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MARKUS KLUTE: Welcome back
to 8.20, special relativity.

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Let's start here
with a short summary.

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We have seen through
experimental measurements

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that there is no ether, that
electromagnetic waves travels

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through vacuum.

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We have discussed the concept of
the relativity of simultaneity,

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meaning that two events
might occur simultaneous

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to one reference-- in
one reference frame,

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to one observer, while
they're not to another.

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In the last two videos,
we looked at clocks.

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And we've seen that
moving clocks run slow.

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We've also seen that moving
objects appear smaller.

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They're lengths contract.

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We found that the
time of a moving clock

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is related to the time
in the clock at rest

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with a gamma factor
because it's time dilation.

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And, for the length, we
have seen that there is a 1

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over gamma dependency,
length contraction.

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The entire discussion was
based on Einstein's postulates.

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We simply used
Einstein's postulates.

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And then we looked at
experiments of clocks.

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Now, you might argue that
the setup of the clock

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is actually what's
tricking us here,

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but I can tell you
that is not the case.

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As you have seen for
the muon, the muon

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doesn't know about
optical clocks.

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It just decays based
on its own properties.

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I think it's fair to say that
Poincaré and Lorentz came

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to similar conclusions about
the same time as Einstein did.

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Delta t at rest and delta L at
rest, the time and the length,

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are also sometimes called
the proper time and length.

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In German-- and I actually
prefer this a bit--

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we use the word eigen,
which means own.

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So it's basically the time,
the own time of the object,

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the object being addressed.

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So we can now ask, [INAUDIBLE]
seen time is suspect.

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What is not suspect?

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What are the observables
which are invariant?

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And, by this, I mean
the observables,

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when we have two
different reference frames

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and we have a conversation,
we do actually

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agree in a conversation
about an observation we have.

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We have seen we
cannot agree on time,

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and we cannot agree on the
length in the direction

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in which we are moving.

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So can I ask, for
example, what happens

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to the width or the height?

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If I put a train on a train
track and it's fast moving,

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is that contracted or even
expanded or changing at all?

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The answer is no.

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Similarly, if I put a
train track on a track

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and I go very fast
into a tunnel,

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does the height of
the tunnel change?

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Also here the answer is no.

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And we can verify this
later quantitatively.

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In summary, transverse
dimensions are not affected.

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They are not suspect.

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We can agree in a
conversation of two people

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in different reference
frames about the height

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and the width of a train.

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That's good.

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But what else is invariant?

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So here I want you to
consider the time and distance

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between two events or
maybe even three events

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observed from different
reference frames.

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And we introduce or reintroduce
our characters Alice and Bob

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and add Carol to this.

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So we're going to have three
reference frames of three

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observers in this discussion.

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So I want you to look
at this one here.

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So assume that Bob
has a clock, and it's

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moving with a velocity v. And
Alice is observing Bob's clock.

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We have done this before.

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Now, we want to also
add Carol to this.

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And Carol is moving with
three times the velocity

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and is also observing
Bob's clock.

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What I want you to do is
look at this property here.

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So we have seen that
the height is invariant.

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So let's look at what
happens if I calculate

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2 times the height squared, and
I use x and t, time and space,

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in order to express
the height, all right?

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Again, this is an opportunity to
stop the clock, stop the video,

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and work this out
on a piece of paper.

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So it turns out
it's not that hard.

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We basically find that 4
times the height squared

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is equal to--

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maybe it is hard--

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c squared times t
squared minus x squared.

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And, since the height
is an invariant,

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this property, c squared,
the speed of light squared,

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times the time squared minus
x squared, is invariant.

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So we can think about them
as delta t and delta x.

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When we look at the difference
in time and the difference

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in space between two events--

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Professor Klute
entered the class.

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And, at the other
end of the class,

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Professor Klute exploded.

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If I do the delta t and the
delta x between the two--

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we square them and
subtract them--

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that is an observation
we can all agree on.

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Whether or not you're
stationary in the classroom

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or you're passing by really
quickly with your spacecraft,

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that observation is
something we can agree on.

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And it's invariant.

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This property is called
the invariant interval.