 to be speaking to so many people who are interested in both geometry and gravity, and not just one of them. So I'll try to take both perspectives. I should say first, though, that I'm a mathematician by training mostly. So my perspective will be mostly mathematical, but I'll try to give the physical intuition behind the notions as well as well as I can. The topic I'll be speaking about sounds rather old. If you look at the second, these are the main references. I wrote them down already. So if you look at the overview book is from 1996, and then the next overview article is celebrating four decades of black hole uniqueness theorems. So they're an old topic. However, recently, there have been many interesting generalizations and also different methods of proofs. Some of them I will mention at the end. So it's, again, a hot topic. So I thought it would be a good topic. So there are rather basic things you can say, but it's also current. And you can do research right away at the end of the lecture series if you like. Of course, these are not the only references I'll use. Most other references I will use, if they're about older stuff, are in there. So I'll just mention names and years. If I mention references that are not in either of these works, I will, in fact, write them down properly along the way. I will also produce lecture notes eventually, but I just had a small baby, so please allow this to take for a while. So this is the plan of the class. In lecture one, which is today, we will talk about static space times and introduce all the important notions, important being defined as we will need them in the course of the lectures, not they're important generally, or not things we don't talk about are not important. So this is a very brief introduction. Then in lecture two, we will talk about static vacuum black hole uniqueness. And then in lecture three, we will again talk about static vacuum black hole uniqueness, but looking at different types of proofs. And then in lecture four, we're gonna talk about generalizations in particular to higher dimensions and to photon surfaces. And I don't expect anybody to know what a photon surface is by now. I'll explain that in the last lecture. So that's the plan. And I presume you've already seen the Schwarzschild space time many times probably in every lecture. So I decided I'll write it down once so that you get used to my notation and then I'm using it as an example, but I embed it into a larger class of examples. So those who are very familiar with Schwarzschild don't get bored and maybe learn something new. If you're not very familiar, just always imagine the example I'm tracking through to be just Schwarzschild, okay? So Schwarzschild space time can be written as the Lorentzian metric ds squared of the form minus a function n squared dt squared plus one over n squared dr squared plus r squared d omega squared on the manifold are containing the time coordinate t cross some interval maximum of zero to m infinity cross s2 and this contains the radial variable r and this d omega squared is the canonical metric on s2 with the function n being a function only of the radial coordinate r given by the square root of one minus two m over r and m in r is the mass, okay? So this you've probably seen a lot, but I've written it in a general form and we're now going to look at space times which have this form for various different functions of n. But let's first recall that the Schwarzschild space time models a vacuum space time in four dimensions exterior region of a black hole as you've probably heard and it's of course spherically symmetric and I wouldn't be talking about it if it wouldn't be static. Static by now for us only intuitively means that nothing moves. We'll define it more properly later. Pardon? In the bracket the max of zero and two m, okay? Because you need the square root to be well defined. So you can't have r less than zero or zero and you need two m. You need it to be larger than two m if two m is positive and this is an infinity. So if m is positive this is the exterior region of a black hole, if m is zero this is a weird way of writing Euclidean, Minkowski space time and if m is negative this is weird altogether but it's still a well-defined space time. It's a special case of the Kerr family that I'm sure you've seen if you pick the angular parameter a equals zero. That's most I'm going to say that if m is bigger than zero then the black hole horizon is at r equals two m. So that's the threshold where this function becomes a zero and notice immediately that r equals two m is not inside the manifold. So it's on the boundary of this manifold because this is an open interval. That's important because we want this function here to be positive, okay? We'll see this occurring again and again. Okay and we will encounter metrics of this form more generally and I'll give you some examples but before I give you examples let me fix notation or conventions and say that we have the gravitational constant and the speed of light set equal to one which I think probably most lectures are doing here. More examples of this form orders called the Schwarzschild space times in higher dimensions or sometimes called the Tangerlini space times or sometimes called Schwarzschild Tangerlini and are equal to three. We can write the function in of r as the square root of one minus two m over r to the power n minus two. This is also positive, bigger than zero and bigger than the one over n minus two root of two m depending if m is positive and this in fact is just a natural generalization of the Schwarzschild space times to higher dimensions. It's also vacuum. It also has a black hole at this threshold if m is positive, yeah. If you're a physicist and you're wondering about the units here of course you would need to have an r over some reference length to make sense of this and a mass over some