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My name is Emil, we are going discuss the general theory of relativity.

So, the goal of this lecture is to understand what means

metric in space time, what means general co-variance,

what different coordinate systems, and what does it mean to

make a transformation from one reference system to another reference system?

And what does it have to do with a change of coordinates?

So, for that reason, let me start the easy

discussion of the Minkowski spacetime and of special theory of relativity.

In special theory of relativity, we are dealing with flat Minkowski

spacetime whose metric is famously known to be like this,

where mu and

mu are indices which are ranging from zero to three, and

metric tens has the falling metrics, which is diagonal.

And, its diagonal components are as follows.

As the result, we have dt squared minus dx vector squared,

where x vector is just x1, x2, x3,

and throughout my lectures I will assume that the speed of light is set to be 1.

So, what does it mean that we have a metric in space time or

in space time in general?

Well, let me start with the idea of Newtonian mechanics and metrics in space.

Suppose we have a two-dimensional space, two dimensional space for

simplicity to draw it on the blackboard, and

we discussed for example behavior of two cockroaches.

One of the cockroaches stays still at some point.

And the other cockroach starts its motion from the same point and moves somewhere.

So, in Newtonian mechanics, we usually discuss projectory, we want to understand

it's lengths how long was the pass of the coke ridge, etcetera, etcetera.

So, in Newtonian mechanics, we have a metric in space and

then separately we have a very naive metric in time,

which has nothing to do with the combined metric on space time.

So, what does it mean?

It means that two cockroaches, both the one who was standing and the one who

was moving spent the same amount of time during their motion.

For example the cockroach which was standing its world line

Its behaving space time was a vertical line.

And their world line of the second coverage was like this.

And one would say that the coverage during this

behavior became all done by the same amount.

It means that they started their motion At the moment of time t0 and

ended their motion at the moment of time t1.

And both became older in Newtonian mechanics by t2- t1.

The presence of the metric in Minkowski space-time tells us

there is a situation in special theory of relativity is completely different.

Because now time and space combine together in one continuum

where we have a metric which means that there is a way to measure distances.

In space time, between any two points in this space time.

It means that we can measure the lengths of this line and

separately the lengths of this line.

Obviously as we know from special the length of this And

world line divided by the speed of light is just the time spent

by the cockroach moving along the corresponding world line.

Say, the first cockroach which was moving along this world line,

which was standing, this cockroach has spent exactly

This time, this proper time.

While the cockroach which was moving now spent a different proper time,

a different proper time which is not equal, not equal to t2- t1.

It's a lens of this world line which is Equal to

integral over d s over this world line.

And this is not equal to t 2 minus t 1.

In fact, one of the is all around the other.

Now this in fact is older than this one because this line in Minkowski,

space time, is longer than this line due to this minus sign in the metric,

so Minkowski metric is given by

a very special bi-linear form Very special by linear form.

Metric is in one to one correspondence with the coordinate net.

What does it mean that we have a coordinate net?

One can draw in space time a key

X one x two t, by the way these pictures of course not

quiet a liquid because I am drawing on there

Two-dimensional Euclidean plane of the blackboard and

trying to plot three-dimension plot of Minkowskian space-time.

This picture of course is not sensitive to this minus sign, and

that one has to bear in mind while describing or considering these pictures.

So, how do I do a coordinate net?

I for example fix planes x 1 constant

in equal dispositions along x 1 line.

So it means that I have this plane, I have this plane,

The same I can do along x 2 Constant and a low T constant,

this way from hyper cubical a coordinate lattice

which is corresponding to this Bilinear form specifying Minkowskian metric.

This bilinear form is invariant under the following

coordinate transformation which is known as the Lorentz boost.

It is given as follows.

We have transformation from t and x1 to t bar.

Which is t hyperbolic cosine of alpha,

plus x1 hyperbolic sine of alpha.

X1 bar t sine of hyperbolic

sine of alpha plus x1, hyperbolic cosine of alpha.

At the same time, we do not change x2 and x3.

And alpha is constant, which means that it doesn't depend on the coordinates.

The physical meaning of this transformation Is that it takes

us from one inertial reference system to another inertial reference system.

And the second inertial system moves with velocity v.

With respect to the initial one.

Then, cosine of alpha is nothing but

the relativity gamma factor, which is as follows.

And sine of alpha is just -v divided by square root of v- v squared.

And v is constant.

So we have two inertial reference system.

So this coordinate transformation doesn't change this bilinear form.

And takes us from one hypercubic coordinate lattice,

to another hypercubic coordinate lattice.

But let us consider an arbitrary coordinate transformation.

for example, let us consider the transformation from

x mu to x bar

mu which is an arbitrary function of the initial coordinates not necessary linear

Under such a transformation the bilinear form will change unrecognizably.

But at the same time, the distance between any two points

In space time will not change, notice that I do not deform the initial space time,

I do not shrink it, I do not expand it, I do not bend it by any means,

I just chose different coordinates in the same space time.

Different coordinate lattice, meaning that I draw these hypersurfaces of constant

x mu along each direction, coordinate lattice will look horribly,

it will not be hypercubical, and also this bilinear form will change unrecognizably.

For example, the metric.

As we will discuss in the next lecture, matrix changes as follows.

We take from g from at the mu

nu to a matrix tensor g mu nu, which is as follows.

dx Alpha dx Beta divided by

dx mu bar dx nu bar.

And because of this transformation, we will discuss it in greater details

in the next lecture, because of this transformation we observe That

the line element, ds squared doesn't change.

Originally the line element was eta mu nu dx mu dx nu,

and now it just given by g

alpha beta Of some function, of x bar,

not necessary constant, dx alpha bar dx beta bar.

Because of this transformation, we have this that this is equal to this and

line element interval doesn't change.

So Now arrises a natural question.

If we have such an arbitrary, not necessarily

a linear transformation of coordinates, what does it mean physically.

We start from inertial reference system.

Where do we go?

what reference system do we obtain?

Do we obtain a reference system?

Now we are going to address this issue.

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General covariance

Introduction into General Theory of Relativity

Meet the Instructors