Black holes in Einstein’s gravity and beyond

Transcript

Thank you, John. Thank you for having, so the thing we’ll discuss about holes in Einstein’s reality and beyond the outline of the top is volunt. First, I will remind you the relationship between reality and the geometry of space time in particular gravity and the metric. We’ll talk about how to quantify this relationship that bring us Einstein situations.

Then we’ll discuss the most fundamental solutions of general relativity and. Such as price amount from cash human metrics. And then we’ll come to let hole discussion with let horizons discuss a little bit go versus true similarities before coming to a more advanced topic, which involves laws of let hole, organics that or not is laws.

Dynamics of the real system then will come to more speculative CLOs, such as photographic principle engaged. So, first of all, we all know to some extent, some people from high school, some people from popular movies or books, we all know the contents of charit, relativity matter and energy curve space time.

They distort the fabric of space time. And because of the, this distortion motion of other objects, for example, or light trace around the big objected for space time. Change they differ, right? So if you have light rate time from infinity, and there is some large body in which works face time, and the light trade will not propagate along line anymore, it’ll be deflected by simply the fact that the propagate along thet and the Trump current.

So this is a qualitative feature, but of course, the question is how would this, and to quantify this, we need to write down equations of motion, the equations of motion, general relativity. We’re written about a hundred years ago and they are Einstein equations written here and we’ll discuss them from some detail in Holland.

So basically the contents of general activity is that geometry means metric will be affected by energy or, and mass and vice versa. So what are the of Einstein’s equations of motion? The main hero of this discussion today will be the magic. So let me remind you the simplest. Facts, uh, such as fibro FIM.

So if you have a two dimensional flat space and you would like to measure distance between points B and C, for example, distance Delta S then the distance is given by the em the distance Delta S squared is simply equal to Delta X squared, plus Delta Y square. Then we can take the distances to be ES small and then the same line.

But now with ES small elements applies, we have DS squared. This is called. One element is equal to DX square plus device square. This is the metric of two dimensional flat space time. We can generalize this two, three dimensions. Now DS square is equal two D square. Plus we square plus not according is X squared.

And in fact, we stop at three dimensions. We can have generalize this in four or five, six dimensions if necessary. And so in inter dimensional in inter dimensional. Uh, flat EFL space time. The line element squared is equal to the sum of squares of this infinite space. We can put this in a more elegant form metrics form by right.

The following expression D a squared is equal to G I J D X, Y, D, and J where summation over I and J is assumed. So this example is given in two dimensions. So I and J go from one and two. Each individual goes one and two. And in the end, we should get what we had in the previous line. Mainly the expert bought the iceberg.

So explicitly this relationship is written here in this line. We have four elements, right? With some of line and train. There are four elements, G one one G one, two G two, one, and G two, two. And these four elements, it’ll be elegantly putting into two by two metrics here, and we can compare this line to this line.

And we see that these two of the elements are equal to zero and G one is equal one. And should do with one as well. So this is a metric comes up in flat two-dimensional space. Suppose now we generalize this. So in the previous example, GI J was independent of X. It was just a metric constant metrics with one and one and zero and zero.

But in principle it may depend on point in space or space time. So for example, Uh, we can have, again, the three-dimensional ingredient space with DX square D way square and use that square. But another one important thing that we can use different coordinates we can use for the same three-dimensional flat space.

We can use either C coordinates or third board to show in this picture. And then the line element will change inch visibly from this to this, right. It’s not surprising because if you changed words into this expression, The distance squared will not change, but of course these burdens change and therefore the metric itself, the metric Tonsor must change.

So in metric form, it means that in first case we have a 3, 3, 3 with 1, 1, 1 on the, in the second case, we see that the metric tons are depends on the point in space. It acquires the explicit dependence on space. It’ll be important. We’ll discuss like. So some examples of, um, flood ethics ideal here. So this is a metric three flood dimensions.

Now we can generalize this more, and this is a tension to the contents of the special relativity with discovery, uh, again, made by, and sign and others that, uh, we actually leave at least locally in four dimensional space time where this is a space show part. D square UI square and D that squared, but the time is not an independent spec.

