The least action (or minimal action) principle (part 1)

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Из курса от партнера National Research University Higher School of Economics

Introduction into General Theory of Relativity

83 оценки

National Research University Higher School of Economics

Introduction into General Theory of Relativity

83 оценки

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

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Einstein-Hilbert action and Einstein equations

We start with the explanation of how one can define Einstein equations from fundamental principles. Such as general covariance, least action principle and the proper choice of dynamical variables. Namely, the role of the latter in the General Theory of Relativity is played by the metric tensor of space-time. Then we derive the Einstein equations from the least action principle applied to the Einstein-Hilbert action. Also we define the energy-momentum tensor for matter and show that it obeys a conservation law. We describe the basic generic properties of the Einstein equations. We end up this module with some examples of energy-momentum tensors for different sorts of matter fields or bodies and particles.To help understanding this module we provide complementary video with the explanation of the least action principle in the simplest case of the scalar field in flat two-dimensional space-time.

The least action (or minimal action) principle (part 1)11:44

The least action principle (part 2)12:13

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Emil Akhmedov
Associate Professor
Faculty of Mathematics

[SOUND] In this complementary material,

we want to discuss least action

principle for the fields.

To simplify the discussion of the least action principle for the gravity,

we will consider least action principle for the scalar fields.

So to understand, and even more, we will consider two dimensional scalar fields.

But to understand deeper field theory, let us consider the following situation.

Let us consider a net of particles, infinite net.

So this is, say, indexed.

This is number 0, this is -1, this is 1 and so forth to infinity.

2, etc.- 2, etc.

So we have such a collection of particles

each one of them having mass m and

the springs having the hook index k.

Hook index k.

It is not hard to find that the action for

this system, which is just the integral

over dt from t1 to t of the Lagrangian which

is just the difference between total

kinetic energy- total potential energy.

And in our case, this is just t1 t2.

Where here, total kinetic energy is just the sum of

all kinetic energies of the particles from minus infinity to plus infinity.

m phi i dot squared over 2,

whose phi I will explain in a second.

Sum over i from- infinity

to + infinity k phi i + 1-

phi i squared over 2.

Phi is just displacement.

So in equilibrium particles positioned at

equal dispositions from each other, along this line.

So but when we move one of the particles in one or

the other direction from its equilibrium, the displacement from

the equilibrium we call as phi i, where i is the number of the particle.

So the displacement if we moved one, or they started to shake somehow.

There are waves running.

We're going to describe this wave with this discussion which we are going

to conduct.

So they are changing in time.

So that's what we get.

And now, it's not hard to see.

I assume that listeners of this course, they know the minimal action principle.

At least in non relativistic situation, and minimal action principle in

non relativistic situation leads from this action to the falling equation.

m phi i double dot is =

k phi i +1- phi i-

k phi i- phi i- 1.

So the meaning of this formula is very simple.

It's a just Newton's second law, m times acceleration of e's

particle is equal to the force acting on it from one side,

from this side, by the spring which is on the right hand side.

And the force by the spring on the left hand side.

So there's two forces acting on this particle, which are I mean,

the same can be plus because this can be bigger than this one.

Depends on whether the spring is shrunken or expanded.

Okay, anyways, now we wonder, what does it have to do with the field series?

Well, let us take the continued limit.

Continued limit means that we want to somehow consider on one side,

on the physical side, we want to consider long

wavelengths fluctuations of this system.

Such that the wavelengths of the waves that are running along this

system are much bigger than the distance between the both, between the particles.

And on the other hand, this is just analogue of crystal,

one dimensional crystal.

And if we look very far at the crystal, it looks completely homogenous.

So we want to discuss this homogeneous situation.

What does it mean mathematically?

Mathematically, it means that this index i becomes continuous.

So it becomes some continuous coordinate x.

What does it mean?

It means that phi of t becomes some field phi(x,t).

What you see before taking the continually limit the field is just,

if we're given given phi i for all the particles.

So value of phi for every particle, this collection

gives us discreet field at given moment of time.

But when we take continuem limit, this guy describes

to you kind of displacement of inhomogenuity compression or

expansion of the crystal nearby the point x.

I hope it's clear what I mean.

So this is two dimensional field.

One plus one dimensional field.

One space, one time.

So then, if this is the case

then f i +1 of t, so to say,

goes into phi(x + delta x of t).

Then, it is not hard to see that phi

i +1 of t- phi i of t goes into that's

not hard to see into f prime of x and t times delta x.

Two linear order in delta x.

So what does it do with this?

Well, let us multiply.

This by delta so divide by delta x here and multiply.

And here, we divide by delta x squared and

multiply by delta x here.

And then, we take out delta x, so we didn't do anything.

If we enter this into the brackets,

it cancels with this and everything cancels out.

Now, we want to take delta x going 0,

delta x going to 0.

But keeping this ratio, and this ratio,

fixed in this limit, such that m over delta x

is fixed constant, and bar, and k times delta x,

is fixed constant k bar in this limit.

So what does it mean keeping this things.

We want to obtain elastic crystal such that it has some elasticity properties.

If we don't fix this constant,

then the density in this limit, density goes to infinity.

Or the hooks, the hook coefficient for

unit area per unit length becomes infinite or zero.

So if we do not fix this constant, we either and

this limit obtain collection of dust of particles consisting this crystal

which are not interacting to shallow then this is not a crystal.

Or we obtain absolutely rigid tenser, either that or that.

But if we fix this custom, fix these quantities, fixed,

we obtain some elastic body and that's our goal.

Now, what do we obtain in this limit then?

Well, this guy becomes phi prime as one can see

and as the result this action goes into, let me write it.

Well, we can write it like this It's t1,

t2 dt integral of a dx.

The sum was delta x goes into integral from- infinity to

+ infinity and we obtain some something like like this.

m bar over 2 phi dot squared where phi is this field, this field.

-k bar over 2 phi prime squared.

Dot means the differentiation with respect to t and

prime means the differentiation with respect to x.

So if we ride this like this k bar over 2,

dt, dx- t 1,

t 1- infinity + infinity.

Here, we obtain 1 over c bar squared phi dot squared- phi prime squared.

What is c bar?

c bar is just m bar over k bar, sorry, c to the- first.

This square root of this.

This some quantity having the dimensionality of velocity

c bar is having dimensionality of velocity.

As can be seen from the dimensional properties of this quantity and

it has the meaning of the speed of sound in this crystal.

So if we introduce the vector xa c bar t and

x, where this is x 0 and this is x 1.

Then this quantity can be written as follows.

k bar over 2.

This k actually can be absorbed into rescaling of phi which is not a problem.

dt, dx and here we have da, phi, da, phi.

So this is the action for the fields.

Simplest two dimensional field and

you will be having the proper sign here.

[SOUND]

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