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MARKUS KLUTE: Welcome back
to 8.20, Special Relativity.

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In this section, we talk
about spacetime diagrams.

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They turn out to be very
useful tools to describe

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events or sequences of
events, in particular

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when observed by
multiple observers.

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So what is a spacetime diagram?

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Here's an example.

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You have an x-coordinate
and a t-coordinate

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for space and time.

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I plotted an event in blue here.

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And the world line of events.

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A world line is just a sequence
of events as they occur.

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In this case,
something seemed to be

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moving with constant velocity.

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The world line is just a
continuous line of movement.

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The velocity of
this event is delta

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x over delta t, which
is 1 over the slope.

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Let's have a look here.

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So the time axis is defined
as those events which

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all occur at the same
space, x equals 0,

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whereas the x-axis is
defined as those events which

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all occur simultaneously
at the same time.

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And then you can
draw additional lines

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into the spacetime diagram
where, for example, all times

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are equal to 1.

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You might want to add a unit.

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I omitted this here.

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Time might be given in seconds,
in days, in hours, in years--

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whatever you like-- similar to
space in meters or light-years.

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So lines here in green of
the same time, meaning time

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is constant.

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All events on that line
happen simultaneously,

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while in blue are those
lines where events happened

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at the same location, so
x is equal to constant

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some specific values.

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Here in red, I add one
additional caveat, which you're

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typically not aware
or considering

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very much in diagrams is
the role of tick marks.

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Here in this spacetime
diagram, our tick marks

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are perpendicular to the axis.

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It's also OK or correct here
to say that they are parallel

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to the second axis, and
we'll come back to this point

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later on.

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An example of a
world line is simply

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drawing all events which
correspond to me, right?

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Professor Klute is
pacing in his office.

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You know, maybe on a line,
just the x-coordinate

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is plotted here, time
passes, and I'm just pacing

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the long-changing direction.

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Each time, each little
segment is constant velocity.

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That's the world line
of me from some time t

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for minus t to term
time equals 40.

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OK, so now our first
concept question.

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Let's consider the set of
world lines, 1, 2, 3, 4,

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and the question is,
which of the objects which

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correspond to the world
line is moving the slowest?

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Let's consider
this for a second,

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and then we look at this.

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As the velocity is
1 over the slope,

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the object with
the steepest slope,

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the largest value of the
slope, is moving the slowest.

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And in this case,
it's object number 2.

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All right, now we want to
actually make them useful.

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Yes, they can be used in order
to describe certain event

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lines, but they're
really useful when

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you describe events happening
for different observer.

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So in this activity, I invite
you to draw Bob's spacetime

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diagram into Alice's, and
then as a second step,

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draw Alice's spacetime
diagram into Bob's.

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The situation is very similar
to previous ones discussed

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in this lecture.

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Alice is stationary
and Bob moving

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in this rocket with a velocity
of half the speed of light,

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or a gamma factor of 1.2.

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All right, try.

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Go ahead.

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Try to show where is
the time axis for Bob

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and where is the spatial
coordinate for Bob.

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OK, so the way to approach
this is the following.

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We want to use Lorentz
transformations in order

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to figure out what is the value
of Bob's time axis and space

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axis for different values of
Alice's spacetime diagram.

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So we start with drawing
Alice's spacetime diagram.

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And then if you want to find
the x-axis as seen by Bob,

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we have to set the time
for Bob to equal 0 and then

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find the corresponding elements
or tick marks on the axis.

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So the first point
we're going to find

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is tB equals 0 and xB equal 1.

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So with the Lorentz
transformation,

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we find that xA 1, so
this point corresponds

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in Alice's spacetime diagram
to xA equal gamma equal 1.2

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and tA equal gamma times v
over C squared equal 0.6.

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So we can make this--

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find this first point and
plot it in our diagram.

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It's right here.

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OK, and then we go move around
and find the second point

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and the third point,
and we do the same

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for the time axis, where xB
is equal to 0 and tB equal 1,

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then corresponds to 4.6
in xA 1 and 1.2 in xA 2,

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where we find these points here.

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I failed to say that the
origin of those two spacetime

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diagrams [INAUDIBLE].

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OK?

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So this is already it.

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So we found Bob's time axis,
where xB is equal to 0,

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and Bob's x-axis,
where tB is equal to 0.

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And I did draw those tick marks
parallel to the second axis.

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So if I want to now find out
the time axis for xB equal 1,

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I just have to follow along and
draw a parallel in the picture

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here.

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All right, so the
second question

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then is, where Alice's axis
in Bob's spacetime diagram?

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So the procedure is
very similar as before.

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We draw Alice's axis--

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sorry, we draw Bob's axis,
and we find Alice's x-axis

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by setting tA equals 0, and then
we find the number of points

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and connect those.

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And the time axis is found
by setting xA equals 0

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and finding them points
for various values of time.

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And you see here, this looks a
little different than before.

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I just zoom in
here a little bit.

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What you find specifically,
because relative

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the direction of motion
changes, the positive values of

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x are in positive value--

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and negative-- so the
positive values of the x-axis

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are in the negative
time direction,

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while the negative values of
x are in the positive time

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direction, so would
be across down here.

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So this needs a little
bit to get used to,

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but you will later
see when I draw

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in any of the two
spacetime diagrams

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specific sequence of events,
I can immediately read off

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how this event is perceived
from Bob's and from Alice's

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perspective.

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And this makes our space
diagram very, very useful tools.