 Good morning everyone. It's a pleasure to be here. It's a great pleasure to see so many young people from all over the world interested in general activity. So as Lorenzo told you I'm from Vienna now but I'm originally from Poland. I have a name which is very difficult to pronounce. If you want to do it right it's Chruszczel. He was almost almost there. Since my predecessor started limb breaking information I'm going to join in and so I don't have many sportive achievements but I like a lot of skiing and that cost me several broken legs so compared to his fingers. Good. So and as you have noticed I'm from University of Vienna. I mean if you haven't then you can read it here and we're very proud to be the second German speaking university ever. The first one was in Prague and it's in the region the third oldest one so it's been actually 654 years now. Good. So I'm going to tell you today a little bit about mass in asymptotically flat space times. So if again my colleague said that his handwriting of the blackboard is challenging well you haven't seen mine I still have some students recovering from my lectures in Vienna but so please don't hesitate to interrupt and ask questions because that's what we are in for here for and so let me start with the first question why mass or energy these things go together and I must admit I will not be very consistent I will often mix use the two names for the same thing things which are not quite the same so in any case that's my first question I'm probably going to avoid this dark spot here so how does it work well if you look at classical mechanics this is a very useful thing to have and in one dimensions then well if you have your conserved energy which has the usual form with the potential you know that this thing is conserved and so you draw your potential so say the potential looks like that you know everything about the orbit I didn't know everything about how the system behaves so for example if you are this energy level so this is your energy one then you know that you have that much kinetic energy here so you have as much mx dot square as this thing allows so obviously if you are at this point in X and this is V then you must be moving right because this measures how much you have this either going to this way or this way so well you keep moving until you can't anymore because here you know that you have zero velocity right and so at this energy level you know that the orbit will just do this and in a periodic way just by uniqueness of solutions of oddies and if you are at this energy level here then you know well they have that much energy well you have much more energy than me right so you have well you have that much energy here so you still have velocity here and you are here you still have velocity so either we'll be going this way and you'll be going forever because this can never get zero or you'll go here and you'll have to bounce back and go back right so this is extremely useful if you just don't need the exact form of the solution but just want to know what the solutions do then then you're done here right when in higher dimensions it's somewhat similar so if so I'm not just going to say anything but in a radial potential in 3d you have the same picture 3d then you go to a you just take the radial part of the motion well you can analyze the similarly similar analysis for analysis for our dot and in fact this potential here is what you will get if you are in a I hope this should be a Newtonian potential with angular momentum right so you have a angular momentum barrier here and this is going to well not quite to zero but anyway so so that's also useful well what about field theory so let's say scalar field so you look at this equation box phi is equal m square phi and well as long as I haven't told you what is my signature the sign here doesn't really matter because I can always change the metric to minus the metric to get the sign but I'm thinking about a sign which gives you an energy which looks like integral over all three of say one half d phi over dp square as space gradient square plus m square phi square so this thing is constant in time because of this equation so one implies that d over dt is zero well actually if you are a mathematician you should should stop start wondering whether this is finite to start with whether you can differentiate it to get d over d theta there's a serum which says that in physics every integral is differentiable so you can certainly do this so the fact that this is conserved is telling you that these things each of these terms remains finite so nothing can go bad with your field and in fact you can use the fact that this is conserved to prove global existence of solutions in this simple case it's a linear problem so proving global existence is not an issue but if you do this for a young mills fields which I'm going just to mention for those who know what they are then if you don't that doesn't really matter you can just do something like that e square plus b square so it's like a Maxwell field but not necessarily a billion so in a Maxwell of course there's a clash of notation here this is the electric field this is Maxwell field maybe so it just like that so again this is for Maxwell or young mills this is conserved by the same theorem as before and and in fact for linear theory there is no really no difficulties to prove global existence but if you take young mills equations the statement that you have global existence of solutions of the young mills equations is a non-trivial one and there's a extremely difficult theorem of Kleinermann and Machadan which based on the conservation of this proves global existence of solution of the young mills equations in Minkowski spacetime okay so I'm not going to write this down because it's just kind of a parenthesis of what I'm doing but so energies are useful for