AdS-CFT conjecture from a GR perspective
- https://www.youtube.com/watch?v=0RzsQOYjjW0&t
today will be much less ranty and less physics II well somewhat less physics II there'll be an increasing amount of geometry hopefully so last time I told you about what adsf tears it's a conjectured correspondence or duality physical equivalence between on the one hand quantum certain quantum field theories which are theories as we discussed that live on some fixed space-time and a quantum mechanical and on the other side of the correspondence certain quantum theories of gravity we could call them but really they're quantum theories where space and time is dynamic that's what one might most generally call a gravitational theory and you know there are conjectured specific examples of this correspondence where you know what both the theories on both sides are and there's a lot of evidence in particular for this sort of canonical example this N equals 4 super yang-mills we talked about there's a lot of evidence that this conjecture is true it's still a conjecture and as I as I said it's very unlikely to be proven because to prove it you would need to know what both sides look like and we don't you know understanding quantum field there is that a rigorous you know level of proof is already very hard for free field theories let alone interacting theories so I don't think it's reasonable to expect this to be proven and as I said before if you believe it's not true I think by now you probably need to have some rather specific reason you believe it not to be true however that said it's still important to stress its this isn't known to be fact yet so I talked about relativistic quantum field theories which may be to many of you is very familiar in to many of you is a little bit alien probably half and half today I'm going to tell you what a CFT is and given what we've said before I think we can we can go reasonably quickly conformal field theory is a huge subject it's it's basically a statement a conformal field there is a quantum field theory with some additional symmetries beyond relativistic invariants of the Poincare symmetry it's a huge subject particularly in low dimensions and we won't we won't be interested in low dimensions by which I mean 1+1 theories where all sorts of beautiful things happen and we'll need very little from the general story I will so I will only say things about what will actually need in order to define a DSC F T so we'll start off by talking about CFDs and hopefully we will also discuss ABS which how many of you are familiar with ABS spacetime how many of you aren't ok so so so hopefully we'll hopefully we'll be able to discuss both of these things today and then I'm hoping tomorrow I'll be able to state actually in detail how the correspondence works we'll see so let's just recap in the usual coordinates on Minkowski we have the Lorentz and translation symmetries of the juan curi group we can think of as being generated in this way and we are going to now consider a theory that is not only invariant under those but it also has a scale invariance so we'll also require our QFT to have scale invariance and scaling Barents is an invariance of Minkowski space-time again as with the punc are a group where I just scale all my coordinates by a constant time and space okay so that's a scale transformation and our fields that we build our theory from if we want our theory to have a scale invariance we should have that the action is scale invariant and that we should have our fields have some definite transformation under this and we require our fields to transform in the following way okay so you know maybe there's other indices on the field that some tensor field but let's suppress those so the field transforms as a as an invariant as a scalar quantity up to some overall scaling where Delta is some constant associated to the field and Delta is called the scale dimension Phi and remember in we've taken mass units so we've taken H bar equals C equals 1 and we're thinking of all the units being given now in terms of mass and this is just this scale dimension at least classically is just the mass dimension of the field okay if you have a little think about how this works okay so so classically it's just the mass dimension and the generator for this transformation we call the dilatation or dilatation operator D and it looks like this so when I press for an infinitesimal scale transformation the infinitesimal shift in my field is governed by this acting on the field yep this is how the field transforms under this transformation so lambda is a constant yeah I'm I think we were butting up against physics versus maths in a very fundamental and deep way and I'm not sure I'm not sure at this point if I try I'm not sure I can raise this to a mathematical level on the fly that's going to be useful and correct okay so we can think let's see how do i how do I think about this is some dippy more I can think of this as some dippy morphism equivalently and then this would be this operator here under this diffeomorphism would be the Lea I mean this diff you morph is amiss generated by some vector field and this is the leader rivet --iv of this field contracted with that that generating vector field okay so that probably still wasn't high enough level okay so then together remember we already have our generators for the prawn curry group which are the translation and Lorentz generators and this new generator has some definite I'm going to be well it has some definite trans you know we can now add it to our algebra this is now our algebra these are all the commutation relations of the lie algebra together with the plan curry algebra that we would require our generators for this