The Principle of General Covariance

[Metric]

Partly to get my mind off some recent absurd arguments generated by race-baiting, I present here an introduction to the principle of "general covariance" -- one of the most fundamental and commonly misunderstood principles of nature.

"General covariance" is a pretty controversial topic in physics, and differing views of covariance have for example been responsible for the rift between superstring theory and loop quantum gravity. Below, I'll give a bit of an introduction to what covariance is about.

One meaning of covariance has to do with Lorentz transformations. A theory is said to be covariant if the equations and predictions transform in a well-defined way under Lorentz transformations (moving to a new frame of reference in relative motion with the first one). Electromagnetism, the standard model of particle physics, etc. is all "covariant" in this sense.

But there is a deeper principle of "general covariance" associated with general relativity. It is this property which made the theory so difficult to understand -- both Einstein and the great mathematician Hilbert struggled independently for years to find the equations of general relativity, and although Hilbert had far greater mathematical ability, Einstein had a unique ability to see and understand the fundamental principles (though he himself had abandoned general covariance for some time in the face of predictions which he misinterpreted).

The principle is this: A generally covariant theory makes no distinction between "dependent" and "independent" observables. To see this, let's consider an example. Traditional electromagnetism is covariant in the special relativistic sense -- the theory knows how to handle Lorentz transformations. But it is not generally covariant. Why? The theory is designed to tell us, for example, the value of the electric field "E" at each spacetime point "x". To test the theory, I need a device that measures "E", but I also need a device to tell me where I am -- what point "x" we are talking about. Without both of these measurements, the theory cannot make predictions. Now, the equations of the theory determine "E" -- but they do not determine "x"! The spacetime points are assumed to live in the background, and are not effected by the dynamics of the theory. They simply exist, and can be measured in order to make predictions of electromagnetism possible. The measurable quantity "E" depends on "x", but the measurable quantity "x" does not depend on "E"!

General relativity, which posesses the property of general covariance, is quite different. Predictions of the theory are not the values of various fields at various spacetime points "x". If you try to make the theory work like this, predictions become non-unique (this is what initially confused Einstein and baffled Hilbert). Covariance is essential. In GR, to make predictions, we can only compare dynamical variables to other dynamical variables. Coordinates have no meaning -- they are used only as a mathematical book-keeping device. To make a prediction, we have to "solve away" dependence on coordinates.

For example, let's say that there are n dynamical variables F_1 ... F_n in the theory, and four space-time dimensions (x,t). The equations tell us how to solve for F_1(x,t)...F_n(x,t) -- but as stated before, predictions of the theory cannot refer to the coordinates, which are arbitrary and not observable. Instead, we have to pick out four dynamical variables to serve as a "material reference frame" and solve away (x,t) in 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 experiment. As Carlo Rovelli has written, "The world is made up of fields. Physically, these do not live on spacetime. They live, so to say, on one another. No more fields on spacetime, just fields on fields."

It is this principle which makes quantum gravity hard -- the idea that predictions should be "background free" in some sense. As I mentioned before, loop quantum gravity theorists take a "hard line" approach to this principle, making it covariant from the very start -- while superstring theorists tend to do perturbation theory about a particular background (though they can still go from one background to another).

My personal feeling is that this is probably one of THE principles of nature. The idea of physically measurable fixed background structure which cannot be effected seems very unnatural to me.

[Borodog]

Very nice.

[Metric]

I suppose I will give this post a bump in the hopes of generating a question or comment before it is inevitably buried under the "meaning of life" stuff.

[bunny]

Well my comment is I enjoyed reading it. Unfortunately, I dont feel qualified to even ask an intelligent question. Perhaps one will come in time...

[cambraceres]

Is GR the only theoretical structure that does not need this "x" coordinate to make a "real" prediction?

What I mean is, in the first description of covariance, the one active in special relativity, there needs to be a spacetime measurement to have a valid prediction. In the second this is not the case, but is there any other artificial structure in science that follows this second line?

Sorry for the noobish question, but I'm noobish ya know?

Much Love

Cam

[Metric]

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Is GR the only theoretical structure that does not need this "x" coordinate to make a "real" prediction?