PDs actually what one needs another parenthesis it's enough to have not conservation but something like DE over DT is smaller than a constant time e and e positive to prove theorems for existence of solutions partial differential equations so some notions of energy are useful for both from a physical point of view and from a mathematical point of view now there's a problem in gr which is as follows so if you have a Lagrangian field theory where you have any kind of collections of fields then there is a energy associated with this which I'm going to come back to later which is essentially some kind of expression in phi and d phi okay so that's a general rule I'm going to review this in my lecture three so Lagrangian field theory you can calculate an object called the energy and what is important that are not only you have a total energy but you have an energy density right so for the scalar field your energy density would be this integrand here for Maxwell fields or for young mills this would be your integrand here and the usual argument that in general activities this cannot work is that well if you think as phi to be the metric then its derivative would be partial derivative of the metric by the way I'll come back to my notation a little later but so this d sigma is the same as dg mu nu over dx sigma and there is a standard theorem in differential geometry which says that suppose you have a manifold choose a point you can find coordinates so that this matrix has at this point constant entries and at this point the derivatives of the metric will be zero so in physics it's called local inertial coordinates in mathematics it's called normal coordinates so at every point you can make g to take a standard form if your Lorentzian it's going to be minus one plus plus plus if your Riemannian is going to be all pluses plus ones and the derivatives will be zero right but then this expression at this point will be zero and so conclusion that people draw out of this is that there is no local expression for the energy of the gravitational field so there is no geometric object which you can integrate and therefore but if there isn't one which you can integrate like that then maybe there is a problem with doing this in general activity as well and well I would think that I would say that this is the kind of conclusion well I don't know how to do some things that I'm trying to I'm going to find an excuse to say you cannot do this right so that's what physicists have been telling us for years and maybe that's correct maybe there is no way to find a good notion of energy density of generative maybe there is one that's people haven't found yet right so this argument only tells you that you cannot find one like that which will have a geometric character as a density at a point but well you know we have a career as a head of you I think if somebody here finds a nice candidate for this that would be a good PhD project but well let me also mention that whatever you do you discover this thing you think it has to be the energy well it has to be useful to something right so there are things like Bell Robin sometimes or things like that you could take the square of this and say well this is the energy you have to be careful with definition definition is good for something if you can prove something with it right if it's useful for something then good that's worse proof thing if you just have a definition and you spend half your verse saying this is the right notion but you can't do anything with it then that's not a good career idea so in any case in GR there's no universally accepted notion of energy density now it turns out that there is one global for asymptotically flat spacetime that that's what I'm going to discuss in well maybe most of these lectures but at least certainly today so there is something called the global energy so there is no well-defined expression like this but there is a well-defined integral okay so there's a well-defined integral before I discuss this so this would be the ADM mass or ADM energy momentum this will be a global integral rather than the local object still the question arises what is it good for right so you define something can you do something clever interesting with it so to my knowledge the most interesting story behind this and is the positive energy theorems which are interesting in their own but but so what right so what something is positive 5 5 is a very nice positive number so 5 is the energy of the gravitational field I have a positive energy theorem without much work so CRM's are interesting if you can do something with them that's what I said before and the positive energy CRM or CRM's have two implications fast first solution of the amabi problem and I'm going to maybe tell you more about this later so at the moment if you don't know what this is then come back to this later so there's a difficult partial differential equation problems geometric which has been solved using positive energy theorem in general activity so this sounds like a good enough reason to present those and another application is uniqueness of static black holes again you use the positive energy theorem to prove that under center conditions well unique static vacuum black holes is far chilled in dimension three or in higher dimensions as well now just a few comments about what I references I may not be giving you references during my lecture a lot of the references which belong to all these topics that I'm discussing you can find them in my lecture notes I don't know if they have been linked on the home page of the conference have they not we have internet here and I just look up my home page does this work can you Google my name so that's how you're going to yes very good okay CH are you as difficult to decide you're almost good good okay very good Patricia from page good so in Google I have a coming