 You'll hear me OK with this, OK? Yeah, so as Paolo just said, that's the overall topic of my lectures. They'll be divided into four parts, although there'll be a fair amount packed into the first two. So if I don't quite finish, I'll have a little bit of time here. So what I'm going to do today is, well, give you basic results on event horizon, well, definition of black holes and event horizons and their properties, completely separate topic of killing horizons, which, however, then will get merged together with this with, well, the Hawking-Rigidity theorem. The next lecture, this will be mostly pictures. This will be mostly general formulas that arise from the Lagrangian formulation of a theory, the main goal of which will be to derive the first law of black hole mechanics and thereby also give a general formula for black hole entropy. That will be valid actually in completely general theories of gravity that are diffeomorphism covariant, not just general relativity. Then I'm going to get into a little bit more applications of this, in fact, showing that dynamic stability of a black hole is, in fact, equivalent to its thermodynamic stability. And then finally, although this is not, of course, in quantum field theory, and I'm not going to get deeply into the details of the quantum physics aspects of this, one can't complete the subject of black hole thermodynamics without talking about Hawking radiation and all of the implications that that has, which, I think, together with everything else here makes black hole thermodynamics one of the more remarkable developments in physics of the last half century or so. So I will get into all those issues in the last of the lectures. So I'm going to be starting with classical black holes event horizons and killing horizons. The most of the stuff that I'll say on that has to do with causal structure would be found in chapters eight and nine. And if not, the chapter on black holes, chapter 12 of my general relativity book. And again, almost everything that I'm going to say is in a living reviews article that I wrote also quite some time ago. OK, so I want to start with the general notion of a horizon. And maybe I should not rush through this too quickly so that people are in the mood of this. So we don't really need to talk about asymptotically flat space times or a special class of observers. If we have any observer, we can talk about the horizon of an observer. But I'm only going to consider observers as represented as in extendable time-like curves. So if we have a spacetime diagram here and the board here is the spacetime, I'm not going to allow my observer to die suddenly of natural causes or whatever without something going wrong in the spacetime. The observer will either live forever or maybe the observer will fall into a spacetime singularity. But I'm going to demand that the observer is, I mean, my demand is mathematical convenience for what I'm talking about. I'm going to demand that the observer is an extendable time-like curve. If the observer just suddenly died here at some event in spacetime, you could extend that time-like curve. His or her child could go on as the observer then. I'm not allowing this situation. And now we can consider the causal past of any observer. That's the observer now is gamma. I minus of gamma is, I'm sorry, the chronological past. It doesn't really matter too much for what I'm about to define. So let me not. So I can consider all events that are connected to the observer by a time-like curve and any event that's connected by a future-directed time-like curve to the observer world line defines what I mean by the chronological past. And now the future horizon of the observer is going to be defined as the boundary of this chronological past. So in this case, where the observer falls into a singularity, there'll be some boundary that I've drawn with the dotted lines that has the property that inside of here you can reach. Inside of here, you can reach the observer with a time-like curve, but outside of here, you can't. So that's going to be the general definition of a future horizon of an observer. And of course, you could define past horizons in the same way. Now, just from this property, not using Einstein's equation or any results of that sort, it follows that each event that lies on this boundary that I've drawn has to, in fact, lie on a null geodesic segment that's entirely contained within the boundary and is future inextendable. I mean, there might be some singular point here. But the idea is that these observers can again, they're not observers, these are null geodesics now, they can't just end suddenly. I mean, this is just immediately following from the definition of the boundary. And I'll talk more about the convergence later in a few minutes, but if we look at the boundary, the boundary will be a null surface. It'll, in four-dimensional space time, what I've drawn here is the line will be, in fact, a three-dimensional null surface. It will be ruled by these null geodesics. There's a notion I will discuss further of the convergence of these null geodesics. And the convergence can never become infinite at a point. You would have a caustic in the congruence then. And the reason you can't have a caustic is if you did have a caustic, then when you go beyond it, you'd be able to connect two points on that caustic by a time-like curve. But if you could connect them by a time-like curve, then you could find a point outside of here or an open neighborhood of this point that you could also connect to that event by a time-like curve. And that would contradict the statement that you were on the boundary. OK, so I just wanted to make sure everybody had a flavor of the kind of pictures that I'm going to draw and this basic result. So just some examples. I mean, if we just have an inertial observer in Minkowski space time, that observer can see everything in his past. And he's also in the future of every event. So there are no past or future horizons of an inertial observer in Minkowski space time. But