scale invariance to satisfy and just something to note now M squared which was a Casimir of the Poincare group is no longer it no longer commutes with everything so no no longer a Casimir and so in particular you can't have you know you can't a field with their mass as we're used to in relativistic field theory no longer is a good representation under this larger one Curie plus scale invariance group and in a scale invariant theory all couplings in the action should be dimensionless but ie that the action should have no mass scales within it because obviously those mass scales won't be scale invariant so action should have mass scales in it so a theory that scale invariant is of this marginal type that's truly scale invariant ease of this marginal type that we wrote down before and in particular roughly speaking it will look the same at every scale so it's a very special theory that's guaranteed to exist at high energies at low energies everywhere you like okay so these are complete theories these are good theories in the sense of if you want a fundamental theory that describes that zero works up to arbitrarily high energy scales or short distance scales these types of theories would be in that class you might wonder whether there's a limit on what Delta could be Delta should be real and greater than some number that's greater than zero and for those of you who know about quantum field theory there's it there are unitarity bounds you know given the we might see this later but depending on the spin of this field there's some different number here that depends on the number of dimensions you're in in the spin of the field okay but in particular Delta had be better be greater than zero or unitarity reasons yeah was there a question sorry yes I've suppressed indices as I said this so these could be some spinner tensor fields yes generally generally that's right so indeed for example the Dirac field transforms in some analogous way okay let's have some examples then so the free scalar that we've discussed so so I'm suppressing the lorentz index there this is a scale in Barrett oh I'm sorry of course what the the free scalar firstly I can't have a master my should have put a cross through that I'm not allowed to master because that will obviously break scale invariance but if it's a massless free scalar that is scale invariant provided my field has its usual dimension Delta is d minus 2 over 2 okay so provided Delta is d minus 2 over 2 this action will be invariant under the scale transformation it's certainly a variant on the Poincare as you all know but it's invariant on the scale transformation the reason being that under a scale transformation this will pick up D factors of lambda the derivatives are inverse you know D by D X so they pick they get will give me 1 over lambda squared and so I'll require my field to transform with the appropriate numbers of powers of lambda to cancel all of that and leave the action invariant yep yes that's at the quantum level yeah exactly yes I mean don't don't worry about this that's but people who we might we might take more about it later but it's not important here's another example classical say Phi to the 4 or yang-mills in 4 dimensions as a scale invariance so if I write down a theory like this let's have some coupling let's call it C I normally call it lambda but I realize I've used lambda so I won't so this thing again for the same this is now in for don't for space dimensions for the same Delta here though Delta would be 1 in 4 dimensions this is also scale invariant so this interaction term remember this is a free Theory the action is quadratic it's sort of trivial theory but this is now an interacting theory classically there's you know if you solve the wave equation sort of in modes the modes would all couple together through this nonlinear interaction here this term is obviously scale invariance because you know the the four lambdas you would get by transforming this under the scale transformation and then canceled by the inverse for you would get from the transformation of Phi itself you may be wondering what this has got to do with yang-mills theory but yang-mills theory structurally looks very similar if i write it out non abelian the yang-mills in terms of the its vector potential you know roughly speaking you'll get a kinetic term and you'll get some interaction terms one of which looks like this and the other one will scale the same way so it's this is a so this is now this is true classically now we can wonder quantum mechanically are these you know it looks like these actions scale as they should one of the key points that I didn't I can't remember if I said yesterday but let me re-emphasize that if I didn't is that just because an action has a symmetry classically doesn't mean when you put it into a path integral and think about your quantum theory that that quantum theory will still have the symmetry it's a very subtle business and it's a very subtle business because in the path integral you have this horrific integral over all field configurations that you have to do and to make sense of it you nearly always have to introduce some way of getting rid of the infinite number of degrees of freedom in the field that you're integrating over and in particular if that when you regulate your theory by for example saying I'm only going to integrate over configurations of my field that are you know that have a have a say a frequency or or momentum less than a certain amount if that regulator breaks the symmetry when you remove it it's far from clear that your theory will still have a symmetry it's far from clear your theory will even exist in a nice way but if it does it's far from clear it will have the symmetry and in