[MUSIC] So we have obtained the following solution, exact solution, of the Einstein Equation. dudv + 2 H (function of u, x perpendicular) du squared- dx perpendicular squared, where H = kappa p delta (u) log of modulus of x perpendicular. So now we want to discuss properties of this solution. So consider, we want to describe particle geodesics on such a background. Consider a particle, light-like particle, which has world line, so it collides, so to say with this penrose parallel plane wave, with impact parameter equal to a. So its geodesic has fixed position in this direction equal to a. It means that dx perpendicular = 0. So because it's light-like, this is equal to 0 for the light-like particle, and as a result, we obtain the following equation. du [dv], for such a geodesic, because this part is fixed, du + dV, H(u,a) times du. So we have two types of geodesics. One type is du = 0 and the other type is dv + H as a function of u and a times du = 0. So these are two types of geodesics. Remember that the world line of the source of this wave is u = 0 and x perpendicular = 0. So let me draw the picture, let me draw the picture. So in the direction t, in the plane t and z, we have the following wave traveling with the speed of light. So the world line of this wave is like this. So this is u = 0, and this is a slice of x perpendicular = 0. X perpendicular = 0. So the direction of x and y is here. Now if there is a wave which travels parallel to this wave, so some way here, it just doesn't feel its presence, so it just goes. This is U constant, U constant. These waves do not feel the presence of the shock wave that we're discussing. But what happens with those which travel along this world line? So that can be described according to this equation, and we have the following situation that, from this equation, we obtain that dV = -kappa p delta of u log of modulus a, because x perpendicular is equal to this, times du. So because of this delta function, before you equal to 0, before this point, the wave travels according to v equals to constant. According to dv = 0, dv = 0. So it travels like this, no, not the way you, the particle, the geodesic of the light-like particle is like this. Then if at u =0, the particle experience the shift along the V direction. Along the V direction because integral of this from 0- 0 to 0 + 0 over du, integral of this expression, of kappa p, delta of u, log of a is just kappa p log of a. So it carried backwards. So it carried backwards along this direction to this amount and the distance depends on the value of the input parameter. Closing this direction, the particle to the cirrus elongate is carried. Farther shorter it is carried. But then it is released and goes, again according to this law, but with a shifted V, the V is shifted by this amount. So that's geodesic for the life-like particle in this shock wave. So their reason for such a behavior can be understood on general grounds. So what we had at the beginning? At the beginning we had the Schwarzschild solution and for the Schwarzschild solution, how you call it, the body is spherically symmetric. We have a field lines, spherically symmetric basically. Then we boost it with some velocity v. And because of the contraction in the direction of the boost, we basically have field lines more densely compressed in the direction perpendicular to the motion. As v becomes bigger, the density of lines in this direction, it becomes bigger and in the extreme limit as V goes to 1, we obtain such a pancake so the source is like this and the field lines are like this. So while this picture is space time, this picture is space-like, so it's just time slice, basically, a photo of what is going on. So this other compression of the field lines. So this is a pancake. The shockwave that we're discovering. Now remember that particle, when it travels light, when it travels in space-time which is curved, the light rays are bended. The main amount of bending happens in the very vicinity of the source, in the very vicinity of the body. So as we shrink these lines, this region becomes smaller and smaller. And as a result here, this bending becomes stronger, and the bending mainly due to the time delay. That is exactly what happens here. We have a time delay. When the particle is very close to colliding with the shock wave, remember that the particle is a bit distance a away from the source of the shock wave from the source. But it is in the field of the shock wave, and the field of the shock wave is such a pancake. So it experiences time delay and then releases and continues its motion. So these are the properties of the parallel plane wave. And well, this is a gravitational field, approximately one can describe by this field the following situation. If you have a break and accelerate it to very big velocities, to such velocities that you cannot ignore the gravitational field, then this a gravitational field it creates. And if you can consider, for example, collision of such breaks what they lead to etc., etc. So basically we have said everything what we can say about gravitational weight. We have studied linearized approximation and small gravitational waves and also we have studied shock gravitational waves. >> [NOISE] [MUSIC]