# The example of relativity of simultaneity given by Einstein

I understand (supposedly) the mathematics concerning the relativity of simultaneity in Special Relativity, but I have a nagging question regarding the original example given by Einstein supporting it (I'm only disagreeing with this specific example, not the concept).

It is normally given as a person on an embankment and a person on a train. There is a relative speed between them (usually presented as the train passing the embankment). Now, when both people are at the same x-position (x=0), there is a flash of light at x = +dx and x = -dx. The argument as I keep seeing it is that the person on the embankment will say that both flashes reach him at the same time, whereas the person in the train will say that the flash in front of him reaches him before the other because he was moving toward it, and thus the observers will disagree on the simultaneity of the flashes.

But given that the flashes occurred at the same distance from each of them, the speed of light is constant in both frames, and either one can claim to be at rest, then won't they, according to SR, necessarily see the flashes as simultaneous (both flashes have to travel the same distance in both frames since at the time of emission, the sources of both flashes were equidistant from both observers). I agree that the person on the embankment will say that the person on the train shouldn't see them as simultaneous (and vice versa) since either observer will see the other moving relative to the sources, but in each of there own frames, they must see the flashes as being simultaneous shouldn't they? Am I just misunderstanding the example?

Thanks.

• "I agree that the person on the embankment will say that the person on the train shouldn't see them as simultaneous" if the person on the embankment did not understand Relativity. – Tuntable Aug 19 '19 at 5:06

I agree that the person on the embankment will say that the person on the train shouldn't see them as simultaneous

Well, then the person on the train shouldn't see them as simultaneous. Some things change between reference frames, but conclusions of the form "in frame $S$, an observer will see..." do not change, since the statement itself specifies which frame you have to be in to understand what it is saying.

The observer on the embankment could easily see the train observer intercept the forward flash before the rear flash. (Of course, the embankment observer couldn't do this in real time; one has to wait until after one's hypothetical grid of rulers and clocks reports back what happened when and where.) One nice thing about SR is that time-ordering is invariant. That is, two events $A$ and $B$ can have one of three relations to one another: $A$ is in $B$'s past light cone (and $B$ is in $A$'s future), the reverse of that statement, or $A$ and $B$ are spacelike separated. Whichever one of these holds will hold for all observers.

So we know, just from the embankment analysis, that "in the frame of the train, the forward flash reaches the observer's eyes first," and this statement is always true for anyone who speaks it in its entirety.

What about the train observer? Indeed, as you say,

the flashes occurred at the same distance from each of them, the speed of light is constant in both frames, and either one can claim to be at rest

Suppose two people, $C$ and $D$, stand equal distances from you and are known to pitch balls at exactly the same speed. With everyone standing at rest, $C$ and $D$ each toss you a ball. You get the ball from $C$ before the one from $D$. This is not a logical inconsistency. It simply means $C$ threw a ball before $D$ in your reference frame. That is, the person on the train, operating under the SR assumption of "the speed of light is constant," and using the data (retroactively obtained from a ruler-clock grid, or maybe obtained in real time based on brightnesses) that the flashes were equidistant, must conclude that the forward flash went off first.

• I see. My problem I guess was that I was imagining the flashes 'going off simultaneously' in both frames, but you can't actually say when they went off until you observe them in one of those frames. Since we say that the embankment guy saw them simultaneously, SR tells us that the train guy must see a difference. I got tricked up thinking that the flashes going off when they are both at x=0 was equivalent to saying they went off simultaneously, which is not correct. – Chris L. Oct 14 '13 at 23:09
• Does this mean that if these same two people see the flashes simultaneously (and are moving at relative velocity), then the sources are at different distances from each of them in each frame? – Chris L. Oct 14 '13 at 23:12
• @user1688944 Basically yes. If you see two flashes simultaneously, you know each flash's distance from you (in your frame) equals $c$ times the time elapsed since it went off (in your frame). Nonzero relative velocity generically implies a disagreement on elapsed time, and thus a disagreement on distances. However, at least in 1D, this would lead to a contradiction if the observers were forced to see both flashes at the same point in spacetime, as the symmetry of the situation means boosts to other frames should keep the distances equal as they change. – user10851 Oct 14 '13 at 23:21
• @Chris White: While I recognize that you gave a fair answer to user1688944's question (such as it is), I disagree with your presentation in a subtile yet surely important point. Please note my corresponding question physics.stackexchange.com/questions/81014/… – user12262 Oct 16 '13 at 21:38
• @Chris White: So is there ever a (timelike) scenario where two observers moving with relative velocities can both observe an event as simultaneous, or is this mathematically impossible? – Chris L. Oct 17 '13 at 12:37

Chris, I haven't read all the responses but I agree with your original position that the onboard observer sees the flashes as simultaneous too, and for these reasons:

1. If the train is at rest relative to the tracks the onboard observer sees the two lightening strikes as simultaneous.

2. According to the conventional (Einsteinian) explanation, with the train in motion relative to the tracks, the onboard observer sees a temporal gap between strikes, the forward strike appearing to have taken place before the rear strike.

3. Therefore, by noting whether the strikes appear simultaneous or sequential, the onboard observer can discern the difference between absolute rest and absolute motion.

4. Point 3 violates SR in that no experiment performed in an inertial frame of reference can determine the frame's absolute rest or motion.

5. Further, the length of the gap can be used to calculate the train's absolute speed. In his 1948 book, The Universe and Dr. Einstein — with an Introduction by Einstein — author Lincoln Barnett says:

"Imagine temporarily that the train is moving at the impossible rate of 186,242 miles per second, the velocity of light. In that event, flash B [the rear flash] will never be reflected in the mirrors at all because it will never be able to overtake the train."

The obvious implication is that the gap between the two flashes indicates the train's speed.

• I agree with you, but you are disagreeing with Einstein. I think because Einstein fudged an example to be easily understood by the common man. – Tuntable Aug 19 '19 at 5:13