particular this scale symmetry is all about scale and the whole point of introducing these cutoff the regulator and then renormalizing the theory by removing the regulator is to kill off very very short distance behavior that is causing the problems with the path integral in the first place and so it manifestly breaks this nice scale symmetry that our action may have and it's in a very subtle question when you remove your regulator whether you recover a theory that scale invariant and in fact in this case for the for the free theories it's sort of it all works and it's fine but in any interacting situation it's very subtle and these theories are not scale invariant okay sort of famously so yang-mills theory in four dimensions is famously not scale invariant the coupling is dimensionless that's true but when you regulate the theory remove the regulator renormalize and so on you secretly pick up some scale dependence in the coupling and this is what's called the running coupling so an example of this a very physical example is that in our theory of sort of hydraulic physics QCD the theory of quarks and gluons the coupling in that is is indeed dimensionless but as you look at processes at different scale feels the effective coupling changes it's a very subtle effect so scale symmetry is not something trivial in you know the quantum level it's not something trivial in the action it's far more subtle nonetheless this example of N equals 4 super yang-mills again looks like some sort of yang-mills theory with some scalars and so on so it it sort of structurally looks a little bit like this it's classically scale invariant that is a theory that is quantum mechanically scale invariant but it's not a it's not a triviality that it's it's by any means that it's a scale invariant at the quantum level anyway so so let me just say scale invariance is often of of an action is often broken but not always by UV regulation and the subsequent renormalization however as I said there are certainly theories that are retained this scale invariance and then it's an interesting fact that in all cases we know of unless have been very recent developments I'm unaware of which there may have been but certainly up to recently in all cases we know of when you have a prawn curry invariant theory that is also scale invariant in fact you inherit an additional slightly larger symmetry group which is the conformal group so you'll end up with a conformal group or a conformal field theory so I to my knowledge has no proof of this well there are proofs of it which require some assumptions about the theory so I think that it's not clear that what the most general proof of this would be but there aren't sort of relevant examples where you have one Korean scale but not this slightly larger group and what is the slightly larger group there's an additional generator of this conformal algebra or conformal group which is pretty odd and looks like this it's called the special conformal transformation I'm not sure why it's special I should say but maybe when you see it you'll see so again using usual Minkowski coordinates the it will be generated by a few morph is immed at looks like this ok so where a is some D vector so a squared is it contracted with itself and so on ok so it's a rather strange action on Minkowski space it is if you if you're brave and plug this into the usual geometric you can check that it leaves the metric invariant the form of the metric invariant and then you can look at its generator which I won't write down it's not particularly illuminating but once you have the generator you can compute the algebra of its generator which we'll call K with the other generators we've written down and I won't write out it in gory detail I'll just be schematic this is the usual Minkowski metric and then there's some other terms there's another term with different index structure and so there's some commutation relations with the other generators oh oh I'm sorry this should be this should be Moo but anyway I mean I haven't you know it doesn't really matter the precise form here because I'm going to write it more nice well more nicely in a nicer way the algebra is actually or the group is so.2d and it's not very obvious from the way I've written it down as it's being written down you know as we've sort of added generators but if you repackage everything it takes a much nicer form so let's introduce indices a B which are 0 1 up to D plus 1 where 0 2 D minus 1 were our usual base time you knew indices so we've got two more indices and then if we write or make the following identifications then you find that the these with I should say J so J a b is anti-symmetric of the other components are determined so these are now the generators of this so.2d in much more familiar form I'm sorry I think I mean I think I mean this I'll leave it as an exercise for you to check the the ordering of the signs here so this looks like the usual rotation generators or Lorentz generators but just with different numbers of minuses here so that's why we have so.2d we've got two minuses in this okay so this is the the group that so the conformal group so if we have prawn curry and scale symmetry we always land up with a theory that has a larger symmetry group this conformal group and it's so.2d that's the important point now obviously when a theory has symmetries it has consequences physical consequences let's just recap some of the physical consequences of the Poincare symmetry for vacuum correlators so we talked about how correlation functions can be computed from this path integral generating functional for them but these are important quantities to remember they look like these are examples of correlation functions they tell