# Relativity of Simultaneous Events

I'm used to relativity of simultaneity situations that are like the Einstein lightning train thought experiment. This problem is different. Say an observer on the space station observes lamps 1 and 2 blinking at the same time. According to an observer on the rocket, lamp 2 goes off first. Why is this?

This problem isn't different. This is the classic illustration of the relativity of simultaneity.

There are pretty many pretty ways to prove that if two events are spatially separated and happen simultaneously with respect to one frame then they wouldn't be simultaneous with respect to any other inertial frame that is moving with respect to the first one. The neat way of seeing this would be via recalling the Lorentz transformations which tell us how space and time intervals between the same events, as observed in one frame, relate to those observed in the other.

So, the time interval between a pair of events as observed by the rocket, say $$\Delta t'$$, relates to the spatial interval $$\Delta x$$ and the time interval $$\Delta t$$ between the same events in the space station frame by $$\Delta t'=\frac{\Delta t-\frac{v}{c^2}\Delta x}{\sqrt{1-\frac{v^2}{c^2}}}$$where $$v$$ is the velocity of the rocket with respect to the space station. In your example, since the blinks are simultaneous in the space station frame, $$\Delta t=0$$. And the thus, $$\text{Sign}(\Delta t')=\text{Sign}(-v\Delta x)$$. In more explicit terms, $$\text{Sign}(t_2'-t_1')=-\text{Sign}(v(x_2-x_1))$$. Thus, in the space station frame, the event who is situated towards the direction of the velocity of the rocket compared to the location of the other event, will, in the rocket's frame, happen before the other event. In other words, if the relative position of an event (say event $$2$$) with respect to the other event (so event $$1$$), i.e., $$x_2-x_1$$, has the same direction as the direction of the velocity $$v$$ (all in the space station frame), then, in the rocket frame, event $$2$$ will happen before event $$1$$. So, that is why lamp $$2$$ will blink before lamp $$1$$ blinks in the frame of the rocket.

• Did you mean space station? You say rocket and spaceship. – user532874 May 20 at 14:44
• @user532874 Oh yeah, sorry I replaced the space station with a spaceship. I watch more movies than NASA videos. ;-) Fixing it. Thanks! :) – Dvij Mankad May 20 at 14:45
• One more thing, what does "Sign" mean? – user532874 May 20 at 14:47
• @user532874 Oh, it just means the sign of that quantity, as to whether it is positive, negative, or zero. See, if the time interval $t'_2-t'_1$ is positive then it means that the event $2$ happened before the event $1$ happened. – Dvij Mankad May 20 at 14:49
• Also, when you say "the event whose location is towards the direction of the velocity of the rocket compared to the location of the other event" what do you mean by a location that is towards the velocity direction? – user532874 May 20 at 14:51

Even the observer on the station only observes them to flash simultaneously if he is equidistant from both.

The rocket observer on that trajectory is never equidistant from both except at a single instant as he passes.

But because observation of an electromagnetic signal is not instantaneous and does not take place in an instant, and because the rocket position evolves in time during the process of observation (because it is moving), and because the Doppler effect changes the observed wavelength of both signals (non-symmetrically - at the point when the rocket is equidistant, one is red-shifted and the other blue-shifted, and thus it takes a different amount of time to observe one cycle), there is no way to time the observation process on the rocket such that both signals are received at the same time.

Relativity provides the tool with which we determine the order and timing at which the signals will actually be observed on the rocket.

• Receiving the light from an event has nothing to do with determining simultaneity of the event with another. With respect to the entire frame attached to space-station, the two blinks happen simultaneously if the light from the two blinks reaches the mid-point of the space-station simultaneously according to the space-station. – Dvij Mankad May 20 at 14:19
• The two blinks cannot reach the rocket simultaneously at the midpoint, because one electromagnetic cycle takes time to elapse (and thus, takes time to be observed, because there is no known machine that can make an observation without receiving at least one whole quantum of energy in full). Even if you jig things so that the cycle from both completes at exactly the point when the rocket reaches the midpoint, any extrapolation backwards will then imply that they were sent at different times - you could only set up such a situation, by applying the principles of relativity. – Steve May 20 at 14:30
• For what it's worth, the exact same relativistic effects are observed with sound waves (with the exact same mathematics), if the timepiece to which the moving observer refers, is implemented in terms of exchanging pulses through the sound medium. At the point when the speed of sound is reached, such a clock also ceases to record any further increments. – Steve May 20 at 14:34
• I think my comment wasn't clear. I just meant to say that "observing" and "seeing" are two different things in relativity. Observation is something that a reference frame as a whole does whereas "seeing" is about some usual local scientist doing his experiments at one point at a time. – Dvij Mankad May 20 at 14:39
• In that schema of understanding, what is "observed" is an abstract concept - closer in meaning to "the intermediary results, after applying the transforms of relativistic geometry, but before applying other further non-relativistic adjustments", and different in meaning to "experimentally observed" (or "seen") which is closer in meaning to "the final results after all adjustments have been applied". – Steve May 20 at 15:06