npnacho said:
in the SR case I've mentioned above, the time elapsed isn't propper time, because we are talking about the trip of a spaceship which is not at rest in our coordinate system... but in that case the change on the coordinate ##x^0## is the time elapsed in my point of view, right? I'm not sure about this, it's a question too.
that is the proper time measured by the guy who drives the spaceship... but I'm asking which is the elapsed time measured by the guy how is sitting at spatial infinity seeing how the spaceship goes from A to B.
There may or may not be something strictly equating to a "point of view", depending on what you think a "point of view means".
See for instance
http://arxiv.org/abs/gr-qc/9508043, "Precis of General Relativity".
A method for making sure that the relativity effects are specified correctly
(according to Einstein’s General Relativity) can be described rather briefly.
It agrees with Ashby’s approach but omits all discussion of how, historically
or logically, this viewpoint was developed. It also omits all the detailed
calculations. It is merely a statement of principles.
One first banishes the idea of an “observer”.
This is basically your idea of "point of view", I think.
This idea aided Einstein in building special relativity but it is confusing and ambig
uous in general relativity. Instead one divides the theoretical landscape
into two categories.
One category is the mathematical/conceptual model of whatever is happen-
ing that merits our attention. The other category is measuring instruments
and the data tables they provide
Misner goes on to describe what the measuring instruments would be. Since this isn't GPS, it's not directly relevant, but you might imainge that measuring instruments would be actual clocks that you carry around with you, and timestamped radar signals, so that you know the time of emission and time of reception (by your local clock) of various radar signals.
You can extend this somewhat to imaginary more clocks and radar emitters that are remotely located (such as the GPS clocks, or the receipt of signals from astronomical reference stars, or perhaps even slide slightly over the line and imagine an infinite array of clocks that don't actually exist, though the later starts to blur the line of what "actual measurements" are, since they are really mental constructs.
The other part of GR, other than the measuring instruments, is the "mathematical conceptual model of whaterver is happening". This is the metric. The metric is sort of a space-time map, it assigns coordinate numbers (which in general are more or less arbitrary) to every event in space-time.
Using the metric, you can compute the path of any desired radar signal, and you can compute the proper time reading of any clock following any specified trajectory (though you need the map-coordiantes to specify the trajectory).
Your question I think relates ultimately on how you assemble a mental picture of what's going on from actual observations. There are some conventions here, but they may or may not match your intuitive idea of what a "point of view" should be. The conventions boil down to the fact that if you have a metric such that it is diag(-1,1,1,1) in some area, that in that area the coordinates used by the metric are a "local" point of view that works while you are in that neighborhood.
For instance, in the Schwarzschild map, the coordinates have this property at infinity. Hence the talk about observers at infinity.
In general, there isn't any unambiguous way to have a "point of view" regarding distant objects. You replace this with the ability to use your map of space-time (the metric) to compute anything you can actually measure (with clocks, and radar signals, as specified eariler).
One final note. Misner notes that this is "A method" to make sure one specifies relativistic corrections correctly. It doesn't necessarily mean it is "the only method". Part of the issues here are philsophical, so you might find people with different philosophies. This particular method is particularly simple and useful, I think, however, and avoids a lot of digressions.