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.  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.  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.  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.  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


Thanks to its status as one of the most developed and stable countries in Latin America, Panama is on many people's radar as a possible destination for teaching English as a foreign language. This guide gives a brief introduction to the country, including some general facts and figures, advice on visas, vaccinations and climate, and some suggestions of popular local food and drink to sample during your stay. It also offers some tips on places to go and things to do during your free time, such as exploring the sun-blossed islands, roaming the streets of picturesque boquete and enjoying a unique party on a converted school bus. Outdoor pursuits including surfing, scuba diving, snorkeling and jungle tracking are also popular pastimes in Panama.


Everybody forced here ripping one box of 2022 Or 2023 actually it is 2023 revolution WWE hobby This one's for Anthony B. Good luck Anthony and thank you for the order JC Jane Goldberg. There's a groove of Finn Balor a Finisher's Austin theory. You've got cubic a lot a lunge ablaze Number 33 out of 49 legends a lunge ablaze On the cubic cold air. There's an astro Becky Lynch rock Lesnar on the stargazing with a That's sick new age outlaws Billy Gunn and a roadhog on the groove and then an auto is Kevin Owens Very solid box here Groove of JD McDonough Rookie that is a brawn breaker on the shockwave Got astro Yokozuna. All right, there's another tag team card. It's iron chic and Nikolai Volkov That is a Brock Lesnar to 149 cosmic Nice Brock. It's a good box Stargazing a Rikishi and an astro of Apollo cruise All right, Anthony that'll do it for the box appreciate it man. We'll get these out to you