you in some sense about the behavior of your theory this in some sense tells you what a field is doing at a given point this object is telling you about how a field propagates from one position in space-time to another if I have more Phi's for example that would tell me about how particles in this theory scatter at least for prawn curry and barian theories and wong korean Barents already tells us that if we're looking in the for the field theory in its usual vacuum state and we look at these correlation functions this must be a constant it can't depend on space and time and that just trivially comes from translation invariance of the vacuum and this quantity here well whatever it is it's only a function of the difference in the space-time positions and in fact it's only a function I should say the norm of the difference in the space-time positions because of translation invariance and because of Lorentz okay so already know that's quite constraining but scale invariance now constrains things further or rather well so for a CFT and not just a relativistic field theory where these would be true it constrains things a little further this has to vanish and that's basically because if the field has any scale and in vacuum this was non zero but constants this would have some scale and it would break scale invariance explicitly so the vacuums not allowed to break scale invariance in a scale invariant theory and so this has to vanish note that if the dimension of this was zero this wouldn't be true but for reasons of unitarity that we said before that's not allowed there's a lower bound deltas should be positive yeah no no no it's under the whole thing this the air the special transformations as well but at least as of I'm not quite sure what the state of the art is but there aren't I don't know of any examples that there is which are not which are invariant under scale but not invariant under this extra symmetry so but in fact some of what we're going to say doesn't depend on on the whole conformal group it actually only just depends on on this scale part anyway the form of the two-point function which is perhaps a little bit more interesting physically is rather constrained you can sort of understand intuitively why this roughly the scale of two Phi's is well so this form here which you know you relativistic cemetery basically just told you it was a function of the norm of X minus y is now entirely constrained to a particular power of that norm so I should say this is the norm of X minus y to the 2 Delta where C is some constant which of course you can set to 1 by just redefining your field appropriately and in fact more generally if I have if I have if you have fields I'm now going to put some indices on these let's say a with scale dimension Delta a and another field B with scale dimension Delta B the two-point function or rather let me just say for some I'm being silly for some ease let's assume I've got a few fields and I've labeled them with some index a it's more sensible and is a corresponding scale dimension Delta a then if I look at the two-point function of two of these things it's relatively easy to see that it will vanish if the dimensions are not the same and it will take a form like this if they are the same though this is nice so conformal field theories are nice because the propagation of particles is then completely fixed well I should should be more careful because these particles don't have a mass it's very subtle whether one can't really interpret them conformal fields there is if they're interacting always have very long range interactions and it's rather subtle the normal manipulations you go through to convert a correlation function like this to a statement about propagation of particles don't really work anymore but nonetheless these these correlators exist and they tell us roughly about how if you disturb the field in one place information propagates physically through excitation of the whatever it is that however you want to think about what this field is doing it's also true that the three point function is is constrained up to some constants so it's functional form so if I have some fields again like this a B and C I look at the three point function which morally speaking although not for a conformal field theory but morally speaking in a normal relativistic theory would tell us something about scattering so this form is fixed in terms of XY and Z I won't write it down because it won't be relevant to us higher point functions are now not constrained they dead in the sense that they have non-trivial dependence that you don't know a priori on the positions of where the fields are living so they are constrained but you they're not constrained enough to know explicitly I suppose maybe it's ID in HoN for sure it is I don't know if I can give you a deep reason maybe think of it like this if I think of the correlation function of two of two fields in relativistic theory with different spin and I think of the two-point function if they've got the same spin they can have a non-trivial two-point function if they have different spin they can't why well that's because of the back of this structure of the vacuum well sorry they they sorry I should be more careful then they may be able to have a non-trivial correlator but they may not depending on the spins I should be more careful but yeah I'm so i mean i can i could show you in more detail after how this works but it's not you have to you have to use properties of vacuum being invariant under dilatation z' and point karere every CFT well if we every quantum field theory we expect to have a we expect to be able to take our theory put it on a curved space and in doing so we can we can vary the metric and