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So we have learned so far what is the physics

behind different coordinate systems.

In spacetimes, what is the physics behind the curvilinear coordinates,

what is the physics behind curved spacetime, etc., etc.?

And finally, we have obtained Einstein equations from Einstein-Hilbert action.

And starting with this lecture, we're going to work with the solutions.

We're going to describe solutions of the Einstein equations.

In this lecture, we going to start with the simplest exact solution of, which

is spherically symmetric Schwarzschild solution of the Einstein equations.

So, that solution appears to be,

it solves the Einstein equations in the case when the cosmological

constant is set to zero, and there is no matter.

Energy momentum tensor of matter is also zero.

So, as we have explained during the previous lecture,

we have to solve this equation.

So this is the equation we're going to address right now.

And now we're going to find non-flat space solution of these equations.

Namely we want to consider spherically symmetric solution.

And to consider spherically symmetric solution, it is better,

instead of using Cartesian coordinates, which are t, x, y, z.

It is convenient to use spherical coordinates in spacial path.

So we're going to use these coordinates, t, r, theta and phi.

So these are the seminal spherical coordinates in three dimensional space.

So the most general metric which respects spherical symmetry has the following form.

ds squared is equal to g tt,

which is a function of only r and

t, dt squared + 2gtr,

which is a function of r and

t also dt dr + g rr, which is again

the function of r and t only.

dr squared + k, which is

a function of r and t only again.

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Times d omega squared.

What is d omega?

d omega squared is a metric, very well known metric,

on the unit sphere which in spherical coordinate looks like this.

So this is a metric on the unit spear.

The range of validity of r is From 0 to plus infinity.

So it ranges from 0.

So this is of of the radii.

The range of validity of phi is as usual from 0 to 2 pi.

And the range of validity

of theta is from 0 to pi.

This is important.

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And why this metric respects spherical symmetry?

Well, let's consider its time slice, meaning that t constant, t constant slice.

It means that dt is 0.

When dt is 0, this drops off.

This drops off, because dt is 0.

What remains is this thing.

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And this is the metric.

This is a partial part of the metric.

So this plus this is a partial part of the metric.

And this partial part of the metric for

t constant is sliced as onion by spheres

whose radii are fixed by this quantity.

So this part of the metric specifies the radii.

Because it specifies the distances in r.

Proper distances in our direction.

And this part specifies the area, how the area of the spheres is changing.

We encounter, a little bit unusual for the first time,

situation that our space is sliced as

onion with spheres for given moment of time.

The radii of these spheres are set by this quantity.

While the areas of these spheres are set by this quantity.

And these two quantities are not related to each other.

So it is not that we have a sphere of radius r and

its area then is four pi r squared.

This is not the relation we encounter here.

So the radius is set by this quantity, the areas by this.

This is somewhat unusual, but otherwise, anyway,

that's the reason we consider this as a spherically symmetric situation.

Of course, if one chooses a different reference system,

one can lose this symmetry, this explicit form.

But for us it's important that there is a reference system in which metric

looks like this.

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This part.

This metric is invariant under the coordinate transformations.

If one takes a vector Xa which is t and

r which has components like this, so a runs from 1 to 2.

So this metric is invariant under the following transformation.

So if one chooses g bar

ab as a function of x bar,

it is g cd function of x,

d xc dx bar a dx d dx bar B.

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And at the same time, we can also transform this one.

This one transforms as a scalar.

So, k bar of x bar is nothing but

k of the function of x which is in its own right is a function of x bar.

So under such transformations, this

part of the metric transforms as a two dimensional tensor with two indices and

this part of the metric transforms as a scalar.

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So, we have two functions.

So basically we have two functions,

r bar as a function of t,r, and t bar as a function of t and r.

So we have two functions.

These two functions can be used via this transformation

to fix two out of these four independent.

So we have one, two, three, four functions.

So using these two functions we can fix two out of the four.

And the convenient choice for future is to put g t r.

So g t r.

To 0.

So which means that we fix this to be 0, and

K, as a function of rt, to be equal to minus r squared.

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Minus r squared.

So then the spheres will have this area, this area, but

the radius of this sphere can be different from r due to the presence of this term.

So after renaming the components into more standard form,

if we name g t t as exponent of the function nu, which is function of.

And g rr as minus lambda t, r.

Then we arrive at the following expression.

ds squared = Nu

(r,t) d t squared

minus exponent of lambda (r,t)

d r squared minus r squared d omega squared.

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But let us stress that this metric is still invariant under the remaining

transformation.

So we can change t to the t bar which is a function of original t.

So we can change this coordinate.

We want to respect spherical symmetry so

we don't want to touch this part of the metric.

That's the reason we transform only time.

So if we change this, this part is changed.

But then we can shift this part as follows.

So nu bar as a function of r and

t bar is just original

nu as a function of r,

t of t bar, plus log

dt dt bar squared.

And lambda transforms as a scalar.