WEBVTT
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MARKUS KLUTE: Welcome back
to 8.20, Special Relativity.
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In this section,
we're going to talk
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about the relativistic
Doppler effect.
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And we make good use of
our space-time diagrams,
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which we discussed earlier.
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So the situation is as follows--
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to simplify this,
we have a source
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which is emitting pulses.
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So the waves are pulses.
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Every now and then there is
a beep, and another beep,
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and another beep.
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And those pulses travel
with their velocity--
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with their wave velocity.
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And they have a world
line represented here
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in the space-time diagram.
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This is pulse number one,
and this is pulse number two.
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The distance between
those two pulses
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is our period, the period of
our wave, which we call tau.
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The question now is, how
is this being observed
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by an observer which is moving
with a relative velocity
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v with respect to the source?
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So let's analyze this.
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So if we want to characterize
or find our position x1 and x2,
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we can do this by saying
x1 is equal to ct1
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or equal to x0, which is
the distance of the observer
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to the source plus c times t1.
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v is the velocity in which
the source is moving.
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And similarly for t x2, we
find c times t2 minus tau.
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And that's also equal
to x0 plus v times t2.
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So the distance in time--
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we're still in the reference
frame as of the source--
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is given by c times
tau over c minus v.
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And the distance in
space is given by v times
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c times tau over c minus v.
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So the question is not
how this observed--
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how this is seen by
the source but how this
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is being seen by the observer.
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So we have to apply
Lorentz transformation.
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So in the s prime frame,
which is the observer frame,
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we find delta t prime is
equal to gamma delta t minus v
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over c squared delta x.
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And then we just fill
in the information
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as we discussed before.
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Tau prime is then gamma times
c tau over c minus v times
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1 minus v square over c square.
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And then you make use of
delta equal v of over c.
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And we make use
of gamma equals 1
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over square root of
1 minus beta square.
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And we find then--
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this is a little bit of
an algebra exercise here--
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that the period now
is given by 1 plus
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beta over 1 minus beta square
root of that times tau.
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And the frequency
is the inverse.
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We'll have 1 minus beta
over 1 plus beta square root
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of that [? times ?]
the frequency.
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So we just calculated
relativistically
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how the period and the
frequency of a wave
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is Lorentz transformed.