reference mass but we'll not bother about units here and the manifold on which this lives is r for the time and then again the centriple max or rather zero or two m if m is non positive or if m is positive to infinity and then cross s n minus two and then the omega squared becomes the canonical metric on s n minus two of volume one. There is also a metric in the literature that sometimes has the name pseudo Schwarzschild and we're not gonna talk about very much but I want to mention it because it has confused some of my students enormously which is for n equals two so that's three space time dimensions n is always the spatial dimension for me that's three space time dimensions and you take n of r is exactly as in the three dimensional case of course this is positive for the zero and two m so this gives you a legitimate space time however the space time is not vacuum it doesn't satisfy the dominant energy condition if you heard about that it has no, besides looking exactly like three plus one dimensional Schwarzschild it has no physical relevance within general relativity I don't know about other theories okay not vacuum I'll talk about it a little in the last class so what it does do it is accept being vacuum and not being asymptotically flat that's something we'll talk about later as well it has all the properties of a Schwarzschild black hole so if m is positive it has a black hole horizon just in the same way yes yes yeah that's a very good observation so if I wanted to use this formula it would be a typo right the power off oh here oh yeah right thanks sorry I thought you meant the power here and so the power here is actually really what's written down because you can't use this power in plug-in n equals two so this metric otherwise is exactly like Schwarzschild it has a black hole horizon it's fairly symmetric and it has nice properties but it doesn't satisfy the vacuum Einstein equations and it's not asymptotically flat so another space-time you might have come across that has this form is the Reisner Nordstrom electrically charged space-time very briefly touched on that in the last class and in that space-time there's a mass m and a charge q think of an electric charge and then the function n can be written as one minus two m over r plus q squared over r squared also positive for suitable r this is again n equals three and there's a higher dimensional analog of this as well and then there are lots of other examples and one relating to the class you're following on the ADS CFT correspondence is a metric called the ADS Schwarzschild or the Kotler metric I won't write it down because it requires to put too much notation I just write it's name down and they also exist in all dimensions bigger than or equal to three so this is a huge class of examples and if you don't put any partial differential equations imposing any Einstein equations imposing matter and all of them are static it's fairly symmetric and have other nice properties that we'll discuss in a second so they'll all be examples for the static space times we may be talking about, we will be talking about later so properties they're static and we will define later what this means obviously it's very symmetric if n goes to zero as r goes to the r star of r in r star infinity which I hope is suggested enough to be self-explanatory in a suitably regular way then there is a black hole horizon at r star and we will define that later as well the asymptotic behavior and you've heard a lot about that I presume in Rick Shane's lecture can be specified just by specifying the asymptotics of this function in because n is the only free function in this metric so that's why they're good toy models so if r goes to infinity we can prescribe this is a question mark what n does for example if we want to be asymptotic to Minkowski we want n to go to one because if we plug in the function n equals one we get the Minkowski space time if we want the manifold and the space time to be asymptotic to the Schwarzschild space time we will request that it asymptotically behaves like the Schwarzschild space time so this function in as r goes to infinity plus lower order terms that we need to be precise how fast they decay and I will not precise on this except in the statement of theorems if we wanted to say we are asymptotically anti-decider like in the last example I didn't even write down we would impose other asymptotic conditions on this function in okay another property they have is that the spatial part being defined as one on n squared d r squared that's r squared d omega squared is the same for every t it doesn't depend on t that's one of the consequences of being static it's t independent and also the second fundamental so g of t is the same g for all t the second fundamental form which I don't know by k of t is zero for all t which we will call time symmetric rick shane may have mentioned that I'll draw a sketch in a second here's a sketch here's time and then for each t set t equals constant in this manifold that I just erased looks exactly the same from its intrinsic geometry perspective and looks completely flat from exterior perspective which is why I'm drawing planes of course intrinsically they're not flat but there's not enough room in the platform to indicate that okay and if you look at the Einstein equation that you've seen many times I presume so the Ricci of ds squared minus one half the scalar curvature of ds squared times ds squared is 8 pi t where t is the energy momentum tensor reduces to ODE's for so to ordinary differential equations for the function n and the components of the energy momentum because everything only depends on the variable r that's why they become ordinary differential equations instead of partial differential equations because of the spherical symmetry and the statisticity and so if you want to do exercises then you exercise one could be compute