The data we need to put also the time elements into the metric and the essentially the content of special relativity is that our space time is locally Inco space time that metric given here again, you can write this metric in different boards, for example, in the cer ordinance. And again is a nine 11 point square squared.

But this part is written in circle ordinance. How do we know that our space space time is actually current? What kind of local experiments or no local experiments can we do to determine the curvature of space, time around us? Some people thought about this for a long time in particular, those, and then essentially, and one local experiment that can do is a foot.

You can draw a triangle and try to measure the sum of the angles in the triangle. If your space on your flat, you discover that the sum of thes is equal below hundred degrees. If it is not. For example, if you draw triangle is here, you can measure these angles and discover that the, some exceeds on hundred degrees, if your space happens to be the subtle point like this one lower, but just the geometry, then the triangle, the sum of the angles of triangle.

So this is a test, one of them. You can do more tests, for example, with the same subtle point feed, which is not its robotic space, but just space, you can do the fully, you can draw a circle and measure the circumference and you discover that on ABO space. This is greater than two by R as it were in, in the flat space, for example, which is not surprising.

Look at the magic line element. Inbo space is even expression with Y square downstairs. So definitely this is not a really a metric. Seems to be, but then you may ask why. So how do I know, do I know for sure if somebody gives me this metric, is it surely the fact that this space is curve or I can do some coordinate transformation, like on the previous slide from certain coordinates to partition coordinates, and actually bring this metric to some complicated, uh, transformation back to the flat space.

Is it possible enough? So we need to have some local test of FENO space is current and such a local test is provided by the. So I’m not going explain what this object is. It’s written down the written down. Uh, you can only notice the following fact that reman tensor contains a derivative of some object, which is known as the CPH connection of total symbol Gama, and the not symbol GA itself depends on the metric.

So ultimately Ariman terms is expressible in terms of derivatives, actually second derivative because we respond third deriv here. And then on top of that, there is another derivative of Ghana. So it is a second derivative of the metric to which agreement data is proportional out Fremont terms. You can build other important ingredients such as reach terms and reach scale.

And the important local statement is that a space is flat even only if its Reza is identical, is. So if somebody comes to you and tries to sell a very complicated metric saying that I found a wonderful solution of science equation, the space is curved, and this is, this is solution that should be named after me.

So you should not believe this person. It, once person, all you should check that not Tesla. This metric is zero by holding these. Formalists and if you discover that 10 zero, then there is that exist. The transformation, which brings this complicated metric back into the reflect space. And therefore the claim is in the sore 10 and it derivatives will play very important role in general.

Relativity of course, as we saw already in Einstein equation of the, these billions, you now have all ingredients to write down Einstein equations and send to the equations to R. so, uh, what is happening on the left side is basically the second order, partial differential equations, back to the metric on the right hand side, there is an object which is known as the stress energy sensor.

This includes energy and matter contained in space. Okay. So, um, important statement is the following, despite the fact that we don’t see anys explicitly entering here, the equations. They are, is you saw on the previous slide, they are second other nonlinear, coupled partial differential equations for the component of G new with GI the metric.

Okay. So you can, I, I try to write thematically. Here is the expression of waste science equations of the left hand side. So you could see that indeed, we are dealing with the second other partial and there is a source of these equations on the right hand side, you can compare with max equations where we have.

So suppose they have distribution of charges and currents in space, and they would like to find electric and magnetic field generated by those currents. Then write down muscle equations and visa, also partial differential equations with back to the four potential, a new from a new you can calculate E and B electric and magnetic field in wall space.

So velocity is the same. The electric market fuel here are sourced. By distribution of charges and cars here, the metric G new is sourced by the distribution of meta and energy in space. That is our fundamental difference in these equations. Maximum of equations are linear. This means that we can find a generic solution if not explicitly, then in the form of some integrals immediately, and people in 19 century did exactly that.

However here Einstein equations are partial. Non-linear differential equations. Generic solutions is not known and is not unique. It is very, very complicated set of differential equations. So even in the simpler case with zero T concept and zero meta source, so supposedly have us a vacuum. We don’t have any store in the right side.