generate a stress tensor in the usual way that's one way to think about the stress tensor for the theory tells us then putting it back putting the metric back to Minkowski that will give us our Minkowski stress tensor and because of this structure we see that the stress tensor itself must always have dimension D okay because the the actions obviously dimensionless there's D X's and the measure the metrics dimensionless in my conventions here and so the stress Center has to have dimension D and that's always true and the stress tensor can have some non-trivial two-point function which again takes a similar form I I should have said what I should have said here these these are really are scalar fields as I've written things I've there are analogous formula when you have indices say tensor indices but now this will this object here will be again determined by some constant but will also have some tensor structure obviously let me let me give you an example of that here the two-point function of the stress tensor always takes this form but obviously we've got some index structure which has been suppressed there which comes out in some object here which I build from I won't write it explicitly it's not terribly illuminating but I build from these objects which are scale invariant so the metric and also the combination X mu X nu over x squared which is scale invariant and so this is a sloppy complicated expression in terms of these things but it's not difficult to write down and this this quantity here in a conformal field theory is usually well it's it's often called the effective central charge it doesn't matter in two dimensions it is a central charge for the conformal algebra but in more than two dimensions it's by analogy it's called this but you can think of it as counting the degrees of freedom this is roughly telling us how energy propagates from one place to another and roughly speaking the more fields you have around to propagates the the more energy can propagate but in particular let me just because this may be relevant if we have time later you can calculate what this is in this theory N equals 4 super yang-mills this example we had before and it's equal to some number which in fact is independent of the coupling interestingly although that's not usually the case so it only depends on n the rank of the gauge group which indeed counts the number of degrees of freedom in your theory that number fields or elementary excitations you could think of but it's some number yeah was there a question yeah well it the dependents it's some it's something light I'm not sure I can reproduce it on the fly it's some combination of these objects well no it's some I'm not sure it there's a this has got various symmetries it's the metric in these it's symmetric in these this is also a conserved tensor one has to building conservation so imagine the most general set of terms you could write down using this sort of object and this sort of object that are compatible with symmetry under these two indices and the whole thing will be conserved if I act with a derivative of MU or nu or alpha or beta and there's only one thing of that sort I'll leave you to look it up or computer yourself it's not very hard but it's not illuminating was there another question okay good so and the last thing then I want to say about conformal field theories and then you're all experts or at least you know enough to carry on with what what we need for ATS EFT is it in our path integral remember we wrote down this generating functional for correlation functions which if I functionally differentiate it will give me these correlation functions and schematically again there may be many fields I've just I'm just calling them all farm suppressing indices the Lorentz indices and there may be many obviously there are many local observables that I could write down let me just write one but you could imagine lots of them and remember we call so this is some local functional of the fields and in this we call a source for whatever operator it is and then we think of this as a functional J and really because there are lots of different local operators really this is a function of lots of different J's one for each local operator I could write down and then if I functionally differentiate with respect to J appropriately I and then set the source to zero I generate these sort of correlation functions we were talking about but the only thing I want to say here is that in a conformal field theory because this will have some dimension let's call it Delta o the source because they when I add these sources the new action with sources should still be scale invariant otherwise I've explicitly broken scale-invariant the source should also have a transformation under scale so that let's say the scale dimension of the source the way it transforms will be correlated so it'll go like D minus the dimension of the operator so that this whole thing is scale invariant that will again be important later and in the very last thing I want to say but I won't I won't dwell on this is that I've really talked about a conformal field theory on a flat space-time on Minkowski space-time just as with any just as with upon carry invariant relativistic theory on Minkowski you can then put it on a curved space-time but this if you're on a curved space time you've manifestly broken pond carry and variance but the action is still constrained by the pond carry and variants although there are potentially new terms you can add to your action that will have vanished in Minkowski terms depending on curvatures in the same sense you can promote a conformal field theory to a curved space-time and then the theory usually will then have a symmetry under vile