Why isn't the light signal from a falling spaceship Doppler shifted?

[sweetdreams12]

Just some conceptual ideas that I don't understand

A spaceship starts falling under gravity with an acceleration g as measured by an observer Barry at rest on Earth. At the instant that the ship starts to fall, an astronaut Harry at the base of the rocket ship sends a light signal of frequency w vertically upward to another astronaut Sally a distance h above.

Barry argues that the light signal reaching Sally ought to be Doppler shifted toward the blue. This Doppler shift Δw is given by (Δw/w)_Doppler = (Δu/c)

where Δu is the velocity of the rocket ship after a time Δt = (h/c)

But Sally does not see this shifted frequency. So why is this? What compensates for the Doppler shift in frequency? And if another observer (let's call her Kelly) was to see the red-shifted light, what is required for her?

 

[Simon Bridge]

Hint: the regular doppler shift equations assume inertial reference frames.

 

[sweetdreams12]

Hint: the regular doppler shift equations assume inertial reference frames.

meaning they are non-accelerating...so Sally isn't accelerating hence she can't see the shifted frequency?

 

[Simon Bridge]

No - doppler shift is noticeable in non-accelerated frames.
That is how it is usually worked out.

I mean Harry and Sally are both following a special path in space-time.
What level are you doing this at?

 

[harrylin]

Just some conceptual ideas that I don't understand

A spaceship starts falling under gravity with an acceleration g as measured by an observer Barry at rest on Earth. At the instant that the ship starts to fall, an astronaut Harry at the base of the rocket ship sends a light signal of frequency w vertically upward to another astronaut Sally a distance h above.

Barry argues that the light signal reaching Sally ought to be Doppler shifted toward the blue. This Doppler shift Δw is given by (Δw/w)_Doppler = (Δu/c)

where Δu is the velocity of the rocket ship after a time Δt = (h/c)

But Sally does not see this shifted frequency. So why is this? What compensates for the Doppler shift in frequency? And if another observer (let's call her Kelly) was to see the red-shifted light, what is required for her?

If I correctly understand your set-up:

x Sally (for simplification: also falling down from t=0)
¦ continuous light signal sent up by Harry
x Harry (falling down from t=0)

x Barry (on Earth)
---------

1. Barry could argue that the light for Sally ought to be blue-shifted, based on the Doppler effect for accelerated objects: Sally is falling faster than Harry's light emitter at the moment of emission, thus a blueshift.

2. Sally is also higher up than Harry. The higher gravitational potential results in a blueshift of her eye's colour reference. According to the equivalence principle, this blueshift due to gravitation is equal to the Doppler effect due to acceleration, and thus Sally does not see a shifted frequency.

Next you ask about "the red-shifted light"; however, it's not clear in what situation. Maybe you want to know in what situation Kelly, maybe also straight above Harry, should be to see the light redshifted.

If Kelly is higher up and not falling, then she'll see the sum of increasing gravitational redshift and classical Doppler redshift, because Harry is falling and accelerating away from her.
PS: I assumed "non-relativistic" speed.

 

[A.T.]

meaning they are non-accelerating...so Sally isn't accelerating hence she can't see the shifted frequency?

The rocket frame, where the astronauts are both at rest, is inertial. So based on their rest frame the astronauts conclude there should be no Doppel-Shift.

The surface observes frame is not inertial. He sees a local gravitational field and gravitational time dilation, which cancels the Doppel-effect, from the velocity difference emitter (during emission) and receiver (during receive). So he also concludes there should be no detectable Doppel-Shift.

 

[Simon Bridge]

Barry reckons that Sally should see a doppler shift because they are accelerating - so Sally intercepts the wavefronts from Barry's signal faster, for a higher frequency. However, Harry and Sally are stationary wrt each other, so they are not at all surprised that Sally reports the same wavelength that Harry sent.

What gives? Either there was a doppler shift or their wasn't!

Barry thinks a bit and slaps his forehead: "of course!" he cries, "the doppler shift was compensated by..." something something something ... which is what the question is asking about. What is the something something that Barry realized?
Barry expects Sally to report a blue-shift ... she didn't... therefore there must have been a compensating red shift. What would cause that?
What is present in Barry's frame that is not present in the ship frame? Starts with "g".

For Sally to see a blue shift - she needs to have a different relationship to Harry. If she broke up with him, he'd get blue quite fast... but I think the easiest answer is to figure what direction shift Barry sees. There's several ways that Sally could see a bluer signal: how many can you think of?

There are a few wrinkles since the description does not mention the relative angular motion.
One is tempted to imagine the ship is falling directly towards Barry - but we remember that Barry's reference frame is rotating wrt the center of mass frame of the Earth.

This isn't the homework section, so we can go a bit easier on you here... but it is still more useful for you if you make certain connections yourself.