 I started collaborating with Thibault at about the same time as Hermann, so 20 years ago, and it has been one of my most fruitful and enjoyable collaborations, fruitful because I learned a lot of things, not only the material, but also the way to address physics problems, and cheerful because, well, it's a pleasure to work with Thibault, he's always enthusiastic and he's never in bad mood. When he does physics, at least. So, well, we worked on the subject that Hermann addressed already in the previous lecture, the Berlinsky-Liftschitz-Kalatnikov of general relativity in the vicinity of space-like singularity, which was in the work of BKL, Cosmological Singularity for Definiteness, but that can be a space-like singularity in the future, and actually the analysis might be more relevant for such singularities. And I will address this problem not by following, let's say, the standard point of view, I will follow a different point of view. Well, after all, as we learned from the previous lecture, we are a bit stuck, so I think developing alternative approaches to BKL and understanding BKL and the implications of BKL, I think, should be welcome. And also, historically, it's actually the way I got interested in BKL a long time ago. So that has to do with the so-called ultra-relativistic limits of gravity. And that can be, so I'm not sure it will be useful, but I think at least it's fun. So if you look at the Poirier algebra, which describes relativistic physics, it has many contractions and there are at least two interesting ones. One which is rather familiar is the Galilean algebra, which you obtain by taking the limit where the speed of light goes to infinity. Of course, you have to be more precise when you define the contraction. You have to say which generators are perhaps after a definition kept finite and which are not. And so this is a bit sketchy, but you take C go to infinity and you redefine the boost generator B, B-I, and the energy E, as I've shown on the slide. Let's see if it works. And so if you do that, you find that the boost now commutes and also the energy momentum relation, so the commutation with the boost of P-I and E are modified. You get a zero here. Well, actually there is also the Batman algebra, I will say one word, where there is a central extension, but I will say that on the next slide. That's the limit which is familiar to us. So velocity is small with respect to the speed of light. And then there is another limit and another contraction of the Poirier algebra that you can consider, which is the opposite contraction where the speed of light goes to zero. And so if you make these redefinitions and keep the new generators finite in the limit, well, you can show that again the boost will commute. Well, there will be a C squared here and C goes to zero, so you get a zero. And the commutation of the boost with the energy and the momentum will be changed, but now it changed somehow in the opposite direction. So E with B is zero, and P with B produces E, so it's the opposite to that. Now, the Galilean algebra, or actually really the Bachmann extension, which has a central charge, so in the commutation, when I say commutation relation, here I'm really, well, you can use the generators and it's really a commutation relation. If you are Hamiltonian minded, these are the Hamiltonian generators and the Poisson bracket relation. So, well, whatever they are, so let me use the word commutation relations of P with B. Instead of having zero, you can actually allow for a central extension, which is the mass, and this leads to the Bachmann algebra. Well, it is well known that this is actually relevant for the non-rativistic limit of Einstein theory, so the Newton-Carton gravity. And the carol algebra is relevant for its ultra-relativistic limit, the opposite limit, which is sometimes called carol gravity, because the carol algebra, well, because it's based on the carol algebra. And the term carol algebra was coined, I should have said it on the previous transparency, by Lévi-Leblanc, who was studying the possible contractions of the Poir-Carré symmetry. And actually, and this is why I'm talking about the carol algebra here, the ultra-rativistic limit controls the dynamics of the gravitational field near a space-like singularity, so the so-called Belinsky-Kalatnikov limit, VKL limit of gravity. And as we heard in the previous talk, it has revealed intriguing connections with infinite-dimensional Katz-Moudi algebras, which in the case of maximum tripogravity is E10. And so I would like to describe in this talk some more carol invariant theories and briefly review the connection with the VKL behavior. And so I will first discuss carol causality, so causality in carol invariant theories, which is extremely simple, and I will compare and contrast it with the causality in Galilean invariant theories. And the main point that I want to make can already be illustrated on a very simple equation, on a simple system, a scalar field in Minkowski space, which is Poir-Carré invariant and described by the Klein-Gordon equation. Here is a Klein-Gordon equation where I kept the factor of C explicit. And we want to see what are the limits of this equation when C goes to infinity and when C goes to zero. Well, in the non relativistic limits, so when C goes to infinity, time derivatives are dominated by special gradients and the equation reduces, well, you drop this to the familiar Laplace equation for phi. If there was a