Hey guys, I am back and today I'm going to be doing a TBR video. If you watched my last video, I did recently talk about the few books that I have read and on that pile was Red Rising. So towards the top of my TBR, I do have the other two books in a series. There is a fourth one that is out now and I do own that but it doesn't match this series so I'm not putting in this video as punishment. But we've got book two, which is Golden Sun and then book three, which is Morningstar. I really enjoyed this. This is a really cool series. If you guys want to look into it, I really recommend it if you want to know more about the series. Go and look at my last video because I don't think people want to hear me talk about this book for probably like the third time in a video and usually I don't stop talking about it. But I did really enjoy it. The first book I gave five stars and it's probably one of my favourites of the year so take a look at it. Next up I have The Love That I Have by James Maloney but basically this is a historical fiction romance that follows a young girl called Margot who is German and she loves Hitler and she goes to one of the camps and she's working in the mailing room and one day her boss tells her to burn all of the letters and she is so confused because she thought that these were good people so why would they burn the letters? And that's when Margot begins to realise that things aren't exactly how she thought that they were and she begins to question things, especially when she does steal a couple of those letters and she starts to read one from a young man who's inside of the camp and he's writing to his girlfriend who has the same name as Margot. And the story follows that and I think I'm probably going to cry during it. I also helped to run the YA book club at Demex in June's Love but I definitely recommend checking out these book clubs especially like because I'll be there but one of them is Hive by AJ Betts and I actually got to hear her speak. She actually signed this book and like as a bookmark I'm just using a lottery ticket. Let's hope that I win at $100 million guys but in this we do have it. It's signed by AJ Betts and I really love the idea of the story. It was not what I was expecting at all. It's basically by hanging her called Haley and Haley looks up to the bees in her city and one day she's walking and she finds a drip from the wall and she's wondering what is this drip because Haley actually lives in an underwater city but she doesn't know that she lives in an underwater city and she becomes curious about this drip and she begins to ask questions that you're not supposed to ask and that is how this story begins. And I'm really am excited especially because this is a love Oz YA author and especially because I've met her and she's amazing to listen to talk to and I recommend her other books as well. I also really feel like reading a trashy romance at the moment and I've heard Colleen Hoover's new book is an amazing but I can't judge it fully yet because I haven't read it myself so I'm going to read Confess. I can speak faster if you guys want me to but I will slow her down. It's one of those typical romance books with a small twist in it but it has artwork involved and characters that are just very typical and I just I love it. I love trashy reads. Like sometimes I want to pick up a book that I know the couple is going to get together in the end and people are going to be happy and there's romance and you know just don't you book that sometimes because that's what I feel like. I just need to read some good trashy romance novel soon. But I also do want to really read some like dark fantasy paranormal books and Maria Lewis is going to deliver that for me. Well you have the main character whose name is Tommy and Tommy's mother has recently died so she must return to New Zealand where her father is and her family where she learned some dark family secrets and she's not just of normal human bloodlines that she assumed that she was so later she now returns to her friends back into normal city and she has learned to keep these secrets closed while also dealing with them and it is very dark and very plot twisty and I hope it's going to be very very good. A second last one that we got is Us Against You by Frederick Bachman. Frederick Bachman is an amazing author. I read the scandal by him at the side of the via and it was probably one of my favorite books so far and this is the sequel to it and I'm really excited and I hope that it's going to live up to the same pedestal that I keep the scandal on because I did really enjoy it and I really want to see what happened to this town and the characters involved in the story as well. Scandal was one that I found in the crime section at work. It was a really pretty cover and I was like I need to buy this because it is pretty and I did and the story was phenomenal. It was a very slow paced character driven book but it's basically set around a scandal that happens in a small town and you have a boy, you have a girl and a scandal that forms and whose side the town decides to take and there was just so much injustice in this book. This story wasn't just words written on a page just a sign that happens to people in everyday life and the book itself just like opens people's eyes and it's just so good and amazing and wonderfully written as well and I definitely recommend it. I really need to work on how I talk about books because a lot of it is just shouting words aimlessly and hoping that they form sentences because every time I edit a video it is hell.  It is called City of Lies. It is a poison war novel by Sam Hawke and I'm actually really excited to read this. I didn't expect this book to come into my radar and the moment that I picked up and read about it I just knew that I would have to read it soon. Even though it is huge and intimidating I'm actually super excited to dig into this. So in this book you have a poison taster whose name is Jovan and his purpose in life is to basically protect the chancellor's son from any possible poisonings and so when the chancellor is taken down and poisoned it becomes a chancellor's son job to step up into his father's role and Jovan then must protect the chancellor and any possible threats and try to discover who killed the original chancellor and I think it's going to be pretty hectic dark and twisty and I hope it's going to be one of those really crazy fancy books where you're just like will anyone survive? I was seven years old the first time my uncle poisoned me. We all have a book that starts with a bit of poison and possible murder. So those are the books that are on my immediate TBR that might change but those are the ones that I'm gonna get through next unless something amazing comes out soon but those are kind of like the ones that I've been eyeing off for a little while so hopefully I'll read them and hopefully you guys will hear about them in my wrap-up video that I'll probably do in like three, four months time but we shall see. Let me know how you guys are doing at the moment like are you guys all okay? Like have you watched any good movies recently? What you've been doing? Tell me about it. Thank you guys for watching and hopefully we'll be back again soon with another video although I don't know when I will be but it'll happen eventually and it'll be something fun I hope. Bye! I think I actually cried in total about six times during the entire book and it was amazing. One time she punched me in the face. It was awesome.