Generic solutions remains unknown and you can call the equations for fluids. Which is also a very complicated set of equations and solutions is not be, however solutions can put into highly symmetric situations can be found by making element assumptions about metric. For example, if the source is very symmetric, we can assume that the solution will be very as well.

If the source of, uh, the left hand side will be hysterically symmetric, such as the distribution of mass and uniform distribution of mass Ander. We can make an assumption about the, uh, line element we can think, uh, that it’ll be stereo. And it means that the a and B here will depend only on the radio, uh, distance from the center, but they will not depend on the animal cause there is a stereo.

Okay. This is a possible assumption. So what will the next we substitute for solutions? This answers into the equations of motion and solve for a and B that the appropriate boundary conditions. And you discover. So that was Carlos Schwar did in 1916, you discover what is known as the metric. What are the most important solutions of journal Soho metric is even here.

It describes thematic of space time outside of the body of mass app. This outside is extremely important. So if you look at the, it’s not so different, different from the. Uh, oscopy one right apart to this little element here to jam over at C square R also here, if you put this to zero, for example, massive zero, then you back INO measure.

This element is actually this piece is actually not very large. So corrections to this one is very small for stars and lemons. So you can think of mean if you think of earth as a thirdly magic body. Then you can estimate what this correction is, how much this space, time is curved right here, where I’m standing.

You can, you can plug in the appropriate distance are, and you calculate this correction to one. This correction will be other of 10 minus nine. So it’s rather small distortion of space time. However, it is still, uh, sufficiently large to effect, for example, GPS for the polarity. So the metric is pathological.

It seems. At a certain point and this whole project becomes zero. That happens at a so-called short radios and R is equal to M square. Then this becomes zero and this becomes zero as well. Therefore this term close up. So something happens to the metric and we need to investigate what exactly well that the metric describes the space that outside of the body of mass M.

So for most object, this pathology, the similarity is no is of no concept. Small signific, but most the swash radio is located deep inside the logic object itself, and therefore is not relevant for the solution because the solution is on the very outside. For example, for the earth, the swash radio is about one centimeter.

So this pathology and the solution is obvious. So opposed to. Is, this is, this is the earth, right? The Schor. This is the earth, the short radi, somewhere deep inside. And the submission is only very outside. So you don’t there. You don’t worry. You want what happens inside, but it’s a different solution that not zero right inside, inside this, uh, inside the distribution of mass for some theor radio is three, the kilometers.

And again, this is a tiny, tiny distance deep inside the sun. You don’t worry about the solution there. Cause the solution is very only helps. However, sudden house, if. Some violent forces would squeeze all the matter of the star or plant into inside dishwash turtles. In that case, we’ll get them the whole, in that case, this pathology is relevant.

So if the matter is S squeeze inside dishwash radios, for example, in the process of the gravitational collapse of the star, then we get that let hole. And this is where, where problems start. So then we have to take this metric and this in similarities very seriously. First of all, you have this, what is known as event horizon that are equal to arch portion.

Then this thing becomes zero. And then there is another similarity in the metric notice here that if R goes to zero, then this whole term blows up and the same happens here. So there’s not a singularity in the metric at our equal to zero. Well, We need to remember that the metric is actually a tens, which looks differently in different coordinate systems.

You change from coordinates to boards, and you can get rid of some pathological behavior. For example, in two dimensions, you have competition coordinates line element in flat face, the square plus device square. You can write the same line element in polar coordinate, and you can say, oh, look at R equals you.

We have a pathology similar to what happens at Georgia radios, right? This whole term disappears our equals zero. But we know that they can make a coordinate automation back to the ion space and points in R to zero is no different from any other point of space. So this is a coordinate similarity. This is, this is a similarity that we can get rid of by changing the change in the cos.

So in a similar way, what can swash that in different coordinates, for example, once, and the magic will disuse that R RS, no pathology, no going up and magic will occur. However, something will happen. The. We discuss in next slide in general, to check whether we have a true or, or similarity, we need to compute something that stay inordinate a change, right?

Because we need something that does not change when the change system or are come from comput to polar and vice and section variant. One of them is called variant. It is, uh, it is basically a product of to fall in, um, uh, some over. For awar metric, this pressure in their end is proportional to one or R to the six.