even if we stick to Minkowski space time, there are nontrivial notions of horizons. And if you have an observer that's uniformly accelerating for all time, in fact, this observer can't see anything beyond this future horizon H plus, which would be a null plane coming out of the board. Again, if we were in four-dimensional Minkowski space time, and similarly, this observer will have a past horizon. These two will be two null planes that intersect in that two surface. And if we consider, again, just to be in the mood of this, Minkowski space time, but now that little open circle layer is supposed to mean I've just removed a point from the space time. It's still a perfectly legal solution of Einstein's equation. Then I can have an observer going into this non-existent point. That will be an inextendable time-like curve. And it will have horizons that look a future horizon that I've sketched. You'll notice in this case, I mean, this will basically just be the past light cone of this would be event without that event. The convergence is going to infinity there, but it never reaches infinity. So it's all completely legal in terms of what I said. OK, so now let's move on to black holes. So if we have, let's consider now, an asymptotically flat space time. Now, one can give precise definitions of what I mean by asymptotic flatness. I do need it to be asymptotically flat in null directions, not just initially in space-like directions. But the precise definition isn't really important for this. I mean, any reasonable, any definition you can come up with of what you mean by asymptotically flat at large distances and also at late times is fine. And we can consider in an asymptotically flat space time the observers who escape to arbitrarily large distances at late times. And we can look at the past of these observers. I mean, we can do these definitions for a family of observers just as well as a single observer. Actually, in this case, I'm really fine, even if I only consider a single observer that goes out to arbitrarily large distances at late times in a suitable sense. I'd have to be a little more restrictive in that. So why don't I consider all the whole family of observers? If the past of those observers is not the entire space time, then we say that a black hole is present. And the definition of the black hole is the space time minus the events apart from the events that are in the past of this family observers. And the horizon that I've just defined here, again, for this family of observers would be called the future event horizon of the black hole. Now, this definition doesn't say anything about bad things that might be happening outside the black hole. In fact, there could be singularities visible to these observers. But an important notion in the theory of black holes and an important belief in terms of it being true, and if it isn't true, then a lot of the theory of black holes becomes irrelevant or not related to physics. But I certainly believe it's true, at least in some form. So to explain what this cosmic censorship that's eliminating singularities is, I have to give a few more definitions. So this is a little heavy with basic definitions here. So a Cauchy surface, by definition, is some, well, it'll have to be a three-dimensional surface. And it has to be a chrono. It could be null as well as spacelike. But let me consider a spacelike surface. So again, this is a spacetime diagram here. So if this is the spacetime, what I've drawn here will be said to be a Cauchy surface if it has the property that if I take any inextendable time-like curve, it has to be now inextendable in the past as well as the future. If that inextendable curve intersects, what am I calling it, C, if that curve intersects C once and only once. Well, only once means the surface is a chrono. But if every inextendable curve intersects that Cauchy surface, that surface C that I've drawn, then C is said to be a Cauchy surface for the spacetime. Now, not every spacetime admits a Cauchy surface. The ones that do admit a Cauchy surface are said to be globally hyperbolic. There are a number of other equivalent definitions of globally hyperbolic. Non-trivially equivalent, it takes quite a bit of work to show they are all equivalent. One result that follows, well, again, not immediately, there's quite a bit of argument involved is that if you have a globally hyperbolic spacetime with Cauchy surface C, then the topology of the spacetime is simply r cross C. So this is sort of time cross C. And in fact, if you have a Cauchy surface in a spacetime, necessarily then globally hyperbolic by definition, then you can, in fact, foliate the entire spacetime with Cauchy surfaces. Again, I'm not claiming any of these statements on this page are obvious, but I am claiming that they're true. OK, so now that's the definition of a globally hyperbolic spacetime. Now what this has to do with black holes and with cosmic censorship is the following. So we have, again, now in this asymptotically spacetime, we have all these observers who are maybe out already at large distances, but at least they make it out to arbitrarily large distances at late times. But in the case I'm interested in, there are events in the spacetime that they can't see. So there is a black hole, and this is the future event horizon, or the event horizon, I don't really have to say future, of the black hole. So this is the case I'm interested in. Since the lectures are all on black hole thermodynamics, we are interested in the case where there's a black hole there. OK, but the black hole is going to be said to be predictable, or in other words, there are no singularities outside the black hole or no naked singularities that would be visible to the observers out here. If I can find, well, if I could find something that is a cosy surface for the exterior, including the event horizon and a little ways, it doesn't have to be far at all, but a little ways into the black hole. So the black hole is, the definition I'm giving is the black hole is said to be predictable