those ODE's just to get more familiar with this class of examples there's one more property that I want to discuss because it will be very important in one of the or several of the proofs of static black hole uniqueness and that is conformal flatness okay this is a confusing topic I'll write down the form of the metric again just so you can see it so of course unless this is Minkowski space this is not flat it's also not conformally flat in general or actually in very few cases it will be conformally flat if you don't know what the notion of conformal flatness means please look it up as another exercise I'll just give intuitive explanation and it's very well explained on Wikipedia so conformal transformations and conformal flatness so intuitively what it means that two space times or two other Romanian manifolds are conformally related or can be conformally transformed into each other it means that at each point if you pick two vectors in the tangent space and put them into the other manifold with the diffeomorphism the angle remains the same and isometry would also imply that the length of both vectors stay the same for conformal transformation we only require that the angle remains the same and this conformal flatness then means you're conformal to flat space or flat space time depending on context where flat space time would be Minkowski and flat space would be Euclidean space of the corresponding dimension okay so what I'm going to write now is that these metrics typically are what is called spatially conformally flat which means that the metric g which is the spatial metric here on any of these fixed slices can be written as a function phi to the power four over n minus two that's the dimensional convention and of course we need to stay away from two to write this down even times the Euclidean metric so delta is my Euclidean metric phi is a function is a function on say t equals zero so they're conformally flat but only at each instant of time so each of these extrinsically flat intrinsically curved time slices is intrinsically conformally flat it's curved but it's conformally flat so the angles are such that it could be flat the lengths could be different and this is true for schwarzschild this is true for schwarzschild in higher dimensions and I'll write down some details in a second and I'll say it again because it happens so frequently this is not to be confused with the whole space time being conformal to Minkowski space time because then there wouldn't be the causality theory for Minkowski and for these space times would be exactly the same and this would be very boring just the spatial slices example schwarzschild M bigger than zero and n equals three we can write the metric either as we've done before as one minus two m over r d r squared plus r squared d omega squared or one plus m over two s to the power four delta or if we spell out the Euclidean metric in spherical polar coordinates ds squared plus s squared omega we're here we needed that r is bigger than two m and here we need that s is bigger than m over two to correspond to the same region and the coordinate transformation reads r is one plus m over two s squared s as you can see if you compare the coefficients of d omega squared so we've never seen this way of writing schwarzschild in different coordinates the angular coordinates say the same just the radial coordinates transforms because we wanted to preserve the angles anyway yeah if you've never seen this I recommend you to do this computation once to believe this that would be exercise three verify if you have seen this and you're curious you could try to come up with what needs to be here in higher dimensional schwarzschild if you have seen this way of writing schwarzschild the spatial part of schwarzschild you've probably seen it with a variable called r here as well which creates also a lot of mistakes in the literature so I'll try to be super consistent and if I write it this way I'll always try to call the radius s okay in these coordinates they're called schwarzschild coordinates because they are the coordinates in which this was originally discovered by schwarzschild so schwarzschild coordinates and on this side the coordinates are called isotropic coordinates because they make the metric look isotropic and again this only applies to the spatial part of the metric okay you can do the same thing for positive mass higher dimensional schwarzschild of course you can do it in all dimensions bigger than or equal to three also for zero m but then nothing happens for negative m you can also do it in a neighborhood of infinity but not all the way down to r equals zero because your transformation breaks down at some point the fee in this formula will start being negative and then this makes no sense okay so in that's why I'm saying here that they typically are spatially conformally flat meaning a lot of them are locally conformally flat spatially conformally flat near infinity and then again on regions but they can be lines of r where they're not okay also Reisner Nordstrom has this property of being spatially conformally flat with a nice transformation globally if the mass and the charge satisfies suitable conditions called sub-extremality then otherwise the regions of Reisner Nordstrom and higher dimensional Reisner Nordstrom will be spatially conformally flat same for ADS schwarzschild it will be a spatially conformally ADS so this is important and one more important thing here is this threshold and it'll get a little color while here the radius threshold two m needs to be there because otherwise the formula stopped making sense this cut off here at m over two doesn't need to be there at all in order for this formula to make sense and in fact we can just allow s bigger than zero and say this gives a well-defined smooth metric spatial metric for s bigger than zero so on either whatever you want