The Principle of General Covariance

[Metric]

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Suppose we compare and contrast GR (pure general relativity with no additional fields) with, let's call it, GRCF (general relativity with some additional classical fields).

So these are purely classical field theories, no quantum anything, that incorporate GR in an appropriate way.

Some questions (sorry for repeats):

Is it true that in GR the metric g on differentiable manifold M, completely determines the solution?

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Yes.

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By contrast, is it true that in GRCF the metric g on differentiable manifold M, does NOT generally completely determine the solution?

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Yes. More fields imply more equations of motion need to be solved for a complete solution.

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Is it true that Einstein's equation gives the stress-energy tensor in terms of the metric, generally in any GRCF solution?

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Yes.

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What constraints are there on the kind of metric g that can appear in a solution of a GRCF?

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General covariance makes distinguishing between "constraints" and "dynamics" very tricky.

To see why this is the case, we have to look at the Hamiltonian form of the theory. In, e.g. electromagnetism, we choose an "initial state" of the fields, and then the Hamiltonian evolves them into the future. But not all choices of the "initial state" of the fields are allowed -- i.e. there are some constraints (e.g. Gauss's law) that the fields must satisfy. In generally covariant theories, though, the entire Hamiltonian is a constraint -- it doesn't generate evolution in the normal way. This is one way that time is a very subtle issue in GR.

This is true both in "pure" GR and in GR+classical fields. It's covariance which forces this somewhat "odd" situation on us.

So I hope this was the sense in which you were asking about "constraints" -- it may be that I have gone off on a tangent because the word "constraints" refers to a very specific issue in GR.

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What further constraints are there on the kind of metric g that can appear in a solution of a GR?

Is there some quantity F(g) that can be computed from g that is identically zero for a solution of GR, but is generally non-zero in a GRCF? (So the answer to the previous question would be `the constraint is F(g)=0'.)

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You could say that the Einstein's equation is such a function. In pure GR, F(g)=0, and in GR+CF you have F(g)=T. But I get nervous calling this a "constraint," since that word has a special meaning in mechanical theories, and both GR and GR+CF satisfy a Hamiltonian constraint.

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In a GRCF, what would F(g) tell you about?

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If you take F(g) to be the "left hand side" of Einstein's equation, then as you already know, it's value tells you about the local energy and momentum density at any spacetime point.

[MusashiStyle]

"the following way:

F_1(x,t)...F_4(x,t) --> x(F_1...F_4), t(F_1...F_4)

Then, we can express the remaining n-4 variables in terms of the first four as follows:

F_m(F_1...F_4) = F_m(x(F_1...F_4),t(F_1...F_4)) (here m = 5...N)

THESE are functions which can be compared directly to "

excellent post but I was slightly confused here.

maybe u could repeat this with more word detail to make it more comprehensible?

[Charon]

Maybe it is enlightening to look at general relativity as an example of a general covariant theory.

In GR the Einstein field equations are 10 equantions, but only 6 of them are independent because of the Bianchi identities. Since the metric tensor has 10 independent elements, it seems that there are not enough equations to find an unique solution for the metric.

But this is what you want, because in GR you want to apply general coordinate transformations, such that the 4 new coordinates are arbitrary analytic functions of the original ones. That means that, if there is a solution to the Einstein field equation, that there exists a 4-parameter family of such solutions. Although these solutions give a different metric, they have the same physical content. So the theory doesnt depend on your choice of coordinates.

This basically is what Metric said in a more general fashion; you want to get rid of the coordinate dependence by expressing them in dynamical variables:

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F_1(x,t)...F_4(x,t) --> x(F_1...F_4), t(F_1...F_4)


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So you need 4 variables for that. Now the remaining n-4 dynamical variables can be expressed in the other 4 dynamical variables:

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F_m(F_1...F_4) = F_m(x(F_1...F_4),t(F_1...F_4))

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I hope this cleared it up a bit.

