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nutgeb
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Alice and Bob are each in their own spaceship, and are traveling radially, initially away from each other, and subsequently toward each other, at some highly relativistic relative speed, say 99c. Each ship emits light flashes toward the other. Alice observes the light from Bob's outbound flight to be redshifted, and the light from his inbound flight to be blueshifted. Ditto for Bob.
I was surprised that, both coming and going, the SR time dilation factor [tex]\gamma[/tex] causes the received light spectrum to be more blueshifted (or less redshifted) than the spectrum shift caused by the classical Doppler effect alone.
This is effect is apparent in the SR redshift equation, which multiplies the classical Doppler shift:
[tex] \lambda_{o} = \lambda_{e} / (1 - v/c) [/tex]
by the [tex] \gamma [/tex] factor:
[tex] \sqrt{ (1-v/c) (1+v/c) } [/tex]
which always results in a decrease in wavelength from the classical Doppler shifted wavelength.
The result is shown graphically in the chart at the bottom of this http://www.fourmilab.ch/cship/doppler.html" at the Fourmilab website.
Intuition might suggest (incorrectly) that a slower (time dilated) clock at the emitter would shift the classical Doppler effect in the red direction, analagous to how the slowing of the emitter's clock by gravitational time dilation affects wavelength.
Presumably the physical explanation for why SR time dilatation instead shifts incoming light in the blue direction is that the distance to the emitter becomes Lorentz contracted due to its velocity, enabling wave crests to arrive at a more rapid rate than they are emitted. In other words, the Lorentz contraction of the intervening distance crowds the incoming waves radially into a smaller space, thereby decreasing their wavelength.
But then, why isn't the shift toward blue caused by the Lorentz contraction exactly offset by the shift toward red caused by the emitter's clock running slower? The emitter's slower clock should cause the frequency originally emitted to decrease (and therefore the wavelength to increase), as measured by the observer's faster clock.
Evidently the increased crowding of wave peaks due to Lorentz contraction must be a larger effect (potentially much larger) than the decreased frequency due to time dilation at the emitter.
I was surprised that, both coming and going, the SR time dilation factor [tex]\gamma[/tex] causes the received light spectrum to be more blueshifted (or less redshifted) than the spectrum shift caused by the classical Doppler effect alone.
This is effect is apparent in the SR redshift equation, which multiplies the classical Doppler shift:
[tex] \lambda_{o} = \lambda_{e} / (1 - v/c) [/tex]
by the [tex] \gamma [/tex] factor:
[tex] \sqrt{ (1-v/c) (1+v/c) } [/tex]
which always results in a decrease in wavelength from the classical Doppler shifted wavelength.
The result is shown graphically in the chart at the bottom of this http://www.fourmilab.ch/cship/doppler.html" at the Fourmilab website.
Intuition might suggest (incorrectly) that a slower (time dilated) clock at the emitter would shift the classical Doppler effect in the red direction, analagous to how the slowing of the emitter's clock by gravitational time dilation affects wavelength.
Presumably the physical explanation for why SR time dilatation instead shifts incoming light in the blue direction is that the distance to the emitter becomes Lorentz contracted due to its velocity, enabling wave crests to arrive at a more rapid rate than they are emitted. In other words, the Lorentz contraction of the intervening distance crowds the incoming waves radially into a smaller space, thereby decreasing their wavelength.
But then, why isn't the shift toward blue caused by the Lorentz contraction exactly offset by the shift toward red caused by the emitter's clock running slower? The emitter's slower clock should cause the frequency originally emitted to decrease (and therefore the wavelength to increase), as measured by the observer's faster clock.
Evidently the increased crowding of wave peaks due to Lorentz contraction must be a larger effect (potentially much larger) than the decreased frequency due to time dilation at the emitter.
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