 naredim nekaj čans, da beri tih tibov, ali nekaj to je počutil. Na zelo sem usteval, da zelo sem stavila, da da modljajo objev však, da tih tibov nekaj je potrebilo. Tručne, kako je tudi objev vse, je to počutil, in nekaj je vse karjere, ker je to da bejte, zelo to je vse vsoje, kako je to bilo. Zelo, najbolje, nekajh dobrovodov, nekaj je odličil na učenke delavne odstavlje, da sem taj pačen. Tih nekaj nekaj dobrovodov, imeš tudi povod, ali so da vstavljali, in ta je doktor Adeta, predstavlja, ...in ležitev, kak je to zradi, z nekaj odložiti... ...zato, da še pačne vzpečne z vsega, še začne, zelo vzpečne... ...srečen, vzpečne z zelo vzpečnji, vzpečnji in nekaj vzpečne... ...zato, da je površena veg, a vzpečnji vzpečnji... začela pogleda v svojih pomembnikov od pupilsu v op račje. Finča v zelo drožici, kar je nagladavo na vsem vrloh, ker namočila roža vzgovorjana vzgovorjana vzgovorjana vzgovorjana vzgovorjana. slabon Trekchez na našobredmarktak elected inaccurente, z giving the total entrop already, total energy but. The local quantities on the horizon could also be given a ptermodenamic or fluid dynamic interpretation, and so the way it was done, I will quickly recall here. If you have this is supposed to be the horizon of the Black Hole and you take a fournier vzijegovati v stavih, na konstantanj stavih, ki se zelo zelo vzijega, je šeklje in evolucija, da je zelo skupil po zelo, načo je zelo kaj je zelo. In tako, če češel je. In kaj ne, nekaj ne, nekaj ne, nekaj ne, nekaj ne, nekaj ne, nekaj ne, nekaj ne, nekaj ne, nekaj ne, nekaj ne, nekaj ne, nekaj ne, nekaj ne, antique. In utenku se Iztavljeje. Danes poslala preziditi kvalitvenje vsepeh tajga prijevanje, kako se lanče bil pohoužen in kraju backyardov, ker je kышbenje. Slavno je malo lovač na kot taj nekominje, splitnega vsepeh vsepeh. Slavno je bil ki ta nekominje taj nekominje, od mnej grači pohruženo obnej. Teta. Zelo je betra tezori. Se je zelo jezmento napijar. Je jezmento tezori. Se je zelo jezmento napijar. Se je to jebe. Teta. Se je tezori. Teta. Se je tezori. Govorilo, kaj je arbila. So tezori je matrik. Teta. Pigučit. To se je ta zelo bars in bil. To se je to, da je teta. To se je ta zelo bars. To se je to, da je tezori. A zelo je zelo še vsega vektor in vsega vsega vektor vzela, kaj je pravda za začelj, da je začelj, da se vsega vektor vsega vsega vsega vsega vsega vsega vsega vsega vsega vsega. A τ looked at the discovery of T-Boh that when we project Einstein equations in this component along the null vector and along the surface, closed equal time surface. This gives some equations that have been dubbed Damoures Navier stok because they have the same form as the Navier stok equations in zelo vseh presežitevajo. Kaj si je vseh rejščiti tez geometričke sredne pristitej, kako ta omega alpha je močelj vseh torturjevcega, nekno vseh torturji, nekako na zelo vseh torturji je zelo vseh torturjevcega in tez sigma in teta je zelo vseh torturjevcega in teta je začeljena. A ki da se tukaj prejestim, v njem jednjen Leaf attempts which comes from the bulk stress energy tensor. 1 over 16 pi is needed to normally, so when you integrate over the whole birds, it reduces to the total angular momentum, this fixes this normalization. In this equation you can read of the transport coefficients na nekaj včasnih vsečnih rečenih, ki je to, kar še v vsekusti in vsečna vsečna, ki je to vsečna, kar je pora 16 pi, tako kaj je vsečna vsečna, ki je minusr, bojte pa se, ki se vz الس. Vsečnjiš, ki je tudi zainem, da je vsečnjaj zredil ni zrčenih rečenih rečenih, da se nekaj neče izvahnih rečenih, in je vse občasno izgleda. Sada, tako, ne bilo vteprniti gravidične in vzpešnje. Ale zelo je vzpešnje. Zelo je negativne vsočne vizkosti. Zelo... Tjega je tibol, je vzpešnje, vzpešnje, which is macht, in the other people have called it membrane. But the fluid bubble is more appropriate because it has a pressure and not a tension. In fact, if you interpret it as a tension, it would be negative. As Tibo explained in the original paper that this is needed because a normal fluid bubble has a membrane tension to counteract the pressure whereas for izor none of this black hole has to counteract a garitte § atr almonds in so negativne tensione so negativne, brez dneva, je zuvak. Ivakovčo pa veliko vyskovstje je negativno raz привodil, kjer nachiče in izgledaj gravidenje, je tai tem, da jel ni značno zazmaitno vsega, in fillo ne. Zdenje, da jel tamo bilo odlič, odliče in izgleda. Vzoire je končne, da je medograv�. kar je generačne tukaj, da počekaj si vse, ki je malo nekaj vziv, že je nekaj pomečen, so vse ne podešte, kaj je tukaj vziv. Zdaj, mi je tukaj vziv, da je objezdačne objezdačne, z kolem delovitej teori, ko je dr, z nekaj delovitej, kaj je zelo, kako je objezdačne objezdačne. Mila tukaj zanjela, da je napravil počet, pri razlar providing you to be available. This is the expansion parameter, so this is related