Question

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.

Answer 1

General relativity is a classical theory. I will restate your dilemma as follows, since this is how Einstein stated it:

We have an abstract manifold consisting of points, vectors that link nearby points, and a metric tensor that tells you the distance between nearby points. What makes these points physical? How can we tell point A apart from point B? Since it seems that the points only get meaning from the stuff happening at these points.

Einstein was very bothered by this question, so he considered the "hole argument". This is the idea that if we give all the points of the manifold names, by using a coordinate system, then these names are arbitrary, and the points are really indistinguishable from one another. So if we change the naming of the points, we change the name of the solution of the Einstein equation, and this seems like it changes the physical behavior.

The resolution of the hole paradox that Einstein settled on (which in modern physics is the central idea of gauge invariance) is that the space-time points are defined by the things happening at these points, not by the names. So he considered filling space with lots of metersticks and clocks, making a grid of measuring devices, and these clocks and metersticks have names that are real, as these are real objects. Then the coordinate system assigns a coordinate to each meterstick and clock, based on the value of the coordinates there.

If you rename all the points, the clocks and metersticks get new names, but so long as the physical relations between them are unchanged, so long as the distance marked out by each little meterstick, and the time ticked out by each little clock are unchanged, then the two situations are identical in the physical sense. This allowed him to define the notion of coordinate invariance in GR: any labelling of the points defined by the grid of clocks and metersticks is equally valid as any other, and the laws should not make reference to the names of the points in their formulation.

This allowed him to make sense of the statement "space time is a manifold M with a metric that obeys the field equation". The statement gets positivistic meaning from the matter in the space, making measurements of local distances and times, and the metric tells you what these measurements are (or would be). The Einstein equation then determines the future metric from the current metric and its time derivative (under suitable constraints) plus the scheme for giving name to the space-time points in the future, which is the condition that tells you what coordinates you are going to use.

With this philosophical position, Einstein resolved the nagging worry of the ill-definedness of manifold points without extra structure imposed--- he imposed the structure by imagining little classical measuring devices everywhere. Since this is classical physics, he can make these devices really little, without affecting anything.

This point of view tells you that when you have an arbitrary labelling that affects the physics, for example, the labelling of whether a given particle (at distances short enough for the Higgs to be irrelevant) is a left-handed electron or a neutrino, the coordinate system that picks out which direction is "electron" and which direction is "neutrino" is arbitary. Then you impose the condition that any choice of coordinates is as good as any other for describing the physics, and this is the local gauge invariance. To make this work, you need to give a gauge field to relate the fields at nearby points, and you need to make sure you don't count coordinate changes (changes in gauge) as physical transformations, since they are just a different choice of name you give to the additional coordinate system you use to distinguish an electron from a neutrino.

This might be a bad example, since at low energies, the Higgs condensate determines which direction is which, and it is good to choose the gauge so that the electron and neutrino don't look related. This is no different than a crystal in space picking out preferred coordinates, based on the atoms being at certain positions--- it doesn't change the fact that fundamentally you have a arbitrary choice of coordinates, one which is conveniently made using the crystal when the crystal is present.

This philosophical shift in the meaning of coordinates is so deeply ingrained now, that everyone does it at some stage and forgets about it. You should understand that the points are given meaning only to the extent that there are observable quantities that are invariant to change in coordinates, the goal of physics is to describe these quantities, and the coordinates are a mathematical crutch to formulate the equations.

The mathematicians define the concepts abstractly, so that the points are a set with a topology and a smooth structure, and the metric is a mild generalization of the notion of function called a section of a bundle. These mathematical definitions are already in a framework where the meaning of the word "point" does not depend on the label you give to the point, and Einstein's confusion is hard to state. But if you come in without this philosophical position, it helps to think about the little grid of metersticks in order to acquire it.