[MUSIC] From the first two lectures, we have learned the physical meaning of arbitrary coordinates formations and a way, a method to distinguish curved space times from flat space times. Now, we are ready to discuss, what is the physics behind curved spacetime? So, experimental observations, set of experimental observations shows that spacetime is curved by anything that carries energy. Perhaps that was, at the beginning, was falling from aesthetic observations, and based on a very few experimental facts, but now the number of experimental facts that have proved this conclusion normal, and nobody questions, nobody among the scientists questions this fact. So, rephrasing this statement that spacetime is curved by anything that carries energy, we can say verbally that metric becomes a dynamical variable, coupled to energy, carried by matter. Our goal here, to show that all we need to formulate, general, serial for relativity, the equations of motions of general theory of relativity, is this guess that a metric is dynamic of variable coupled to energy momentum tensor. General covariance and minimal action principle. We would like to find equations of motions for this, for the metric tensor, such that which relate on one side there is geometry, on the other side there is energy, so we expect this equation of motion to be generally covariant. It means that the action from which this equation will follow will be invariant on the general covariant transformation, so we have to build up invariant from the metric for the beginning. So, from the metric tensor, we have at our disposal metric tensor, and we want to build up an invariant, invariance, which will compose the action. Well, simplest invariant that one could, yeah, first of all, the action should be integral of a spacetime of some density, and this integral should be invariant. So, the simplest invariant that one can write in these circumstances is d 4 x square root of the modulus of g, where the modulus of g, let me define the notation, is the modulus of the determinant of the metric tensor. And d 4 x is just dx0, dx1, dx2, dx3, so this quantity, this integral is invariant under general covariant transformation. This is just the volume of space time. One can straightforwardly check that the square root, the transformation of the square root of the metric. Well, that's easy to see like. If you make a general covariant transformation, this thing is transformed by a Jacobian of this transformation, and this thing transforms by the inverse of the Jacobian, and they compensate each other. As a result, this quantity is invariant. The problem with this quantity that if one will vary it, one will apply minimal least action principal to this sort of action, to the action proportional to this quantity. One will obtain algebraic equation for the metric because this quantity doesn't contain derivatives of the metric, so as a result we don't expect to get anything but algebraic equations, well we will see that, in fact. And it'll be later. But one would like to obtain differential equations, containing differentials of metric along space and time directions. And this equation will describe dynamics, how metric changes in space and time. While algebraic equations don't do that. While this is the simplest invariant, we need to go a bit farther and invent some other thing Which contains derivatives of the metric and is invariant. That invariant, we know from the end of the last lecture and that is Ricci Tensor. Ricci scalar. Well there is a Ricci Tensor And we need Ricci's scalar R, which is just g mu nu times R mu nu. So this is invariant. We have to integrate it over the volume to obtain the action. So the proposal for the action is as follows. Is as follows.S is equal to a integral d four x square root of g. Well this is not the only thing. We have to add b integral of d four x square of modulus of g times R. These are two simplest invariants that one can write and this invariable already contains derivatives unlike this one. These two together can give us something that describes dynamics. A and B are some constants which follow solely from experimental data and cannot be Obtained on the symmetry observations that we have been applying so far. Well this action describes only gravity. It doesn't describe any matter. Anything that carries energy, et cetera. Well gravity can carry energy but that's a separate story. So we have to add to this some matter action. Matter action. Some action which contains matter, fields, or particles. And describes interactions of them with the metric, with the gravity. Well, the simplest case that we have already encountered. During the first lecture, is the action for the particle. So the simplest example of this matter interaction is the following one. Minus m integral over ds. And that is minus m integral over d tau, square root of g mu nu of z Zed dot mu zed dot nu, and that exactly described coupling of the particle to the metric. And that is an example of which we have in mind, but we will encounter it by the end of this lecture A few more examples of this action. But at this point, all that we need to know is that this action is invariant under the general covariant transformation, nothing else for the derivation of Einstein-Hillbert equations. Einstein equations from We will need that fact only. What remains to be done is to fix these quantities. And in total, the total action that we are going to work with is the following. It is minus 1 over 16 pi kappa integral over d 4 x square root of modulus of g r plus lambda. Plus S matter which contains energy metric tensor and matter fields or particles. So, here kappa is the seminal Newton's constant, and lambda is so-called Cosmological constant. We fix these constants to agree with experimental observations. So about this quantity everything is known, I hope to those who listen to these lectures everything is known about this quantity. But what is lambda? Well first of all, quantum field theory predicts this To be very big. Due to so called zero point fluctuations. While experimental observations in modern Cosmology, show that it is not zero but very small. This point is the essence of so called Cosmological constant problem but it has nothing to do with our course. All that we need to know at this point is just some kind of parameter, and we go in to consider this as an arbitrary parameter. So our goal is now to derive this action with respect to the metric. 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