transformation or scale transformation of the metric where I change the metric by some but this is a bit a bit like a scale if this was a Minkowski this would look like a scale transformation but more generally for any space I can do a local scale transformation but like this a vile transformation and if I if I write if I take my CFT action and covariant eyes it appropriately and I can usually ensure that the theory will have some invariance or at least covariance under transformations like this it won't be it won't be important for what we have to say but it's just worth pointing out ok so now that's enough so many of the things that I've written on this board or these boards we will see coming up later in very different contexts so do bear certain things in mind at these the form of these two point functions for example just just keep this in your mind okay that this goes like this we'll definitely see this within a lecture or so in in a completely different context and it'll be very relevant yeah oh I see good I mean of course so now let's talk about the second part of ATS CFT the other bit of jargon which is ATS so many of you will know about ATS space-time or a scene aspect of it so let me but many of you won't so it's a peculiar space I'm going to consider D plus 1 dimensional ad s so we were talking about D dimensional field theory this will be D plus 1 dimensional abs and it embeds as which I'm gonna take to have some radius let's say L in a sort of two-timed Minkowski space I'm not sure what to call it our 2d as so if I take these coordinates on this two-time Minkowski space then ad s can be embedded so that the induced metric is the correct ad s metric in the following way as this hyper surface so just note that whilst the the ambient space that we're going to embed the metric in has to times because of the way the embedding works the induced metric will only what will constant x.i our time like closed curves in this embedding so if I sit at constant X I can still move essentially around this circle parameterised by U and V and if I ask what's the proper distance I move or the proper displacement I suppose I actually move a proper time right it's a time like direction so I'm moving I mean a hyperboloid obviously looks something like this but I'm not sure I can not sure I can imagine the two time directions and in a nice way but anyway the point is you're moving around the hyperboloid but both the directions you're moving in a time like so this is a closed timelike curve and so ad S is not this embedding alone it's the covering space of this okay so 80s it can be embedded as that but AD s is the I'll use the words Universal cover but I'm not I'm not sophisticated enough as a as a geometry to really know what that means so if you know what that means that's great I open s trian the universal cover of the hyperbola and as physicists we can be more explicit if I parameterize the hyperboloid like this where these theatres parametrize a d- one sphere a unit round d- one sphere you can verify that if you plug that into there using you know basic trigonometry it satisfies the embedding condition and furthermore covers all points on that hyperboloid and if we were just covering the hyperboloid tower would be an angle it would be the angle around this hyperboloid but because we actually want a space that doesn't have closed timelike curves abs face time for a TS face time we think of tau being a real not an angle so as tau goes around i wind round around my hyperboloid infinitely many times and the induced metric if you now substitute this into here to compute the induced metric so the metric and this is a global chart usually what's called the global chart of 80s looks like the following okay so it's got a simple form we can also write it in a short childlike form just where this function f looks like this so you see the relation between Rho and r is just some simple relation like this and you see that when Rho or R is small it's a version of spherical coordinates but when R becomes large this deviates from flat space this term becoming dominant over the usual one here that is all I would have in flat space reflecting the asymptotics being different now what are the isometries of this space the embedding we've use allows us to see the isometries immediately and their so.2d okay they're obviously so.2d the isometries are just you know the the equivalent of the prong of the lorentz group for this two-time metric is so.2d not sure what you call it if there are two times but - well anyway so.2d and it also you know this it leaves this this is the inner product of two vectors if you like and that's left invariant again by this so it leaves the embedding invariant and therefore it these will be this is the group of isometries of ad s it's not manifest you know this larger this group isn't manifest obviously in this particular chart which manifests only some rotations there so d minus one and an so2 if you like or just translation symmetry in tau but from this we can see that there is a larger isometry group hiding now let's try and understand the so as row or are equivalently goes to infinity this metric is dominated by an exponential blow-up of the time and angular pieces like this so as row becomes very large you can see the space is becoming very big in these directions and Rho goes to infinity okay so this becomes obviously arbitrarily large now what what does that mean what sort of asymptotics is this well this is a space which is conformally compact so meaning as an asymptotic region which can be compactified in the following sense so a by the way the word conformal here is not the same confer I mean in geometry this is called a conformal eek in fact space but