[MUSIC] So we're going to end this lecture with a vague discussion of three, rather complicated subjects. These are the Kerr solution, rotating black hole solution. The so called no-hair theorem and so called cosmic censorship hypothesis. Why are we going to vaguely discuss them? Because they are rigorous and complete discussion goes way beyond the preliminary lectures on general. So what we do at best, we will just present some formulas and give physical intuition for those three subjects. So let me start with the care black hole. In so called Boyer–Lindquist coordinates, the metric for the Kerr black hole looks as follows. It's rather complicated and I going to give the definition for the notations in a moment, but let me write them. Sine squared / rho squared dt squared + 2 kappa Ma / rho squared R sine squared theta d phi dt- rho squared / delta dr squared- rho squared d theta squared. And finally -A sine squared theta / rho squared d phi squared. So we have to specify the notations. First thing, we are using the following coordinates in space time. The spherical coordinates in spatial path. That's the first thing. Then Delta is just r squared- 2 kappa M times r + a squared. Rho squared = r squared + a squared cosine squared of theta and A = (r squared + a squared) squared- a squared delta sine squared theta. So, one can see here that there are two parameters, M and a, on which the black hole depend. So if we put a to zero, a to zero, if we put a to zero, this black hole metric reduces to the one, as one can see. It's not hard to see. Second thing, as r goes to infinity, we encounter the following metric. As r goes to infinity, the metric then there approximately becomes (1- (2 kappa M) / r), M and a are constant, unlike the previous solution, where M was frequently a function of r, 2 kappa MaSine squared theta / r d phi dt- (1 + 2 kappa M divided by r) dr squared- r squared d omega squared. So again, from this asymptotic form, one can see easily the following situation. First of all, M defines the mass. Because it was all this term. This just reduces to the same form as the SWASH does reduce because this, when a equals to 0, this reduces total. So M is the mass and a gives the presence of this term And also notice that this term because of the presence of this term, the metric although it is invariant under the time translations but it is not invariant under the time reversal exactly because of the presence of this term. This term makes this metrics stationary but not static. So the components of the metric are time independent but they're unknown diagonally terms. Which this term correspond to the rotational terms. They describe the rotation. So the space time describes the rotation, so this M times a is a total momentum, rotational momentum. So, the physical meaning of a is, momentum attributed for your angular momentum attributed to the unit mass. And how to see that this term corresponds to the rotation? Well, the first thing is that one can do the following, you can take Minkowski space time and make a transformation to the rotational reference frame. Another way is to study close this metric and that goes beyond the discussion of this, our discussion. Where one can find this is in textbooks for example, Landau Lifshitz, more careful discussion. Now, let us see whether this metric has any horizons or not. First of all, to do that explicitly, one of course has to draw a Penrose diagram, Penrose-Carter diagram for this metric. But that also goes beyond the scope of the present lecture, so we'll vaguely give an idea of what's going on. So one can see that this part of the metric which is similarly to the case blows up when delta is equal to 0 which corresponds to the, it's a square. It's a quadratic equation as follows from here. And it corresponds to r +- two solutions of this equation now like this. +- square root of kappa m squared- a squared. So as one can see as a goes to 0, r+ goes to rg, Schwarzschild radius. So that is one of the hints that r+ is the radius of the horizon. So the horizon of this solution is r equals to r+. One way to see that is to draw a Penrose-Carter diagram for this solution. Another way to see that is to consider that the tangent vector to this normal vector to this surface. To this surface r = r+ is light like. Normal vector, normal, sorry, n mu squared = 0. Then it means that this is a horizon. Because to escape from this surface, one has to have light velocity. So, this is an event horizon. Another thing is the following. Look when this term is vanishing. When this term is vanishing This term is vanishing for the following radius, r0(thtea)= kappa M+. So this is one of the solutions of the fact that this is vanishing. Plus kappa M squared- a squared +sine-squared theta. So this is so-called, The surface of Ergo region. Notice that this r0(theta) is greater or equal to r+. So we encounter the following situation that we have a sphere horizon at given moment of time. So this is a sphere of the horizon, this is R plus. And we have Ergo region, which has this form. It's touches in the poles of this thing. And this is r0(theta). So this is Ergo region. One can escape from this part, one cannot escape from here, but one can escape from here. So, to see that one cannot escape from here one has to consider normal vector to this surface and find that there is like light. And to see that one can escape from this region is to consider one has to consider normal vector to this surface and again see that it is not like light. So one doesn't have to have speed of light to escape from this region. Then, that is a way to make these observations. Now, one can notice that, so when r equals to this, g 00 = 0. Now, one can see that not always this has real solutions. One has real solutions for this case and for this case, if for this case specially. If kappa m is greater than a otherwise, there is no real solution for this guy. And also one can observe that this metric has closed time like curves. So this is unphysical situation, and we have a mate singularity, because there is no solution for this equation. There is no real R+, it means that no horizon. But at the same time, we have a singularity. So, this metric, this spacetime has a singularity which is not surrounded by the horizon. The explanation why goes beyond this lecture. I do not give explanation for that, but it is un-physical to have such a situation that there is a singularity which can be reached with infinite time from the point of view of an observer which stays always outside of the body, of the black hole. So there is so-called cosmic censorship. Physicists old school study black holes, believe in so-called cosmic censorship. That in real situations during collapse one cannot create such an object. So this guy by this metric was cut by M greater than zero. When cut by M is equal to a it's called critical solution, but one cannot create such a metric which corresponds to this situation. Or