Anamitra said:
We can always set up a coordinate system in an arbitrary manner[for example we may think in terms of spherical or rectangular systems as three dimensional time-slices]. Then we can find out metrics that match against the physical aspects of the problem[this should include gravity and perhaps other factors according to the nature of the problem].
The coordinate system is of course arbitrary----it does not have to have a definite physical meaning.But once we use the physical aspects of the problem to impose the metric coefficients on them,the whole thing becomes meaningful.
This in no way serves as any impediment to my suggestions/thought experiments.
JesseM has suggested several times you think about what it means to set up a coordinate system. I doubt I can do better, but I'll try again.
Suppose you want to label an event B 3 units in x direction from event A (events are points in space time; they have no history - they are specific events somewhere, sometime, in the history of the universe). This labeling has no meaning at all until you know the metric and can express it in terms of x and other labels. Depending on how you do this, 3 in x direction can mean 3 hours later on a clock, 3 kilometers east, 3 degrees counterclockwise, whatever. It is only the metric that gives x any meaning at all. If the metric says x direction is timelike, than x has the character of time for some clock; if the metric says it is spacelike, then it is distance for some path of simultaneity.
More naturally, you can set up coordinates by (perhaps idealized) measurements. Then the measurements determing the nature of of the coordinates. Measurements, of course, take full account of the metric. If you define x by a mechanism for measuring distance, it will represent distance no matter where or when in the universe you do it, no matter what the gravitational field.
If you are 'thinking' about the the interval from (t,x)=(5,5) to (5,7), where 5 is in the future, and you have don't know the metric for this region of space *time*, and don't define any measurement you will do,, then you cannot have any expectation of what they mean. I cannot fathom what you mean by 'expecting' a meaning for this separately from a measurement procedure or defining the metric.
Note, if you define this, for example, by saying that when my watch says 3, I will define my spacetime position to be (3,5), then I will send out a rader signal and if I get it back when my watch says 7, (still calling my postion x=5), then the event of its bouncing off something I will label (5,7). Using such a procedure you would know, at all times, and all gravity situations, that you would be defining a spacelike interval between (5,5) and (5,7), and you could call them 2 lightseconds apart (thus calling your watch units seconds, and x unit lightseconds). An observer elsewhere in the universe might disagree radically on how far apart these events were, but they would certainly agree the separation between them was spacelike.