So, I realize I only have a little bit of time, and that's an ambitious subject, but I think I can handle this, because for those who don't know me, Yosef Mendelssohn, there's a quote from me in the back of your brochure, thanks to Ben at Hash Rocket. I don't even know what that's all about. So, what I need to talk about is what is truth, we need the nature of truth. This guy, I'm not going to say his name right, it's Epimidides, if you've ever heard of the liar paradox, this is somewhat attributed to him. He was a Cretan, he said something like, the Cretans are always liars, it's a little confusing. This is Immanuel Kant, who looks angry and talked a whole lot. This is Michel Foucault, who has an awesome name. He also had some stuff to say about truth. This is apparently truth, according to this dude, who also has a very strange name. It's Jules Joseph Lefebvre. He was Fredge. I can't actually go over the nature of truth in 20 minutes, half an hour, five years, but I can talk a little bit about truth and ruby, which confuses some people, especially people like me, like Brian, I came from Pearl, awesome. I wrote code exactly like that shit, years ago, embarrassed still. So I did a quick Google search and people seem to want to talk about and explain what truth and ruby is. I think it's a little confusing, I mean, there's true, that's very confusing. That's totally true. That is where the pro guys start getting confused. This means ruby is smarter than any philosopher you've ever heard of. It has no problem with that. It's true. That's true. That pisses people off a lot and I can't really go, that slide could take forever if I really wanted to fill it out, but there's some things that are not true like false and nil and that's it. That's the part that really confuses the hell out of people, which is why I'm giving this talk. So when you have some code, people have done that sort of thing. Maybe it's like ARR, empty, hash.empty, that sort of thing, but then you start writing stuff like this and this isn't a Rails conference, but people do Rails a lot, so you end up with stuff like that. But I don't really want to talk about all that, I just want to focus on this. How do you know if you want me to go back a couple slides? We'll have a private showing later. So how do you know? It's really not that hard, but people still get confused. So that's what I'm here for. Remember this. So here's some examples. The first couple, it's like these are the ones, if you don't know, I can't really help you out, because everything's just going to come back to that. So if you're having any trouble, you might want to leave now, but I want to talk about whether things are true, whether they have truth, whether they have truthiness. So the other day just came to me. I decided to write something to help everyone to solve the problem once and for all. So GitHub, of course, you can't actually install it from GitHub. You have to use Rubyforge, because that's the way I roll. I don't want to make gem specs. I just rake release. So we have this. And if you're confused, no, the specs come, hold on. I wrote the specs first, but I'm showing the specs.  So we have this example again, but truthy helps you figure it out. There you go. This is why if you don't know these two, you're screwed. So speaking of the specs, this was actually, it took the longest, and it's not complete. If anyone wants to actually fork this project and add more specs for other things that need to be truthy, I mean, I'm willing to accept all of this. There's also, if you want to donate to the project, I appreciate that, too. And I just want to leave you with one more thing. I mean, thank you.


We had a patient by name Ramalinga Reddy who is 64 year old Jaitumran who was working in his firm who had an accidental fall from height following which he sustained injury to his neck and immediately after the injury he was not able to get up and move his limbs and he was taken him to a local hospital where they met multiple doctors and they are diagnosed to have cervical spine injury in the form of C6 C7 fracture dislocation with para quadriplegia and from there they were referred to a higher center to Manipal Hospital Whitefield. There is the emergency department following which our emergency team doctors with all accidental protocol they followed and got an x-ray MRI and CT scan which was suggestive of C6 C7 fracture dislocation. After obtaining written consent explaining the risk pros and cons involved in the surgical management of the fracture dislocation we have taken the patient into an operation theater within 24 hours of the injury which has happened to the patient and by using advanced technology in the form of intraoperative neuro monitoring to assess his neurological status intraoperatively and whatever the fracture dislocation with was there it has been reduced and it has been addressed both anteriorly and posteriorly by fixing his fracture dislocation with screws and rod. So next to Manipal Whitefield we took a look at the emergency appointment and after that Dr. Rajekumar came and explained the entire procedure and he explained the injury severity also. Next to SARA the procedure was completed so after surgery he is very good. Before surgery he felt very pain every pain is not able to tolerate pain. So after surgery he is really doing very well and even he is recovering very fast. The patient has recovered from the weakness immediately after the surgery he could able to move his all four limbs immediately following the surgery. For an observation we have shifted him to an ICU in ward with the help of our physiotherapy team we able to walk make him walk with on his own feet and he could able to walk with minimal support and he has regained from quadriplegia and now he has regained his power and getting back on his own feet. So I would like to say thanks to Dr. Rajekumar as spine surgeon for just a phone call away is not too far for me so I'm happy with my doctor. Even last Sunday also we just called he reached to the hospital and helped us within a matter of 30 minutes I was really happy with that and thanks to entire Manipal admission department and emergency department and nursing and supporting team and everyone one second very thanks. We at Manipal white field have good supportive team like emergency team, ICU team, operation theater with advanced technologies including neuro monitoring, interoperative neuro monitoring and all advanced technologies are available here which we could cater the complex cases like our Ramalinga Reddy who had very bad cervical fracture dislocation which all such kind of major surgeries with good results we can cope with all our efforts with multidisciplinary approach. Thank you.