if I can find some region of the spacetime while there are too many dotted lines here. So let me get rid of that and use some other color chalk here. So if I can find, it doesn't end up here, so I'm just, but if I can find a region of the spacetime, it might end down here because this black hole may have been formed by gravitational collapse. If I can find, and it doesn't end here, this is sort of infinity, but if I can find a region of the spacetime that includes the entire past of all these observers and the event horizon of the black hole such that that region itself is a globally hyperbolic spacetime. So again, there might be all kinds of nasty things going on inside here. That's fine. These nasty singularities that I'm drawing are not visible to these observers. This would be called a predictable black hole spacetime. And that is a precise expression of the idea that there aren't any naked singularities present. OK, so what does this have to do with cosmic censorship? Well, if we start with asymptotically flat initial data maybe way down here, before there's been any gravitational collapse to form a black hole or anything, the cosmic censorship conjecture says that if I consider the maximal Cauchy evolution of, I'll make this my initial data surface C. So this is in line with Piotr Kruszel's initial data surface. And I would imagine he will talk about maximal Cauchy evolution as soon as I'm done with this lecture in his second lecture. So if this maximal Cauchy evolution, which automatically by definition is a globally hyperbolic spacetime, generically, you have to always put in generically because there are known counter examples and you have to also put in appropriate matter fields because you can do things with dust or fluids and so on that are not nice. But if the maximal Cauchy evolution extends long enough to give you, well, a complete null infinity. So these observers who are staying far away from all the gravitational collapse, well, of course, they could accelerate out to infinity. They could accelerate also so fast that they only live a finite amount of time. I mean, they can't do that with a finite amount of fuel and all that, but if these observers near infinity who are essentially inertial, if we consider observers near infinity who are essentially inertial, if they live forever, then cosmic censorship is said to be satisfied. It's formulated this way because what we're trying to, what one is trying to eliminate in here is that you get some singularity outside the black hole. But if you got some singularity outside the black hole, then the domain of dependence or the maximal evolution of this data would not include anything to the future of this singularity. And then these observers would not live forever even though they're behaving inertially at large distances. So this idea eliminates is a precise and convenient, mathematically convenient way of formulating the idea that there are no naked singularities outside the black hole, but nothing is being restricted in terms of bad behavior inside the black hole. OK, well this, I mean, Mahales de Firmos will be talking for a whole week on this and a somewhat related strong cosmic sensor hypothesis next week. So I won't say anything further here about that, but I'm going to assume that this is valid or if you like, restrict consideration in the rest of the lectures to predictable black holes. But I believe that every black hole that you'd form in nature is predictable, at least generically. OK, so it might be worth spending a minute or two before getting into more. Sorry, yes. Asymptotically ADS, there would be no difference. Asymptotically Desider, then you'd really have to decide what observers you were considering with in terms of defining the black hole. I mean, the Desider spacetime has a compact Cauchy surface and it's sort of not obvious how you divide into what's inside the black hole and outside the black hole. But ADS, if you, I mean, since I didn't give you the precise definition with null infinity, if you replace null infinity by the ADS boundary, there'd be no change whatsoever in the definitions. OK, so it's worth having a little bit of a spacetime diagram picture in mind of what is going on in gravitational collapse that results in a spacetime like I've sketched here. Though it's a little, I think this way, drawing it this way gives a bit of a clearer picture of what the black hole or sort of the localization of the black hole than this diagram, though this diagram, which I'll show you in the next figure, gives a much clearer picture of what the horizon and singularity look like, at least in the spherically symmetric case. So I'm showing the simplest example of spherically symmetric collapse, which one really knows everything relevant about exactly because exterior to any spherically symmetric collapsing matter is the Schwarzschild solution and we know how the Schwarzschild solution behaves. So here I've shown collapsing matter, these are spherical, this semicircular looking line there is the spherical outer surface of a star or some other collapsing matter and that matter just collapses down to zero radius where I possibly should put radius in quotes. It's not in quotes in terms of telling you area of spheres. It's certainly in quotes in terms of sort of suggesting that it's an origin of coordinates. I'll show you that in the next diagram. And all the matter falls into a singularity is created when r equals zero is reached and all the matter falls into that singularity and disappears forever. The only way in this diagram you can tell that there's a black hole here is if I draw in the light cones which in this way of representing things are all tilted over toward the singularity when you're near the singularity. And what this is telling you is that any observer at this event is gonna fall into the singularity and any light ray at this event is gonna fall into the singularity. So any observer out here is not going to see anything coming from an event well inside this dotted cylinder as I've drawn