more generally, it states that, during real physical situations, one cannot create spacetimes with the singularities which are not surrounded by horizons, event horizon. Why? Well, one believes that in real physical situations, starting from generic conditions which are smooth. We have at our disposal something smooth initial metric, smooth initial matter distribution, smooth forces, etc.. And then we're using various devices or tools. want to create from this matter some black hole like object. And physicists believe that using all possible tools we cannot create something which will have a singularity, which is not surrounded by event horizon. Well, this is just generic belief of sounding like possible. Although, as far as I understand, there is no rigorous proof of this statement from first principles. And the proof should rely on the fact that we shouldn't go beyond the general theory of relativity. Because if we can prove that using some other higher theory which includes general relativity. It's a separate story. The question is whether one can prove this statement with an only, general serial for relativity. So, and last thing which remains to be said at this point is the following. We want to describe so-called no-hair theorem. No-hair theorem says the following thing. That if we have only general theory of relativity plus matter, no electric charges, etc. So within this theory, any generic collapse process will lead eventually to such a metric. Which is described by only two parameters, by mass and angle of momentum, and nothing else. So, the reason for the belief in this no here of cerium is, actually, very simple. It can be explained in simple terms from my mind,l and let me just give you the idea why it is so. First of all, as I have explained in the previous lectures, the Schwarzschild horizon, or the horizon of a black hole, is a surface of infinite redshift. Why? Let me explain that on the Schwarzschild Penrose diagram. So, on the Schwarzschild Penrose diagram, it can be obviously seen that if we have anyone who is staying very close to the black hole horizon,it can stay fixed at the fixed radius. If something inside the black hole creates radiation or makes some crazy motion, like masses or charges, any radiation will go toward Towards the singularity. It will not go outside of this surface, so we have a very funny situation. Whatever happens on the horizon, or just behind the horizon, whatever crazy motion happens, whatever crazy radiation is created, just inside the horizon. An outside observer, at whatever radius it stays, will never see any radiation, it will see only static field or stationary fields. It is very important. So, the Schwarzschild horizon, the horizon of the black hole, is the surface of infinite redshift Infinite red shift. Let us study it from the point of view of a collapse. If we have a collapse, the matter of the black hole, folds down the creation of the horizon, fold approaches its horizon. And of course during the collapse process, there are a lot of radiation processes happening so many things are radiated. But as the source of this radiation, the collapsing matter approaches the horizon. The radiation influence is stronger and stronger red shift. So you see, if there is a radiation which is created very close to the horizon, it performs a work against gravitational force. Performs a work against gravitational force, and because it performs a work, again, gravitational force, its energy is reduced. And because of the reduction of energy, the frequency which is proportional to the energy for the photons or for the gravitons, the frequency is red shifted. So we have a stronger and stronger red shift. Related process to this fact is the following. That during the creation of the black hole, during the collapse process, everything that can be radiated is radiated away. And all the higher multiple momenta during the collapse process are decaying with time. It means that the body, which remains during the collapse, after the collapse process, has all the momenta. Which are responsible for the radiation reducing in time, are radiated away. So what remains at the final stage of the collapse can carry only stationary fields. Notice that to have a electromagnetic radiation, one has to have deep dipole moment changing in time. So changing in time, multiple moment or electric charge. Sorry, not monopole moment, electric charge. Doesn't create radiation, electromagnetic radiation. As we will see in nine's lecture, to have the gravitational radiation, one has to have changing in time quadruple moment even more. So rotating angle of momentum doesn't create radiation. Gravitational dipole moment doesn't create radiation. And gravitational monopole moment mass doesn't create radiation. Again, let me tell it in a different way. Suppose you have an electromagnetism, you have a sphere, ideal sphere, with homogeneously distributed electric charge over it. This sphere is breathing. So it goes outside. And so it shrinks and expands. Shrinks and expands, respecting spherical symmetry. And homogeneity of the charge distribution. Independently of the radius of this sphere, it always creates outside of it a coulomb field. So electric field is not 0 but magnetic field is 0. So it means that such a sphere, despite the fact that the charges are moving with acceleration in this motion. This breathing sphere never creates electromagnetic radiation outside of here. It's known by those who are building antennas, they know that spherically symmetric antenna cannot be created. So that is related to the fact that to have electromagnetic radiation, you have to have dipole movement changing in time. While for the breathing sphere, dipole movement with respect to its center is always 0. And actually all multiple momenta, with respect to its center, are 0. So similar situation we have in gravity and even more refined, in the sense that spherically symmetric mass. When it breezes doesn't create gravitational radiation, because outside of it, in Newton gravity, it creates Newton field. In Einstein gravity, it creates due to bulk of CRM, creates Schwarzschild field. So these are very interesting situations. So again, let me say what happens. During collapse process, everything that can be radiated is radiated. What remains is a object which is characterized only by three charges. By mass, angular momentum and if there is electric charge by electric field, by electric charge. So these are the only charges which cannot create electromagnetic or gravitational radiation. All the rest of the multiple momenta, can create radiation. And they are radiated, these are not. So this fact relies in the basis of the no-hair theorem. In the absence of the electric field, when we don't have this, we have the solution which is characterized only by two parameters, by mass and angular momentum, as a result of the collapse process. That is a statement of the no-hair theorem. [SOUND] [MUSIC]