source, it would be the Poisson equation, which implies instantaneous action at a distance. If I have a source here, rho, and modify the source, immediately the phi will feel it and will adjust to the change in the source. So this is the familiar non relativistic limit of the Klein-Gordon equation. And in terms of geometric structure, one can understand this as the fact that the line cones completely open to the hyperplanes x0 equal to constant. C goes greater and greater in the limit the aperture of the light cone gets maximum and light cones become hyperplanes. And so causality is very simple. You have to move forward in the future light cone. That means just forward in x0. Okay, so any motion with x0 is causality. Possible. Now let's look at the opposite limit, Karol invariant theories. So in the ultra relativistic limit, it's just the opposite. So if I go back to the Klein-Gordon equation, now this term will... So time derivatives. So this is meaningful when time derivatives dominate spatial gradients. And so you drop spatial gradients in the Klein-Gordon equation. Interestingly enough, you can still get something that depends on time. So it's still a dynamical theory. It has actually the same number of degrees of freedom as the relativistic expression. But you see that... Well, it's a very simple equation. It's just an ordinary differential equation with respect to time. And that is the opposite limit in which the light cones, instead of opening completely, they completely collapse. So they collapse to what? To a line. To the lines. At each point, xk is equal to constant. So there is a tangent vector d by dt. And so with Karol causality, the field at time t... Well, it's very... These are ordinary differential equations with respect to time. It will depend only on the field and its first time derivative at time t is equal to zero. But at the same spatial point. So nothing is... The equation goes from one spatial point to another one. You have to propagate along the line cone, which in this case is just the line xk equal constant from which you start. And so for that reason, sometimes the word ultra-relativistic or ultra-local is used for ultra-relativistic because you stay at the same point. And more generally, if you consider a more complicated system and I will illustrate that with p-forms, which is relevant to the analysis we did with Thibault, dynamical equations in Karol invariant series will reduce to ordinary differential equations with respect to time, so that the field at one point and some time is completely determined by the field at the same point at the initial time and some of its time derivatives. Okay, now actually when I was discussing the Klein-Gordon equation I wrote things in such a way I could discuss easily the electric limit well what is now called the electric limit of the Klein-Gordon equation there is another way and there are probably many other ways to take by appropriate rescaling limits where c goes to 0 or c goes to infinity and another one actually is called the magnetic limit which has also similar causality features but the Klein-Gordon equation will reduce to some other equation compatible with Karol causality and this actually has been discussed and goes by many people and goes back to the work of Le Bellac and Lévi-Leblanc of 1973 where they show that for electromagnetism there are actually two limits, electric and magnetic and then that was extended to Karolian electromagnetism by these people more recently well, interest in Karol group actually arose also because it's connected with the BMS symmetry since it acts on null hyposurphases and on null hyposurphases till the intersection with the light cone is aligned like here and the generalization of this construction to general theories in arbitrary dimensions has been considered more recently by almost simultaneously by these two groups of authors ok, so I will actually illustrate the other limit, the magnetic limit in the case of P-forms and the best which, so I'm now considering P-form gauge theories in arbitrary dimensions and the best way to discuss these limits something is happening with the microphone the best way to discuss these limits is to write the theories in Hamiltonian form and then one clearly sees what is kept in the various limits and why one is called electric and the other is called magnetic now the Hamiltonian action for a P-form reads as follows so the P-form is described by the P-form potential which is a P-form in the Hamiltonian description only the spatial components and they conjugate momenta define the phase space A0 with a time index are Lagrange multiplier for constraints so the Hamiltonian action or the action in Hamiltonian form will contain the spatial components of the P-form conjugate momenta that's a characteristic PQ dot term of telling us that P and A are conjugate pairs and then there is an energy density which I called H and there is a constraint which is a multiplier which is A0 it's a gauge theory and we know that in any gauge theories there are constraints at least some of the constraints the first class one generates a gauge symmetry but that's the only thing we have here and so here is the explicit expression the energy density first well it's really the analog of the electromagnetic case that we know very well where the energy density is E squared plus B squared which is the square of the electric field is the electric energy density the electric field is is really pi so