up first so now you go to publications you go all the way down so just you know I thought it was showing here that that's not very productive can you do it again good so start again start it says Google me again go go back yeah go back yeah okay we don't want to go yeah go back okay so Google me okay you'll find home page good you click on it good then you go in publications okay again publications you go all the way down you're going sorry it's too long yeah all the way it's just you know unpublished energy in general activity okay so this one you click here and good so the lectures here will be a part of the of what's in here so you can find details of what I'm talking about and references and stuff good thank you good so at this stage we know that there is something useful there is some interest in studying ADM energy because it allows you to prove these things that's at least so far you have also there's a warning coming from with these energies because physicists tend to think that if you have a positive energy theorem then your system is stable and everything will be fine and you don't need to do anything more you understand the world this is actually wrong so the ADM positive energy theorem I don't think it plays such a big role in the proof of stability of Minkowski spacetime you have to work much much more to prove this while proof of global existence for for this scalar field is just trivial based on this equation right so the energy can be useful but doesn't have to you have also to be careful there is a positive energy theorem for asymptotically hyperbolic spacetime so that's something that I'm going to talk about in my last lecture and so one has a positive energy theorem thanks looks well but there is a famous instability of anti-disseter spacetime discovered by Bison and Rostrovsky which so that no matter how small is a perturbation of asymptotically and your asymptotically anti-disseter spacetime it's going to generically form a black hole right so it's completely unstable no matter how small your perturbation is even though you have a nice positive energy theorem so positivity energy and stability are just brothers but maybe very very distant relatives in some cases good so this is so far for the introduction now let me now continue with something which is the simplest possible version of energy in generativity if you look at post-Newtonian gravitational fields or even before if you look at Newtonian and post-Newtonian fields right and and ask again later and post-Newtonian right mass or mass good well so what's Newton's theory of gravity well if you're interested in a model perspective on this there's a nice book by croissant and wheel about this well part of it has a lot of information about this but the simple you just take Newtonian potential let's see so I have to put some pies and I have to put a G which is the and I have to put a mu so this is the Newton potential Newtonian potential and this is the thing which governs the equations of motion so I guess that would be d2 x over dt square is equal mm-hmm oh gosh it's probably with this convention this is this is my a nightmare point if all my lectures when I'm talking about the Newtonian potential I've lectured a lot in France and they use a different convention to the Newtonian potential as everybody else in the world and so I never know which like which convention I'm going to use so I think maybe some US participant can help me with this so if I define the potential with this sign then I have minus gradient or plus gradient right so this is the mass which cancels out by the future of the of the equivalence principle anyone can help me yeah plus minus right well one of those works so this is a Newton's constant which is obviously one from now on this would be the mass density which is of course the energy density and that's if the confusion starts coming in here right but so that's the energy density and there is a factor by c square probably and c is obviously one so this c will disappear my equation in other words you can think of this at the energy density as well good and so so that's Newton's gravity right you have a distribution of mass you solve the Laplace equation so this Laplacian is the usual one d2 over dx square plus d2 over dy square plus d2 over dc square good and then you have a sign which is plus minus depending upon your convention there but let me just choose this sign so with this convention here I think and I hope that phi will be m over r rather than minus m over r and that's why the I think that the the minus is the correct sign here so with this convention if mu is 0 outside a ball of radius r so this is a ball of radius r center of the origin doesn't really matter but that's what it is then phi will be equal to m over r for r larger than r where m is total mass so so what is this total mass then this is an integral over r3 of mu the mass density but of course because my mu is zero outside of the ball it's actually the same as integral over all over r of mu but then I can use my equation here I'm just wondering if I have my my signs right at the end but let's hope I will so this is a integral of let's see so g is one right so we just don't want to worry that 4 pi is unfortunately not one so we need to divide by a minus 1 over 4 pi Laplace phi and use stock serum to make it an integral over a sphere of radius r of the gradient of phi times psi well this are the surface forms if you want to think of this thing as di phi times the normal right times d2 s but this is a normal yes but again yeah this this is something is beyond me should we take a vote I've put a minus here right so the potential should be then now this is this sign is impossible so you know I but let's agree we can leave with equations up to a sign or at least this equation yes no thanks a lot I'm happy to see that some students are