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Killing vectors and conservation laws

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Introduction into General Theory of Relativity

National Research University Higher School of Economics

4.6 (104 Bewertungen) | 30K Teilnehmer angemeldet

General Theory of Relativity or the theory of relativistic gravitation is the one which describes black holes, gravitational waves and expanding Universe. The goal of the course is to introduce you into this theory. The introduction is based on the consideration of many practical generic examples in various scopes of the General Relativity. After the completion of the course you will be able to solve basic standard problems of this theory. We assume that you are familiar with the Special Theory of Relativity and Classical Electrodynamics. However, as an aid we have recorded several complementary materials which are supposed to help you understand some of the aspects of the Special Theory of Relativity and Classical Electrodynamics and some of the calculational tools that are used in our course. Also as a complementary material we provide the written form of the lectures at the website: https://math.hse.ru/generalrelativity2015 Do you have technical problems? Write to us: coursera@hse.ru

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Bewertungen

4.6 (104 Bewertungen)

5 stars
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4 stars
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JQ

Jul 18, 2017

One of the most important courses ever taken and yet. still difficult to understand really well.

AB

Jan 29, 2017

Demanding course with good overview of general relativity content.

Aus der Unterrichtseinheit

Classical tests of General Theory of Relativity

We start with the definition of Killing vectors and integrals of motion, which allow one to provide conserving quantities for a particle motion in Schwarzschild space-time. We derive the explicit geodesic equation for this space-time. This equation provides a quantitative explanation of some basic properties of black holes. We use the geodesic equation to explain the precession of the Mercury perihelion and of the light deviation in curved space-time.

Killing vectors and conservation laws8:03

Test particle motion on Schwarzschild black hole background11:29

Test particle motion on Schwarzschild black hole background(part 2)8:12

Mercury perihelion precession24:48

Light ray deviation in the vicinity of the Sun8:58

Unterrichtet von

Emil Akhmedov
Associate Professor

Skript

[MUSIC] In this lecture, we provide a quantitative approval of those observations that have been made qualitatively in the previous lecture. And also, we derive from general serial frailty some effects which approved by classical experiments. To do that, we find the geodesics in Schwartz's space time, in that which was found in the previous lectures >> And a convenient way of finding Geordie's axis, to use conservation laws. To use integrals of motion. Now we will provide the integrals of motion, for the motion of particles and Shcwartz's space tag. Well, let us start from a generic observation, and explain how to find a conservation laws. As we have explained in one of the previous lectures, under the infinitesimals coordinator, transformation which is like this, Metric tensor transforms as follows. Plus covariant derivative epsilon mu. And symmetrized over the indices of x. This notation was introduced in one of the previous lectures, so it's symmetrized covariant derivative. So if for some vector epsilon nu, which we will denote k nu, the metric doesn't change under the coordinate transformation, doesn't change at all. It means that D mu k nu is equal to, which is by definition, is D mu k nu plus D nu k mu. So, if k mu is such that this is equal to 0, then the metric doesn't change. And, such a k is called Killing vector. So, for example, for the Schwartz's space-time, which has a matrix ds squared, which is 1 minus rg divided by r dt squared minus dr squared, 1 minus rg over r minus r squared D omega squared. As you can see, this metric doesn't change if we make time translation. And, if we, because this is independent of phi. Because this is a d theta squared plus sine squared Theta d phi squared. So metric is not dependent on time, and is not dependent on phi. So the components of the metric are not dependent on these things. So the metric is invariant, and as such transformations. I don't know, for arbitrary a and b. These are exactly the Killing vectors, which I explain in a different way just by looking at the metric. So they are seen explicitly in this metric. So at least we have these Killing vectors. And they, in the notation that k mu has the following coordinates, k t, k r, k theta, k phi. This one corresponds to k mu equals (1,0,0,0). While this one corresponds to k mu equals (0,0 0, 1). So these are the Killing vectors, at least we have these Killing vectors in the space time. They are explicit in this form of the metric, explicit in this form of the metric. And let us see that the presence of the Killing vectors leads to conservation laws. For example, consider a particle moving along z mu of s, which is a curve, which has a low line is mu of s. Velocity for vector is dz mu over ds. Now, consider the following combination, k mu times u mu. And let us take it's derivative. D over ds, which is a proper time of k mu times u nu. Then this is just d mu of k nu times u mu d z nu over ds. Well this is just u nu. And now we take the derivative of this. Derivative of this is the following. So by Laden's rule, first of all we have this new mu, but then we use Laden's rule which means that we get k mew times d new. U mu, so we apply to this thing. Plus we apply to this vector, which is u nu, u mu times d u k mu. Now, let us continue and represent this as k mu u nu D nu u mu, plus remember that one can see that this is symmetric under the exchange of indices u and mu, so this can be represented as one-half u mu u nu u mu D nu k mu symmetrized, which means represented like this. So no one can see that if this thing is geodesic, then the motion goes along the geodesic. This is 0. Because this is the equation for the geodesic this thing is equal to 0. And if K is Killing vector then this is also 0. As a result, we obtain the following conservation law. So this quantity doesn't change. Doesn't depend on proper time for the motion along the. So this gives us a conservation law. So, our goal is to achieve, we have shown that if there is a Killing vector in a space time, then that generically leads to conservation law. Now we going to consider this conservation laws for the Scwartz's space sign in greater detail. [MUSIC]

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