it here. On the other hand if you're far enough away from the singularity then you don't have any problems sending light signals out to an observer so this event is not inside the black hole. The boundary shown here is the event horizon. Now I think that gives a sort of clear physical picture of sort of the collapse but it doesn't, but the light cones are all tilted over. We can redraw this and it'll look much more like what I've drawn here but only if we suppress, I don't know of any nice way of drawing this without suppressing all the angular directions but if you do that, so we're down to just the sort of radius and time in quotes, a two dimensional spacetime where each point in this spacetime that I've drawn here is actually a two sphere. Then I can straighten out the light cones and you can see easily in this picture that the event horizon is a null surface as it better be, it's ruled by null geodesics. What you can see much more clearly in this picture is that the singularity inside the black hole, though I'm not gonna be worried too much about anything going on inside the black hole in these lectures, but that's space like in the case of this spherical gravitational collapse and you can sort of see why you can't send light signals out to distant observers here because all the observers and their light rays will eventually fall into the singularity, the singularity is a kind of end of time. Okay, so now I wanna get onto, first major result that will be interpreted as the second law of black hole thermodynamics, the analog of entropy increase and I already alluded to the convergence or expansion, I guess is what I'm defining here, the expansion is minus the convergence of the null geodes, well this could actually be an arbitrary congruence of null geodesics, they don't even, at the moment for what I'm writing down, they don't even have to be surface forming but I just have null geodesics filling up spacetime and if I, so I'll let their tangent, I'll let them be affinely parameterized and their tangent with that affine parameterization be K and then the spacetime divergence of K is what I'm defining to be the expansion, so if they're diverging in this precise sense, then the null geodesics are expanding. Now if we consider, well, an area element that let's consider now, since I'm gonna restrict to that almost immediately anyway, just the null geodesics generating a null hypersurface, this doesn't take any derivatives off the hypersurface, in fact, so this is still perfectly well defined even if we have null geodesics only generating a hypersurface and if I imagine, so again in four spacetime dimensions, the null hypersurface is three dimensional, a cross section will be two dimensional and if I look at an infinitesimal area element that is carried along by these null geodesics, then in fact, the expansion as I've defined it here is just the derivative with respect to affine parameter of the log of the area carried by these, area of an element carried along by these null geodesics. Now again, if the null geodesics generate a hypersurface, the so-called twist, which would be the anti-symmetrized derivative, so if I had these indices be different and anti-symmetrized, that would essentially define the notion of the twist, but if they're surface forming, the twist will in fact vanish and when the twist vanishes, there's a remarkable equation that underlies 90% of global arguments in general relativity, the Rachiduri equation, which is just basically a version of the geodesic deviation equation for the congruence, which it's just a formula that relates the Ricci curvature, which is what's gonna come in when you average over the geodesics, I mean the Riemann tensor comes into the geodesic deviation equation, but when you average over them to get the area element instead of just the geodesic deviation vector, the Ricci tensor comes in and the Ricci tensor is related, this is just, this is not using Einstein's equation or anything else, this is just basic differential geometry of null congruences, it's related to the expansion and the shear, there would have been a term if I had the twist in there that would come in with opposite sign and spoil any argument, but we're interested in the case where these are surface forming, so you get this formula, well it's really a formula, well it's the geodesic deviation equation averaged over the null geodesic surrounding the one that you're following and what's absolutely remarkable about this equation is if Einstein's equation holds and this is one of the few places where I'm directly using Einstein's equation with a stress energy tensor that satisfies positive energy, in fact all we need, I mean positive energy would, positive energy density as viewed by any observer would be saying non-negative energy density as viewed by any time like observer would say this for any time like curve, we don't even need that, we just need this to be true for null curves, yeah, oh okay, yeah or maybe I'll write it more toward the middle, anyway the condition that, so if this were true for any time like vector by continuity it would be true for any null vector but we only need this in fact to be true for null vectors, in fact if you have a cosmological constant it's quite important that you're only needing this for null vectors because then the cosmological constant contribution which is proportional to the metric doesn't enter at all so this holds for both signs of cosmological constant so if that holds and Einstein's equation holds then this term is non-negative, this term is manifestly non-negative, well this term, yeah with a minus sign here, okay we get an inequality immediately from this, this also has a definite sign but I wanna keep that term, we get this remarkably simple inequality which you can also write as the lambda derivative of one over theta is less than or equal to or yeah is less than or equal to one or one half or whatever the right algebraic expression is I guess it'll be a half so you can integrate that inequality and get this result