So this is completely non similar than R is equal to Rho, but it is similar. It blows up R is equal to zero. So out the two similarities only one is the true one on the non similarity that are equal to school. We don’t know. We need to modify physics in order to understand what is happening direct you, but we don’t need to modify physics in order to understand what happens at.

That’s a difference. So what happens at, at the, at the event horizon, something interesting happens at the event horizon. Uh, we can study the motion of articles and light trace around the holes, just like you study the Orbis around in the star or, or any other us source. So let’s do about what happens to light.

So, uh, here is what happens in speciality, right? The speciality, if the observer is located here at the point, Let’s say right somewhere in the center, then there are two cones, the past light cone and the future light column, which means that if the particle, for example, the massive particle does not propagate does not move in the retraining on this observer just stays there.

Then it simply propagates along the time X time goes up. If this particle has some velocity, no matter how small it’ll not stay on at the same space point, it’ll move in space. And therefore it’ll move on this graph. The sum end with some slope, right? The slope will be equal to the speed of the motion of this particle.

So the maximum speed can count especi relativity in the speed of light. And this response to this, they have an emotion along the surface of, of the light. So problems. Light trace will move in the future along the surface of this, of this moon. And that’s therefore it’s called the future light bulb. So let’s see what happens to problems in the, of the blood hole, in the

So here, these light poems drawn locally, light bones outside of the horizon, they look more or less like in the special relativity, but then you come closer to the horizon. You launch that they become a little bit squeezed and a little bit tilted. This is a proper short space time. And then, so you can follow these, this, uh, this angle of the light palms and light it from this point, for example, will only propagate inside the light palms.

So let them only propagate inside this little, little vision here, which eliminates from this, from this point. But now if they come to the ized, you can see that example, the, which these light bulbs are tilted, uh, towards the RS becomes 0.5 degrees and therefore. All thoughts, which, which, uh, emanate from, from the beginning of it, like poem here and here, they will only propagate inside this region, which is bounded by the event horizon.

They will never come outside. They can only come inside the, inside the horizon. And eventually they will end Dari equal to zero for the similarity east and where we don’t know what kind of physics it has. So this is the property of the event horizon, which is compatible to the, our understanding of a plate hole in sort of the normal terms, right?

It’s a, it’s a body who’s gravity, so strong that nothing even might state that’s that’s more. So let’s see what happens in terms of the more advanced physics in the limits of one of physics post. So again, this is the short metric and you may ask by myself, what do you know about the similarity? The answer is unfortunately after hundred years of not so much, we know a great deal about the behavior of particles and fields around horizon, but we don’t really know that it’s having size similarity.

One thing to remember is that general volatility is an approximate theory. It’s not the last word it is. So here is an example of Aran general relativity, uh, where Einsteins equations follow from these two terms in. But you can easily write down other terms which are layering and therefore they can enter the action.

And on a quantum level, they must enter the action according to the quantum democracy. And if you vary these, uh, these Ang, you discover are corrections to Einstein’s equations. We don’t know at the moment, what kind of IANS enter these corrections, but generically, we can say that genetic corrections to the metric are expected.

So Schwar method is not the last word. There will be connections to the metric coming from the total high firms and also coming from quantum, uh, quantum whoops, quantum, uh, quantum operations. Again, we don’t know the theory quantum at the moment, you can think of gravity as an effective theory at large distances in time, compared is again to fluids, right?

We have now yes. Situation, which describe the motion of essential any fluid, but we know that on a very short distance. The fluid is distinct. We have molecules, we have Atos, we have perks and so on. So it has a very complicated structure, which is not captured by equations. In fact, you can generalize the equations to come closer and closer to this, uh, to this, uh, distribution of these, uh, molecules and the discreetness of fluid.

Right? So similarly gravity, if you want, you can think of gravity as a separate equation. equations, separate equations, which describe this. Fluid develop sort of looking or fluid or space time without looking at the fine details at the very, very short business. We don’t know those details, and this is the frontline of the modern technical.

So let’s, uh, briefly, uh, discuss other solutions of maintenance equations. One of them is right from not solution. This is amatic distribution of mass M and non charge Q. So look at the metric. It is a quashed metric. That is an exception that is additional term here, which depends on the charge. So now, uh, this ball is, is, is massive, but it also has one zero charge Q.

