This is a recapitulation of a subject that has incidentally appeared in another thread (Light shone in a train bouncing off mirrors…) that started over another issue (why does light take the direction of the source but not its speed?), although there may be a connection between the two things. I have already received some helpful comments, but I still have doubts. If you find this introduction too long, please jump to the next post, which is where I place the question that worries me now. The subject is to judge if the duel described by Brian Greene in the Fabric of Cosmos (when explaining the relativity of simultaneity) is “fair”, in its original version and with a variation. JesseM confirmed that the answers to questions (a) and (c) is: And to question (b): We still discussed for a while whether the question “is the duel fair?” is purely legal or also physical, but I think in the end we agreed that: - The legal problem is the same as the physical problem. The role of the law is to solve problems, just as the role of phsyics is to solve problems. - The legal problem is not to establish some arbitrary conventional rules for the game. The rules, on top of being agreed upon, must ensure that both duellers have “equal opportunities” to shoot and avoid being shot. For example, after receiving the signal for shooting and before being shot, the duellers may stand aside, duck down or try any other tricks. Do they dispose of the same time for this purpose? - This legal need can be translated into a well-defined physical expression: . (I initially contested the definition, but I was wrong. In the end I agree that it was perfectly appropriate.) In other words, the law requires that the duellers can do equal number of “tricks” to win. Physics answers that that means that their clocks register the same number of “ticks”, but in particular SR specifies that the relevant ticks are “proper” ticks, those of the duellers or of assistants situated where they receive the signals and the shots. These are the calculations assuming that the relative velocity of the train wrt the ground is 0.5 c and that the train is 2 ls long in its rest frame (1.732 ls in the ground frame): - In the train frame: Back and Front get the light signals simultaneously (after 1s) and their shots take 2 s to reach their opponents. So the proper time interval is 2s for both duellers. The duel is fair. - In the ground frame, the duel is also fair because the proper time interval, although another one, is also the same for both duellers: * Back got the light signal earlier (0.577 s) because he was heading towards the signal (thus making its path shorter), which looks like an advantage, but then it also happens that he is shot earlier (2.886 s), for the same reason, because he was heading towards the laser (thus making its path shorter), which is a disadvantage. The interval is 2.886 - 0.577 = 2.309 s. * Front got the light signal later (1.732 s) because she was racing away from the light (thus making its path longer), which looks like a disadvantage, but then it also happens that he receives the shot earlier (3.732 s), for the same reason, because she was escaping from the bullet (thus making its path longer). The interval is 3.732 – 1.732 = 2.309 s. Then we engaged in a challenging discussion about what it means that, as observed in the ground frame, the duellers get the light signals non-simultaneously. My final understanding, as referee in the ground, is very prudent: the physical fact about it is that, if the clocks of my assistants (situated by Back and Front when they received the light signals) had been synchronized following the Einstein convention, they would show different readings; but this is just a piece of the puzzle; if I want to rule, I have to compare the clock readings at reception of the light signal with the clock readings of other assistants witnessing the reception of the shots; the latter are also non-simultaneous; the difference is the same for both duellers = they disposed of the same proper ticks to do their proper tricks. Thus I rule, in accordance with the advice of SR experts, that the duel was fair. Now we face a variation of the same duel: the signals for shooting are still light signals but the duellers shoot mechanical bullets. What I have studied about SR tells me that the solution must be identical, since mechanical objects obey the same laws of physics: - In the train frame: just like the light signals travel equal paths at the same speed (c) and arrive simultaneously at the duellers, their bullets also travel the same paths at the same speed and hit simultaneously the duellers. - In the ground frame, you can reach the same conclusion through two routes: * If I apply the relativistic formula for the addition of velocities to the light signals, I will discover that they still travel wrt me at c. And if I apply it to the bullets, I will find a differential element. Each bullet travels at a different speed wrt me, but this fact does not change the judgment: the bullet from Back travels wrt me faster than the bullet from Front, but less fast than it would if we had applied the Galilean addition formula (which compensates for the fact that it departs earlier) and the bullet from Front travels less slowly than it would with the Galilean formula (which compensates for the fact that it departs later). * If, with the aid of my assistants, I measure directly the speed of the light signals and the bullets, I will find that they travel at exactly the speeds that stem from the above formulas. However, I have doubts: due to some reasoning about the difference between what is “real” and “conceptual” (which I will not repeat here, it’s just too long), I fear that, in this new case, the proper time intervals (during which the duellers must do their tricks) should be different. I don’t know who would have the advantage and, in any case, the difference would be tiny. But this would make the duel, by definition, unfair, in the opinion of both referees. Nevertheless, I conclude: if the experts tell me that the solution to the thought experiment I describe in the next post is “yes” (and maybe you give some sort of explanation for that, it doesn’t have to be very profound, I want to finish my job!), I will rule as proposed by SR. Please give me a hand!