guys liquid just came in the mail the gold play button from YouTube dude this thing is enormous it's like my whole torso it's pretty insane I just want to say thank you guys so much for the support on the channel those that comment those that like the videos all the good stuff you know that boosted the channel and I watched the videos just to watch them for me you guys are awesome I appreciate it and this play button is ours not only mine because I wouldn't have it without you guys and one more time thank you for the bottom of my heart now today I have one of the craziest crossovers ever since the scooby-doo and courage the cowardly dog we got FNAF X scooby-doo I feel like this crossover is gonna be awesome because I mean it's FNAF man FNAF and not only scooby-doo scooby-doo he's like legendary like Zoinks dude alright so with that being said shout out to Egan Tealman and we're gonna watch scooby-doo where are you in springtrap grab your popcorn your snacks everything right now we'll pause the video going right now hurry up hurry up filming home video dude this is so nostalgic this is awesome I'm gonna snatch the weave boogie ghost crazy baboon dude this is awesome man it's in the pizzeria it feels like an actual scooby-doo episode which is which is pretty insane alright scooby-doo where are you in springtrap oh Freddy Fazbear Pizza of course he's already savoring the food security guard that's strange I'm not supposed to lock up for another hour someone must have closed shop early that can't be possible I'm the only one here sounds to me like we have a mystery on our map alright so we've already seen that in a metronica walking inside the pizzeria I don't even know if the security guard is aware of it here you guys can set up camp in my office you sure run a tight ship around here burglars must really like pizza it's not the burglars I'm after it's what's already inside what do you mean inside here I'll show you creepy camera he just doesn't want to step on it oh no I'm sure scooby can help us sniff it out come on scooby where are Shaggy and scooby dude he's whistling the song man it's a dream come true locked in a pizza parlor overnight like this is so much better than a haunted house best book they are let's find the kitchen scooby springtrap oh 19 June it's me like scooby check this out okay now is that blood or is that pizza sauce pizza sauce it's like whoever to call the pizza walk straight through this enough to go dude this is so cool man can't get over it he has that chica's beak oh yes freddy's top hat oh no he's wagging his tail dude hilarious man oh come on scooby the door oh they didn't close the door oh the iconic scene where they get chased and they pop out of the door this is iconic right here he took her glasses is that the other half of the picture oh they're cornered I think we got him he busted himself through the wall wow what's going on with that William was jealous of Henry's success so he created the ghost of springtrap to scare customers away from freddy's but I thought the bunny was a robot but with his patented springlock technology he was able to make the snow gear to be a robot again and like a totally spooky ghost by night he's got to say it I want to have spouty springlocks I would have gotten away with it too and these meddling kids come on oh he didn't say it freddy that's got to be scooby at the bottom half freddy it was scooby alright oh is that the end dude that was amazing scooby-doo and for the half crossover I have to say shout out to Egan he took like the old scooby-doo concept he like kept it alive it kind of felt like an original episode as if you're watching it back back in what I don't know when scooby-doo would air like in the 80s I think something like that well it was awesome overall I enjoyed how he kept like how he integrated like the fnaf with like William Afton like being the villain at the end when he gets unmasked I mean you had to see it coming but it was just it was cool alright you guys well I'll link the original video down below in the description and one more time shout out to Egan for his awesome videos animation he is a one man team so I'm pretty sure this took him like a bunch of time to create on his own yeah see you guys in the next one peace


antimicrobial exposure during curative intent treatment of triple negative breast cancer, TNBC, has been linked to poorer survival outcomes. In this study, researchers examined the association between antimicrobial use and peripheral lymphocyc count during TNBC treatment with survival. They found that each additional antimicrobial prescription was associated with worse overall and breast cancer specific survival. Additionally, this association remained even after adjusting for neutrophils, which are white blood cells that help fight infection. The authors suggest that these findings could inform future studies on the effects of antimicrobial use on TNBC survival and potentially guide clinicians when prescribing antibiotics to patients with TNBC. This article was offered by Julia D. Ransohoff, Victor Ritter, Natasha Purington, and others.