[MUSIC] This lecture is rather formal because I'm going to introduce the notion of tensors, explain the geometric meaning of Christoffel symbols, which have appeared at the end of the previous lecture. And I'm going to explain what is the difference between flat space and curved space, how to distinguish what is just to flat space in curvilinear coordinates in comparison with curved space. And so, I assume that those who listen this lectures are familiar with terms in flat space time. And so, my introduction will not be very basic. So I will assume this knowledge. So, let me just remind you that coordinates, if I have a coordinate transformation, x mu are going to x bar mu, awhich is a function of x, original x. Then the coordinates themselves transform tautologically. Tautologically means that dx mu is equal to dx mu over dx bar mu, dx bar nu. Well, this is a tautological transformation because if I divide both sides by dx bar mu, I just obtain equality between the same thing. So, but it is important to bear in mind that the vector is referred to as contravariant if this transformed in the same fashion. So, the vector is contravariant, so it carries upper index. If it transforms as coordinates do, so like this. So this, such a vector is referred to as a contravariant. Covariant vector is transformed differently. It transforms as a one-form. So it carries lower index, lower index. And transforms according to this rule, so this is a function of x. This is A bar mu function of x bar already. Multiplied by dx bar mu. And as follows from this relation, we obtain that A mu of x is just dx bar mu, divided by dx mu A. So, mu bar of x bar. So, this kind of vectors are referred to as contravariant. This as covariant. So now, if I have a metric changer, it means that I define line element in the spacetime like this, phi mu. And since the last lecture we're familiar with the fact that metric transport can be in principal arbitrary function of x. So, if we have a metric tensor, and we have its inverse tensor with upper indices, where this is a Kronecker symbol, which means that it is 1 if mu equals 2r for nz and zero otherwise. So, so if we have two tensors, metric tensor and inverse metric tensor, to every contravariant vector, with the use of the metric tensor we can define corresponding covariant vector. And vice-versa. If I have covariant, but multiplying by this, I obtain contravariant vector. It's important to understand that due to this relation, the components of this guy coincide with the components of this guy. So, the separation is the reversible. Now, what is a tensor? What is a tensor? Tensor is the following quantity, n-tensor. Is the following quantity. It carries n indices, some of which are upper case, And the others are lower case. The order of indexes is important, but in this formula, to simplify it, I just ignore this fact. I will stress it later, why the order is important. So, but it’s important that l+k here in this formula is just equal to n. So the total number of indices is n. So, such a tensor is a concrete quantity which transforms appropriately with respect to the coordinate transformation, which means that these indices, upper case indices, transform according to this rule, nu1/dx bar alpha1 dx mu l/dx bar alpha l. And vice versa for these ones. So, it's dx bar beta 1/dx mu 1...dx beta k bar / dx mu k this tensor is a function of x. Multiplied by T bar of correspondingly beta 1, beta k. And here, alpha 1, alpha l. As a function of xbar. So, if there is a quantity which under coordinate transformations transform like this, we call this as n-tensor. So, vector is just a variant of n-tensor with only one index. So, it's a one-tensor. The simplest tensor is a scalar, a quantity which doesn't have any index, and which transformed trivially under the coordinate transformation. Trivially in the sense that it just transforms from being a region, it was a function of x, and became some new function of x bar. So this the rule for the transformation of a scalar. Vectors are like this. Now, why does one need tensors? Why does one need tensors? As I explained in the last, in the first lecture, we need to write laws of nature, I mean, dynamic eloquations of motion, describing motion of particles, or dynamic eloquations of motion describing spacetime changes of the fields. We have to write them in a covariant fashion. In the way that the equations of motion do not depend on coordinate systems. They change. So in the bar system, they're expressed through the bar quantities, bar tensors, bar coordinates. In the regional through unbarred quantities. So, for defining these kind of laws of nature, we need tensors because tensors are quantities which are transformed appropriately with respect to the coordinate transformations. [MUSIC]