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Originally Posted by publius
So that clicked the old light bulb about "space".
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Right, that's why I came to reject the ds^2 notation in favor of a dtau^2 notation, with dt^2 positive, like you used. Because I feel proper time is the only dynamical variable that is actually observable, and all the rest are another level of abstraction invented to understand why the things that "matter more" to us also require shorter light delays to get the right time ordering. But relativity means that physics as we conceive it is purely local, and so there is not "space", there is a prescription for mapping our physics "here" so that it looks like someone else's physics "there".
Thus I don't think of space as existing for one observer, but rather as a connection between a real observer and a hypothetical observer if we imagine ourselves somewhere else watching some local physics unfold. In other words, I think of relativity as something extra-- physics is local, relativity is how we conceptualize space to connect to somebody else's local physics. In that sense, Newton got the local physics right, but not the action-at-a-distance spatial connections. That's also why I like writing the metric as
ds^2 = f(s,t)*(dt^2 - dtau^2)
because it's supposed to be a construct for how to make distance based on how one observer measures time, relative to an observer who was at "both events" in question. I suppose I'll never succeed in getting it this way in all the textbooks, and indeed the "high priests" might have a reason for not liking my way.
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And this mess is why I sort of keep my mouth shut about "cosmological stuff". Cosmologists have a language about distances, velocities and stuff involving this "space" thing. Which comes from a particular coordinatization. And which is utterly arbitrary.
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Not
utterly-- indeed I would say cosmology is the only place where the concept of space makes sense in a non-arbitrary way. It's all based on the cosmological principle-- that's what selects comoving coordinates as the "special" ones, the ones where you don't need to explicitly treat a global (universal) concept of gravity. Why should we need universal gravity to understand our local physics? We shouldn't-- it should only come up when we connect to someone else's physics. Again, universal gravity isn't physics at all, it's in the connections.
Your comments on this thread have been highly insightful, thank you.