r3 without the origin or zero infinity cross s2 whichever way you prefer to write it okay so this way of writing the schwarzschild spatial slice metric allows you to extend the spatial slice beyond where we can look before and then the surface the two surface in this three space time in this three space sitting here any two surface sitting inside here can actually um... sorry what did I want to say so I can extend through this threshold and speak of this in a regular way just in the sense that I listed before that if you take n to zero is suitably regular way now here in these coordinates it's hard to say whether I can take n which was the square root of this to zero in a suitably regular way but if I change to these isotropic coordinates then the surface which used to be called r equals 2m as the boundary of my spatial slice is now the surface s equals m over 2 which is a perfectly regular surface in this manifold okay and I'll draw a picture so if your usual schwarzschild spatial slice equals zero schwarzschild looks like this where r to infinity is here and r 2 2m is this boundary which is the horizon which is why everything goes in there vertically then in the s coordinates actually what happens is I have two copies of this and then this corresponds to s m over 2 this corresponds to s to infinity so this is s bigger m over 2 and this is s bigger than zero and less than m over 2 and this is s going to zero okay so now I have two copies of a spatial slice of schwarzschild glued together completely smooth through this t equals zero section of the horizon and the whole thing is conformally flat as you can see by this formula okay so now in the spatial slice there is no singularity anymore at all in these coordinates of course the singularity is still there in the spacetime I'm not going to talk about that otherwise I wouldn't be boring again but in the spatial slice it's gone and you can do the same thing with suitable choices so in higher dimensions for m positive in schwarzschild for what is called sub-extremal horizon or Nordstrom so suitable condition between the charge and the mass and also in higher dimensions and for ADS and ADS schwarzschild you can do something similar but not precisely that and what you can see in this picture and which is actually true fully and can be proven rigorously is that this double is geodesically complete okay spatially geodesically complete this is just the t equals zero slice I'll write that over there and the other thing I want to write down excuse me is that it's reflection symmetric through the surface S equals m over 2 so if you look at this picture I already tried to suggest it in this picture you can take something along the lines of an inversion in the sphere in the sphere or something like S going to 1 over S but you need to be a little bit more clever than this to take the upper part to the lower part and vice versa which is why this is a totally geodesic hyper surface so this is then totally geodesic and if you don't know exactly what this term means it means it also has vanishing extrinsic curvature in this curvature I'm going to denote by H in particular it's minimal this is something you've probably seen in Rick's lecture so the mean curvature is zero this is mostly to fix notation and also what we've seen is that n of S m over 2 is zero so if you're very listening very carefully you notice that n was a function of r and not of S and S is now a new coordinate so I'll write a little twiddle over there meaning the same function written in a different coordinate variable okay so it's a different formula but it's the same function and we won't need the explicit form that it will be zero there okay so much for this class of examples that you can have at the back of your mind now that we introduce general static metrics before I go there though is under any questions yes maybe I should write that that way that's a good way of saying it n r of S and S equals m over 2 thank you so here comes the definition definition one the spacetime r cross mn ds squared i.e. a Lorentzian manifold which is time orientable mn is orientable and connected hello is unless I say something else all manifolds will be connected in this in this lecture the spacetime of this form r cross mn with a metric ds squared is called standard static and most people never use the additional term standard but still use this definition if ds squared can be written as just connected doesn't matter just not several pieces ds squared is minus n squared dt squared plus a metric g where t is the variable in r n from mn to r and positive is a function called the lef's function and g is a Romanian metric on mn so we have this product manifold r cross mn r is the time coordinate and mn is the spatial manifold as before and now we have a term here that looks exactly like before but our spatial metric g can be different from spherical symmetric and n also doesn't only depend on some radial coordinate but can depend on the whole spatial coordinate set but of course we still want n to be positive and now g is a Romanian metric meaning it's also time-independent and n is also time-independent they're just on mn they don't depend on t that's what's called standard static and of course all the examples we've looked at before are of this form and i want n bigger or equal to three why is this called standard static let's make a few remarks so first of all dt is a time-like killing vector field in the spacetime and in fact if you were wondering what the physical meaning is of n it's the length of dt with respect to the Lorentzian metric if you draw the picture then dt is a field here and also and this is a more tricky condition to write down the orthogonal complement of this field dt is for venues integrable that's how the mathematicians would say it the physicists would say and this actually means the same thing dt is hypersurface