Transition to the homogeneously accelerating reference frame in Minkowski space–time (part 2)

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Introduction into General Theory of Relativity

90 calificaciones

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National Research University Higher School of Economics

Introduction into General Theory of Relativity

90 calificaciones

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

De la lección

General Covariance

To start with, we recall the basic notions of the Special Theory of Relativity. We explain that Minkwoskian coordinates in flat space-time correspond to inertial observers. Then we continue with transformations to non-inertial reference systems in flat space-time. We show that non-inertial observers correspond to curved coordinate systems in flat space-time. In particular, we describe in grate details Rindler coordinates that correspond to eternally homogeneously accelerating observers. This shows that our Nature allows many different types of metrics, not necessarily coincident with the Euclidian or Minkwoskain ones. We explain what means general covariance. We end up this module with the derivation of the geodesic equation for a general metric from the least action principle. In this equation we define the Christoffel symbols.

General covariance12:28

Сonstant linear acceleration16:37

Transition to the homogeneously accelerating reference frame (or system) in Minkowski space–time8:22

Transition to the homogeneously accelerating reference frame in Minkowski space–time (part 2)13:23

Geodesic equation8:26

Christoffel symbols14:23

Conoce a los instructores

Emil Akhmedov
Associate Professor
Faculty of Mathematics

[SOUND] So, let me now describe

the physical meaning of this

transformation that we have made.

So again, originally we had the Minkowski metric.

Dt squared- dx1 squared-

dx2 squared- dx3 squared.

And we have transformed to a new metric, which by the way has,

is called Rindler's metric.

D rho squared- dx2 squared- dx3 squared.

So, in all my drawings and discussion I will ignore these coordinates,

which will remain untouched during this coordinate change.

Now, the first thing that I wanted to explain why I say that this method

describes non-inertial reference frame corresponding to constant eternal

acceleration.

Let us recall that originally our coordinate lattice was built as follows.

It was built as follows.

We had original coordinate t, original coordinate X1.

And lines of constant X1 were just vertical lines.

And lines of constant t were like this.

And what is the physical meaning of these lines, of constant X1?

These lines are just world lines of observers or

particles, which perform inertial motion, in this case, just stationary.

So these are world lines, vertical lines,

are just world lines of observers,

of which correspond to dx1 = dx2 = dx3 = 0.

Now in new coordinate system, coordinate lattice looks as follows.

It's original t and original x1.

I just, for convenience, I draw them.

And new coordinate lattice looks as follows.

So we have lines of constant tau.

These are lines of constant tau.

And these are lines of constant rho.

Rho constant.

So who are these hyperbolas?

These hyperbolas are nothing, but the world lines of those observers or

particles which move with constant acceleration.

And the value of this acceleration is nothing but just 1/rho.

Different lines correspond to different acceleration, but

these are world lines of accelerated observer.

So that's the reason I refer to this metric,

metric describing what is going on in constantly accelerating motion.

So, now there is an important issue corresponding to this metric,

that it degenerates when rho is getting to 0.

What is the physical meaning of this degenerating?

To explain it, consider the following situation.

Suppose we discuss, first of all, I have to say that motion with constant

eternal acceleration is impossible, physically impossible of course.

No one can eternally accelerate, because it demands infinite energy consumption.

So real motions are that someone

is moving with acceleration only during a finite period of time.

Let us describe that motion first of all.

That motion could be described as follows.

Suppose someone was static, didn't move until t=0, until this moment of time.

Then accelerated for a while, turned on acceleration, accelerated for

a while, and then continued its motion with the gained velocity.

It means that it continued its motion along the line which is tangential

to this hyperbola.

It is important that any tangential to this hyperbola at any point has

the angle which is degree less than 45 degrees with respect to the vertical line.

Even if you accelerate eternally, you never exceed the speed of light.

Your speed approaches the speed of light asymptotically if

you accelerate eternally.

But if you switch off acceleration at any finite moment of time,

your gained velocity will be less than the speed of light.

And the world line will be straight,

after the switching off of acceleration, will be straight line,

which has angle less than 45 degrees with respect to the vertical line.

So you see, if you accelerate during finite period of time,

your world line arc coincides with this hyperbola only during

some finite period, but then it is different from hyperbola.

It is important that this motion,

if you accelerate for a finite period of time, is homogenous.

It is this moment of time when you switch on eternal acceleration and

switch it off, is of course different from any other moment on this world line.

And that moment, any moment of time on this part of the world line,

is different from any moment here, and different from any moment here.

So, this motion isn't homogenous,

and that is the reason we do not consider this kind of motion.

Because in this case we do not expect to get such a nice looking metric.

