WEBVTT
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MARKUS KLUTE:
Welcome back to 8.20.
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In the previous section, we have
seen the relativistic Doppler
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effect, and now we want
to study how light--
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in this case, a
monochromatic plane wave--
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transforms in the
Lorentz transformation.
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In other words, we have,
for example, a distant star
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emitting light at a
specific frequency.
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The question now is, how
do we observe this light
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when the star is traveling
away from us or towards us?
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So here you see our
monochromatic plane wave.
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We have an amplitude A and
then just a simple cosine,
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which is a function
of x, y, and t, time.
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This is the solution
of the wave equation,
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and we have already seen
this as part of the p
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sets, but also
discussed in class.
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So the wave is characterized
by so-called wave numbers
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in x direction and y direction.
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The squared sum-- the square
root of the squared sum
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is the so-called wave number.
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The frequency omega
is equal to 2 pi f,
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where f is the frequency and
omega is the angular frequency.
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And if you divide the angular
frequency by the wave number,
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you get the speed of light.
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Similarly, you can multiply the
frequency and the wavelength.
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OK, so as a first
activity, I will
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ask you to see that how
does this solution, how
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does this specific wave,
transform on the Lorentz
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transformation?
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As a reminder, we have seen
that the equation which
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governs how this
light propagates
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is [INAUDIBLE] the
Lorentz transformation.
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But now we want to
investigate what
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happens to the wave itself.
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OK?
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So we have to investigate this
specific solution and Lorentz
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transform x and t.
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And I just do this
here in this equation.
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So you see that now we
have, as part of the cosine,
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Kx gamma x prime plus beta
ct prime plus no change in y
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direction as we look at
the Lorentz transformation
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in x direction, and then we have
the transformation of the time
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axis.
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OK?
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So now this looks very
cumbersome or complicated,
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but we can try to refind the
very same characterization
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of the wave as we had before.
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How does now the transformed
wave number look like or does
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the frequency look like
after Lorentz transformation?
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And so we want to identify
the individual terms Kx prime,
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where we label Kx
prime as the parameter
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you find here in the solution,
in this Lorentz transform
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solution, and we do the
same for omega prime,
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and you find a solution here.
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So now there's
this angle cosine I
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defined as the angle with
respect to the line of motion.
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Omega prime is now the baseline
frequency and omega the one
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which is detected.
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That's just a matter of
changing the direction of beta
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with the plus and minus sign.
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But if we use that definition,
we can now discuss the result.
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So as part of the discussion,
we can look at the specific case
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where the wave is
moving towards us.
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OK?
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So zeta is equal to 0
and beta is positive.
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In this case, omega is
larger than omega prime.
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And so the frequency
is going to be higher.
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So the detected frequency is
going to be higher blueshifted.
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So if you have a situation that
a star is moving up towards us
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and emitting light, the light
is detected by us, maybe
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by our eyes or by a
telescope, that light
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is going to be blueshifted.
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It's going to go to
higher frequencies.
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The opposite scenario is where
zeta is equal to 180 degrees
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and beta equal to--
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greater than 0, or
the other way around.
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We could have defined
this also as zeta
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equals 0 and beta negative.
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In this case, omega is
smaller than omega prime.
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So the frequency
is lower, meaning
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that the light we
observe is redshifted.
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And therefore,
this term redshift
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is a measure of whether
or not the source of light
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is moving towards
us or away from us.
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And the larger the redshift,
the higher the velocity
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is of this object
moving away from us.
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So we can define this redshift
as the relative change
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in frequency omega prime
minus omega over omega,
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or we can define 1 plus
Z, 1 plus the redshift
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is omega prime over omega,
which is square root of 1
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plus beta over 1 minus beta.
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All right?
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So if you now observe
the stars in our galaxy,
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and you can do
this, for example,
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by its specific spectral form.
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There are the specific
spectral lines,
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lines of specific
frequency, which
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we can observe
from stars as they
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are in certain distance
from our solar system.
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And if we do this, we basically
see all stars being redshifted,
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meaning all stars are actually
moving away from us, which
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is a measure of the fact that
the universe is expanding.