not trying to to point out please do ask questions and find out mistakes yeah yeah so so this is the and of course actually already writing this is I'm assuming some boundary conditions because I could have a solutions of this equation which have wide behavior for a distance good but so this is asymptotically that's it good so I'm going to insist on this this sign here because I want a minus here so the the bottom line is that the mass is 1 over 4 pi with a minus integral of the gradient of phi at the boundary right times the dsi that's what you do in Newtonian gravity and by the way here I've integrated over a sphere of radius r but I can pass with the because the Laplacian of phi is zero outside of matter is the same as lemurs as r goes to infinity of this integral so minus 1 over 4 pi rad i phi dsi and let me call this operation of integrating of a sphere of radius r and going to infinity is an integral over a sphere of infinity maybe I put the index apple down depending my mood but open my mood but put it here so df dfi dsi okay so interrovers for infinity actually means you take a sphere of a fixed radius and pass to the infinity right and this follows that this this is the same as the limit follows either from asymptotics or just from the fact that Laplace phi is 0 for for large distances so so this is my formula for mass and good and I have a positive energy theorem is trivial here right I mean this was equal of me integral of mu so if mu is positive then energy is positive right so mu positive positive and actually when it's 0 so let's go back this calculation right this is equal to the integral here this is the integral Laplacian but this is equal integral of mu so if we started from 0 we have that this is 0 but if mu is positive and the whole integral is 0 then mu is 0 right so you have and m equals 0 is only in vacuum only in vacuum and of course this is going to be wrong in some sense if you have black holes but and if you have to relativity then that's going to be wrong right so the gravitational field has has mass good so this is Newtonian gravity what about post Newtonian yes right so that was my point here right that you can pass to the infinite the question was does this number depend upon the choice of the sphere so well I think that you can probably see it from here right that if mu is 0 outside of a large ball then it's also 0 outside of a larger ball right so I can do this calculation with any R larger than so maybe I should do this R1 where R1 is larger than R right if I do this calculation here I'd get R1 here right so R1 here so I get the same number no matter what the radius is as long as it's larger than the support of of mu okay but thanks for pointing this out right that's actually the probably simplest way of seeing right so if we take any R1 larger than R then integral of 3 is the same as this because mu is 0 outside use the Stokes theorem on this and you get the result no matter what R1 is so passing to infinity is then again trivial because it does stop good so let's go to post Newtonian matrix so if you have some background in general activity you know how post Newtonian matrix look like if you haven't well that's the formula so you just take a weak gravitational field gravitational field small velocities so you need both for this formula then you get the metric is good that's the difficult part so again this sign is going to show up here I think it's minus 2 phi I think that's correct thanks good and there might be some G's involved it's a G's one and there might be some C's involved and C's one so so that's an approximate form for the metric very weak gravitational field so assume that phi is very small and its derivatives have to be very small and second derivatives have you small and so forth right so and not only this but the time derivatives have to be even smaller because I've also forgotten time derivatives so so this is a kind of introductory GR exercise maybe at this stage I should tell you a little bit about monetization so here I understand our physicists are mathematicians and a various school of thinking so vectors well either while mu is for me typically in zero to n and I'm not especially attached to three-dimensional space times but n is always the space dimension so you can if you like string theory maybe your n is 10 or whatever your favorite number here well obviously at this stage I was in three dimensions for Newtonian gravity so then the indices ij will be in one n a vector for me is a differential operator so if you don't like this notation then think of a collection of numbers the vector is an object which has several numbers and I like to write this like that this is just that if you don't know what's the identification between vectors and differential operators the important thing here is this coefficient and so this is in space time and in most of my lectures I will be in space so this would be a vector scalar products the mathematicians would write this g of xy so a metric two vectors take the scalar product if you are a physicist coming from the space time you probably want to write something like that where now because the indices are repeated they are summed over right so there's a all everywhere summation convention this is the same as saying x0 0 plus x1 plus xn dn and well if you do it in space then you have a similar formula this is for space time and in space I will write so say gw z is gij the same thing in index notation or something that mathematicians prefer let me also write my convention for the Riemann tensor which probably I will not really use because I will not do any many explicit calculations but just to make sure so the Riemann tensor is well that's the formula I like so x, y, z are vector fields so if you are a mathematician you've probably seen this one and you're a physicist you wonder what this means but this