# Doppler shift of radio signals to an accelerating observer

Suppose a man leaves from Earth to a star which is 1000 light years away. He accelerates to a velocity such that the entire trip lasts a year, from the reference frame of the rocket.

Now lets pretend the person in the rocket wants to have a transmission of the radio to him.

Due to time dilation, when a day passes on earth only a few seconds pass on the rocket ship, so from the travellers frame of reference, as he accelerates the frequency of transmissions goes up.

As he arrives at his destination the frequency of transmissions should go down.

Is this correct?

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I know this is just a wording issue, but be careful of using the term "accelerate". An accelerated reference frame is outside the realm of special relativity. –  Ataraxia May 1 '13 at 5:04
@ZettaSuro There's nothing wrong with an accelerated frame in SR - that is a common misperception. It is gravitation - i.e., tidal forces, i.e. curvature, which is incompatible with SR, not simple accelerated motion in an ordinary flat spacetime. Otherwise accelerator physicists would have to use GR... and they don't. :) –  Michael Brown May 1 '13 at 8:44
But after the trip is complete shouldn't all of the signals have reached the ship? If the rate of signals arriving is going to slow down as the ship speeds up then the rate of signals should go up as the ship slows down and reaches its destination –  frogeyedpeas May 1 '13 at 12:13
@frogeyedpeas There are two effects here. One is the simple delay (retardation) for the signal to propagate (which behaves as you suggest) and the other is an actual change in the relationship between passing time and the distances between points (which does not). It is very easy to conflate the two effects so you need to be careful. It is also the combination of these effects that can giver a rapidly approaching object an apparent velocity higher than $c$, something that has been observed in pulsar jets. –  dmckee May 1 '13 at 14:59

You are not quite correct (see edits). Except I wouldn't say that "in reality it takes just slightly over 1000 years" - the rocket frame is no less real than the Earth's frame. As far as the actual numbers go: at one gee acceleration it takes about a year in the rocket frame to accelerate, and a year again to deccelerate, so two rocket frame years altogether. You can achieve a one year trip at a higher acceleration but your passenger might feel squished. :)

Edit: Oops, read it wrong. From the rocket frame he receives fewer messages per second, not more, since he's moving away from the Earth. I'll put up a spacetime diagram to illustrate this later. Lagerbaer gets it right in his answer.

Edit2: Okay. Here it is:

I made it a shorter 2 ly trip than your 1000 ly one just so we can see what's happening on the plot. Nothing essential changes because of this, things are just easier to see. The blue curve is the rocket accelerating away from the Earth at $x=0$ until it reaches the halfway point and begins deccelerating. The total proper time for the rocket is 1 year, but about 2.4 yr elapses on the Earth.

The red dashed lines are regular messages sent from Earth at 0.05 yr intervals. Notice that in the middle of the journey the ship receives very few messages. Here are the arrival times:

$$\begin{array}{cc} \text{Earth time signal sent} & \text{Rocket time signal received} \\ 0. & 0 \\ 0.05 & 0.0574144 \\ 0.1 & 0.137814 \\ 0.15 & 0.273035 \\ 0.2 & 0.623843 \\ 0.25 & 0.815773 \\ 0.3 & 0.912568 \\ 0.35 & 0.977809 \\ \end{array}$$

Note that between the fourth and fifth messages 0.35 yr elapses in the rocket frame, compared to the 0.05 yr between them in the Earth frame!

Spacetime diagrams like this are the only way to get intuition about relativity. Learn to love them. :)

Edit3: With a minor tweak of the code I can run the numbers for your design journey - 1000 lyr in a proper time of one year. The plot is unreadable, but the stats are:

Acceleration: $20\ \mathrm{g}$

Earth time: $1000.1\ \mathrm{yr}$

Max speed: $0.999999995\ c$

$$\begin{array}{cc} \text{Earth time signal sent} & \text{Rocket time signal received} \\ 0. & 0 \\ 0.01 & 0.0111428 \\ 0.02 & 0.0254581 \\ 0.03 & 0.0455048 \\ 0.04 & 0.079222 \\ 0.05 & 0.229245 \\ 0.06 & 0.915324 \\ 0.07 & 0.95177 \\ 0.08 & 0.972725 \\ 0.09 & 0.987495 \\ 0.1 & 0.99891 \\ \end{array}$$

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Nicely drawn diagram - I like how the rocket is just under the speed of light. –  twistor59 May 1 '13 at 11:35
Thanks. By the way, the acceleration is just a sliver under 5 g. For a year. Passenger would not be happy. –  Michael Brown May 1 '13 at 13:11
I wonder if a person would 'get used to' 5g, ha. –  user12345 May 1 '13 at 16:43

This is not correct. From the rocket ship's point of view, time on earth is passing more slowly, so the frequency of radio transmissions received is going down. You say "Due to dilation when a day passes on earth only a few seconds pass on the rocket ship" but that is the view that earth has. The rocket ship has the opposite view: When a day passes on the rocket ship, only a few seconds pass on earth, so you'd have to wait even longer for the news.

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