orthogonal which in formulas means dt index alpha the covariant derivative in the spacetime beta index gamma anticommute so this is the vector field dt index alpha covariant derivative in the direction of coordinate beta take again of the vector field dt in coordinate gamma and cyclic anti permutation so what it really means if you know nothing about these things is it means that there is a right angle here so i can have surfaces of constant time and the vector field dt is perpendicular to them everywhere if you think about Kerr this cannot be done in Kerr there's no time function on Kerr that does this the dt in Kerr is a time like healing vector field at least far out the way from the black hole but this is not for venues integrable or in other words this is non-zero so the vector field is not perpendicular to the t equals constant slices and typically the statisticity is defined to be static is defined as just this and this not what i wrote this implies that locally it has this form so let's call this form three i forgot a number two on the equation for the conformal transformation of Schwarzschild if you want to add that to your notes so i can refer to that later so locally it has to be of the form ds squared has to be of the form but what we're assuming is that this is globally true that's why it's called standard static typically people are very sloppy and use these two notions interchangeably even though they mean global and they would mean standard static but they don't say it so be careful when you read the literature there also we can alternatively represent the standard static space-time as the system which has just the remanian manifold m and g and the function in and such things are called static systems so then let's have definition two we want to talk about black hole uniqueness so we need to say what a black hole is in this context then m and g and your static system and then if n minus one sits in m and in in the boundary of m and as a comp is a closed surface so compact surface and so maybe you're confused surface in if any close three it's a surface otherwise it's a hyper surface but I'll be sloppy and has and is tricked in to the surface equal zero where we implicitly presuming that and extends to the boundary I'll give an explanation of this in a second we call it a killing horizon or the cross section of a killing horizon the why this the laps function in was the length of the killing vector field so if and zero the killing vector field is at least now or even possibly zero on the surface this which is why it's a surface characterized by the killing vector field the surface cannot be inside the manifold but has to sit in the boundary because we said the function and has to be positive in the manifold so then it can only be zero on the boundary and not inside because it's positive yes no so so if you think about short yield and you have your usual short yield on which and is positive it will be zero on on the horizon so the horizon is not part of this static system okay and then the horizon will be this exactly the surface sigma n minus one which sits in the boundary of your special slice of short yield at the quad that the T equals your slice of the killing horizon in the usual sense okay and then the mean curvature age of sigma n minus one in mn union with the boundary is zero so if it's a minimal surface which I'm sure Rick Shane discussed it's a horizon and if you've heard about marginally outer trapped surfaces or mots this is exactly a mots here because the second fundamental form of the slice in the space time vanishes so the part coming in a mots from the second fundamental form is not there if you've never heard about this ignore the comment and we will say and say sigma n minus one is a static horizon and extended to n minus one is zero and age is zero so both conditions hold true and we will call it a static horizon it's there a question does everybody yes does it depend on the coordinates in which I define my metric no yes if a coordinate if a time function T exists that you can write it like this there is no assumption on spatial coordinates just on the time court court yes in some sense you may be afraid that these conditions are a little bit redundant and if you assume Einstein equations with pseudomilm matter fields in some scenarios if you have vanishing laps function you get mean curvature vanishing for free or vice versa but it depends on the matter okay so in general it's not redundant and of course in order to even say that age is zero here I need to extend the geometry of the manifold and then to its boundary assuming g extends sigma n minus one as well so what does this mean again in the short you context let me draw again this picture so this is our men with metric and laps function from short shield then on this surface it's a minimal surface h equals zero and the laps function vanishes but the surface let me repeat is not part of the space which is why it's the boundary of the space once we doubled it this is a nice smooth judezically complete manifold however the laps function of course still vanishes on this surface so this surface cannot be part of the standard static space time because we requested that and the positive and even worse well and his positive here you reflect and down here and will be negative here necessarily okay and then equals zero in this so only the upper part of this is a standard static system those of you who know what this means this is the special section of the domain of out of communications okay inside here you're inside the whole in the special section of being inside the whole so only here you're in the stat in the static system this is a little bit complicated to to speak about that's why i'm making sure everybody understands this in the short example so let me say once more in the in the part above and not including the boundary here not