it's not really related to conformal transformations well directly there is some relation but it's a slightly confusing terminology I mean or rather it's a fine terminology but don't be confused by the competing terminologies I should say so a conformal compact space can be written as in the following way so we have a we have a metric here a regular metric on a manifold M with boundary that encloses N and Z is what's called the defining function and so said is greater than zero inside M and vanishes on the boundary of M linearly let's say vanishes linearly ie DS ed is not equal to 0 on the boundary and such a space Shoji is a perfectly regular metric with a real boundary and the full or space-time and the full space-time then has an asymptotic region due to this defining function one over it blowing up okay but it blows up in a sort of controlled way so the full space-time has what's called a conformal boundary so and it lives let's say on M - its boundary so the the boundary now is some asymptotic regime region so the full space doesn't have a bounder anymore you can only think about its asymptotic behavior we say it's got a conformal boundary although there's no boundary at all it's a boundary only in the sense that you can conformally by multiplying appropriately you can then turn it into a real boundary of some other metric space space-time and we say it has a conformal boundary metric which is equal to that induced on the boundary of this regular space by this regular metric okay so this regular space has some real boundary with a real induce metric on it and we say that's the conformal boundary metric of this full space so really there's no boundary at some asymptotic region but there's some notion of it having a boundary you know having some metric that describes that that that asymptotic region but and it's not going to be relevant for us but obviously it's inheritant if what i've said i have a freedom in how i can formally compactify a spacetime that has a conformal boundary give one positive smooth function on this space and recover a new defining function with a you know with a different regular metric so this whole construction is only defined up to multiplying this defining function by some positive now strictly positive function and therefore the conformal boundary metric is defined only up to a vile transform so again it won't won't play much role in what we're going to do but it's just worth noting and now we can see let me see I started at 2:15 right so ten minutes now we can see what this is okay this is a conformal boundary this is a conformal II compact space and the conformal boundary metric at least a representative for it ie it's only defined up to vile transformations so we talked about conformal classes of the boundary metric as in the equivalent elements within the class being generated by vile transformations so this thing that the conformal boundary metric of this is actually just or a representative is just this it's just minus D tau squared plus D Omega squared so that more so four 80s over there if we take e to the row e to the row equals one over Z then asymptotically just as simpatico we could we see the metric tends to this form and this I can write as 1 over let's call it it's big Z squared this is our defining function it certainly goes to zero in a nice this is a regular metric I can say take Z to be from 0 to infinity and then or in fact I don't sorry I don't even need to I'm only doing an analysis locally anyway but Zed has a boundary at 0 is what I want to say and extends some to some positive value here this defining function indeed goes to 0 at the boundary and is positive away from it so this is a conformal boundary and so the conformal boundary metric let me write it as the the boundary of let's call it this here this space m and right it is the boundary of M even though it's not a boundary it's a conformal boundary let me just write it like that so this is what people would usually call the Einstein static universe but it's basically just in a time cross a sphere around sphere so that's the boundary of this ad s spacetime and we can in draw a we're in our picture we've conformally compactified it so we're sort of drawing a picture of it in this conformally compactified sense we should think of it as an infinite cylinder solid cylinder where the boundary of this cylinder so the ad s faces the interior the boundary of this cylinder if you like in this regular space is the is what will give you the conformal boundary of the full space-time so this is the conformal boundary it's not a real boundary remember it's an asymptotic region and the alias space-time lives in the interior and our tau coordinate goes up and our angular coordinates go around and if you like our radial coordinate there's a we've drawn it there's a sort of center of spherical symmetry and then our radial coordinate goes out and reaches infinity of this conformal boundary tends to infinity there and one of the key points that's very easy I mean it's a simple calculation one of the key points about ATS is that whilst this is an asymptotic region this can form a boundary is time like the Lorentzian manifold and in particular one can ask supposing I send a null ray in my space out towards the boundary does it reach it and in fact as I've drawn things this isn't really a conformal Doug I mean it's not a Penrose diagram but some morally speaking light rays would travel at 45 degrees in this diagram so in particular light rays really do reach the boundary if I emit them from anywhere in the interior with my flashlight they travel out and they meet they they hit the boundary at a finite time Tao and that's a key key physical point about a TS space it's actually