well what's remarkable about this result is that it says that if you're initially have negative expansion so you're converging then in fact one over theta is gonna have to go to zero within a finite affine parameter that means the expansion, I think I may have assigned the sake in this, no it's this thing is negative so this goes to zero that means the expansion goes to minus, it's correct as written, the expansion goes to minus infinity at a finite affine parameter, okay so one immediate consequence of that well you can do better by improving this argument as I'm indicating here is if we look at the event horizon of a black hole it's a horizon it can't have infinitely negative expansion at any point along that so if one were to assume which there's no reason really to do but we can do better than this as I'll explain in a second if we assume that the horizon generators are complete it means that all of these generators have to be expanding because if they started, if they were converging anywhere they would converge infinitely in finite affine parameter which would contradict this being a horizon well that argument is not really optimal because the horizon generators might not be complete the predictability assumption doesn't guarantee that the horizon generators are complete but you can actually make a nicer argument by arguing if the expansion were negative somewhere then you could deform the horizon there outward so that it would be visible at infinity but you'd still have negative expansion but now because it's visible at infinity you'll run into a similar contradiction in this manner so that leads to the following result which I should probably draw a picture of I think I can just do it under here so if we have the event horizon of a black hole here as I'm drawing and we have some initial Koshy surface for the exterior region including the horizon which we do assuming we have a predictable black hole which we will have for sure if cosmic censorship is true so this will intersect the horizon in some cross-section which I don't know if I gave the name S1 to and if we go to some later time later Koshy surface in the foliation and look at the intersection of that Koshy surface with the horizon that will be some surface S2 and now there are two things going on that go in the same direction the generators of the horizon are sorry I shouldn't have said future complete I should have said future in extendable so every null generator that was on the horizon here will continue up and still be on the horizon there now there may be new null generators that form in between but the consequence is that the area of this surface here A2 right that big because this is important the area A2 of the surface here is gonna be bigger than the area of the surface down here for any two surfaces well for two reasons there are the null geodesics that came from here but those have positive expansion so they're carrying at least as much area and then there may be new null geodesics that arise here that's only gonna add to the area even more there aren't any null geodesics that were down here that didn't make it up to here on the event horizon because of these basic properties of horizons that I discussed in the first five or 10 minutes of the talk okay so that gives the area theorem the surface area of the event horizon of a predictable black hole never decreases with time okay now I'm gonna switch gears to what ought to appear initially like a completely different topic but everything will get tied together I hope by the end of this talk I guess I still have another half hour so there's a reasonable shot at that so I'm gonna now talk about symmetries and well an isometry is just a diffeomorphism that when you carry the metric along with that diffeomorphism the metric goes into itself so it's a symmetry and if you have a one parameter group of diffeomorphisms you'll have a vector field generator of it and that generator is called the killing vector field with a capital K for the name of the person it's not murdering anything particularly even though it is kind of killing the metric and so the change of the metric in some sense and the equation of a killing vector field is just that the symmetrized covariant derivative of the killing field with its index lowered by the metric is zero now yeah let me denote the so the symmetrized first derivative is automatically zero so the only thing for first derivatives that comes in as the anti-symmetrized first derivative and let me use capital F to denote that now a very important property of killing vector fields and isometries is that they're completely determined by well the killing vector field in particular is completely determined by knowing at a single point P the value of the killing vector field and the value of its first derivative now that may sound remarkable but if you know if I give you a I can figure out where the isometry is gonna move every other point from this information one can see because I can connect well let's imagine just for the sake of simplicity that I can connect every other point to P by a geodesic that's not necessarily true but that but you can modify this argument slightly to so if I wanna know where, how does an isometry so here is my fixed point P here's my other arbitrary point if I wanna know where does this point go under the isometry all I have to figure out is what happens to the geodesic that connected them but to figure out what happens to the geodesic all I need to know is what happened to the tangent to the geodesic at the starting point but to figure out what happened to the tangent to the geodesic well okay let's consider the case where I don't move point P either so the killing field vanishes at P sorry for garbling this argument if the killing vector vanishes at P then where the geodesic goes is completely determined by the derivative of the killing field because that's sort of telling you what happens at the tangent space at P and then you can figure out where the isometry goes anyway that was a bit garbled