I get pi squared for the electric energy density then the magnetic energy density is the square of the magnetic field which is B squared but here we have P-forms in dimension so that's spatial components of the curvature F so that's for H it has two E squared plus B squared that's electric magnetic and then there is a constraint which is really Gauss's law for these P-forms that tells us that in the absence of sources which is the case I considered the divergence of the electric field is equal to zero of course if they were sources that they would be sources okay whoops how much time do I start it late no 15 now well maybe I should have stressed C so you see that the velocity of light so I naturally with this choice of variables naturally appears here C squared and there is no velocity of light here so if you take the magnetic click-mit is actually the one that you obtain by the direct C goes to zero limit of that expression and you cross out then this term the magnetic energy that survives what is the gauge group here sorry what is the gauge group here the gauge group is the group it's adding to the P-form the exterior derivative of a P-1 form and you can show that if you take the bracket of the variable with the constraints sandwiched with the gauge parameter of that transformation I just described so a P-1 form is the gauge transformations so that's the gauge symmetry it's a billion now the electric limit is well you have to do first some rescaling of the fields to put actually the speed of light to transfer the speed of light from the electric energy density to the magnetic energy density so you do this rescaling and this opposite to rescaling A with P should be one and not C or one over C so you want to keep that this is the way you normalize things but then you can make this transformation and then take the limit and in that case instead of dropping the electric energy density you drop the magnetic energy density and you remain with the electric energy density both limits are compatible with gauge invariance because the electric field and the magnetic field are the same invariance so you are not spoiling gauge invariance by taking the limit and it takes the same form there is no C in the constraint and actually it's so it's compatible with carolin invariance that's easy to check you can write the transformation of the field and verify gauge invariance but a more direct way to check gauge invariance sorry carolin invariance in the Hamiltonian formulation there are invariant theories so in both case limits that I considered the electric field sorry the electric energy density only contains the momenta so momenta with momenta is 0 so the bracket here is clearly equal to 0 and in the magnetic limit the magnetic energy density only contains the Q's and Q with Q is 0 so you get this bracket relationship the energy density at two different special points commute in carolin invariance theory so which is in contrast with Poichard invariant theories where you would get here the momentum density as was explained long ago by Schringer and Dirac the commutation relations of the energy density and momentum density reflect Poichard invariance well in our case they reflect carolin invariance and it's simpler it's 0 here instead of having terms proportional to the momentum density and it's clear that this will imply straight forwardly when you compute the total energy and the boost generator which are obtained by integrating the energy density that's for E and integrating the energy density but with the first moment with XK to get the boost generator because E with E is equal to 0 E being either E electric or E magnetic when you compute these brackets you clearly get 0 which is indeed what you should get in a carolin invariant theory so this actually implies the characteristic brackets of carolin invariant field theories in the case of Poichard we would not have that and so the brackets here would be the ones of the Poichard algebra now I described the limits in the case of electromagnetism and P form theories so I will start with the and again it's easier to do it in the Hamiltonian formulation and I will only describe the electric limit and so I will start with Dirac and Hamiltonian action for Einstein gravity which reads well let me as follows so you have the spatial metric and its conjugate momentum the laps and the shift which are really the temporal components of the metric which acts as a Lagrange multiplier for constraints there is a surface term we talked about it yesterday so I don't have to talk about it today it depends on the boundary conditions and I am not going to discuss the behavior of the field at infinity in this talk so it's not right the boundary term that might be there if necessary now the Hamiltonian constraint and this is a so-called momentum constraint and they have the following expressions and well I have already simplified the expression of H2 by redefining the fields and redefining the laps actually this will be shown on the next transparency but I want to make the point first and then I will discuss that issue if you rescale appropriately things and redefine variables this is what you get for the Hamiltonian constraint and the momentum constraint the Hamiltonian constraint has two terms one which is quadratic in the momenta where the width super metric is the inverse of the super metric since we are working with the momenta here which was discussed in the previous lecture and then we have