to the entropy production when you associate the area of the horizon to the entropy. You get such an equation and this also is slightly unexpected because you would normally expect only these terms which tells you that the entropy increases with time in a way dictated by the hydrodynamic, so with the square that is built tako v zelo, da je vse občasna, da je vse občasne, da je vse občasna, da je vse občasna, da je začučnja. In je tudi druga drivati, ki je vse občasno, da imamo izgledati tudi, na zelo, na zelo, ki je zelo, teologijan koncept, ki je tudi, da jaz sem tudi, ne bo občasno, da je ne bo občasno, zdaj je to temne, da je nekaj tudi vse skupaj, in začeljno je tudi povolj, da je kaj je dobro, in začeljno, da je nekaj, ko je dobro, iz vsej izvrstok in povrstke. Tudi tač, in tudi tač, teči, da je tudi kaj je povolj, začeljno je povolj. I take, zelo je, da imam inthropidens, z zeločenem vsevjej vsev, zeločen sem zeločen, vsev je bolj 1 over 4, in je zeločen jazel je 1 over 16 pi, bo je 1 over 4, bo je 1 over 4 pi. Tato resulta je tudi v 1979 dobro. Prvno, nekaj dobro so se da se izgleda, da smo izgledali 3, nekaj, 2, in 3 poslustvene projekcije v nadi stokšnjih, tudi na 10. Jedno komponent ne je zelo zelo vzelo in je to začel na vzelo, da ne je vzelo v nadi stokšnjih. In dogodno v držemo je vsega izgleda ta pejba, že nekaj ta ojev svoj izgleda, da so želite na površenju, nekaj Manhattani buti, da se na površenju postavil na tento bev, bo kaj mogu, da ne lepo sledajte boj, sledajte, če bobe ima doga delovljava, da je tkaj nezavručuje, priko ne bo bila, če dozbje se tako dodalje, tajči vsega prihvorilje, kako bila pošla kakva vsega vših odtah, tako je nekaj, da se učite in vsega. In tajči od ovega tega prihvoril, ko se pogledo, da je nalivno vsega para, način je vsega vsega renj, ki jaz glasba vsovaj praktivno, načo vsega zvonirajovaj problema očetku, in vseč vseč da zelo lahko naredajte, nekaj je pačnja na različenju. In tudi neh nekaj je nekaj nekaj pravda, zelo lahko nekaj, nekaj je nekaj traj, zelo lahko nekaj nekaj traj, zelo lahko nekaj traj, nekaj nekaj traj, nekaj je nekaj traj, nekaj nekaj traj, in vsega sezaj načinje. Zato ne zelo vsega aplikacije, ker je zelo, da bo vsega zelo, kako toga gravidina korrespondentja nekaj zelo, ali je zelo, kako je izgleda z vsega gravidina z vsega gravidina ades stoveje, kako je byla nisem nerejo veliko tudi očeska, nekaj Valdasena in Kvita in so in vs. In do pravi tomo prezentacije, mi dala vse večjelovite carton veče delo. V blocksupske, ki si drobimo v tem roha, ki se je zubo naprejvalo prezentem, in ukr sustainability. V tem spasih je radijel koordinec, in je konformal boundari at infinity, in zato, da je metrič nekaj, da je infinity v koordinec, in zato, da je interija, in da je interija, in da je tukaj površen, na radijelosti. Basically the statement of the correspondence, if there is a field theory that lives on the boundary of the space, you can compute the correlators in this field theory by doing some computations of fluctuating fields in the bulk, so in particular if you are interested in stress energy densor you can consider metric excitations, so gravitone excitations and then this computation ko vživo, da te dobenoaueno. If you have a black hole, it is supposed to tell you that belt ratio is not in the vacuum. But it's in a thermal state at the temperature, given by the black hole temperature. And then in the late nineties and early 2000, people have just discovered this and we're very busy computing all in z naprej collator, in se skupaj, ko bil v koncentru in starih taj, je zelo vsečeno, da se však naprej lačnega z učenju izručnega, poštela s prejo kaj je Igor Klebanov, kaj je poštela za vsečenje in poštela za način, pa prišli smo način poslijevače do način. Tudi bilo vzumeno, da se poslijevače za korilator, tudi tako, kaj je včasnja korilatora hidrodynamica, tako to je včasnja hidrodynamica, predičenja, pa vsega vsega vsega, ne, ne, ne, to je vsega vsega, vsega vsega, pa vsega vsega, pa vsega vsega vsega. Vsega vsega vsega je, ki tega vse zelo, da je vsebe. To je zelo, da se je to svoj. Tudi je zelo, da se je kojefizijen, ko je valje in z tjetjo. In to je obtajeno, kde bilo, da inšljaj se stresenarji, tudi in tudi. Je to prejflujetne in korrektur, dove tez kojefizijenje vsebe. To je vsebe, da sem odložila. konstant, ki se vsegače v pol, je vsegačenja z eta, da je vsegače. In komputacija je, da je konstant iz 1 over 4 pi t, a potem eta over s, ki je 1 over 4 pi, zelo v komputaciju, ki je vsegač, ki je vsegač, ki je 0, je ni minus 1 over 16 pi, ki je vsegač. In vsegač je vsegač, ki je vsegač, ki je vsegač, ki je vsegač. V zelo vsegače, zelo vsegač je, da je to ne kaj vsegač, saute nekaj, pa je z da работa iz vsegačОvj trg vizant, oz. zama, da jeh bih dolužit, por. z simplicity, in oz, kako je nožno 4 super ještje, naja dar vzelo da je vsegač, z atadi je 18 leave. In bilo to prihamajne, boje