Time Dilation, the Hubble Shift and God's Eternal Universe

What observer's proper time does t refer to and what class of observers does the coordinate time t_1 correspond to?

Let me make this simpler. What class of observers does the coordinate time correspond to?The coordinate time has to set up in reference to some class of observers, defining their time and space coordinates as the ones used for calculation.Innately in relativity or physics in general, you don't have coordinates until you introduce them via a set of observers.

This isn't true, you can have motion in Minkowski spacetime (the flat space whose metric you are referring to as "the Fundamental metric") it just doesn't have gravity.

What Captcass is saying makes no sense, so this is more for the interest of others. What exactly qualifies as an observer is currently not entirely resolved. We do have some results suggesting that there are issues with applying QM to other observers directly, but this is an ongoing area of research.Electrons don't have a position, so really you aren't resolving that property when you observe it. It's more that you make the electron report a position to you, or as some physicists like David Mermin say "give you a position experience". QM doesn't actually describe what microscopic objects are like, it describes what reactions they give to observers when observers probe them.

To give a fuller exposition.Quantum Theory predicts probabilities for a how a system will react to certain kinds of questioning/probing/measurements/way of looking at it given:
(a) What way you are measuring it
(b) The results of previous measurements, along with what exactly those measurements were.Once you get a reaction to your measurement it then provides rules for updating your expectations of future reactions (those rules were once called "collapse", but today we say "state reduction" or "Bayesian updating")However note that the predictions depend on what you've witnessed before, so different users of quantum theory can disagree on the probabilities. Probabilities are subjectiveAlso the different ways of looking at the system don't cohere or make sense together. It's just a collection of incompatible subjective "impressions" of the system that can't be made sense of together. Impressions are subjectiveObviously these two forms of subjectivity are not what one normally wants in a fundamental theory, but it seems to be the way the world is. Quantum theory deals with your probabilities for reactions you might receive from micro-systems.What happens for classical things like a stone is that Quantum Theory says that every way of looking at it makes sense together and that they can be combined into one cohesive picture. The probabilities are also definitive and don't depend on the observer.Thus it says there will be inter-subjective agreement. All observers should expect to see the same reactions for each given way of probing and all ways of probing make sense together. Thus you can detach the observer from the description and speak of what the stone is "actually like" independent of any observer.Since you can now speak of the stone without reference to anybody, i.e. as it is in itself, you are then able to have equations for the stone that predict what it will be like in the future and was like in the past. In other words its history and predictions for its future.Since we can't detach observers from the description of micro-systems we can't really speak about their past which is a bit of difficulty for cosmology in some areas. It also means statements like "the protons in your body were once in stars" are not strictly true.Edited by Son Goku, : Tidying