but that was just giving a explanation of why it's not completely insane that killing vector fields are completely determined by their value at a given point and the value of the derivative at that point so let me now consider the case that I was already jumping ahead to consider what I was just talking about to the case well let's restrict to two dimensions I'll go up in dimension easily in a little in a moment but let's consider the case where the where a killing field vanishes at some point P here's the point P then the anti-symmetrized derivative in two dimensions has to be proportional to the volume element at P so in fact the killing field is unique up to scaling there can't be more than one killing field that vanishes at P and you can easily figure out what goes on I mean you can figure out easily what goes on in the tangent space this corresponds to an infinitesimal rotation but that means that also goes on in the space time if you follow along with geodesics so if we have a Riemannian metric I'm interested in the Lorenzi in case any killing field that vanishes at a point is going to have an orbit structure like what I'm drawing here it's going to be a rotation and this will be the rotation axis a rotation point since we're only in two dimensions I think everyone is familiar with this what I find somewhat outrageous is in the Lorenzi in case most people I mean who have not gotten deeply into general relativity I mean this should be you know a major if there's going to be like a course or several weeks on special relativity I mean it's hard to imagine a course on Euclidean geometry where you wouldn't introduce the notion of a rotation and draw this kind of picture for what a rotation looks like well this is the corresponding picture in Lorenzi in geometry so if you have a killing field that vanishes at a point then in the tangent space this is again just two dimensional the symmetry is a Lorenz boost and this is the orbit this is what I find outrageous that people don't automatically know what the orbit structure of Lorenz boosts are so it's the Lorenzi in analog of a rotation but now these orbits are not closed they're not compact or anything they have an orbit structure where they're for the sign of the killing field that I'm choosing they're future directed in this sort of right wedge past directed in this left wedge space-like in the direction shown in these future and past wedges and the killing field is null on these horizons that I'll they're not at the moment horizons in the sense I introduced them they are gonna be killing horizons in the sense that I so the killing fields are null and generate two intersecting null surfaces that vanish at this point where I you know where which was the special point where the killing field vanishes so now if we go up to four dimensions or whatever dimensions you want if we make the field vanish on a co-dimension two surface we get similar results and the case I'm interested in is where we have a Lorenzi in metric and this co-dimension two surfaces space-like then again the orbit structure near that surface is gonna be just like this and the pair of intersecting null surfaces that you automatically get from this is what I'll call a bifurcate killing horizon the killing field is then normal now normal means tangent for a null surface I mean normal implies tangent in fact tangent precisely implies tangent to the null geodesic generators of the horizon so this in this bifurcate killing horizon case the killing field is tangent to both of these horizons more generally if we have any null surface that has the property that a killing field is normal to it I will call that a killing horizon okay so now for a killing horizon an arbitrary killing horizon it doesn't have to be a bifurcate killing horizon one can introduce the notion of surface gravity it's convenient to introduce an affine parameterization of the null geodesics I mean an arbitrary affine parameterization of them and I'll let capital U denote the affine parameter of them and we can also parameterize the null geodesics by the killing parameter I don't seem to have defined little u anywhere here but the killing parameter is defined by derivative I mean this is also a derivative along the null geodesics but I've normalized the null tangent to be the killing field here and this isn't equal to zero this would be equal to one so that defines the killing parameterization the affine parameterization again let's have that be one would be defined by the null geodesics would be one would be defined by moving along uniformly in time with respect to the affinely parameterized tangent well both of these both the affinely parameterized null geodesic tangent and the killing field are tangent to the same null geodesic so they're proportional to each other by some function of proportionality on the horizon the surface gravity is defined well as I've written it here as the derivative of the log of this proportionality function with respect to the killing parameter a better way of thinking about it is that the surface gravity is telling you how non affinely parameterized the null geodesics parameterized by killing parameter are kappa I mean if C was the affinely parameterized tangent then the geodesic equation would be telling you this is zero but if it's non affinely parameterized then it's proportional to the tangent and kappa is the function of proportionality well because C is a killing field it immediately follows that kappa has to be constant along each null geodesic generator but in principle kappa could vary from generator to generator now because it's constant on each generator there's you can integrate well F is that and kappa is that and kappa is independent of little u or capital u so you can immediately integrate to get the relationship between the killing parameterization of the null geodesics and the affine parameterization and you get this exponential relationship so actually notice that when the killing parameter runs over its full range of minus infinity to infinity the