# Doppler Effect and absolute simultaneity

1. Mar 11, 2008

### 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.

2. Mar 11, 2008

### DrGreg

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.

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

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

Last edited: Mar 11, 2008