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One can construct other models or new theories that have this property, but GR is of course the one that people believe is a realistic theory. I should also point out that other field theories like electromagnetism etc. become generally covariant when combined with GR. So strictly speaking general covariance is not just a property of the gravitational field -- it is a property of all of physics the moment that gravity is included.

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What I mean is, in the first description of covariance, the one active in special relativity, there needs to be a spacetime measurement to have a valid prediction. In the second this is not the case, but is there any other artificial structure in science that follows this second line?

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One can actually formulate ordinary mechanical theories etc. in a covariant way, but the essential point is that they don't really need to be formulated in this way for us to use them effectively. Their dynamics takes a particular form which effectively says "you can pretend that there is a background structure without any loss of generality." But with GR, we can't do this anymore -- it is a theory in which we are forced to come to terms with covariance in order for the theory to make any sense at all.

But to answer your question a little more directly, no, I can't think of another well established physical field theory that explicitly requires general covariance quite apart from being connected to GR (though of course many "cutting edge" theories want to include it due to the lesson that GR has taught us).

[cambraceres]

But would you consider one of these cutting edge theories to be viable?

Cam

[evil twin]

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Very nice.

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[Metric]

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But would you consider one of these cutting edge theories to be viable?

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Sure... Superstring theory and loop quantum gravity are two examples, though as I mentioned before they have a subtly different take on covariance. LQG is TOTALLY background free -- states are made up of abstract "spin networks" that sort of build up spacetime from nothingness (the downside is that it is extremely difficult to make predictions starting out this way). Superstring theory lives with a background spacetime, but the theory doesn't exactly care which background is used (within reason) -- it retains the freedom to go back and forth between different ones (this approach is easier to live with from the point of view of making predictions, but at the cost of living with an arbitrary background).

But anyway, these are both certainly viable and they are both desperately trying to make testable predictions, though without much success.

[Skidoo]

Bump to the present.

Well written post. It chimes in with my own general observation of how many absolute frameworks have fallen in recent years mostly due to unexamined assumptions about their empirical nature being discovered to be incorrect. Even space itself isn't a sure thing to hang your hat on anymore.

[Charon]

Very good article.

As far as I know all the quantum field theories are not generally covariant, but are treated in a pertubative manner. Even for quantum electrodynamics and quantum chromodynamics we don't know if the full pertubation series converges, only that the first few terms of the series do agree nicely with experiments. Besides the fact that general relativity is non-renormalizable it seems contradicting to the classical view of general relativity to treat it in a pertubative manner. So do you think that pertubation theory is a likely method to achieve a unifying theory? (do you think that string theory is the best option to achieve unification atm?)

I don't know much about unifying theories, but somehow it seems to me that a unifying theory is not likely to be build in a pertubative manner. I wonder what you think is the most likely way to achieve unification. Start with the principle of general covariance and build up from there?

[Metric]

I think perturbation theory is a good way to jump right in and get some experimental feedback (at least it worked that way for most of the 20th century -- not so much these days), but as far as quantum gravity goes, I personally like to think about some of the main "problems of principle" that perturbative approaches are designed to ignore.

So I think that there should be a place for both approaches, even though the superstring approach completely dominates in terms of research resources being spent these days...

[thylacine]

I came across this (SMP) forum more recently than this original post, so I'm glad I've seen it now.

I've had a look at "Spacetime and Geometry" by Sean Carroll, so I have a rough idea about GR, but there are many gaps in my understanding, so I have some questions.

The book develops the theory in the first four chapters so that at that point if you understood it all, then you know what GR says, or at least understand it as a piece of mathematics without yet delving into the physical consequences. Unfortunately, I would have liked to see a compact summary of what all the relevant quantities are, what conditions are imposed, what equivalence relations are there, and what has been assumed without-loss-of-generality in situations where a different WLOG-based convention could have been chosen.

Firstly, can a solution to Einstein's equation be completely described (perhaps non-uniquely) by a differentiable 4-manifold M with a metric g_{\mu\nu}. It seems that once you have g then you have \Gamma, R (with 4,2, or 0 indices) and the energy-momentum tensor T, so all these quantities just come along for the ride once you have g. Is that right?