the curvature which come and they come with different powers of C and you can adjust these things to be that way by appropriate rescaling and so you see that if you take the limit C goes to zero you are just dropping the spatial curvature so spatial gradients and this is a consistent limit because it does not spoil the number of gauge the number of gauge symmetries it will take a different form but you don't spoil it because all the constraints which are first class in Einstein gravity remain first class in the limit and actually these ones become even simpler because they are just they just commute in Einstein gravity this would contain HK but here we get in the limit zero so they are first class carol electric limit of Einstein gravity is just the kinetic term in the Hamiltonian constraint now as I said you have to do some rescaling so if you put back all the constants and don't rescale so N is the unrescaled laps and H would be the unrescaled Hamiltonian if you rescale things which actually I was already doing this is the expression you get I mean if you keep everything and I have rescale to make so you get we know that in Einstein action we have the gravitational coupling constant G we have C and there is some numerical factor and this is actually the expression of the Hamiltonian constraint once you have rescale the laps in this way so I pulled out some factors in order to have one here but then this is what you get really in front of of the special curvature you see that actually you get indeed the C squared but you get also a factor of 1 over G squared and there is this factor epsilon which has been called by the Hamiltonian signature of spacetime because he was asking the following question suppose that I want to write the Hamiltonian formulation of Euclidean gravity so I change the signature of the spacetime metric what would be the expression of the constraints some people thought that this might change the signature of the DeWitt super metric which has Laurentian signature but it doesn't so this stays unchanged it remains a Laurentian actually it's a metric on the symmetric space a GLN over SON so that you are not going to change but the signature comes here and actually it will reflect there is an epsilon which comes here but of course when you take the limit and so this is really the Hamiltonian signature of spacetime for Euclidean signature you get a minus sign for Euclidean signature you get a plus sign so this is how you can tell from the constraint whether you are dealing with Euclidean gravity or mid-constant gravity and so because of all this you see that the limit C goes to 0 which I was considering the Karol limit is actually equivalent to the limit where G the gravitational coupling constant goes to 0 and actually that was considered by Isham even before and he called it the strong coupling limit because of that reason it was just setting G goes to infinity and that was the terminology used by Claudio, epsilon is equal to 0, you go halfway between Euclidean and mid-Coskian but actually that's precisely where the light cone collapses to 0 before even completely disappearing and so that was also called the 0 Hamiltonian signature limit of general relativity in all cases which is consistent system but that's also equivalent to the Karol limit I just wanted to point that out This K in the previous Friedrich metric, is it the same as N? Which? D I J K M and there is a generalist momentum by M N Oh I'm sorry you are very you are really reading everything so indeed K M should be K K M here I mean thus it summed let's see did I yes so it's really a copy and paste of an incorrect formula all the way through okay I'm sorry so indeed all indices are saturated now it turns out this is just an aside remark that this action that I just wrote in Hamiltonian form in the Karolian limit can be written co-variantly but the objects are not Riemannian metric they are actually degenerate metric but then is the metric is degenerate is determined is zero you don't have a volume element so you have also to bring in a volume element and you get the same number of degrees of freedom of independent fees as in Einstein gravity in the Hilbert action formula and you can write the action in a covariant way in terms of the lead derivative of the degenerate metric along the null vector I mean defining the light cones normalize this way so this is an aside remark but that's actually my interest in Karol from some time ago now there is also a magnetic limit in which actually you keep the special curvature and drop the kinetic term in the Hamiltonian constraint and that has been considered recently but also by other authors so why is this relevant to the BKL limit it turns out that in the vicinity of a space like singularity 5 but now I am on known territory and Hermann covered part of it the description of the gravitational field coupled to P forms as in extended supergravity models can be described in very simple terms doesn't mean that the motion itself is simple but you can tell which motion it is in simple terms because time derivatives dominate and so if you in the limit where you drop special gradients the description of the dynamics will be just ordinary differential equations with respect to time and at each special point you would have the generalized customer solution I am saying generalized because there will also be some expression for the P forms in which special gradients are set equal to zero so you get a