Relativity of simultaneity

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Question:


Two people are on a high speed, constant velocity train on opposite ends. One person is standing on the platform. The two people aim a laser beam at the center of the train at the same time in their reference frame. In the center of the train, there is a special gun that will shoot and kill whoever's laser beam reaches it first; but if the two beams reach it at the same time it will be disarmed. From the person on the platform's standpoint, one of them dies due to the relativity of simultaneity. From the two people on the train's standpoint, they both live. Later, the two people get off their high speed train and meet with the person on the platform. Is this not a paradox? The platform person clearly saw one of them die.


Answer:


Let’s call the person in the back of the train A and the person in the front B. In their reference frame, which is also the rest frame of the train, they fire simultaneously and their laser beams travel equal distances to strike the device at the center of the train. Therefore the two beams arrive at the device at the same time and disable it.


How do things look from the viewpoint of the person standing on the platform? In the “platform frame of reference”, A and B do not fire simultaneously, because of the “relativity of simultaneity” mentioned in the question; but rather A fires slightly before B. But in this frame, the train is moving and therefore so is the device. And so, as the laser beams are traveling through space, the device is moving away from the point where A’s beam was emitted and toward the point where B’s beam was emitted. So A’s beam has a greater distance to travel to reach the device than does B’s beam. Therefore it takes longer for A’s beam to reach the device than it takes B’s beam. This exactly compensates for the fact that A fired earlier than B in this frame, and the two beams reach the device at the same time and disable it. So, the story has a happy ending in both frames.


Actually, a standard way of showing the “relativity of simultaneity” of events separated in space is to think about a situation opposite to this one. Suppose that the only thing doing any shooting is the “device” at the train’s center, which shoots at A and B simultaneously. Imagine that in the train's frame of reference the device emits two beams simultaneously, one towards A and one towards B. In the train’s frame they hit A and B at the same time and kill them simultaneously. In the platform frame, however, the beam traveling towards A has a smaller distance to travel, because A is moving toward the point where the beams were emitted by the device, whereas B is running away from it. Since light travels with speed c in every frame, A gets struck and killed before B does in the platform frame.

For more information, see Wikipedia's article on relativity of simultaneity.

-Stephen Barr