What conditions are imposed on M and g that qualify them as a solution? Certainly g has a certain signature, it is symmetric, there are certain differentiability conditions, but what else? I saw in the book certain identities, e.g. Bianchi's identity, but I lost track of if this is just true always, of if it is imposed for some reason, or if it can be assumed without loss of generality, or what.

Are there some conditions imposed, that did not need to be imposed but which can be (and which in practice are) for without-loss-of-generality type reasons. For example suppose that a certain tensor could be allowed to be not symmetric in a certain two indices, but that a physically identical situation would be described when the tensor is symmetrized in those two indices. Then it is natural to impose the appropriate symmetry condition, but it is not a condition designed to exclude non-physical solutions, but rather, it is a natural convention for choosing one representative solution from an equivalence class of physical solutions. So which conditions play such a role?

Is there actually any other information other than M and g needed to fully describe a solution, where that other information could take the form of specifying some convention that has been chosen (e.g. the covariant derivative or the connection)?

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? How does this go?

What ways other than (M,g) can a solution be represented, e.g. Ashtekar variables(?) and what are the answers to all the above questions in that case? (You can treat the latter part as rhetorical, I am asking too much by now.)

I guess one way of keeping track of the effect of imposing various conditions, conventions, and equivalences, is to count the number of degrees of freedom at each point, right? Perhaps there is an exception when diffeomorphisms are brought in, which can map any point on M to any other point on M, but it still seems there should be some way to count degrees of freedom. Is there? Does it simply take away 4 degrees of freedom?

And is it still nevertheless true that GR has a spacetime consisting of (utterly pointlike) points? Or have I utterly missed the point?

[Metric]

Yipes, that's a swarm of good questions there -- I don't think I have the energy to answer all these in a single post (some of which I'd have to check up on to make sure I don't say something mistaken).

Suffice it to say that there are many equivalent formalisms describing GR -- e.g. some involve a metric compatable, torsion free connection, some do not. But all these formalisms are related. Here's an excellent review paper on different GR formalisms and their relations to one another -- see the figure on page 5 to get a feeling for the various creative ways people have done this.

http://arxiv.org/abs/gr-qc/9305011

[thylacine]

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Yipes, that's a swarm of good questions there -- I don't think I have the energy to answer all these in a single post (some of which I'd have to check up on to make sure I don't say something mistaken).

Suffice it to say that there are many equivalent formalisms describing GR -- e.g. some involve a metric compatable, torsion free connection, some do not. But all these formalisms are related. Here's an excellent review paper on different GR formalisms and their relations to one another -- see the figure on page 5 to get a feeling for the various creative ways people have done this.

http://arxiv.org/abs/gr-qc/9305011

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Thanks, that paper looks good and seems related to what I was asking. It probably assumes the reader starts with a better understanding of GR than what I have, but I'll see when I (try to) read it.

Actually most of my post (except 3rd last paragraph) asks all the questions it asks with respect to one formalism. But certainly understanding one formalism in the light of my questions, is very closely related to understanding the connections between the different formalisms.

[holmansf]

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What conditions are imposed on M and g that qualify them as a solution? Certainly g has a certain signature, it is symmetric, there are certain differentiability conditions, but what else? I saw in the book certain identities, e.g. Bianchi's identity, but I lost track of if this is just true always, of if it is imposed for some reason, or if it can be assumed without loss of generality, or what.

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I don't have much of a clue about GR, but I do know mathematics reasonably well. The Bianchi identity is always true. It is a consequence of the symmetries of the curvature tensor.

I don't know if this addresses your other questions, but my understanding is that GR says that a physically realistic spacetime (M,g) is one where g is a pseudo-Riemannian metric with the correct signature that satisfies Einstein's equation. Einstein's equation is a good one, because it is the variational equation for critical points of a certain functional (the Hilbert action, I think) on metrics with a certain signature (Lorentz metrics). If you choose different actions, you get different variational equations.

Ok, I better stop, because as I said above, I really don't have much of a clue about GR.