And so we have this term here and this term here, if a good charge to zero, we have back to swatch metric. We not, then it is a generalization notice that it has two horizons as we have one R square. And that equation will give us to horizons and horizons can side the solutions of the Extremo solution.

And, uh, the very important. Other. All right. Then another solution is even more important. It was found by Roy care in 1963, not so long time ago. It is a solution, uh, which describes data in T thematic uncharge distribution of mass its parameters are mass and or momentum. J so there’s a complicated metric also.

So the two horizons and region here in between them, I will not go into the detail of the paramedic. Just notice that it is not a solution. So you can say. Why not aid in charge to that? Indeed, the solution was found so you can have data. A symmetric charge distribution of mass and parameters described completely described is by Paul would be mass charge and behind momentum chain.

Now let’s come to more advanced topics. The four loss of black hole mechanics is noted by.

Of the study, the metrics such as metrics and more generic graphics, they tell us follow. There is a zero flow that says horizon by coal has sponsor surface. K. You can think the surface gravity as the analog of acceleration of three, four G 9.8 of per square, but now not here, but neglect that’s is important for the of black poles.

The first law tells us that in of, in pole. The change of mass is related to the change of charge and momentum chain and horizon area by the following quantitative equation. So if you, if you modify the blackhole suppose you take a brick and probe into black hole, right? You increase the mass and other parameters will increase a per the second law tells us that the horizon area a of the black hole is a non decreasing function of time.

And the third law, it is impossible to achieve zero surface gravity by a. So, what you can do is to compare these four laws of gains with the laws of thermodynamics and you find it remarkable, uh, a remarkable coincidence if you want, or maybe it’s not a coincidence, right? So the zero four FNAs tell us the system internally equilibrium has a constant temperature team, right?

It’s very similar to the zero four games. The first law of Ana is the law of conservation of energy. It tells us that if you change the energy of that pole, then the, of the system changes and the less work terms, the second law, which is more important tells us that the Py of Ana system is non decreasing function of time.

And the third law says that it’s impossible to achieve due temperature, very physical process. Is it a coincidence of we have is a deeper connection between the laws of black gangs and the laws of. You are tempted perhaps to identify papapa to some competition with temperature and area, uh, and area to be entropy of some of some system.

But it’s not clear how in 1972, Jacob Becken Stein suggested that the left hole should have a well defined entropy proportional to the horizon area as pretty slide. And he was actually lasted by, by actually many researchers because black holes are black. They do not radi do not meet any irrigation and therefore cannot have a temperature associated with them.

So what, what, what, what is about, however, in 1974, Hawkin demonstrated the let pole actually doing the radiation at on level. And so one can in fact associated temperature. So that was rather dramatic advance. So here are some Pharmac. So how showing basically the black holes, IIT radiation with the black Bodi spectrum at the temperature given by this expression.

Not is the present of the H bar. So it is a process which happens at the quantum level. And you can see from this formula that the hawing temperature is adversely proportional to the massive of black hole. So if you have a super massive black hole, its temperature, the hawing temperature will be very small.

You can look at these numbers and, and top the cell. However, if the black holes are extremely small, a more by full bang, then the temperature would be sufficient Tolu. So for example, one solar mask is a temperature of about. So this fix as to proportionality in Stein conjecture, and the Stein is given is indeed proportional to be to the horizon area, given horizon area.

And again, not presence of H R downstairs. So that’s an immediate consequences and immediate problems, which are related into this identification. First of all, let elaborate with time. You see, from this impression, it’s almost obvious that they have negative. To the heat. Right? So what happens if, uh, there is a temperature with black hole?

I suppose it radiates some of its mass mass decreases mass decreases. This end becomes smaller. The therefore temperature becomes large, right? And eventually the black hole will Eva. The question is what happens after it operates. If in select body, the spectrum it’s completely featureless. There are no imprints of any information which was ever thrown into a black hole.

So you can throw in the black holes, any sort of complicated systems, which books, but so well, some complicated systems, right? Which contains some, some zero information and in the end and let falls operate all the irrigation feature. This irrigation remains. It seems maybe we don’t understand something at the very end of this.

