Aberration and Doppler shift in Special Relativity

[RafaPhysics]

Homework Statement

I'm studying special relativity and I tried to solve this problem

Relevant Equations

$$\gamma_1=\frac {1} {\sqrt{1-r_1^2 \Omega^2 / c^2}}, \qquad \gamma_2=\frac {1} {\sqrt{1-r_2^2 \Omega^2 / c^2}}$$

A large disk rotates at uniform angular speed $\Omega$ in an inertial frame $S$. Two observers, $O_1$ and $O_2$, ride on the disk at radial distances $r_1$ and $r_2$, respectively, from the center (not necessarily on the same radial line). They carry clocks, $C_1$ and $C_2$, which they adjust so that the clocks keep time with clocks in $S$, i.e., the clocks speed up their natural rates by the Lorentz factors I wrote above respectively. By the stationary nature of the situation, $C_2$ cannot appear to gain or lose relative to $C_1$. Deduce that, when $O_2$ sends a light signal to $O_1$, this signal is affected by a Doppler shift $w_2/w_1 = \gamma_2/\gamma_1$.
Note that, in particular, there is no relative Doppler shift between any two observers equidistant from the center.

 

[anuttarasammyak]

In my sense Doppler effect is said with motion in IFR. O1 and O2 are at rest in the rotationg frame of reference. On which frame the IFR or the rotating FR are you considering ?
Don' t you have to consider their angles $\phi_1$ and $\phi_2$ on the disk for your question ? Say $\phi_1-\phi_2 =0$ they are moving same direction. Say $\phi_1-\phi_2 =\pi$ they are moving opposite direction.

 

[RafaPhysics]

I haven't considered any reference because I've not been able to solve it, may need a little help

 

[TSny]

Suppose baseballs are thrown at a regular rate from the adjusted clock $C_1$ to the adjusted clock $C_2$ (always thrown in the same manner). Let $\omega_0$ be the rate at which baseballs leave $C_1$ according to $C_1$-time. Formulate an argument for deducing that the baseballs must arrive at $C_2$ at the same rate $\omega_0$ according to $C_2$-time. This should not require any significant calculation, just reasoning.

Next, imagine that there is an unadjusted clock $\widetilde C_1$ sitting at the same location as the adjusted clock $C_1$. $\widetilde C_1$ measures the proper time of observer $O_1$ at $C_1$. Likewise, there is an unadjusted clock $\widetilde C_2$ located at $C_2$ that measures proper time for observer $O_2$. Let $\omega_1$ be the frequency at which the balls are thrown from $C_1$ according to $\widetilde C_1$-time and $\omega_2$ be the frequency at which the balls arrive at $C_2$ according to $\widetilde C_2$-time. How are $\omega_1$ and $\omega_2$ related to $\omega_0$? What is the ratio $\omega_2/\omega_1$?

 

[RafaPhysics]

View attachment 329398

Suppose baseballs are thrown at a regular rate from the adjusted clock $C_1$ to the adjusted clock $C_2$ (always thrown in the same manner). Let $\omega_0$ be the rate at which baseballs leave $C_1$ according to $C_1$-time. Formulate an argument for deducing that the baseballs must arrive at $C_2$ at the same rate $\omega_0$ according to $C_2$-time. This should not require any significant calculation, just reasoning.

Next, imagine that there is an unadjusted clock $\widetilde C_1$ sitting at the same location as the adjusted clock $C_1$. $\widetilde C_1$ measures the proper time of observer $O_1$ at $C_1$. Likewise, there is an unadjusted clock $\widetilde C_2$ located at $C_2$ that measures proper time for observer $O_2$. Let $\omega_1$ be the frequency at which the balls are thrown from $C_1$ according to $\widetilde C_1$-time and $\omega_2$ be the frequency at which the balls arrive at $C_2$ according to $\widetilde C_2$-time. How are $\omega_1$ and $\omega_2$ related to $\omega_0$? What is the ratio $\omega_2/\omega_1$?

Don't certainly know, but I think I have to express the metric interval $ds^2=c^2dt^2-dx^2$ but in polar coordinates, because like you showed me, It's a disk. What do you think?

 

[TSny]

Don't certainly know, but I think I have to express the metric interval $ds^2=c^2dt^2-dx^2$ but in polar coordinates, because like you showed me, It's a disk. What do you think?

You don't need to work with the metric. The reason for adjusting clocks $C_1$ and $C_2$ such that they always agree with time in the non-rotating inertial frame $S$ is that you can get the answer for $\omega_2/\omega_1$ with very little calculation.

I introduced baseballs to show that the result applies to any periodic signals sent between $C_1$ and $C_2$ where the signals travel at a fixed speed relative to the inertial frame $S$. You can replace the baseballs with wave crests of electromagnetic waves.

Does each baseball that travels from $C_1$ to $C_2$ travel the same distance relative to the inertial frame $S$?

Does each baseball take the same time to travel between $C_1$ and $C_2$ according to $S$? (We're assuming that all of the baseballs travel at the same speed relative to $S$.)

If 10 baseballs per minute leave $C_1$ according to frame $S$, how many baseballs per minute arrive at $C_2$ according to $S$?

If 10 baseballs per minute leave $C_1$ according to frame $S$, how many baseballs per minute leave $C_1$ according to clock $C_1$?

How many baseballs per minute arrive at $C_2$ according to clock $C_2$?

How many baseballs per minute leave $C_1$ according to the unadjusted clock $\widetilde C_1$ if $\gamma_1$ happens to equal 3?

How many baseballs per minute arrive at $C_2$ according to the unadjusted clock $\widetilde C_2$ if $\gamma_2$ happens to equal 2?

($\gamma_1$ and $\gamma_2$ are the gamma factors that you introduced in the first post. Gamma factors as large as 2 or 3 for baseballs and rotating disks are ridiculous, but it's just a thought experiment.)

 

[RafaPhysics]

In my sense Doppler effect is said with motion in IFR. O1 and O2 are at rest in the rotationg frame of reference. On which frame the IFR or the rotating FR are you considering ?
Don' t you have to consider their angles $\phi_1$ and $\phi_2$ on the disk for your question ? Say $\phi_1-\phi_2 =0$ they are moving same direction. Say $\phi_1-\phi_2 =\pi$ they are moving opposite direction.

Excuse me, what do you mean with IFR and FR?

 

[PeroK]

Excuse me, what do you mean with IFR and FR?

Probably (Inertial) Frame of Reference.