[MUSIC] So we have explained the physical meaning of the transition to a curvilinear coordinates in flat space-time. The physical meaning is just transition to another not necessary inertial reference frame if in the coordinate transformation we mix spacial and time coordinates. Now what is the physical meaning behind curved space-times will be explained in the upcoming lectures. But at this point let me make some exercise here which will be relevant for the discussion of curved space-times. But let me stress that even if we consider curvilinear coordinates, metric in space-time can look unrecognizable. So it means that it can be some function of x dx alpha dx beta, and this can be some. It won't be obvious from the form of this metric whether we are dealing with flat or curved space-time. How to distinguish flat and curved space-times we will describe in the next lecture. But once we are considering generic metric terms here, let us describe a motion of particle in such a generic metric. The motion of particle we can describe using minimal action principle. To use minimal action principle, we have to write an action for a particle in curved space-time. If we have a particle in curve space-time, suppose that we have an arbitrary curved space-time. It can be flat space-time, and with curvilinear coordinates it can be flat space-time and flat coordinates. It can be curve space-time. When we have a particle and its worldline, the unique invariant characteristic one can write for this worldline, it's length. So it is natural to expect that the action will be proportional to the length of the worldline of the particle. The reason why we are looking for the invariant action, invariant means with respect to the coordinate transformation. The reason why we are looking for the invariant action, because we expect that from the minimal action principle for this action, we expect to get equations which are covariant under the coordinate transformations. Covariant means that they look the same in different coordinate systems, the equations of motion. The reason why we are looking for the equations of motion which are covariant under the coordinate information is this principle of general covariance which I have just explained to you. We want to have equations of motion which are looking the same in different coordinate systems. So the invariant characteristic is a length, and also in flat space-time it is obvious that the extremal action will correspond to the straight line. So it's natural that the action is just the length. So let us write the formula for the length. To write the formula for the lengths, we have to approximate the worldline of the particle by chain of straight intervals. And when the intervals are short, we can approximate the lengths of each element, each segment. For example, the east segment, the length of the east segment is just g mu nu (zi)[zi + 1- zi mu mu] [zi + 1 nu- zi nu] square root. So this is just ds. Ds, and ds squared under the square root. Along the worldline of the particle, along this segment of the worldline of the particle. As a result the lengths of the worldline is just the sum of these guys. Sum over i from 1 to N. Now if we want to obtain the smooth worldline we have to take N to infinity, number of segments to infinity, and the size of these segment to 0. As a result we obtain that the length is just the integral of ds over the worldline of the particle. And at the same time, this is just tau1 to tau2 d tau square root of g mu nu [z(tau) z mu dot z nu dot]. Where dot means the differentiation with respect tau. Tau is just a parameterization of this worldline. Notice that we can choose any parameterization of this worldline. We can change to tau to f(tau) as long as f dot (tau) is greater than 0. Because then if we just use this coordinate change we will obtain that this is equal to df f1 f2 square root of g mu nu dz mu / df dz nu df. If this is valid, f dot (tau) is greater than 0, then we can extract this from the square root and cancel, and change to a different parameterization. Tau and f are just different parameterization of the same worldlines. And this thing just says that we have to respect the order of points along the worldline. So the action then is just- m integral over tau1 to tau2, square root of g mu nu [z, which is a function of tau in its own right]. Z mu dot, z mu dot d tau. So this is the action. The reason why we have minus here is very simple, because actually the longest line is a straight one in Minkowskian signature, the longest one. Among all lines, those which are having. Now but first of all, let me, okay. Minus m here is just due to complimentarity. If for example this metric is just flat Minkowskian metric, we obtain just the standard action for the relativistic particle in Minkowski space-time. I just remind you why there is a minus sign here. In Minkowskian metric, minus sign here is necessary because actually among all lines whose tangential have angle less than 45 with respect to the vertical line. Among all these lines, the straight one has the longest length. Unlike the Euclidean geometry, in Minkowskian geometry the straight lines have longest length among those which have angle less than 45 degrees with respect to the vertical line. So that explains why we have minus sign because we want to have, for the longest line, we have least action. We want to have least action. [MUSIC]