Process, but it seems to way this is known as information lost paradox, which has not been result until, until, until now. In fact, it’s not clear, you know, what is happening, but doesn’t mean about like the most important question. However, is all, what are the microscopic degrees of freedom underlined by?

But remember the is also an effective hearing, like the usual system. We have normally, you know, microscopic level, we have statistical mechanics with notion of temperature, entropy, and so on emerging from the dynamics of the microscopic quantum systems. So the example of this, the simple example of this is the system, uh, of particles of the spin, right?

So I suppose we have, we have one particle, the spin, it has to microstate the spin can be up or down. If you have two particles, is it two particles, then there are more options. Both can just spin down or spin up or, or, uh, in the, in the opposite directions. Right? So together in one particle, we have two micro states for two particles.

We have four micro states. Generically, if you have N particles to spin, the number of microstates is equal to two to power N and Bolton told us that the enter the usual on this microscopic level can be calculated as the low of a number of micro. And in this case, the entropy you can easily calculate is proportional to the number of degrees of freedom of this quantum system.

What happens with black hole? We know that black holes have he in backin, haw calculated this , but this ENT, what are the micro states, which we want to count in the similar fashion here to get, to get this Py in the microphone. Can the micro, the, and we can start this off. What are the micro? This is a fundamental.

Question and people attempt to, for, for many years, they attempt to solve this problem. And there is a partial progress in, in this direction in particular, this progress problem probably used by stronger in 1996, we are able to count to microstates of a very special, super the call in five. Yeah. It’s a hard cry from Washington, which sits in the middle of our gala song, but.

but nonetheless, it is, it is a model which achieves this, uh, uh, success, that result sites exactly. That you Stein talking from America, the reason why they had to, uh, resort to special also mathematic like, oh, is the following. What they do basically is there is a construction of things, theological feeling on his brains.

You can count at peak coupling, you can count microscopic states of the system, just like you do in statistical mechanic system on spin. Then you squeeze this whole thing beyond dishwash radios. It becomes a like whole, and you can compute as you usually do. And then you compare this to results, but you must be sure that in the process of squeezing the, you don’t lose in states.

So this is protected by super symmetry and this other special problem. The value special properties, you cannot be sure you hope it is case, but you, there are no thes Inda. This is a major difficulty here. So strong WOL has been generalized in many ways since 1996. However, still do not know how the micro with the normal, for example, the simple short of dimensions global less, it has profound consequences.

Withs. Well, let’s talk about the photographic principles. So if systems about gravity, the entity is extensive proportion to volume. You think of gas or, or, or some fluid. And the of the system will be proportional to, to the volume that gas enterprise. But notice that Stein talking, which judge discussed is proportional to the area horizon area to a hole, not the volume, which device the, it is promotional to the area.

And it seems that the gravitational degrees of freedom in de dimensions generically are effectively described by some theory in D minus one dimensions. This is normal as the telegraphic principle, which was first discussed by faulting in 1992. So again, gravity, the system of greatly occupying some volume.

The effective degrees of freedom seems to leave on the boundary of this, of this region and in lower dimension. So this development resulted in what is known as the G duality sometimes also is known as ad respond. And it’s basically the same statement and it grew up. It’s important to understand that a correspondence grew up out of a chemical understand, like whole enter.

So in the open sync picture, you have dynamics of things and brains and low energy. And these dynamics are described by filt without credit. Some case theory or some point of theory, you squeeze this whole thing beyond the device and radios, and now you have a closed string picture dynamics of things.

And brain deploy actually is described by gravity and other fields in higher dimensions. And you, you conjecture the ATS C pons is the conjecture by the fact that it’s bio many, many, many tests, but it is a, Conject not proof. The, the conjecture is that this side is exactly equivalent to this side. It’s exactly forwarded by by in 1997.

So that’s all good. So we know that. Alright, so there is some microscopic system, it describes our black holes. It describes them infer as we saw the laws of black from the laws of are identified to and pot temperature. But what about what falls beyond the. So uncertain, the calls are characterized completely by global charges, such as mass, total electric charge and total and momentum.