affine if the kappa is non zero the affine parameterization is only going to run over a half line there's the reason for the name surface gravity comes from the following formula that's not takes a few lines to show with manipulations using killings equation and so on if you look at the if you were to normalize the okay outside I guess let me back up even one more slide so for at least for a bifurcate killing horizon although now we're talking about general killing horizons but I'm gonna relate that to bifurcate horizons very soon the orbits are time like in a wedge outside the horizon I can look at the I can look at unit norm observers following these orbits and look at their acceleration that acceleration is gonna diverge as I approach the horizon it's very hard for observer time like observers to move like null objects on the horizon without the acceleration going to infinity but the norm of the killing field also goes to zero and it's not hard to show that the product of these two is the surface gravity now the acceleration of a time like object like this I'm not gonna drop it on the surface of the earth well only the people familiar with general relativity understand that this thing is accelerating right now if I were to drop it it would not be accelerating so the acceleration of this object is if it were unit mass or if you divide by the mass is what in freshman physics you call the surface gravity surface gravity defined that way for a black hole without the redshift factor would be infinity but the surface gravity is what I the finite answer that I was talking about well we haven't connected black hole horizons with killing horizons yet either so I am kind of jumping around a bit but anyway that product of the redshift factor in the acceleration is acceleration of the killing orbits times the norm of the killing field limit as to the horizon does give you the surface gravity okay so as I said in principle the surface gravity could vary from generator to generator the zeroth law of black hole thermodynamics which is the analog of the zeroth law of thermodynamics zeroth law of thermodynamics in case your course started with the first law or something is the statement that the temperature is uniform for a body in equilibrium okay well the zeroth law is that is basically the statement that the surface gravity is constant well on the event horizon of a black hole but we haven't gotten to that yet because that isn't a killing horizon yet it's about to be sorry for the disjointed versions of what I'm telling you so there are in fact three separate independent results that talk about the constancy of the surface gravity on the horizon so the first one is that if you have any killing horizon at all in a space time where you have Einstein's equation with the dominant energy condition now so that's a stronger energy condition than the null energy condition if that is the case then the surface gravity has to be constant on this killing horizon this is actually a fairly non-trivial I mean this takes a pager to prove it's not entirely straightforward it's so any killing horizon that arises in general relativity where matter satisfies dominant energy condition is going to have constant surface gravity there are a couple of other versions of this law that actually don't use Einstein's equation but make other assumptions if you have a killing horizon where either the killing field is hypersurface orthogonal the static case so this is in four dimensions that I'm stating this for at least with respect to the second condition or you have a second killing field you have axi symmetry that second killing field commutes with the original one and well you have a reflection isometry with respect to the two killing fields then in fact you can show the surface gravity has to be constant so that's a little more complicated but it doesn't use Einstein's equation I think it doesn't use Einstein's equation the third one definitely doesn't use Einstein's equation which is if you have a bifurcate killing horizon then it's actually fairly straightforward to show that the surface gravity has to be constant that does not use Einstein's equation at all okay but now with one additional result hopefully I'll have time for questions if I'm confusing people with the array of things that I'm throwing at you right now but hopefully this is reasonably clear if you have a bifurcate killing horizon as I just said the surface gravity is constant but it's also possible to show again not using Einstein's equation that if the surface gravity is constant so you don't know that it's a bifurcate killing horizon but if the surface gravity is constant and non-zero then in fact the horizon must be essentially a bifurcate killing horizon I mean maybe somebody only gave you one portion of the bifurcate killing horizon not including the bifurcation surface so maybe for starters somebody just gave you a space time that contained this fourth of a bifurcate killing horizon but the claim is here that it really was a bifurcate horizon you could always extend the space time if necessary to make it a bifurcate killing horizon so since by using Einstein's equation only killing horizons with constant surface gravity will arise this statement says that given that the surface gravity is constant the only cases that really need to be considered are the cases of a bifurcate killing horizon and the case where the surface gravity is zero that will not correspond to a bifurcate killing horizon so only these so-called degenerate horizons are the only ones that we really have to concern ourselves with respect to killing horizons in general relativity okay so now what does this have to do with the first half of what I was talking about with event horizons because I've only been talking about killing horizons well the major result that ties the two together is what's often referred to or usually referred to as the Hawking rigidity theorem so if you have a stationary black hole so more precisely a stationary asymptotically flat space time again