Relativity of Simultaneity: possible scenario?

[peio]

Hi, I've been reading some literature on Special Relativity, and even if I think I have more or less understood the relativity of simultaneity, some of its consequences (in the way I have understood it) appear too surprising to me. The problem is similar to the Andromeda Paradox, but seems even more exotic to me. I'll describe the scenario, I'd like to know if I go somewhere wrong to invalidate the conclusion; if not, well... then this is a real consequence of the theory.

A crime has been committed on top of mountain X. The police arrives some time later and want to know what happened. A time t1 after the crime they send an unidirectional signal to mountain Y (in the speed of light c or nearly) which is a distance L from X and where there are more policemen. The signal is uni-directional, and it asks to the people in Y to check what happened in X. Of course, X and Y are stationary to each other, they're in the same reference frame S.
The world police is organised in a way so there are patrols going around the world in a speed v close to the speed of light. The police in Y wait until one of these patrols passes beside them in the direction contrary to X, they wait t2. When the patrol passes beside them, the folks at Y give the coordinates of X to the patrol. This information exchange is instantaneus as they are both in the same place. The patrol doesn't stop, there has not been any acceleration or decceleration, so we stick into SR only. Now the patrol needs the time t3 to adjust its telescope towards X and so on, and now they look. As the patrol moves away from X, I understand that the patrol, in its reference frame, is seeing the "past" of the X point in the S reference frame.
So, considering the desynchronization factor v*L/(c^2) is greater than t1 + t2 + t3, I suppose the patrol could actually see what happened in X at the time of the crime, as even if this is the past in S, it is simultaneous with the patrol in its reference frame.

Is this a valid scenario and correct? Can you know what happened in the past asking to check it to a guy in another reference frame?

 

[A.T.]

As the patrol moves away from X, I understand that the patrol, in its reference frame, is seeing the "past" of the X point in the S reference frame.

No, you confuse two things:
- What the time at X is in the patrols reference frame.
- What the patrol sees looking at X.
They differ by the delay due to finite light speed.

 

[peio]

No, you confuse two things:
- What the time at X is in the patrols reference frame.
- What the patrol sees looking at X.
They differ by the delay due to finite light speed.

Exactly. But still if he accounts for the delay for light speed, his "now" for the location in X is the past for someone in the S reference frame by the factor of desynchronization v*L/(c^2), so he could be seeing something that happened before the crime instant. I mean, the things he sees in X happened, in the S reference frame, before the things a stationary policeman in Y sees in X (in this case both suffer the delay due to speed of light). So if v and L are large enough this difference would be enough for him to be seeing the crime. Am I wrong?

 

[Janus]

Exactly. But still if he accounts for the delay for light speed, his "now" for the location in X is the past for someone in the S reference frame by the factor of desynchronization v*L/(c^2), so he could be seeing something that happened before the crime instant. I mean, the things he sees in X happened, in the S reference frame, before the things a stationary policeman in Y sees in X (in this case both suffer the delay due to speed of light). So if v and L are large enough this difference would be enough for him to be seeing the crime. Am I wrong?

The light that patrolman sees at any given moment is the same light that someone in the S frame that he is passing would see. So for instance, at the instant he passes the stationary policeman, they "see" the same light arriving from X. So they "see" the same events occurring at x at this moment. They will agree as to what they saw. What they will not agree upon is when the light they are seeing left X.

So, no, the traveling patrolman would not see the crime.

 

[peio]

Mmmm, that breaks every understanding I had about simultaneity :((

I'll add some more events to clarify everything and make it closer to the typical basic simultaneity scenario. I make t1 = t2 = t3 = 0, so the patrolman starts looking when the crime light arrives the stationary police in Y. At the time of the crime, a lightning hits X. At that same time (stationary reference frame, S) another lightning hits another mountain that is in the coordinate -X. Clearly, the police in Y sees these lightnings at the same time, simultaneusly, cause he is in the middle. You said:

they "see" the same light arriving from X. So they "see" the same events occurring at x at this moment

So are you saying the patrolman also sees both lightnings at the same time?

 

[A.T.]

Mmmm, that breaks every understanding I had about simultaneity :((

I'll add some more events to clarify everything and make it closer to the typical basic simultaneity scenario. I make t1 = t2 = t3 = 0, so the patrolman starts looking when the crime light arrives the stationary police in Y. At the time of the crime, a lightning hits X. At that same time (stationary reference frame, S) another lightning hits another mountain that is in the coordinate -X. Clearly, the police in Y sees these lightnings at the same time, simultaneusly, cause he is in the middle.

So are you saying the patrolman also sees both lightnings at the same time?

Yes, the patrolman will see them at the same time. But when he calculates when they actually happened, assuming constant light speed in all directions in his frame, and considering that X and -X are moving relative to him, he will conclude that they did not happen simultaneously.

The guy at Y in frame S, by making the same assumptions will conclude that they did happen simultaneously.

That is relativity of simultaneity.

 

[peio]

I think I understand it now, I was thinking of simultaneity as something more complex that what it actually is. As far as I can see now, an observer who is getting closer or further from the location of an event sees that event earlier or later than what he would expect, so he thinks that it happened on a different time compared to a stationary observer.

I misunderstood the Andromeda paradox and that tricked me: I thought it said that a standing man and one in a car see different things happen in Andromeda, but now I see they don't; they just think that what they see happened at different instants.
Thanks for the answers.

 

[A.T.]

Yes, it's mainly because everyone assumes he's at rest, and light moves at the same speed in all directions relative to him. Unfortunately observation supports this egocentric world view.