[MUSIC] So, let us continue with the intensity of the radiation and the consideration. So, in the previous lecture, we have seen that if the radiation is traveling along the said direction, then we have energy flux as follows. So it's a flux in the said direction. And it is 1 over 32 pi kappa times hij dot times hij dot. And ij here run from 1 to 2. But, in generic situation, if we have some region V where body is moving randomly, well generically randomly, so they create radiation going in all directions, in all directions. Hence, our problem is to find hij, which is, three is less and tangential to unit vector in the direction of radiations. So we want to find such, twiddle hij. We have at our disposal hij, from the previous part of this lecture, which is approximately -2 kappa over 3 modus of x. qij dot dot. So we want to construct from this vector something which is, well it is traceless already because qij is traceless. But, we want to construct from it tangential to direction of radiation, other direction of radiation, to other unit vector ni. It should be symmetric of course, And traceless. So we want to construct such from arbitrary hij. We want to construct such a hij twiddle obeying these conditions. So, if we have at our disposal, If we have at our disposal hig, the that is tangential to n has the following form. It is hig- ninkhkg- njnkhik + ninjnknlhkl. This one, can easily check that this h perpendicular obeys, so, h perpendicular ij obeys the condition that it is, Perpendicular to n. Then,transfer traceless part of it, h twiddle ij is the falling. It is hij octagonal minus one-half delta ij minus ninj times hll orthogonal. So, now having this, and using the fact that ni squared = 1, we have the following situation, that hij times hij twiddle is equal to hijhij- 2ninjhkihjek + one-half ninjhij squared. So, this is what we obtain. Now, we want to take the average. Well, if there is a random motion, random motion, the radiation more or less goes in all directions homogeneously and we have to average all of the possible directions of this unit vector ni. So, the averaging is done with the use of the following formulas. Well, the definition of the averaging, the definition of the averaging is as follows. So, if you have a bunch of vectors ninilnin. Well, big N. This average is by definition is just integral over the solid angle or 4pi of the product of ni1nin. Where, the components of n, vector n I understand as follows. It's sine theta cosine phi, sine theta, sine phi and cosine theta. So, you block every component over this here, and take this integral, this way you get this average. This is a definition. Well, the result of this calculation say for, to take the average of this quantity, to take the average of this quantity over all directions of n, we have to bear in mind that the result of the averaging for ni, nj is just one-third of delta ij. And the average of ni, njnknl is 1/15th delta ij, delta kl + delta ik, delta jl + delta il, delta jk. Well, anyway those have been studying electromagnetic radiation and the motion of charges in homogeneous electromagnetic fields In principle should know this procedure and should understand the meaning of this answers. Well anyway, as a result of this averaging hij twiddle, hij twiddle, average is nothing but 2/5th hij times hij. So this is a result of the averaging. As a result, hence, the total intensity, which is the integral over d sigma iti0 Is approximately after n average, is approximately equal to 4pi times x squared, so this is actually the area element of the distance x squared, times 1 over 32 pi kappa times average of hij twiddle dot hij twiddle dot. The fact that there are dots here doesn't change this fact, well just leads to dotting here. And in all, we obtain the following expression, kappa/45 times qij triple dot times qij triple dot. So, this is a total intensity of the gravitational radiation. In the none relatively seek quadruple radiation. Well let me just explain this formula, write what is d2 sigma i is just x squared times ni d omega. So, to understand what's going on, we have a source of radiation here, we have a solid angle and we have this is d2 sigma i. This is ni unit vector in the direction of the radiation and this is just models of x with distance. So, that's it, that's a formula we were going to obtain. And that's a formula we have obtained. [MUSIC]