means exactly the same thing as di dj zk minus dj di zk is equal r chi l ij dl okay so that's probably what physicists are used to and if you are a mathematician you're wondering what this means but these are the same equations and so that's my ordering of indices various people put indices in various orders so that's the one I'm used and the Ritzy tensor is the so this is the Ritzy tensor is the this contraction then once you have the Ritzy scalar Ritzy tensor you just do the Ritzy scalar and you have noted that I'm using space indices here but I might as well have used space time indices I'd be mostly using space indices anyway in these lectures but good so these are my conventions and so so this is yeah this is my way of writing a metric which means that this is g00 with the sign included or sometimes actually I will write GTT so that it's clear that this is the p-coordinate right and this is gij good so now we gravitation of small velocities phi satisfy exactly the same equation as before minus or plus did I have a minus minus minus 4 pi actually rho which is the energy density but we said that this is the same as mu right so it's minus 4 pi mu so this is the same equation and therefore the total mass is going to be the same right so that's we know what the mass is the mass is minus 1 over 4 pi integral of grad phi on the boundary at infinity right so that's normal so good so positive energy theorem as before there's no issues here and now what is intriguing now is that phi appears both in the space part of the metric so you can then read the mass from the space part of the metric but you can also read the mass from there so and what's the right way is this the mass or is this the mass well this metric doesn't matter but this metric is very special so if I take a more general metric can I just use this as a hint so so we will see that actually the right thing to do is to use this to obtain the ADM mass which will be essentially this formula in a more complicated way and but you can use this to define something called the common mass however the common mass is something which works well in stationary space times stationary space time and there is a beautiful theorem of a colleague from Vienna Bobby Byg which says that for a stationary space time these are equal so if you happen to have a killing vector stationary space time you can measure the mass from this or you can measure the mass from this the formulas are not going to be as simple as this because this is a very special situation but that they'll be equal and this common mass business which was already mentioned by my colleague before is actually needs this right so if you see common mass you should say well careful it has to be stationary right so stay common mass is only defined for well you can define it in general but you want a killing vector in there and why this ADM mass works in general for asymptotically systems good so now this is the simplest situation you can think of Newton theory or post-neutral matrix to continue yes I should be going to zero infinity at some rate and the rate well if I have this equation the rate is just coming from something called potential theory one knows everything about solutions of Laplace equation which go to zero infinity so you don't have to think much it just mr. Laplace tells you what it is actually I think the first paper where I've seen a nice analysis of the asymptotic is here is probably Murray a very old paper to Rick would you know where how how for back this goes asymptotics of solutions of Laplace equation yeah I think so this yeah it's very classical but yeah so Murray in the 50s but maybe before anyway so so this is you don't have much choice once you've given mu and say phi goes to zero I don't have to tell you anything more and and and my calculation for the mass was just based on the equation so I didn't really need to to have the asymptotics good so so we go to the Cauchy problem in general activity because this is going to be a key to to set up the framework to understand what the mass is so this is going to be a very short introduction to this usually I give several lectures just on this again you'll find the information in the lecture notes well on the same web pages notes on the Cauchy problem I wanted to try to download those good so the question is how do I construct solutions systematically how to construct systematically solutions of Einstein equations well vacuum and or otherwise so the answer is Cauchy problem well one answer is of course that you just start making various answers is so you open the exact solutions book like that with this group symmetry this group of symmetry try to solve them we was mentioned that if you just take axis symmetry five functions try to put it on maple maple will crash so it's not something that you can just what you don't do it reasonably by hand without a lot of thinking about even then you can do it with computers without thinking and so here this is the heart of mathematical generality mathematical generality is essentially about the Cauchy problem for Einstein equations right so either you solve you study evolution problems or you study initial data for those and this is what mathematical GR is about right so Cauchy problem and this means what so give initial data at t equals zero and find a solution the equations right so give initial well I forgot to write data that's the kind of things I often do in my lectures so I say something but I write something else data this is somebody said about Julius Schauder Julius Schauder was one of the fathers of partial differential elliptic equations that he was the worst lecturer ever so he thought one thing said another one and wrote a third one so I'm trying to too much so yeah initial data for find a solution so so the question is what does what are initial data and what is t equals zero and