Relativity of simultaneity question. Two spectators, one moving relative to two explosions at equal distances on either side of them?

Question

and the other standing still relative to two explosions at equal distance on either side of them, Einstein said the moving spectator would experience the signal from the explosion first. Why is this? I understand the basics of the first principle of relativity. Non-inertial reference frames have the same laws of physics. Two things move relative to each other, not to space. There is no origin of space or point in space relative to an origin to move relative to. But either spectator experienced both explosions in the same instant at the same place. This leads to believe the explosions actually happened at the same time. Please explain my error.

Answer 1

I think you are having problems with what simulaneity in relativity actually means. If you were looking at a star a 1000 light years away and somebody turned on a light bulb next to you, a photon from the star and a photon from the light bulb next to you can arrive at your retina at the same time. However, the emission of the photon from the star (an event) and the emission of the photon from the light bulb (an event) are NOT simulataneous in your frame according to relativity. One EVENT happened 1000 years ago and the other EVENT happened picoseconds ago in your frame.



So, what do we actually mean by simultaneity in relativity? It’s actually easy. Define a standard Cartesian x-t coordinate system. Two events are simultaneous in that coordinate system if they have the SAME value of t in that coordinate system. That means for a given value of t, all of the events that occur from x=-∞ to x =+∞ at time t are SIMULTANEOUS in that frame. For example, draw a line at t=1 parallel to the x axis. All events on the line t=1 are simultaneous in that frame. Now chose two events A and B anywhere on the line t=1 (e.g. one located at x=a and one at x=b)



Now define a new observer moving with respect to that frame with an x’-t’ coordinate system. As you should know, the Lorentz transforms are equivalent to an axis rotation. That means the t’ axis will be rotated with respect to the t axis. The moving observer clearly has a different definition of simultaneous - the two points A and B do NOT lie on a line of constant t’. If you picture time for t’ progressing into the future, a constant time line has to intersect A and B at different times. Whether it intersects with A first or B first depends on whether the ‘ observer is moving toward or away from the ‘stationary’ observer. The different directions correspond to rotations in different directions of the t’ axis.



The above will become abundantly clear if you simply draw this out on a piece of paper. Those are called spacetime diagrams and are incredibly useful for getting a firm intuitive grasp of what is going on.

Answer 2

Excellent question and nicely set up but in the very setting up of the question you have highlighted why it is so hard for most people to make sense of special relativity.



Have you ever wondered why all these thought experiments have explosions? It is because they are instantaneous. So think about this; After the explosion has happened, where did it take place? Bear in mind that space has no preferred reference frame.



So the deal is; I am in empty space, I am inertial and I think I am stationary and I saw an explosion 1 km away in some direction with respect to my personal reference frame.



You are inertial and moving at half the speed of light with respect to me but you consider yourself to be stationary. As you pass me, we match up our direction reference frames just as we see the flash.



One minute later, where will you say the flash occurred? Remember, you think you are stationary and when you saw the flash, it was 1km away in a certain direction that we both agreed on. So you will point in that direction and claim it was 1km in that direction.



But clearly, 1 minute later we are millions of miles apart and yet we both think the flash happened 1 km away from our own current location. You see, when they said there was no reference frame in space, they really meant it. There is no true location where the flash occurred.



And thus you can't move relative to an instantaneous event. You can only move with respect to a persistent object.



I'm not going through the full answer to your question, my goal here is just to impress on you that "space has no reference frame" is a far more tricky idea than you realised. And part of the reason is that when you visualise an explosion or draw a cross on a page, you have imposed a personal reference frame. And it is easy to forget that the page (mental or paper) on which you place your explosion is not "The Truth", it is your observation of events in your frame.



Here's a thing; Imagine in space there is a long straight railway track and a train riding along the rails at a constant velocity, all inertial. Is the train travelling along a stationary track or is the track sliding under a stationary train?



Answer: Either. It depends on your reference frame. There is no preferred answer. To grasp Special Relativity you have to be able to see it both ways and see that each is equally valid at the same time



The front wheel of the train crushes a bug and leaves a mark on the line, 1 minute later, where in space is the place the bug got crushed? Is it the bottom of the front wheel or the mark on the line?



Answer: Either. There is no right answer unless you give a reference frame, then it could be the wheel or the mark on the line, or indeed some other point with respect to a third inertial observer.

Answer 3

Observer A is moving to the right at high speed. Observer B is a rest. The flashes of light from both explosions reach B at the same time. He says the explosions happened simultaneously. But B is moving toward the light from the explosion on the right and away from the light from the explosion on the left. So B sees light from the explosion on the right first. He then measures the distances from his spot on the train to the marks on the train made by each explosion. He finds that those distances are equal so he must have been halfway in between the explosions. Since he saw the light from the right explosion first, he must conclude that it happened first. (The right one is the one in front of him.)

Each observer can say that he is at rest and the other is moving. So both their conclusions are correct. Did the explosions happen at the same time? It depends on who you ask.

Answer 4

I assume both observers are next to each other when the explosions occur and the explosions are displaced from that point by some distance x in the direction the moving observer is moving in. So call x = 0 the point when the two observers are next to each other and the time it occurs t = 0. It will take light from the explosion a time t1 = x/c to reach the stationary observer. But in that time the moving observer will have moved, relative to the stationary observer a distance x1 = vt1 . Clearly the moving observer sees the light first.



In terms of the moving observer, the distance to the explosion is x' = x*sqrt( 1 - (v/c)^2) and the light reaches them in a time t'1 = x'/c = (x/c) sqrt(1 - (v/c)^2) = t1*sqrt(1 - (v/c)^2). This means t'1 < t1 so the light reaches the moving observer first.