se prihmala, koščenja, in je, da zelo vljena teori izgleda, that the transport coefficients have to be positive because in fact, they can be expressed from the coupe formulas in terms of filter correlators and if the filter theory is well defined, unitary, and has all the nice properties, then this formula will tell you that this coefficients have to be positive and so, in fact they are positive or zero. Also the aerodynamic equations that we get in this way are not exact ker this is just the first order in an expansion, but you could continue with the cars in terms that have higher, more and more derivatives. And you will get more and more transport coefficients. So, at first order there are just two, but at next order there are already ten and they increase very fast. All these coefficients will depend on the details of the theory, just the shear viscosity universal, at least in Einstein gravity, it won't be universal anymore, ko vsečne modifikacije u zelo. Tako hrešanje zelo pravajnje. To je odstavil v te dve režite in in vseša del del ovečko vikbalen. Proste pa vzelo podnečno kar v te spremene paradagmi. Ko je se, da vseče vseče je vseče vseče vseče vseče vseče vseče. Taj je konfutat, da to je bavne. Zelo se konfutat ki v veselji ekel, Even you can transport it to the horizon and read it there. And so, yeah, it could be defined as a horizon quantity or a boundary quantity indifferently. Basically because it can be obtained from the equation of a massless field in ADS. If you try to do it for other coefficients, however, you don't get the same equations. And so there is a modification depending if you read it in the boundary or some other point. So, yeah, the difference with the membrane paradigm is that in this case the fluid does not live on the horizon, but in fact is dual to the entire bulk. So the dynamics of the fluid is dual to the entire bulk dynamics. And so a solution of the hydrodynamic equation can be lifted to a solution of the Einstein equations. And this was implemented very nicely in this paper in O7 by these authors. Where they implemented this scheme, they start from an equilibrium solution of gravity, which can correspond to a black brain and you have to start from a boosted black brain from reasons that will become clear because then this gives some solutions which depend on a number of parameters. And then you promote this parameter. At first they are just constants in the original solution. Just like a static black hole will have temperature or chemical potential or when you boost it, you also add the velocity. Then these parameters are promoted to spacetime dependent quantities, which you can expand in derivatives around some point, say around the origin. Then what happens when these parameters become spacetime dependent, the solution is no longer a solution of the equation, but can be corrected in order that it remains a solution. And this procedure that you can do order by ordering in epsilon, which counts the number of derivatives. And so in this way you get the hydrodynamic solution of Einstein equations. Then these solutions, if you are in a simtotically ABS space, they will all have the same behavior at the boundary. And if you read the boundary behavior, you extract the stress energy tensor. And since the metric is corrected order by order, also the stress energy tensor will be corrected order by order. So you get the procedure to systematically derive the hydrodynamic equations. Then later it was even realized that actually this scheme can be applied in full generality. In fact, you don't even need to be in ADS. You could be also in flat space. In flat space there is no boundary, but you don't need that either. The boundary can be imposed on a generic time-like hypersurface. And then there was this period when people talked about the rich flat fluid correspondence. So why you can do that? Because it seems that I have been telling you before that it's important that there is this duality, which there is a real fluid, which comes from the fact that there is a real field theory dual to the bulk. In the case of flat space we don't have holographic duality. There is no real theory. But this as a formal scheme it can still be applied and it leads to some results. So you can ask what does it lead to. So the reason you can do it is that basically you know that you just take a hypersurface in any space and the momentum and Hamiltonian constraints imply the conservation of the Brown-York stress energy tensor with some equation of state. And then there is one remaining equation, which is used to solve for the radial evolution of the metric. So in this scheme you use all of the Einstein equations. And this is automatically conserved. In fact it's automatically conserved for any constant c. So you can always add the term proportional to the induced metric on the surface. So there is an ambiguity actually in this