[thylacine]

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What conditions are imposed on M and g that qualify them as a solution? Certainly g has a certain signature, it is symmetric, there are certain differentiability conditions, but what else? I saw in the book certain identities, e.g. Bianchi's identity, but I lost track of if this is just true always, of if it is imposed for some reason, or if it can be assumed without loss of generality, or what.

[/ QUOTE ]

I don't have much of a clue about GR, but I do know mathematics reasonably well. The Bianchi identity is always true. It is a consequence of the symmetries of the curvature tensor.

I don't know if this addresses your other questions, but my understanding is that GR says that a physically realistic spacetime (M,g) is one where g is a pseudo-Riemannian metric with the correct signature that satisfies Einstein's equation. Einstein's equation is a good one, because it is the variational equation for critical points of a certain functional (the Hilbert action, I think) on metrics with a certain signature (Lorentz metrics). If you choose different actions, you get different variational equations.

Ok, I better stop, because as I said above, I really don't have much of a clue about GR.

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But I don't see how Einstein's equation places any contraints on the metric. Instead it seems that it just defines the energy-momentum tensor in terms of the metric. If you just wanted to consider GR solutions (M,g), and didn't care what the energy-momentum tensor is then you could simply disregard Einstein's equation. And if you ever did want to know the energy-momentum tensor, then that's what you would use Einstein's equation for (and that's all you'd use it for).

Is this right? Somewhat right in some qualified sense? Wrong? Not-even-wrong?

[holmansf]

Your questions are probably best left to more of an expert. I thought the energy momentum tensor represented in some way the masses (and their momentums) present in a system. This is the sense in which mass "curves space time" (it determines the metric).

[Metric]

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But I don't see how Einstein's equation places any contraints on the metric. Instead it seems that it just defines the energy-momentum tensor in terms of the metric. If you just wanted to consider GR solutions (M,g), and didn't care what the energy-momentum tensor is then you could simply disregard Einstein's equation. And if you ever did want to know the energy-momentum tensor, then that's what you would use Einstein's equation for (and that's all you'd use it for).

Is this right? Somewhat right in some qualified sense? Wrong? Not-even-wrong?

[/ QUOTE ]
You're right in the sense that if you have the metric, and you know it's a solution, you can solve for the stress-energy tensor. But rarely are you just handed a metric that happens to solve Einstein's equation.

Einstein's equation comes from a variational principle -- the stress energy tensor happens to be the term you get when you vary the matter Lagrangian with respect to the metric tensor. So one of the equations of motion (in more variables than just the metric tensor) happens to be Einstein's equation. A complete solution to your theory involves not just the metric, but all other fields as well (for example, the electromagnetic field, in terms of which the stress-energy tensor will be defined). Of course once you have a solution, information about the stress-energy tensor is also encoded in the metric itself as you noticed.

[thylacine]

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But I don't see how Einstein's equation places any contraints on the metric. Instead it seems that it just defines the energy-momentum tensor in terms of the metric. If you just wanted to consider GR solutions (M,g), and didn't care what the energy-momentum tensor is then you could simply disregard Einstein's equation. And if you ever did want to know the energy-momentum tensor, then that's what you would use Einstein's equation for (and that's all you'd use it for).

Is this right? Somewhat right in some qualified sense? Wrong? Not-even-wrong?

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You're right in the sense that if you have the metric, and you know it's a solution, you can solve for the stress-energy tensor. But rarely are you just handed a metric that happens to solve Einstein's equation.

Einstein's equation comes from a variational principle -- the stress energy tensor happens to be the term you get when you vary the matter Lagrangian with respect to the metric tensor. So one of the equations of motion (in more variables than just the metric tensor) happens to be Einstein's equation. A complete solution to your theory involves not just the metric, but all other fields as well (for example, the electromagnetic field, in terms of which the stress-energy tensor will be defined). Of course once you have a solution, information about the stress-energy tensor is also encoded in the metric itself as you noticed.

[/ QUOTE ]

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?

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

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

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

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'.)

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

[Metric]

[ QUOTE ]
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. /images/graemlins/smile.gif