you've got to have appropriate matter it really it certainly needs hyperbolic equations and so on describing the matter that contains a black hole then in fact the event horizon of the black hole so the thing I was drawing here and showing you pictures of early on when the black hole is stationary but one would expect it will asymptotically approach a stationary final state so this should be describing state final states asymptotic final states of black holes the event horizon has to be killing horizon so there has to be a killing field orthogonal to the event horizon of the stationary black hole okay so right and given what I just said because of the constancy of the surface gravity it further implies it follows immediately that the black holes we need to consider in general relativity are ones whose asymptotic final states can be described by bifurcate killing horizons together with these black holes with zero surface gravity most of what I'll be talking about in the next lecture and so on will be considering these asymptotic final states of black holes that have a bifurcate killing horizon okay so an important implication of this is while there are really two cases when I say the event horizon has to be a killing horizon of course that means there's a killing field orthogonal to the to the horizon well I assume that there was a killing field already when I said we had a stationary black hole so there are two possibilities maybe the killing field is in fact the killing field that's normal to the horizon must be maybe one possibility one is it's the killing field that we already started with in that case by further arguments one can show that in fact that killing field is hypersurface orthogonal meaning the black hole is in fact static there's a time reflection symmetry associated with it as well as the time translation symmetry and that's important because for black hole uniqueness Israel had before this theorem before the Hawking result had already proven that the Schwarzschild black holes are the only static black holes if it's if the original killing field is not normal to age to the horizon that means there's another killing field around and it can further than be shown that some linear combination of the killing fields has to have a close space like closed orbits so in other words if the black hole is rotating because of the stationary killing field isn't the one that points along the null generators one would naturally say the horizon is rotating uh... then in fact the uh... rotating black hole has to be axi-symmetric and so you can choose the normalization of the horizon killing field so it's the stationary killing field plus some multiple constant multiple of the axial killing field and this is the angular velocity of the horizon that Nico uh... that came up uh... well in several contexts in Nico's uh... talk this morning so this is what the idealized kind of continued general black hole stationary black holes state well general except for the case of the zero surface gravity black holes uh... you'll have a region outside here of time translation uh... uh... with the orbits of the time translation symmetry the original time like uh... killing field may go space like near the horizon as Nico also discussed giving an ergo region here and then in this continued space time you'll have the black hole region but you also can continue it to a white hole region and uh... and a new universe i mean if you've seen the crustal diagram for schwarzschild that's uh... this had the general black hole with a bifurcate killing horizon will have a similar structure not necessarily up here with the singularity but with the horizon uh... that may look weird but of course this is what's going on in minkowski space time relative to lorenz boosts i've already shown you the orbits of lorenz boost symmetry they look the same and one thing to very much keep in mind with black holes is if you consider a very large black hole the curvature picking a particularly large one uh... curvature at the horizon is smaller than the curvature in this room so an observer uh... who falls into a black hole would hardly be able to tell he or she is not in minkowski space time falling if you like through a killing horizon uh... perhaps uh... i mean we're all falling through killing horizons right now right at this moment if we approximate our space time as minkowski i mean really should not be able to tell by any local measurements that you're not in minkowski space time of course uh... a few weeks later this observer is very much going to notice uh... that he or she is not in minkowski space time and it's going to be far too late to do anything about it okay so i and that the summary slides i looks like i'm actually going to finish this on time so let me since there are a few parts of this that sounded a little incoherent to me let me uh... uh... summarize uh... everything together so cosmic censorship holes starting with any non-singular initial data at least generically if you have gravitational collapse that is you know will form a singularity i mean penrose theorem and other singularity theorems will guarantee you that under conditions where you form the trap surface which i haven't talked about here all the singularities uh... will be hidden in a black hole and your space time outside the black hole will remain perfectly predictable the area of the event horizon of black hole is will be non decreasing with time if you form a black hole it would be natural to expect that in fact rather quickly i mean timescale of milliseconds uh... for a solar mass sort of black hole uh... will quickly asymptotically approach what you might call an equilibrium final state in the thermodynamic analogy i think the equilibrium is a good term but a stationary final state this final stationary black hole the event horizon of it will be a killing horizon will have constant surface gravity and if it has non-zero surface gravity it can be extended to have bifurcate killing horizon structure okay that's it for what i wanted to say