scheme. In the true holographic correspondence this is fixed by requiring that this remains finite when you go to the boundary because otherwise normally it's divergent. But if you are just in a finite surface there is no reason to fix it to any value so it remains a bit undetermined but most of the time you set it to zero unless you have a specific reason to do otherwise. And so you just take the canonical Brown-York stress energy tensor so this t is scalar and if you find the problem. What is the t square here? Yeah, this is the trace of t squared and this is t squared trace. This t is the bulk t that enters in the Einstein equations. So you can write the generic answers for this metric in this form. This is written in Eddington-Fingerstank coordinates so corresponding to incoming object trajectories into the horizon. And this parameterized by three functions whose form has to be determined on the static equilibrium solution by looking at the Einstein equations. But these g and f are functions which contain in some way the parameters, the mass and all the parameters of the black hole. And then you boost it with the velocity ua so this gives the boosted black metric. And we want to be at a specific time-like surface so sigma c, which lives at a fixed constant radius. And we impose Dirichlet boundary conditions at this point. So we fix that these functions are equal to one at rc. Then you can read off from this form the fluid parameters. So the pressure will be given in some way the energy density from some other way at the cutoff surface. Whereas the entropy and temperature as usual are read off from the horizon values of these quantities. And so the scheme then you can apply it. You can again make this time-dependent, space-time-dependent and iterate the Einstein equations. So what do you get from this? For example, if you are in ADS then these functions take this form but you see the usual blackening factor of a black hole. And rh is the parameter which will become space-time-dependent. And when you run the machinery, you find that eta takes this form. So it's the horizon radius divided by the cutoff radius whereas zeta is still zero. So you may be surprised at this point because eta is the radius, so you may be surprised at this point because eta was supposed to be not changing. In fact, eta over s is still not changing but eta itself is running. So it is a function of this rc. And this can be interpreted as a kind of rg flow of the transport coefficients because if you take this surface at a constant it reduces not infinity in the ADS-50 correspondence even though now we are in flat space, but no, okay, this is in ADS, sorry. In ADS the radius is dual to the energy scale of the dual theory. So if you are at finite radius it's like you are looking at, you are doing some kind of rg flow. You are supposed to have integrated out the degrees of freedom outside the surface and this integrating out will generate an rg flow which can induce also some rg flow of the transport coefficients and this is a simple example but one can do more complicated examples and this has been studied also in numerous situations. Five minutes? At most. At most. Okay, no, I'm almost done. Actually so as I said this was in ADS but you can apply it to flat space. It's even simpler because the functions are simpler. What you find strangely is that this, if you apply the formulas from before the energy density evaluated from, in this case gives zero. So this is a fluid with zero density but still it has pressure. This has been called the Rindler fluid because the Rindler, it's a Rindler spacetime, just the local horizon. So, okay, it's not very physical maybe but nevertheless you still find sensible transport quantities and sensible hydrodynamics. The eta is equal to one in this case and zeta is still equal to zero even though its fluid is incompressible so there is no zeta but there is another quantity which plays the same role so I call it zeta as well and these are the equations of this fluid which are just the Navier-Stokes equations and in fact here an important point is that when you go near the horizon you can take different limits. In fact you can rescale the variables in a relativistik way so this just gives the normal derivative expansion because each derivative will be rescaled by an epsilon independently fit space of time and then to promote it to a symmetry of the gravity situation you have to rescale the parameters in this way so the rc and rh both go to zero but in the same way so you are going near the horizon both with the horizon itself and with the surface that you consider is going to zero but the ratio remains constant but you can also do a non-relativistik scaling where you rescale time and space differently then you also need to rescale the velocity this is the symmetry the scaling symmetry of the non-relativistik Navier-Stokes equations and this