We expect to get such a nice looking stationary metric,

only if we consider homogenous eternal acceleration.

That is the reason we consider this a physical situation,

which allows us to draw some interesting conclusions.

That's the reason we consider this.

So, now what is important, that if you eternally accelerate,

if your acceleration is constant, let me remind you that

here you decelerate starting from the speed of light, you eternally decelerate.

Then here you accelerate actually,

because three-dimensional acceleration is always directed along this line.

But what is important that if someone decides at this moment of time, t = 0,

emit a light ray which tries to catch up with you, this light

ray is always parallel to this asymptote and never intersects with the hyperbola.

It means that the light ray emitted from any point behind the asymptote,

never catches up with the eternally accelerated observer.

But it does catch up with someone who stopped the acceleration

at finite moment of time.

Of course, because if you stop accelerating at finite moment of time,

you continue your motion along the line tangential

to the hyperbola because it has angle less than 45 degrees with respect to this line.

It is not parallel to this line, it intersects with it.

So, only in case if you externally accelerate,

you have this particularity that the light ray doesn't catch up with you.

How does this fact reveal itself in this formula?

It reveals itself as follows.

Suppose you have a motion with constant velocity along this direction.

Then it means that dx1/dt, in our annotation, is equal to c which is 1.

As a result, ds squared = 0.

But if ds squared = 0,

here we obtained that d rho/d tau = rho,

which is not necessarily equal to 1.

It is equal to 1 only if rho equals to 1.

So now we obtained the strangest situation, that in non-inertial reference

frame, we obtained that the speed of light does depend on the coordinate.

And that has to do with these particularities that I

just described to you.

It becomes even 0 when rho goes to 0,

when the hyperbolas degenerate to these straight lines.

So if light ray is emitted in the very vicinity of this asymptote and

tries to catch up with someone who accelerates, it takes very long time for

this light ray to catch up with the acceleration.

That reveals itself through this formula.

And this degeneracy of the metric is the price we have to pay

to consider such unphysical situation of eternal acceleration,

which has this property that light rays can not catch up with it.

So, this is the first thing I wanted to stress.

And another interesting observation that one can make here is that, for

example, consider an observer which is just stationary in the Minkowskian metric.

Its world line in this coordinate looks like this.

Of course it's world line in this coordinate looks as follows.

But it is important to stress that for this particle,

which is just stationary in original Minkowski spacetime,

it takes infinite time, tau, to get from here to here.

In fact, this line corresponds to tau =- infinity,

and this line corresponds to tau = + infinity.

In new coordinate system it takes infinite time, tau, to get from here to here.

So if we for example try to draw this coordinate system in the rectangular

fashion, so it means that we can see the, let me stress that we can see the rho > 0.

So we can see the lines of constant rho like this.

These are just lines of constant rho.

And lines of constant tau like this.

Tau constant.

Then, the world line of this guy who is just static in the original

Minkowski spacetime, in this new way of drawing rho and

tau coordinates, will look very strangely.

It will just asymptotically approach this line, rho = 0.

Asymptotically approach this line at past infinity, and

asymptotically approach this line at future infinity.

But will never intersect it during finite period of time, tau.

So, all particularities of this Rindler metric,

which will be necessary for our further discussion in upcoming lectures,

it is important to stress just one thing,

that this line is called past event horizon of the Rindler observers,

of those observers which are at constant eternal acceleration.

And this line is referred to as future event horizon of Rindler observers.

This is just for notational reasons.

Now, what conclusions do we have to draw from all these considerations?

I gave you just an example of a coordinate transformation, which is not linear,

not linear because t and X1 have a nonlinear relation to rho and tau.

And which transforms us from inertial reference

system to a non-inertial reference system.

But we basically drew

a different coordinate lattice in the same spacetime here.

This is the same spacetime, but different coordinate lattice.

Of course, the geometry of this space, we didn't deform it,

we didn't shrink it, we didn't I don't know, expand it.

We just drew a different coordinate lattice.

Of course, the geometry of the spacetime didn't change during this transformation.

Also, it is obvious that if one would consider, for example,

a motion of a cockroach in this spacetime, and

the same motion of the same cockroach in this coordinate system,

this will be the same motion governed by the same physical laws, but

observed in a different fashion, a new coordinate system.

Just in different fashion, but the same physical situation.

So, it is important that we now are observing a very

important thing which lies in the core

of general theory of relativity and which is referred to as general covariance.

The general covariance is a statement that the physical laws and

geometry of spacetime shouldn't depend on the coordinate system that we are using,

and shouldn't depend on the type of observers that we are using.

Different coordinate systems correspond to different observers, that's it.

[SOUND]

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