[MUSIC] So, we have made the following transformation. Originally, we have the following metric, which is, sorry, plus dt squared minus dx1 squared minus dx2 squared minus dx3 squared. And then we have made the transformation that t is rho hyperbolic sine of tao. And x1 is rho hyperbolic cosine of tau. Rho and tau are new coordinates. After this transformation, we obtain the following metric as we observed a few moments ago. And rho squared d tau squared minus d rho squared minus d x two squared minus d x three squared. The two second insert coordinates were not touched during this coordinate information. So, now we obtain a new metric tensor, whose matrix, g mean nu looks, it is diagonal. And time independent as promised. It depends on spatial coordinates, it depends on rho, but it doesn't depend on mu times tau. So, as we expected, we obtain a static method. Static means that it is invariant under time translations, under the change of tau by tau plus a and time reversal. Tau minus tau. So, as expected, due to the fact that we transform to a homogeneous moving reference system, we have obtained a static metric. So let us first explain why this is a transformation to non-inertial reference frame. And second, let us describe the properties of this metric. First of all, here we have to specify that rho is positive. We will see in a moment why. Then one can see that if one divides t by x1, one obtains that this is just hyperbolic tangents of tau, which means that constant tau just straight lines passing through the origin, and as tau changing. When tau goes to minus infinity, this goes to minus 1, as tau goes to plus infinity, To plus infinity, this goes to plus 1. So this constant tau lines are just straight lines passing through the origin in the coordinate system t and x 1. And at the same time, one can see that x subscript 1 squared minus t squared is just rho squared. And constant rho aligns, just hyperbola. So we have the following coordinate in eight transformation, originally we have the lattice, code in lattice, t X1. Of course, we take a section of constant x2 and x3. So we just consider a section of constant x2 and x3. In this section, we had originally this lattice, coordinate lattice, or rectangular coordinate lattice. And now, we obtain a new coordinate lattice, which looks as follows. So we have this line in terms of regional coordinates, let me draw how it looks like. We have a line which corresponds to, This line corresponds to tau minus infinity. This line corresponds to tau plus infinity. These lines of constant tau, let me draw them by a different color. So, this tau constant. And the lines of constant row are just hyperbolas. So these are lines of constant rho. Rho constant. For all positive, and for all negative. This is for row constant and row greater than 0. And for all negative, we have, also, Hyperbolas, but this corresponds to rho less than 0. So from rectangular coordinate lattice, we have obtained this kind of strange lattice, which covers only these parts of entire space time. The reason for that is when rho is greater than 0, it is not hard to see that x1 is greater than models of t. As a result, for greater than 0, these coordinates cover only this quadrant of whole entire space time. And for all less than 0, these coordinates would cover this quadrant of entire space time. For the reason that x1 would, in that case, will be less. Well anyway, it doesn't matter. So it is important to stress that all these hyperbolas for different rho have the same asymptotes, which are just these lines. And when rho rates become smaller, hyperbole becomes closer to the asymptote. And as rho goes to zero, hyperbolas degenerate to the two lines that x one squared is equal to t squared, which is that x one is equal to plus minus t, which are exactly the same line. So, these lines are corresponding simultaneously to tau minus infinity, and the row equals to 0. Tau plus infinity and row equals to 0, and intermediate values of tau are here. Intermediate values of row are here. So we have these kind of strange coordinate systems. Now, one should notice that this metric does degenerate when rho is 0. When rho goes to 0, this element of the metric vanishes, and this is a degeneration of the metric. This degeneration is very similar to the standard degeneration of the two-dimensional metric in polar coordinates. If you have this metric d phi squared. Obviously, that generates when r goes to zero. But the generation of the metric doesn't mean that the two dimensional space where this metric is drawn has any peculiarity at r goes to zero. The space at r equals zero space itself is completely irregular, it's just the origin of the space. Similarly, here, when rho goes to zero, the matrix does degenerate, but the space-time itself doesn't have any fixed peculiarity along these like light lines. This is the same Minkowski flat space that are on these like light lines. This is just the generation of this method. And now we're going to discuss the meaning of this degenerating of the metric. [MUSIC]