homogeneous solution of the Einstein P form equation at each special point and this is the BKL limit and it leads to it turns out that this dynamics can be re-expressed by change of variables to the dynamics of a ball moving in a billiard the billiard itself being in a region of hyperbolic space okay but this is precisely the ultra relativistic carolian limit where time derivatives dominate special gradients so the limits of the Einstein equations close to a space like singularity is precisely controlled by the limits which is compatible with carol invariance now I should say that for pure gravity so it's important to have the P forms to have this statement 100% correct so correct in other words for pure gravity the special gradients cannot be set equal to zero for all times as you go to the singularities there is always a bounce to some curvature walls so the curvature plays an important role it can be replaced by an effective term that depends only on the fields but nevertheless it cannot be completely set equal to zero however in the presence of P forms it turns out that the bounce is due to the so-called curvature walls while the curvature walls are subdominant with respect to the electric walls so if you keep the electric energy density of the P form which is what the carol limit does you can consistently drop the curvature walls and the collision with the electric walls take into account the dynamics to leading order you get a complete carol description now actually it turns out that for the for gravity you have always to keep the it's always the electric limit which is relevant near a space like singularity if you couple to P forms sometimes it is the electric limit of the P form Lagrangian that you have to take sometimes it's a magnetic limit it depends on the spacetime dimension and the rank P of the P form but in both, in all cases you can have a carol description so in the particular case of 11 dimensional supergravity which has a particular interest so maximal supergravity the bosonic field content is 11 dimensional well is a metric and a three form and turns out that it is the electric energy of the three forms that dominate in that case the limit is equivalent to taking the electric carol limit for both gravity and the three form so in which you keep in the Hamiltonian constraint where now we have also the energy momentum density but you take so you keep the kinetic term here and the electric energy density of the three form now there are also terms in general in the full theory there are terms which contain the curvature as we have seen actually it's multiplied by C6 the curvature and I think B squared is multiplied by C squared and the moment the magnetic energy density but in this carolian limit they disappear and this is the correct limit as you go to the singularity and you can actually also keep the chance I must term in the carolian limit that it survives in fact now and this dynamics is in that limit as we have heard has been shown to have remarkable properties because the billiard that you get is not just any billiard but is a fundamental value chamber of E10 and this exhibits well at least points to remarkable connections with hidden symmetries but this is already this is true in the carolian limit I mean you are not losing this important feature on the contrary you see it more clearly in the carolian limit of the theory so let me conclude so we know very well since the early days and from our teaching that the non relativistic limit of gravity is quite useful in general relativity and very powerful methods exist to control developments in powers of 1 over C the opposite of travesty limits as I try to argue is also useful because it is connected to the B and so physically relevant because it is connected to the BKL analysis in the vicinity of space like singularity actually it appears also in other gravity related context and cosmology for instance in that recent preprint where they argue that the energy momentum tensor of a perfect grid in the carolian limit should have the equation of state P this rho is equal to 0 which is relevant for dark energy but a systematic expansion in powers of C remains to be more fully investigated there is some interesting work by dotcour from some time ago but I think that this deserves to be better understood in the context of the BKL limit and before I thank you maybe I can close by saying actually this last transparency shows the range of interest of Thibault in terms of the speed of light it goes all the way from C goes to infinity the non relativistic limit to C equal to 0 the BKL analysis so thank you very much thank you very much for the nice talk and keeping the time for a lot of questions there is a question please if you just replace this true by non-Avelian if you impose some duality between magnetic and electric component then we get SL2z and then the solution of this wheeled gravity equation psi as in in the previous talk then if we impose SL2z invariance then we get an automorphic form is it true or something I would think that this is true too because you see the curve here we have the same wheeled super metric just the dynamics is different but I would think that all these constructions based on symmetries remain valid so just imposing a non-Avelian will be set I'm not sure there are probably two things involved in the questions we can talk after if you want last quick question there is nothing on the chat so thank you very much thank you