so in general activity you don't have t equals zero so cosmologists are trying to tell us that the we are living in a boring universe where there is a preferred time function that's terrible I think that this is going back to either ideas and really the anti-generality vts as much as you can but so if you're not doing cosmology you don't have a cosmological time function you don't have a preferred time so this t equals zero is actually means any well well behaved because there is still some conditions to to be put but well behaved base like surface and so we're going to call it sigma and the well behaved is all only if you want to have some uniqueness properties if you don't then actually don't care about well behaved can just take any space like surface and you have to give some data on the surface right so and find a solution what is the solution well a solution is not only a metric but you want to manifold right so you want to manifold M is an interval times sigma so somehow solutions of the Cauchy problem are boring as far as topology is concerned because so I is an interval of time right so you have a your initial data surface here and you construct a space time around it right so this is but not right so this is sigma and your space time is a something around it yes yes I see so as I said if you just want to talk about constructing solutions anything works I didn't tell you what the remaining initial data so I will tell you in a second if you want to get uniqueness theorems and stuff like that then you have to to add maybe some conditions right so but at this stage anything is good so so what are the initial data well the equations are kind of hyperbolic so hyperbolic equations this is not quite true and actually is completely wrong if you just want to think about it as written here certainly the equation the equations are second order right so equation second order well what are the equations so g mu nu is equal say zero in vacuum let me just put everything in here I think 8 pi g over c some power of c some random power of c 4 is it 4 4 yeah let's go for 4 right so this is the energy momentum tensor of matter and this is the Ricci tensor minus one half the Ricci scalar times the metric and there will be the my last lecture a cosmological constant so lambda is a cosmological constant good right so so the second order obviously because let's see so how does this work so the Ricci is a contraction of Riemann and the Riemann is first derivative of the Christophels plus their squares but the Christophels are g minus the inverse of the metric times the derivative of the metric plus gamma square so you're going to get g minus 1 d2g plus that's right so second order equations and there is a problem here if you have a solution g mu nu is a solution and if y mu of x alpha well in a coordinates x in coordinates x is any any change of coordinate then well you change the coordinate you still get a solution because everything is tensorial right so g mu nu dy well it was in an x coordinate so dx mu over dy alpha maybe you can see here right then g mu nu dx mu over dy alpha dx nu over dy beta is also a solution fine that's on one hand that's very good because the idea of general activity was to have a theory which is coordinate invariant so you have this here but on the other well if you have well posed evolution equations well posed new uniqueness so if you have a and in fact if you are familiar with this a little bit with theory of partial differential equations you often use uniqueness to prove existence or think like that these things go together right so energy estimates are used to both prove uniqueness and existence so if you have a set of equations which don't have uniqueness that's already bad and that certainly cannot be a wave equation or any that cannot be an elliptic equation that cannot be an a wave equation that cannot be a parabolic equation because these equations have uniqueness properties right so maybe globally there could be problems but locally this this classes of equations elliptic hyperbolic parabolic have unique local solutions short in time and helmet break man might break lose if you take large times but for short times everything works right so I send equations cannot belong as such to any well-posed class so there is a wonderful trick invented by madame even shakib Rua which says well you have to fix coordinates and there is one gauge condition which works is something called how wave coordinates or harmonic coordinates or the donder coordinates or something like that right same pose useful coordinate condition to get rid of this freedom so let me show you one condition which is not useful so bad idea is just to say well let's try to find coordinates where in which t looks like minus dt square plus gi j of x t dx i dx j so we have second-order equations so initial data for this would be gi g of x 0 and dt over this right so second-order equation initial data good no way I mean there's no theorem which gives you existence in this coordinates in a direct way so you can go around I'm going to give you a good way to doing this so you can do the following you solve your problem in a good way and then change coordinates to get here this works which by the way tells us that there's something we don't understand about PDs because if you can do it like that then this means that you could do it directly somehow right but we can't so that this must be a mechanism so that you could work directly with this without doing these rounds again an interesting topic maybe for you to look at there's one case in which it works and this has been I think a darn why is the first one to have seen this in the long time ago it's a darn wine maybe in the 50s or something like that if gi j if the initial data are real analytic then you can use Koshiko Valeska to solve the problem if G and at 0 t equals 0 