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I have a two-fold question about the light-cone structure of spacetime, specifically about space-like separated events.

As far as I understand it, any two events that happen at the same time in a given reference frame are space-like separated. If so, any two simultaneous events occurring on my arm and leg are space-like separated.

What confuses me is this: If I am not mistaken, all the events that we perceive are in our past light-cones. This is because we only perceive events that have emitted light that has reached us, and thus has causally affected us. If so, what happens when I look at my arm and leg? It seems to me that for any two simultaneous events A and B, where A occurs on my arm while B occurs on my leg, A and B must be time-like (since I have perceived them) and not space-like separated. In a nutshell, if simultaneous events are space-like separated, while I only perceive time-like separated events, how can I perceive simultaneous events?

And second, could someone recommend me an article or a book explaining how the light-cone structure relates to ordinary perception? I struggle to connect the light cone structure to real-life events, so some kind of graph or an explanation of this would be useful.

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You are missing the point that in your setting there are three entities:

  • arm
  • leg
  • brain as perceiver

The light-cone you're interested in is the one centered at the brain at some given instant of time. The cone divides space-time into three regions:

  • past
  • space
  • future

And the "past" region of this cone tells you what can be perceived.

So, if at a time T1 two events happen simultaneously in your arm and your leg, they will not be perceivable to your brain at that moment T1, because the light-cone centered around your brain at T1 will have both of them in its "space" region. Only when time proceeds to T2 (a few nanoseconds later), they will have entered the "past" region of the new light-cone, the one that moved forward in time, and thus expanded its "past" region.

Some results of the light-cone concept:

  • At T1, your leg will not perceive a T1 event from your arm, because the leg-T1 cone has an arm-T1 event in its "space" region.
  • A T1 event from your arm can only be perceived in space-time points inside the "future" region of the cone centered around your arm at T1.

Up to this point, all this can also be explained from Newtonian physics, just as a result of finite travelling speed of light.

Special relativity adds the fact that, whatever super-fast-moving inertial system you select, the cone stays the same. If in one inertial system, an event E1 lies in e.g. the "past" cone region of some reference E2, then this E1-E2 relationship is the same for each and every inertial system you can imagine.

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The notion of 'perceiving an event' is misleading.

Perception happens here and now, so all perception at a given time for a given observer takes place within its own single event (the observer here-and-now as a point in spacetime).

What is perceived is causally determined by whatever happened in other events within that observing event past light cone.

So while we can say that we observe what happened elsewhere in the past, in fact we observe what is here and now with us and that is a more or less direct consequence of what happened elsewhere in the past, which is not quite the same thing.

In other words, one must not confuse the common notion of 'event' as 'something happening' with the relativistic notion of 'event' as simply 'a point in spacetime'.

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You are correct to think that all the events we perceive are in our past light cone.

We perceive, more or less simultaneous events, since for all intents and purposes, for us, light is more or less instantaneous.

To actually see how Einstein goes about constructing a synchronisation of clocks - that is simultaneity - you could try his original paper on special relativity, On the Electrodynamics of Moving Bodies. It is, unlike many modern papers, short and uses only school level algebra and quite straight forward to follow.

You can also try the first chapter of d'Inervo's book on general relativity to see how simultaniety is to be understood. He explains it on the basis of Bondi's k-calculus, and which has nothing to do with calculus. Here k stands for the radial Doppler factor.

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