can also be realized but just by taking a different scaling of this quantity in which case the horizon radius goes to zero faster than the cutoff surface and in fact going to the horizon is not by itself sufficient to give a non-relativistik limit it depends on how you take the limit and in more general space times one can also realize other symmetries like Schrodinger symmetry which is the symmetry of the Schrodinger equations or leaf sheets scaling symmetries or Karolian symmetries that we have heard about in the yesterday actually and so this has also generated a lot of work and it is not enough time very quickly there is a canonical way here to define an entropy current because the lines of constant are not geodesics in this parametrization of the metric so they can be used to transport quantities from the boundary to the horizon it is a canonical map from the boundary to the horizon so if you pull back using this map the area element of the horizon you get a current on the boundary or on the surface that you are looking at and this current will have this form it is equilibrium current plus corrections you can determine the location of the horizon in this scheme order by order in the derivative expansion so it is even though the horizon as I said it is a teleological concept but order by order in the derivative expansion it is actually determined locally and you find that this divergence of the entropy current is always positive as expected so it is again sensible hydrodynamics however when you go away from the derivative expansion you are far away from equilibrium it is still not unclear how to extend this because then you have to really deal with this issue of what is the horizon and should I take the apparent horizon or the true event horizon it has not yet been completely understood so to conclude I would say that now thanks to the gauge gravity duality we have a more systematic understanding of the various ways in which fluid dynamics is related to gravity and that despite the fact that these two things are tantalizingly similar but the membrane paradigm is not exactly the same as the fluid gravity correspondence but the relation between the two is not yet completely clear so notice that in fluid gravity the fluid parameters are given by the radial evolution of the metric which is not the same as the time evolution they are completely different things in the membrane paradigm of the more the two are linked because it's both the evolution along the horizon gives both the time evolution and the definition itself of what are the fluid quantities and the parameters also are not the same if you take the pressure as defined in the membrane paradigm will not be the same pressure as in fluid gravity but it will be corrected by higher order corrections order by order so we might conjecture that perhaps membrane paradigm is a sort of resummation to all orders of the fluid gravity correspondence but it shouldn't be a standard resummation to try to resumm that in hydrodynamic expansion first of all it's an asymptotic series so it's not so obvious but to the extent that you can do it in some limited situation it was shown that what it gives you can recover the quasi normal modes of the black hole this was this paper in 13 but then on the other hand it was also shown that the membrane paradigm using the stretched horizon approach at least the very high excited quasi normal modes of the black hole so in some sense this resummation should not yeah I don't know should be some other scheme of resummation also it's not clear why you could get negative bulk viscosity perhaps it should be just a formal formal way of resummation but it's just speculation I don't have an answer for this point so I will finish by saying happy birthday to Tivo Are there questions? Tivo in another part of my PhD which was not the PhD I was not test data but I have discussed also that some of the field equations are equivalent to minimizing an entropy production on the black hole a la privogen is this a thing that appeared in ADS-CRT minimize entropy production? Not exactly there has been some some discussion about the fact that if you try to construct an entropy current that its divergence is positive then this is equivalent to it puts the same constraints on the theory as requiring that there is an equilibrium partition function in a generic background metric so it's not the same thing but yeah it's vaguely related Okay there is another question so there are some old works by Arnold that formulates fluid dynamics in terms of geodesics and the group of people more physical things like that is there some relation with this number rather than for them it's a good question not directly not that I know this was for the statistic for the Navier stocks I think yeah no I don't know how to relate Okay there are more questions If not then we will go on to the next speaker Thanks again