Doppler Effect and absolute simultaneity

[bernhard.rothenstein]

The Doppler shift formula relates two proper time intervals measured in I and I' respectovely
(tau)=D(tau)'
D representing a Doppler factor that depends on the relative speed of I and I'. By definition the events involved in I and I' respectively take place at the same point in space. If the two involved events are simultaneous say in I they are simultaneous in I' as well and vice-versa.
The time intervals measured by clocks synchronized using the "everyday clock" synchronization procedure are related by a Doppler shift like formula. Could that be an explanation of the absolute simultaneity: two events simultaneous say in I and taking place at the same point in space are simultaneous in all inertial reference frames.
We find the same situation in the case of Selleri's approach.

 

[DrGreg]

The Doppler shift formula relates two proper time intervals measured in I and I' respectovely
(tau)=D(tau)'
D representing a Doppler factor that depends on the relative speed of I and I'. By definition the events involved in I and I' respectively take place at the same point in space. If the two involved events are simultaneous say in I they are simultaneous in I' as well and vice-versa. The time intervals measured by clocks synchronized using the "everyday clock" synchronization procedure are related by a Doppler shift like formula.

If you look at the absolute simultaneity equation for "left-to-right" Leubner coords

$$t_S(I) = \gamma \left(1 + \frac{v}{c} \right) t_S(R)$$​

which I obtained in this post, you will see that

$$\gamma \left(1 + \frac{v}{c} \right) = \sqrt{\frac{1 + \frac{v}{c}}{1 - \frac{v}{c}}} = D$$​

which is precisely the doppler factor between the frames I and R. This is no coincidence.

Imagine light being sent event e at the origin of the R frame at time

$$t_S(R)(e) = \tau(R)(e)$$...(1)​

(remember Einstein-time, Leubner-time and proper time are all the same at the spatial origin).

It is received event r at the origin of the I frame at time

$$t_S(I)(r) = \tau(I)(r)$$...(2)​

But because of the way S-time is defined we must have

$$t_S(I)(r) = t_S(I)(e)$$ ...(3) and
$$t_S(R)(r) = t_S(R)(e)$$ ...(4)​

Combining (1) with (4), and (2) with (3), and using the doppler equation

$$\tau(I)(r) = D \tau(R)(e)$$​

we get

$$t_S(I)(e) = D t_S(R)(e)$$ and
$$t_S(I)(r) = D t_S(R)(r)$$​

the same absolute simultaneity equation (evaluated at both events e and r) as before.

Could that be an explanation of the absolute simultaneity: two events simultaneous say in I and taking place at the same point in space are simultaneous in all inertial reference frames. We find the same situation in the case of Selleri's approach.

I don't really understand this. The general absolute simultaneity equation

$$t_S(I) = a(I) t_S(R)$$...(A)​

is measuring the same event in two different coord systems I and R, whereas the doppler shift equation

$$t_S(I)(r) = \tau(I)(r) = D \tau(R)(e) = t_S(R)(e)$$​

is measuring two different events e and r in (effectively) two different coords systems I and R. It's only in Leubner coords we can link the two.

(NOTE: the absolute simultaneity equation (A) could also be written

$$\frac{\partial t_S(I)}{\partial x_s(R)} = 0$$​