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MARKUS KLUTE: Welcome back
to 8.20 Special Relativity.

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In our quest to understand how
we get to general relativity,

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there is two things to consider.

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The first one,
this lecture is not

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meant to give you
a full description

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of general relativity,
but just a view into where

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this might lead, where
this discussion might lead.

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So in this quest,
we can understand

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the theory of general
relativity as a theory on how

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to patch together the different
reference frames which each can

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be described in special
relativity, in the framework

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we discussed up to now, and
it's valid in short intervals

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in spacetime.

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Consequences of
general relativity

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are that spacetime is curved.

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So we have modified geometries.

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We learned that, because
of gravitational effects,

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matter curves spacetime.

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As a consequence
of that, there must

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be modification of gravity
based on matter distributions,

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and so there must also
be gravitational waves,

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gravitational lenses which
bend light, black holes,

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and there's cosmological
predictions coming out

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of general relativity.

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So let's have a
discussion first.

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What does it mean to have a
changed or modified geometry?

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What could that mean?

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So you are all used to
Euclidean geometry, where,

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when you draw a
triangle, you add up

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all the angles to 180 degrees.

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If you try to parallel
lines that never cross,

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they also don't diverge.

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But if you have a
modified geometry--

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for example, the geometry on
a sphere, like on our globe--

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the angles do not add
up to 180 degrees.

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Actually, the sum is
larger than 180 degrees,

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and parallel lines will cross.

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We will call this kind of
space positively curved,

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but you can have the opposite
example, like on a saddle.

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So you can have other spaces
and other curved spaces,

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and they can be
negatively curved.

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In this example, if
you add up all angles,

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you find they add up to
less than 180 degrees.

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Parallel lines do not cross,
but they will diverge.

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Mass changes the
geometry of spacetime.

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We just talked
about light bending,

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and because of the
change in geometry,

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light will not go on a
straight line anymore,

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but will bend around
massive objects.

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Spacetime is curved.

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Geometry of spacetime tells
us how the mass is moved.

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You can think
about a trampoline.

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When you put a heavy
object on a trampoline,

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all the other objects
on the trampoline

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will gravitate towards
the heavier object,

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and that's kind of a picture
on how spacetime actually

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looks like.

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Einstein used those
findings in order

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to redefine Newton's
first law and found

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the so-called Einstein
field equation.

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So on one side of
the equation, there's

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a description of spacetime
and its curvature,

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and on the other
side of the equation

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is the energy momentum
tensor, the description

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of how energy and momentum
of object is distributed.

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And those two things, spacetime
and energy and momentum,

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they're kind of interlinked
in this equation.

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So if you read this
description, you

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can read it from one
side to the next.

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Spacetime tells
matter how to move.

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Or you read from
the other direction,

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say matter tells
spacetime how to curve.

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That is an equation,
and you can just

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read it from the
left to the right

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or from the right to the left.

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Our understanding here.

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It says space and time
are not fixed things

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through which matter and
energy moves through.

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The matter and energy
themselves define spacetime.

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And matter, because of
spacetime, is dynamical.

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It's changing.

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It's interacting with the
matter and with the energy.

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This is a super exciting picture
from Hubble, the Hubble Space

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Telescope.

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And you see galaxies,
but what you also see

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is those structures which
looks like the light has

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come through lenses.

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Those lenses are actually
matter distributions, galaxies,

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which actually lead to
the bending of the light

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and those lensing effects.

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OK.

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If you want to summarize
general relativity,

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you can first say that
spacetime is curved

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and it follows the
pseudo-Riemannian manifold

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with a specific metric.

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We have seen the metric before.

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It's minus, plus, plus, plus.

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And the relationship
between matter and curvature

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is given by the
Einstein equation,

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and here I give you a
slightly different form

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where there is the
dynamics, again, on one side

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and the energy momentum
on the other side.

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Let's just look at
one example here.

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So we discussed, in
special relativity,

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invariant intervals, and
we have this delta squared,

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or we have a
different name for it.

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I given by minus dt squared
plus dx squared plus dy

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squared plus dz squared, and
we could have just written

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this in polar
coordinates as well,

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where you find that dr
squared and r squared

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d theta squared, and
then r squared sine

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squared theta d phi squared.

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OK.

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Same thing.

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It's just a different
coordinate system.

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So as a solution to
Einstein equation,

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we find something which
looks very, very similar.

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That's not a surprise, as
we find general relativity

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as a patchwork of small
spaces of special relativity.

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So the solutions
might be very similar.

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And the solution found here,
the so-called Schwarzschild

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solution, which is a
unique solution in vacuum

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with spherical symmetry
of a matter distribution.

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So you have a spherical matter
distribution like our sun,

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and this is a solution
which describes

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spacetime around this.

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You find this
invariant interval here

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has two interesting features.

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There's two
singularities in here.

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This should be a minus 1.

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You find those
two singularities.

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One is at r equals 0.

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That's kind of expected.

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In the middle of the
mass distribution,

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this thing is not
defined anymore.

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There's no mass left.

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But there's also a second
singularity at 2GM.

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This is called the so-called
Schwarzschild radius,

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and if you get to
the singularity,

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you basically don't
define anymore

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this invariant interval.

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You can think about the
surface of a black hole

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as this singularity.

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At this r value, at
the singularities,

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everything becomes timelike,
or everything within the radius

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becomes timelike.