EDIT: I think I can pinpoint my confusion a bit better. Here comes my updated question (I'm not sure what the standard way of doing things is - please let me know if I should delete the old version). The major change is that I removed focus from the third question which probably is a purely mathematical question (in the notation below, it asks what properties of (M,T) together with (M,T) being consistent, forces (M,T) to be unique.).
Say that a pair (M, T) is consistent if it satisfies the Einstein field equations. Here M is a manifold with a metric g (from which one can define its Ricci tensor among other things) and T is a map from M to tensors living in the same space of tensors as the Ricci tensor (I ignore units for now). I put absolutely no other restrictions on (M, T).
Now relativity makes perfect sense to me as a mathematical statement: it distinguishes some pairs (M, T) as 'consistent'.
Therefore, if we had a way to map "our world" to a pair (M,T) at least we could theoretically check whether (M,T) is consistent or not. My problem is that I do not at all understand how to do this.
To begin with, which set do I choose for M?
I think I can answer this question myself. I take this set to be the set of intuitive descriptions(that I can make intuitive sense of) of events in the world. For example, E := "(a particular point in) Stockholm on 08:00, Jan 24, 2013". This description I could understand intuitively, and at least theoretically (if time permits) I could go there to check the theory if it makes a statement about E. Another kind of description, given already another description F, could be G:="the event I get to by using rocket R, travelling for time T according to this watch I bring with me, from G", where R and T are intuitive descriptions. Please let me know if this choice of set is inappropriate.
In this case I have no problem of turning the set M into a manifold, not yet with a metric.
Finally (and here is my confusion): I am at a point p (constructed as above). What (intuitively described) experiments do I perform to find the metric tensor at p, respectively the stress-energy tensor at p? I cannot come up with two different answers for these two tensors - and in this case the theory is not a very interesting one, since then it just predicts that two identical (in the intuitive sense) experiments are the same.
If I try to get an answer to this from e.g. Wikipedia I get lost in a deep tree of coordinate-dependent definitions which in some places appear to assume I already have intuitive sense for both mass and metric, and that I have intuitive sense for these being the related as relativity predicts they should be. I'm hoping there are two distinct intuitively described experiments I could perform, which relativity predicts should have the same result.
FINAL EDIT: I have received many useful comments, and the answer by Ron Maimon answers my initial question, which was "what is a suitable set to choose for M when trying to map 'our world' to a pair (M,T)?". It seems a definition such as I suggest above "should work", as should that described by Ron in his answer. Furthermore, Ron points out(I think) that it is an assumption of the theory that any such labeling should give the same results.
Since my initial question is answered I accept Ron's answer and will possibly come back with my further question "how to intuitively understand the metric and stress energy tensors in terms of experiments any person with sufficient degree of common sense and superhuman abilities (by which I just mean, can reach high accelerations, is not so heavy as to affect the stress-energy tensor in significant ways et.c. Equivalently, superhuman abilities would not be needed in case the speed of light was something like 10 meters per second) could perform?" if I am able to formulate it in a precise way.
OLD VERSION (not needed for the question): As far as I understand, general relativity states that
the world is a manifold M, and M is completely described by the Einstein field equations.
This already appears as an incomplete statement, and I'll explain why I think so. Before that:
- what is a complete statement of general relativity, possibly including undefined terms (so in my attempt above, the "is" in "the world is a manifold" and "Einstein field equations" are undefined terms)?
Now to why this does not make sense to me. The Einstein field equations state that two tensors (it is not necessary to define tensor for my confusion to arise) agree at every point.
This seems to presuppose that the set of points making up the manifold is already given. Thus:
- what is a good description of the set of points of M?
For clarity, my definition of manifold M says for one thing that M is a set.
For the first question, it appears to me that there need to be some extra assumptions, since one could conceivably think of a 'world' without matter which should be completely 'flat', and also of our world which is not. These cases should clearly be different.
- Exactly what data determines a 'theory'? (meaning that M is completely determined from this data - again I would like to have complete description but am happy with undefined terms as long as it is clear that they are such)
Ideally, the second and third questions should be answered by any answer to the first question, but I added the latter questions to indicate what confuses me particularly.