including the minimal surface we are a standard static system if we add the time direction and the laps function incorrectly to have a space time on the boundary we are have a horizon because we can smoothly extend both the metric and the laughs function through here however the whole thing will not be the special slice of the standard static system and is zero here and negative here so if you want to do another example but in my book a little another exercise number four verify are equals two m and i'm writing a quotation marks because this is not part of the system is a static black hole closed and all i'm assuming it's closed i'm assuming it's closed and smooth and everything but the part of the boundary and then if the laughs function and assume the laughs function extends smoothly and the metric extends smoothly and then these conditions and uh... and a horizon in the sense of initial data sets now comes a definition you may have seen in a similar form in rick shane's lecture deciding on uh... defining the asymptotics the static system and three g and and then yeah let's do it in three dimensions here it's easier now we can do it then and and bigger equal to three is called asymptotically short shielding were also asymptotically isotropic if there exists the number m and r if m and can be written as a compact set in the disjoint union with the manifold e and which is called an end such that e and if your morphic are and outside some close ball the idea look like one plus over two as forward to the four over n minus two delta ij plus lower order and n looks like one minus power missing as n minus two n minus two plus lower order terms ss goes to infinity i presume you've seen asymptotic flatness in rick's lectures so asymptotic flatness of course here you just write delta ij and lower order terms and lower order in rick shane's lecture lower than even this so this is in particular asymptotically flat with very very vigorously prescribed asymptotics and of course to make this precise i would need to say what i mean by lower order sorry yes disjoint union so there's a compact set and an end that and they don't have any intersection and so of course i would need to make precise what i mean by lower order okay but i won't because it won't matter too much i will write it in the theorem and in particular keep in mind that lower order terms also means derivatives have prescribed behavior so that the first partial derivatives of this also have look like the first partial derivatives of this plus terms of lower order and and second ones as well and depending on context third and fourth ones as well possibly okay so lower order terms can include derivatives and you will find this in the literature mostly under the name of asymptotic short being asymptotic to schwarzschild or asymptotically schwarzschildian however i wrote it in as variables because it really looks like the isotropic version of schwarzschild and this is not equivalent to being asymptotic to the schwarzschild space time and schwarzschild coordinates and this order of lower order that we will be talking about you can perform the same change of coordinates of course to transform one into the other but then you will have a different different morphism here so be careful it's really should be called asymptotically isotropically schwarzschild or something like this and i'm going to call it asymptotically isotropic throughout so i'm going to go where is green i'm going to call it asymptotically isotropic of course rick shane has introduced the ADM mass and maybe some of you have already been wondering at least in three dimensions does this M have anything to do with the ADM mass and yes n equals three M is nothing else but the ADM mass of this G in high dimensions two if you know of higher dimensional notions of ADM mass and in particular this means that it's unique you cannot have the same static system having two different mass parameters for different different morphisms and have these expansions it's an invariant not important and we're going to call it of course the mass of the system so the last thing i want to talk about today is look at what happens to the Einstein equations and the Einstein equations of course are an equation for general space-time with matter field and if i specify the matter then it becomes like a clear equation for example if i say it's vacuum the energy momentum tensor vanishes then it's an equation which equals zero so if i have this equation which equals zero and i have a space-time of the form where the metric is of the form minus n squared dt squared plus this three metric G or n metric, the Romanian metric G i can plug in this formula and simplify the equations so in vacuum plugging in this into Einstein and did the Einstein equations we obtain the following system the Hessian of n equals n times the Ritchie where this is with the Hessian with respect to G and this is the Ritchie with respect to G and the scalar curvature with respect to G vanishes these are the equations that remain scale or flat and the Hessian of the lapse is the lapse times the Ritchie now there are a couple of nice things about this this equation doesn't involve the lapse at all and this equation is linear in the lapse this will be useful later and together if you trace the first equation the Laplacian with respect to G or n vanishes automatically so n is harmonic necessarily as a consequence if you trace this it says Laplacian of n is n times scalar curvature, scalar curvature is zero so Laplacian of n is zero so n is harmonic so the last homework exercise for people who want to do homework will be exercise five verify this and of course if you choose another matter model then vacuum say electro vacuum or something else a fluid you will get other equations they will keep the property though that they're all linear in n so thanks for today and I'm taking questions