and DT G equals 0 is analytic I can solve the problem solve using Koshiko Valeska but Koshiko Valeska and analyticity well these are suspicious theorems for once one never knows whether this Kovaleska or that's already a hurdle here anyone knows I know that her name was Sophie but anyway and so maybe there's an eye here or something like that so what's the right way to do well the right way to do has been discovered by even Shakebra that lovely lady who wrote a very nice book of memoir you can find them in French and I think they're being translated to English now and so then she writes that she was she saw her father was a professor I call normal in Paris I was actually the director there and she she knew every mathematician from that time because they were all friends of his father so darmois was one part of the crowd of his friends and by that time he actually when she was in high school or something like that he proved the serum and somehow was discussing about this with her father and she got interested in this then she went to Lichnerovic who was a PhD advisor and he told her well why don't you study the constraint equations and things like that so Lichnerovic had a version of understanding the constraint equation in general activity I'm going to tell you a little more about them shortly why don't you try this and that's going to be a thesis and then while she was tried to do this and she met Lure and he told her well you know if you do this it's going to be a little improvement of something that Lichnerovic already did you should try a really hard problem a real hard problem is the Cauchy problem for general activity so she tried and she managed to do this and the reason I'm telling her about this is because well you have maybe some of your PhD students too so I don't go for the easy problem go for the hard one that's how you I mean her paper is a milestone it just created mathematical GR as it is and so don't go for the easy problems do like even did anyway so the good idea is well I probably won't need this now so so to use harmonic coordinates or weights coordinates wave coordinates so good idea use let me put this in parenthesis generalized wave or harmonic and so by the way next week Harvey real I don't know if he's already here he's happy here we'll probably tell you more about this Cauchy problem so so this is going to be very sketchy in any case use so impose the condition impose box x mu equals zero that's what she did so in other words don't use any coordinates when trying to solve your equation require that the coordinates satisfy the wave equation so this this is called harmonic because if you replace the wave operator here by the Laplace equation Laplace then this would be harmonic coordinates and so somehow people didn't make much distinctions originally between these so this was called harmonic now this is the original harmonic the generalized one so this is the original idea but in fact the modern way of doing this is actually to put here any functions which depend upon G the derivative of G and X okay you can the existence theorems work either in this framework this what she did but you can put any functions as long as they don't depend upon second derivatives here right so that's the no D2G here good so if you do that then you get an existence theorem and the existence theorem is the following well so first then if you ask for say one then one plus Einstein equations are equivalent to an equation which looks like g mu nu mu nu d mu d nu g alpha beta is equal f alpha beta of g dg and x for some functions which might be very complicated but who cares I mean as far as if you're doing PDs you just say this is junk doesn't really matter the important part is here so this is really clearly a wave operator and so this is a wave operator acting on the metric the funny wave operator don't try to put something like that g mu nu d mu d nu g alpha beta which would be covariant but would be of course completely useless because covariant derivative of the metric is zero so so this is a bad idea to try to do this so you really need this breaking of covariance by introducing some coordinates here and that's the equations you get right so you get a wave operator acting on the coefficient is awesome lower order terms so yes so this box here very yes thanks so this box is actually right so written like that well written like here this box would be one over square root that g mu d alpha g alpha beta d beta acting on whatever it wants to add right so this is oops and there's a determinant so this is this box here this one is a bit similar differs by some lower order terms or something like that right so this is a scalar scalar wave operator here is a funny one it's really a in coordinates right so so it's it's obviously not coordinated invariant but it's meant not to be good so so then you solve yes five minutes okay I've put my timer on just going to check yeah I think four it's what's left five minutes 20 seconds good so you look at that and you think now this cannot be correct because now I have well posed wave equations I can solve them I have unique solutions but what guarantees that these conditions will be satisfied that's a problem what guarantees that the solution of so that the solution of two good and the answer is the constraint equations okay so the argument that proves this is long one and I don't have time to do this but the answer is constraint equations and I'm not going to write them down that's going to be my lecture this afternoon so the constraint equation so there is a miracle again discovered by Yvonne Shokevara which says that if the constraints are satisfied and if you have a solution of these equations then this will also hold and you'll get a solution of the action equation so this is a complicated argument which deserves a good PhD thesis and so I stop him thank you