I Relativity of Simultaneity: What Einstein's Contribution Was

[e.chaniotakis]

TL;DR Summary

A silly question about the relativity of simultaneity

Something that I consider very silly, yet I try to solve relates to the relativity of simultaneity. According to SR , two evevnts taking place in different positions along the line of relative motion of two inertial observers are not simultaneous in both frames.

Now, I wanted to see how this relates to the speed of light and its independence of the motion of the source of the light ray.
If we suppose that we have three rockets, A , B and C all moving towards the positive x-axis with velocities equal to u each with respect to an inertial observer O. Let us suppose that the rockets start moving at t=0 with respect to O from positions x=0 for A, x=x for B and x=2x for C.

At a time equal to t with respect to O, B emits two light rays back to back towards A and C. According to B, the rays will arrive at A and C simultaneously.
However, according to O , the ray towards A will arrive first and the ray towards C will arrive last. Ergo, relativity of simultaneity.
So far so good.

Now let us suppose that for a moment we go back to Galileo's time and we perform the same experiment while unaware that c=invariant with respect to observers and we believe that as with any other projectile, the projectile emitted will obey the Galilean transformation of velocities.
From B's point of view the rays will be emitted with velocities -c and c and aill arrive simultaneously to A and C.

From O's point of view the rays will be emitted with velocities u-c and u+c towards A and C respectively.
The ratio of arrival times to A and to C will be a function of the speed of the projectile and of the velocities of the rockets. This ratio will be equal to 1 when u=c ( if I did the math correctly).
This implies that a (wrong) Galilean treatment of the problem predicts that the only way that two events can be considered simultaneous in the frames of O and of B is when the relative motion equals the speed of light.
Now as Galilean relativity doesn't have an upper value for the speed of a projectile this could be interpreted by saying that in Galilean relativity there can exist two inertial frames which observe two events taking place at different positions along their line of motion simultaneously.

Is that correct?

Therefore, Einstein's contribution here is that by introducing the fact that the speed of light is not affected by the speed of the source, he secures that there cannot exist two inertial reference frames observing two spatially separated events as simultaneous.

What is your view on this treatment?
Thank you for your time.

 

[Pencilvester]

Let us suppose that the rockets start moving at t=0 with respect to O from positions x=0 for A, x=x for B and x=2x for C.

It's bad notation to label a specific point using the same letter as the coordinate itself unless you have, say, a subscript to distinguish the coordinate of the specific point and the general coordinate. It's possible that this is causing you to confuse yourself and make errors. Let's just simplify this and say at t=0, A is at x=0, B is at x=1, and C is at x=2.

the projectile emitted will obey the Galilean transformation of velocities.

If we're assuming a universe where the Galilean transformation governs the behavior of light, then you could just as easily set up a scenario where B rolls bowling balls instead of shooting light beams, in which case, the answer is obvious and intuitive: if the bowling balls reach A and C simultaneously in the moving frame, then those two events will also be simultaneous in frame O. If you want to work it out mathematically, see below.

The ratio of arrival times to A and to C will be a function of the speed of the projectile and of the velocities of the rockets. This ratio will be equal to 1 when u=c ( if I did the math correctly).

Assuming simultaneity of A and C receiving the light beams in the moving frame, then the ratio of arrival times in frame O will be equal to 1 regardless of u and c. All you have to do is find the intersection of the two lines $x=ut$ and $x=(u-c)t+1$ and then the intersection of lines $x=ut+2$ and $x=(u+c)t+1$.

 

[x-vision]

Now let us suppose that for a moment we go back to Galileo's time and we perform the same experiment while unaware that c=invariant with respect to observers and we believe that as with any other projectile, the projectile emitted will obey the Galilean transformation of velocities.
From B's point of view the rays will be emitted with velocities -c and c and aill arrive simultaneously to A and C.

This is incorrect.

If rocket B emits two light rays towards A and C, the light ray towards A will arrive first and the one towards C will arrive second.
The time intervals will be x/c+u for B-to-A and x/c-u for B-to-C.

According to Galilean relativity, time is absolute, not relative.
This means that the clocks for O and B are always synchronized, even when B is moving.
So, both O and B will see the arrival of B-to-A at the same time (and then B-to-C also at the same time - but later).

 

[Pencilvester]

If rocket B emits two light rays towards A and C, the light ray towards A will arrive first and the one towards C will arrive second.

Not if you’re assuming the light gets an additive velocity boost from the motion of B. Of course this isn’t the way the universe really works, but it was the OP’s hypothetical scenario.

 

[x-vision]

Not if you’re assuming the light gets an additive velocity boost from the motion of B.

True. I automatically assumed an "undular" light propagation rather than corpuscular ;).