not true if you take a space like curve and take it out to the boundary it's an infinite proper distance very easy to see that but a null a null geodesic travels out to the boundary and hits it in a finite time Tao the key physical point there is that ad S acts like a box like a real physical box so even though this isn't a real boundary it's a conformal boundary it's not some asymptotic region actually for null Ray's they really get to the boundary in a finite proper time for observers in the interior so imagine someone sitting arbitrarily far out you know in a finite time in the interior you could send a message arbitrarily far out get your friend to send it back to you and you will reach you know you'll recover it in a finite proper time so what that means physically is if I want to define dynamics in this space I must think impose some sort of boundary conditions out here you know all my fields all my dynamics high energy modes a short distance wave mode or you know high frequency wave modes will propagate out to the boundary and I'll have to deal with the boundary when I deal with dynamics I can't ignore it and in some sense I will expect stuff to go out and with appropriate boundary conditions it may come back at me okay so it really acts like a box yeah there's a couple of questions yeah so if if I got a timelike curve here could be a geodesic oh this this is some null ray I just mean it will reach the boundary at some in the conformal boundary at some finite tau and then you or if you like it will go what do I mean well sorry for what how do I answer the question should I say so I think it is a I actually can't remember if it's a finite a fine time along the curve you can you can do the calculation and tell me but it's not the important point and I think it's not a finite a fine time along it's not a it's not a finite affine parameter along the curve and for the physicists that's because the affine parameter basically tells you about the blue shift or red shift of a photon if you like which becomes infinite so it isn't a finite affine parameter but nonetheless it will reach in a finite time tau and therefore someone arbitrarily far away can catch this before it escapes and send it back and for me in a finite time tau I will then see that see some information return sorry was there another question okay oh yeah maybe I I'm not sure i I'm not sure what you mean it's not a physical boundary no it's an asymptotic region it's an infinite proper distance spatially so if I send a spacial curve out I mean you can just you know do that do the calculation for for the metric I gave you if you if you look at a curve that's on a constant time slice and ask how long is it it's got a proper an infinite proper distance length to the to infinity yeah so it's it's really not a not a not a boundary in the usual sense of boundary but the important point is it is a boundary for null rays let me just finally in the one last 1 minute write down another chart which we will be using really or this is the way we'll think about ad s and it's the Poincare a chart and in terms of this embedding if I did everything right but you can go away and check there's a rather complicated way of parameterizing not all of the hyperboloid but half of it using a different parameterization than we had before so this is only covers half the hyperboloid and what half is it or well or rather there's a there's a complicated map when you think about what portion of this space you're covering so you only cover half of the hyperboloid but remember abs is the cover of the hyperboloid so what bit of this solid cylinder are we covering and basically we're taking a null wedges I think this is the way to think about it you take know a null wedge that intersects the boundary or slices through this cylinder and then take another null wedge that just intersects the boundary here at the same point and basically the interior of this wedge is what's covered by this and so if we thought about a conformal diagram for what these what we're covering really there's a boundary or a conformal boundary and then we've got some caoxi associated to the these null surfaces but the induced metric now takes a very simple form and just to be clear here a ran from one up to D minus one and now you know Mew is our usual 0 1 up to D minus 1 ok so this is the metric of AD s in this pawn car a chart it doesn't cover it's not geodesic Li complete obviously I can extend it through these cow horizons but note I'll say this again next time this manifests some interesting isometries in particular there's a plonker a like isometry which is the usual one associated to these coordinate system in cows key metric so clearly if I do a Poincare transformation on these X's I generate an isometry of this metric and there's also an interesting an interesting one here where I scale my Z coordinate and the X's in the same way ok that's obviously also an isometry it leaves this metric invariant but think about the conformal boundary now Zed is a defining function in its own right now what's the conformal boundary it's it said is zero it's just the Minkowski metric and these isometries therefore have an action on the conformal boundary and what's the action on the conformal boundary it's prawn curry and scale and in fact whilst this doesn't manifest this special conformal transformation that's also there it's just not manifest in these coordinates it's a more complicated isometry you know in these coordinates is something nasty okay so as well say Nick you know as I'll just re-emphasize the beginning of next time we see that the isometries of ATS have an action on the boundary that looks like it's the conformal group action okay so sorry I've ran over