Formulation of general relativity

Question

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 an intuitive sense for both mass and metric, and that I have an 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 labelling 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 a 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 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 a 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.

Answer 2

General Relativity assumes that the universe is a differentiable manifold. The Einstein equations calculate the metric that allows us to assign a meaning for distance on the manifold. So you're correct that you just have to assume a manifold exists.

The same manifold can have many different metrics. The metric is determined by the stress energy tensor, so the manifold+metric if there is no matter will be different to the manifold+metric when matter and/or energy is present.

I think this answers your questions, but please comment if you want me to expand on this.

Response to comment and edit of question: bearing in mind that I am that most pragmatic of scientists, a physical chemist, I'm not sure I understand your concerns as they seem a bit philosophical to me.

The metric tensor describes the curvature of space and you'd measure it by parallel transporting a vector round some convenient closed path and comparing the direction of the original and transported vector. The stress energy tensor you determine simply by measuring mass/energy density, momentum flux and pressure. The fact that Einstein tensor and the stress energy tensor are required to be identical everywhere doesn't seem trivial to me - actually it seems quite remarkable.

Incidentally, the measurements you make will always be done using some convenient coordinates because we need coordinates to measure anything. However this just affects the representation of the tensors we end up with. The tensors themselves are coordinate independant.

Response to response: Erik makes the point that you can't actually transport a vector round a closed path in spacetime, however you could start at a point, travel two separate paths and meet at a second point and make that your loop. This seems to me a perfectly good way of measuring the transport round a loop.

In terms of actually doing the measurement, at every point on the loop spacetime looks locally flat, so all I have to do is locally parallel transport my vector along a straight line, and I know how to do this. Admittedly I have to move an infinitesimal amount at each step, which isn't possible in practice, but then no-one's suggesting we'd actually measure the curvature this way. The point is that I don't need to know the metric or how to do covarient differentiation in order to (conceptually) do the measurement.

Answer 3

Your assertion that "M is completely described by the Einstein field equations" is not correct in any universe with spatial dimension greater than two, since in larger dimensional spaces we have to worry about topological constraints on our manifold that are NOT determined by the metric properties that are provided by solutions to Einstein Field Equations. (In two spacial dimensions, the metric and topology are determined by one another)

In addition, your third question seems to ignore an important aspect of the einstein field equation, the stress energy tensor (the source term). Below are the einstein field equations (without a cosmological constant and with geometrized units).

$G_{\mu\nu}=8\pi T_{\mu\nu}$

That $T_{\mu\nu}$ term is the stress energy tensor and will define the difference between your empty, flat universe and the one we have, with lots of various matter and energy.

Answer 4

Too long for a comment, so here some remarks and questions. Might well be that I understand you question wrong though...

Firstly, general relativity is a physical frameworks which permits to model different situations like any other theory. In Newtonian mechanics you can ask how one particle behaves in a field or you can ask how two particles behave with a field and with each other. And so on. The theory contains differential equations (Newtonian equations of motion) and you can plug things into them, like the number of particles or the mathematical expression which is the force. As you're dealing with a differential equation, you also need initial conditions. Like where the particles start out and what the initial velocities are.

So the first thing to clarify is that Einsteins field equations are differential equations as well, there are clearly many mathematically permissible solutions and so it's not like general relativity is equivalent to one and one manifold M. Instead of saiying "M is completely described by the Einstein field equations", I'd say M (or rather the field $g_{\mu\nu}$, i.e. the metric tensor, on M) obeys the field equations.

I'm not sure to what extend you're technically aware of this, but you also describe the problematic of having to model spacetime while it interacts with matter, which needs spacetime to be somewhere too. Yes, this really makes it difficult to deal with the equations. In any case, you decide what features you want to have in your model and solve the field equations in a selfconsistent manner.

There is certainly some extra thoughts in coming up with a description of the world than postulating the equations. Similar to the Newtonian exmaple, you can go on and search for the general relativistic description of one particle exposed to the pull of the gravitational force field of another. You might find the Schwarzschild metric, construced to be constent with well estabilshed measurements, whos analoges where known in Newtonian mechanics too, see e.g. correspondence principle. Or you might really be interested in an empty universe and yes, you put in that emptyness by hand. And if you want no matter, i.e. make the right hand side of the Einstein field equation zero, then there is still some playground, see e.g. the Einstein-manifolds. The theory doesn't say "You can't model the system whos solution is contains Schwarzschild metric, because hey, there are alot of other masses in the universe". No, you ... ah, thankfully there are some elaborations on wikipedia. The cosmological solutions, i.e. the ones concerting more global question, also make some convinient assumptions, like I don't know, isotopy. Things that are "plausible" within the language of the theory.

Btw. are you a mathematican? I like your way of being skThen I have a small question regarding the formulation "the world is manifold M", because to me that word "world" is open to at least interpretations. If by world you think of the four dimensional manifold M in typical general relativity solutions, then matter is not part of the world, but "in" the world. On the one hand you say "at every point" but then you also say "a 'world' without matter".

I also don't really know what is ment by "what is a complete statement"?

So after this, some of your questions might be clarified or you might at least be ale to ask differently, in case others have problems too.