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Exercise 6.4 of Misner Thorne and Wheeler is:

A radar set measures velocity by emitting a signal at a standard frequency and comparing it with the frequency of the signal reflected back by another object. The redshift measurement is then converted, using the standard special-relativistic formula, into the corresponding velocity, and the radar reads out this velocity. How useful is this radar set as a velocity-measuring instrument for a uniformly accelerated observer?

(a)Consider this problem first for a special case when the object and radar set are at rest with respect to each other at the instant the radar pulse is reflected. Compute the redshift $1+z=\omega_{e}/\omega_{o}$ that the radar set measures in this case, and the resulting (incorrect) velocity it infers. Simplify by making use of the symmetries of the situation.

(b)Now consider the situation where the object has a non-zero velocity in the momentary rest frame of the observer at the instant it reflects the radar pulse. Compute the ratio of the actual relative velocity to the velocity read out by the radar.

I'm really asking about part (a). As I read the exercise, part (a) is asking me to bounce the radiation off the interior walls and ceiling of Einstein's elevator. Suppose the signal is emitted from the floor and bounced off the ceiling. That is, assume the reflecting object lies in the direction of acceleration relative to the observer who emits the signal. I believe the radar will report that the object is at reset. If that is the result the authors expect me to obtain, then I can work out the case in which the signal is not propagating parallel to the acceleration.

There are various arguments to support the conclusion that the radar will indicate zero velocity when the signal is bounced off the ceiling. But the simplest is the energy argument which says that light climbing out of a gravitational potential well loses energy, and light falling into a gravitational potential well, gains energy. If the uniform acceleration is simulating near Earth gravity, then symmetry and the conservation of energy argue that the return should have the same energy, and therefore the same frequency as the emitted radiation.

Is this correct? Is it what they are asking for?

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  • $\begingroup$ I've added the homework-and-exercises tag. In the future, please add this tag to this type of problem. This is one of the things that we ask you to do in our homework policy: physics.meta.stackexchange.com/questions/714/… $\endgroup$
    – user4552
    Dec 24, 2019 at 1:57
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    $\begingroup$ I sometimes forget that there are people who actually learn physics in schools. I signed up for a couple physics classes, but never finished one. $\endgroup$ Dec 24, 2019 at 7:54

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I believe I now understand the intent of the authors. I believe they intended to source and reflector to be at rest at the time of emission, and to begin accelerating at the time of reflection.

I was reading the question to mean that, for example, the pules is emitted at the first point where the vectors are depicted. Both the radar set and reflector and are uniformly accelerated at the same rate relative to their respective proper times so that the pulse is reflected as both cross the space axis of the diagram. The pulse will then travel an equal interval while the radar set continues to accelerate. By the symmetry of the problem, that will result in no net Doppler shift. Which, to my mind, is the correct measure of velocity.

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I'm trying the same exercise and it is not fully clear to me what MTW had in mind in this exercise, but my thought was: You have your accelerating guy sending off a signal and the signal reflects in a wall placed, say, at $x_0$. The signal bounces and comes back to you, so we can evaluate the ratio of the shift with the usual formula (the coordinate free ratio which MTW used in the centrifuge and the photon exercise).

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