How does the relativistic Doppler effect model change if I want to include general relativistic effects also?
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Derivations for the relativistic Doppler effect abound, which extends the description of the Doppler effect to include the effects of relativity in cases where the source or observer is moving at relativistic velocities. This is all pretty well-described in many sources that I have found.
But what if I would like to include the effects of general relativity instead? Assume the following scenario:
There is a stationary source on the earth emitting light at frequency ff (with that frequency measured on the reference frame of the source).
I launch a rocket that travels to orbit with some velocity profile v(t)v(t).
The rocket has a detector that continually observes the light source during its flight.
I would like to know the observed frequency of the light wave from the perspective of the rocket, as a function of time, f′(t′)f′(t′) (where t′t′ represents time in the rocket's reference frame).
I see three separate effects that would alter the observed frequency aboard the rocket:
The classical Doppler effect
Time dilation on the rocket due to special relativity
Time dilation on the rocket due to general relativity
All of these will change as a function of time as the rocket accelerates and as it moves within Earth's gravitational field. However, all analyses of the relativistic Doppler effect that I've seen only encompass the first two effects. Is there some model that includes all three?
hope this helps
But what if I would like to include the effects of general relativity instead? Assume the following scenario:
There is a stationary source on the earth emitting light at frequency ff (with that frequency measured on the reference frame of the source).
I launch a rocket that travels to orbit with some velocity profile v(t)v(t).
The rocket has a detector that continually observes the light source during its flight.
I would like to know the observed frequency of the light wave from the perspective of the rocket, as a function of time, f′(t′)f′(t′) (where t′t′ represents time in the rocket's reference frame).
I see three separate effects that would alter the observed frequency aboard the rocket:
The classical Doppler effect
Time dilation on the rocket due to special relativity
Time dilation on the rocket due to general relativity
All of these will change as a function of time as the rocket accelerates and as it moves within Earth's gravitational field. However, all analyses of the relativistic Doppler effect that I've seen only encompass the first two effects. Is there some model that includes all three?
hope this helps
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The normal Doppler effect in general refers to how a wave's detected frequency changes when the source moves relative to the observer. I think you want to compare the Doppler effect of light in vacuum with that of sound.
In both cases, the effect is small until the relative velocities get close to the speed of the wave in the medium. But since the two waves behave differently, so you have different behavior. One of them is sound, a mechanical disturbance traveling at a paltry speed of sound. You can measure measure the supposed wave travel speed to be different if you move in the air. The other is light, moving at, well, the speed of light, and is measured to move at that speed no matter what. The closer you move to the speed of light, the more relativity becomes important, while moving close to the speed of sound has no such problems.
Now, on to the actual differences.
The Doppler effect for sound actually acts differently depending on whether the source or the observer moves (relative to the air). They can both move, and both effects will occur simultaneously.
When the source moves, the sound physically looks different in the air. A stationary observer directly in the path of the moving emitter will see the frequency of the sound change by a factor of cc−vcc−v, where c is the speed of sound, and v is the speed of the source, positive when moving toward the observer. This means that when v gets very close to c, the factor grows enormous, while when v gets close to -c (so the source is moving away), the factor is 1212, not too bad. You can also consider speeds higher than sound, which is completely feasible. When a supersonic source approaches, the change in frequency is negative, so the sound is played backward. If moving away, the frequency gets closer to 0. You can do something similar for a source that isn't purely approaching the observer, by calculating the frequency change of sound emitted at a particular instant.
When the observer moves, the detector receives the sound differently. The new detected frequency changes by c+vcc+vc. Again c is the speed of sound, but now v is the speed of the observer against the wave, positive when moving toward the source. This means that when v = c, the frequency is only two times higher. But when v = -c (so it's moving away), the frequency becomes 0. If the observer approaches faster than sound, the factor is something bigger than 2. If receding, the factor is negative (note: this only works if the sound was already heard before the observer began receding faster than sound. Otherwise, the sound can't reach the observer in the first place). Transverse motion itself doesn't matter, though it might change the speed measured against the wave over time, by messing with the angles.
For light, any observer has the right to be called stationary. Light doesn't seem to change speed when moving. Instead, when the observer moves, it only makes the source seem to move differently. So the only situation that matters is a moving source. Taking effects like time dilation into account, the frequency change factor looks like c+vc−v−−−√c+vc−v, where c is the speed of light and v is the speed of the source, positive when approaching. The result is somewhat of a combination of the two cases for sound Doppler shift. above. When v = c, the frequency is infinitely increased, while when v = -c, the frequency is 0. The equation fails when the source isn't directly approaching the observer, for even if the source is moving perpendicular to it, time dilation still decreases the frequency of the sound emitted.
In both cases, the effect is small until the relative velocities get close to the speed of the wave in the medium. But since the two waves behave differently, so you have different behavior. One of them is sound, a mechanical disturbance traveling at a paltry speed of sound. You can measure measure the supposed wave travel speed to be different if you move in the air. The other is light, moving at, well, the speed of light, and is measured to move at that speed no matter what. The closer you move to the speed of light, the more relativity becomes important, while moving close to the speed of sound has no such problems.
Now, on to the actual differences.
The Doppler effect for sound actually acts differently depending on whether the source or the observer moves (relative to the air). They can both move, and both effects will occur simultaneously.
When the source moves, the sound physically looks different in the air. A stationary observer directly in the path of the moving emitter will see the frequency of the sound change by a factor of cc−vcc−v, where c is the speed of sound, and v is the speed of the source, positive when moving toward the observer. This means that when v gets very close to c, the factor grows enormous, while when v gets close to -c (so the source is moving away), the factor is 1212, not too bad. You can also consider speeds higher than sound, which is completely feasible. When a supersonic source approaches, the change in frequency is negative, so the sound is played backward. If moving away, the frequency gets closer to 0. You can do something similar for a source that isn't purely approaching the observer, by calculating the frequency change of sound emitted at a particular instant.
When the observer moves, the detector receives the sound differently. The new detected frequency changes by c+vcc+vc. Again c is the speed of sound, but now v is the speed of the observer against the wave, positive when moving toward the source. This means that when v = c, the frequency is only two times higher. But when v = -c (so it's moving away), the frequency becomes 0. If the observer approaches faster than sound, the factor is something bigger than 2. If receding, the factor is negative (note: this only works if the sound was already heard before the observer began receding faster than sound. Otherwise, the sound can't reach the observer in the first place). Transverse motion itself doesn't matter, though it might change the speed measured against the wave over time, by messing with the angles.
For light, any observer has the right to be called stationary. Light doesn't seem to change speed when moving. Instead, when the observer moves, it only makes the source seem to move differently. So the only situation that matters is a moving source. Taking effects like time dilation into account, the frequency change factor looks like c+vc−v−−−√c+vc−v, where c is the speed of light and v is the speed of the source, positive when approaching. The result is somewhat of a combination of the two cases for sound Doppler shift. above. When v = c, the frequency is infinitely increased, while when v = -c, the frequency is 0. The equation fails when the source isn't directly approaching the observer, for even if the source is moving perpendicular to it, time dilation still decreases the frequency of the sound emitted.
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