamiltonian plays a role but it's not directly there at least in some sense Okay so you had a question not entropy I'm saying that smoothness of the Horizon directly leads to the evaporation so in that sense The Horizon is directly responsible for the information paradox yeah if the Horizon is not smooth then we have to somehow see yeah so there are the things that I told you this entropy this temperature all of those things they have they have found you know they've always been fairly compelling because they have a certain niceness and coherence to them even though I would say that none of the calculations of uh you know the original calculation of Hawking Etc there are various things one could criticize in principle but nonetheless at the end of the day you know there are very many reasons to think that they are there is a certain coherence to the entire landscape of ideas here and they have been done for like it's very robust like any black hole any Dimension any Theory you get similar things so and but even more importantly I don't think we can sacrifice these results because why are the ads safety correspondence these results can immediately be translated into properties of the quantum field theory that I mentioned so ads CFT correspondence is basically a statement that quantum gravity is equivalent to some Quantum field theory in D Dimensions this is in D plus one dimension a Quantum field Theory with the negative cosmological constant is equal to some Quantum filter in D Dimensions so by these this this temperature entropy all the thermodynamic and even you know fluid Dynamic quantities that you can compute on the gravitational side you on black holes they have direct translations to things in qft and they are extremely reasonable and in some cases these are the only ways of at least getting reasonable results for certain strongly coupled Quantum field theories so these results it's I think it is you know like of course the easiest way to resolve the information Paradox uh is to declare in some way so so yeah the easiest way to declare the uh you know the the I mean you know yeah I mean you could just I wouldn't say to the easiest way but you know if you if you're willing to tolerate you know so the challenge in some sense eventually I think is that we also do not like the singularity even though I don't see really same I didn't really say much about it because Singularity is also definitely a place where which is the r equal to zero part of this metric we expect everything to break down we definitely need something more than generativity in order to explain the information you know the The Singularity of a black hole but the Horizon is some place where we feel like generativity should work but again we run into some problem but ultimately I want to really emphasize that in order to have a fully satisfactory theory of quantum black holes we need to both understand singularities and Horizons so people oftentimes make this trade-off of oh you know so so for instance there is one simple strategy for getting rid of information Paradox is to just declare that you chop off the job there is some magical thing that happens which cuts off your space time before the Horizon you know what I'm saying so if you believe that then of course your Prime of AC Horizon okay but then all the problem the problem is basically to explain entropy all the thermodynamics all of these other things so those things are successes of black hole the idea that the black hole is a thermal system you know what I'm saying so we we cannot throw them off throw them out so another way in which people often Resort the information Paradox or at least attempts of certain kind which resolve information that is declare that this is smooth and then let's deal with various consequences but the problem is that the singularity is sitting very much there yeah and you have no mechanism for understanding it you know what I'm saying so like yeah so there are there are you know these are all fairly yeah I mean I you know there are many things to be said about all of these things so but I'm just saying that there are you know all that is I wouldn't say that there is a completely satisfactory resolution of the information Paradox yet that's I I you know it's just I I there's nothing so not nothing we have made progress in the last uh let's say 10 15 years and uh or even 10 you know maybe starting for the last 30 years maybe starting from a paper of Toft I would say we have steadily made progress but I don't think there is a fully satisfactory resolution so it's kind of like we are kind of seeing a blind man seeing the elephant kind of thing various pieces which seem interesting but none of them are kind of have fit together in a completely coherent satisfactory way the black holes are an open problem you know so any other question okay if there's no question we'll see on Tuesday we don't know that so I'm just I'm saying that Hawking assumes that that is the case and then he gets the and you know then he comes to a yeah