And themic system is also characterized by conserv charges, total energy, total electric charge, total and momentum. So this is not, that is surprising. Now what happens if one per are trips, mutational system, for example, pretty like this one. it’ll oscillate. The ioma is normal nodes, right? Usual. right. So, and these I frequencies dependent on the system itself.

So for example, for this object here, the I frequency will be simp as good, obviously constantly divided by mass. And of course in more, it’ll be not just one frequency. It’ll be the full spectrum of discrepencies, but it can be found. And what is crucial? It depends on this, on the system itself. It does not depend on the force terms of the system.

It is a characteristic. The system. So what happens to black holes? Suppose I want to do the same with black holes. I want to procure my hole. I, I want control some stuff and it make it oscillate. Can they find the spectrum? What happens if want returns? Can they find the spectrum, which characterize these black holes and how the spectrum is encode into this microscopic that I was talking about?

So if has, if you could follow is engaged in duality, gives a quantitative way of comparing and, and starting the spectrum and identifying the ingredients. This spec. So look at the left side of this slide. So you have some black hole of that brain. You can dry by these concern charges and it has fucking temperature Stein, fucking Py.

Now, suppose you return of that hole. The original metric is returned by as little age new and gives Einstein’s equations. You can write down precise equations, which describe these operations age new together with appropriate boundary conditions. The important difference. Between the normal Mo like a pendulum on and the black hole selections is the presence of a horizon.

Classically. If we don’t think about quantum effect, like ion, classically, any stuff that goes inside the horizon disappears forever. And therefore our system is not a normal mode old system anymore because information, part of the information in the system is lost. Once it cross the horizon, you can think of some quantum.

Uh, well, for example, right, internet, well with semi transparent wall. So part of information is leak out. So suppose this is horizon part of information. So the spectrum will not be real in this case, it’ll reflect the fact, the boundary revenue problem is not your mission. It is known as a quite a normal spectrum.

It’ll require some non zero. Imaginary part is exactly the same. What happens is for digital, this is the same as happens. If we take a usual non gravitational system and deviate it from. What will happen from system goes back to right, will be some diffusion or it’ll be a sound wave, for example, right? If it return the air in this room, some more important part will be a sound wave.

If the sound wave will not propagate forever, it’ll at it’ll die. Eventually it’ll die because the dispersion relation from the sound wave has an on zero mention. The part are proportional to discuss it. There is a friction in air. And here is a dispersion relation for the sound. You can see the, uh, part which describes the application of sound.

And then there is accumulation part, which is proportional to the viscosity. And so from this previous slide, it is clear that if you identifying the microscopic system, which black hole, we must find the quantitative education between price, normal spectrum or black holes. And the spectrum of microscopic system and that has been done.

So one simple example is the following. This is the spectrum of the black hole, the honest population from, uh, from, from variational side. And this is the dispersion, which is expected from the point of fuel of the fuel feeling of freedom by comparing the two. You can see that the speed of sound is speed of light divided by the square of three.

And you can compare these two terms and read off the ratio, Vico ratio, OFS of the microscope. Moreover one can go on and relate, not just toxic equations of this fluid of these microscope of freedom. And Einstein’s take, this has been done around thousand seven eight. So this is a very recent right coming to the end.

So we have some with black holes hope, uh, you also doing the launches, uh, from the threshold point of view list, they test a little bit power knowledge, any dis candidate for a few must be able to explain the properties. This is a mask black hole, the have entered the temperature and behave like. I think we know why, because we talk about principle.

We know that in some models, in some cases, we are able to identify the precise piece of freedom that govern the behavior of the whole, let whole spec of expectations in code non properties of these two microsystem, such ASCO on the previous slide and many other properties. We do not know how to resolve the blood.

This is the puzzle between the past and many, many, many years that requires from everything. Einstein is an approximation. It is an effective theory. The amount of this analogy with your equations and fluid, we do not know the full theory at the point. This is both frustrating and also interesting. And, uh, we should also look forget about the practic.

All this was about the theoretical concept of a drive, what the communities are from a theoretical point of view, but. We should not forget about practical applications and real web holes. For example, real work holes in our very long universe and docs by next will introduce these real.