[Metric]

Basically, in a generally covariant theory, you may find yourself solving differential equations for a field F as functions of the coordinates (x,t), but the result F(x,t) cannot be compared directly to experiment. Experimentally, one can only compare fields F to other fields F' (more generally, substitute the word "dynamical variable" for "field").

Misunderstanding this principle has lead to some embarrassments in the literature -- for example, I have heard that following the Apollo moon landings, some ultra-precise distance measurements were made. But for a while it was a meaningless "coordinate distance" that was reported, rather than a physical length which depends on the value of certain fields (e.g. the metric).

[Metric]

Thank you, Charon, for helping with the explanation.

[thylacine]

What is the equivalence that fully describes when two solutions (M,g) and (M',g') described physically identical situations? Presumably this is what your post was about. Are they given exactly by diffeomorphisms? Do the infinitessimal diffeomorphisms correspond to (all or some) vector fields which are in (and are the only things in) the Lie algebra of the Lie group of diffeomorphisms? Do the 4 dimensions of the vector field (at each point) correspond somehow to the 4 fields you "solved away" in your post?

If a vector field V gives an infinitessimal diffeomorphism, transforming g to (say) g_V, then what kind of expression gives g_V in terms of g and V? Is this basically enough to consider for general covariance?

Also why do diffeomorphisms need to be expressed in terms of analytic functions, or don't they, in which case what are the conditions on what kinds of functions can be used, and what are the effects on the mathematics and the physics of such a choice of conditions.

BTW in an earlier post I used the word `constraint' just as a general synonym for `condition', `restriction', `imposed equation', etc. I guess I was just asking what conditions are required to be satisfied by (M,g) to be able to answer "yes" to the question `is (M,g) a solution?' without asking to what extent, and in what sense, does partial information about (M,g) determined not-so-partial information about (M,g).

[Metric]

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What is the equivalence that fully describes when two solutions (M,g) and (M',g') described physically identical situations? Presumably this is what your post was about. Are they given exactly by diffeomorphisms?

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Yes, provided all predictions can be gotten from (M,g).

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Do the infinitessimal diffeomorphisms correspond to (all or some) vector fields which are in (and are the only things in) the Lie algebra of the Lie group of diffeomorphisms? Do the 4 dimensions of the vector field (at each point) correspond somehow to the 4 fields you "solved away" in your post?

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Yes. You will have four spacetime vectors which generate infinitesimal diffeomorphisms, and they correspond to the four spacetime coordinates that you have to get rid of to make physically meaningful predictions.

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If a vector field V gives an infinitessimal diffeomorphism, transforming g to (say) g_V, then what kind of expression gives g_V in terms of g and V? Is this basically enough to consider for general covariance?

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g_V can be thought of as an "infinitesimal variation," written down in terms of the Lie derivative with respect to V.

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Also why do diffeomorphisms need to be expressed in terms of analytic functions, or don't they, in which case what are the conditions on what kinds of functions can be used, and what are the effects on the mathematics and the physics of such a choice of conditions.

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I'm not quite sure, actually. A lot of the time we physicists tend to restrict ourselves to "well behaved functions" in order to avoid "pathological" situations which are artifacts more of the use of crazy functions than the underlying physical principles. In physics, the game is often to minimize such issues for the sake of conceptual clarity.

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BTW in an earlier post I used the word `constraint' just as a general synonym for `condition', `restriction', `imposed equation', etc. I guess I was just asking what conditions are required to be satisfied by (M,g) to be able to answer "yes" to the question `is (M,g) a solution?' without asking to what extent, and in what sense, does partial information about (M,g) determined not-so-partial information about (M,g).

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I see that you're just trying to get all the general little details (such as the fact that the metric is Lorentzian, rather than Riemannian), but as I mentioned before some of these are formalism-dependent. The only real "imposed equation" after the variables and formalism are selected is the dynamics itself -- i.e. Einstein's equation.

[thylacine]

Thankyou very much for all your answers. [img]/images/graemlins/smile.gif[/img]