Reply #49: The question is somewhat ill-posed

[caraher]

The good news is that you can use Lorentz transformations to answer any reasonable question about "how long it takes" for the signal to arrive.

The bad news is that you need to pose your question with more precision. Anytime you ask a question about time, you need to frame it in terms of what an actual clock would measure. And you need to tell me what starts and stops this hypothetical stopwatch.

One tip on working with relativity: Always frame the problem in terms of "events," where by an event you must specify a location in space and moment in time (here, you do not need to specify who is measuring time). In this case, there are two events: A. Signal transmitted B. Signal received by spaceship. Each occurs at a well-defined location (A: the location of the transmitter; B: location of the ship) and time (A: when the message is sent; B: when the message is received)

The answer will not be 20 minutes. I'll come back later with the correct answer; the most precise way to get it involves using Lorentz transformations (simple equations to use, but you need to be very careful to understand what everything means). But one can use graphical methods to develop intuition... a very nice introduction we use in our first-semester course is in Tom Moore's "Six Ideas That Shaped Physics": Unit R. A used copy or other edition should be fine... anyway, it's chock-full of examples like this (this would be a very easy problem at the end of a chapter) and is extremely readable as physics texts go.

And yes, there will be a relativistic Doppler shift. The formula is different for light from other kinds of waves. In fact, this relativistic Doppler shift is the basis of many everyday applications of radar (you can thank it for many speeding tickets!).

<edited to clarify what I found ill-posed>