Are there controversies surrounding the principle of general covariance in GR?

Question

I'm a physics graduate now working with computers. I study GR in my spare time to keep the material fresh. In the Wikipedia article about the mathematics of GR, one can read the following:

The term 'general covariance' was used in the early formulation of general relativity, but is now referred to by many as diffeomorphism covariance. Although diffeomorphism covariance is not the defining feature of general relativity, and controversies remain regarding its present status in GR, the invariance property of physical laws implied in the principle coupled with the fact that the theory is essentially geometrical in character (making use of geometries which are not Euclidean) suggested that general relativity be formulated using the language of tensors. [My italics.]

Do anyone know what kind of controversy the author(s) may be aiming at? Isn't general covariance, ehrm ... diffeomorphism covariance, a founding principle of GR?

UPDATE: Evidently there is no "right" answer to a question like this (unless you happen to be the author of said article and thus could share with the world what you where aminig at). Anyway, it seems as there isn't a widely known, heavily debated controversy regarding general covariance. Even so, I've chosen to accept Ron's answer.

UPDATE 2: I've retracted the acceptance due to the linked article by prof. Norton. I think that, for all practical purposes, Ron's answer still stands, yet I want to review said article first. However, nobody should hold their breath waiting for me to figure this out. :)

Answer 1

The controversy is due to the fact that the statement "the laws of physics should be generally covariant" is very striking and meaningful, it includes the equivalence principle, relating accelerated motion and a local gravitational field. But when formulated as "The laws of physics should be invariant under coordinate changes" it becomes trivially easy to fulfil--- any laws of physics can be described in any coordinates, just by changing the coordinates! For example, if you have Laplace's equation, you can change the coordinates and reexpress Laplace's equation in elliptical or spherical coordinates.

So people who believe that the physics is like mathematics would like to give a mathematical axiom which corresponds to the physical principle of "General Covariance", and they identify this axiom as "The laws of physics must be expressible in arbitrary coordinates", and since this statement is trivial and content-free, they conclude that General Covariance is content free, hence the controversy.

This controversy is not so interesting. The statement of General Covariance starts with the equivalence principle, which states that a locally accelerated frame is equivalent to a gravitational field. Since the dynamical quantity that determines the local acceleration is the metric and the associated connection, you conclude that the connection and the metric are the gravitational field and potential. Then you formulate laws of motion for the field and the potential. The equations of motion have to be sensible--- no unstable runaway solutions, positive energy of small perturbations. Then the principle of fewest-number-of-derivatives (scaling most relevant terms) picks out GR plus perhaps some topological terms.

The principle of General Covariance is just that there is no a-priori preferred metric, that the metric is determined by local equations of motion, like the electric potential. You don't have any "restoring force" pulling the metric to +1,-1,-1,-1, or any other value. So that the following ridiculous action violates general covariance:

$$ S = \int ( R + (g_{\mu\nu} - \eta_{\mu\nu})(g^{\mu\nu} - \eta^{\mu\nu}) ) \sqrt{g} $$

There is not much more to General Covariance than disallowing explicit tensors, like the above.

The principle is simply that the theory must be geometrical, with no preferred background geometry. This is emphasized today by the people who like loop quantum gravity.

Answer 2

There has never been a water-tight, rigorous, cut and dried formulation of what one means by « general covariance,» hence the controversies ever since Einstein's day. See http://philpapers.org/rec/STATMO-5 and other papers about the philosophy of physics. Mr. Maimon's answer also evidences the existence of a controversy, since he criticises many people's ideas of what it means as « content free,» i.e., vacuous. And with some justice...on both sides. There are also differences of opinion about what « the equivalence principle » says, again Mr. Maimon has pointed out this in other answers to related, linked, questions such as « How does this thought experiment not rule out black holes?» Prof. Geroch of Chicago ( http://arxiv.org/abs/1005.1614 ) has pounted out misunderstandings of even Special Relativity, showing that the laws of Spec. Rel. would not be violated by faster than light communication ... so one should be cautious about listening to only one point of view here.

There is certainly controversy about whether Diff(M) is any kind of a gauge group (see prof. Streater's http://www.mth.kcl.ac.uk/~streater/lostcauses.html , and there seems even to be a lot of confusion about the difference between an element of Diff(M), which is global, and local changes of coordinates, which are the ones relevant to the principle of general covariance. (After all, Einstein always stated the equivalence principle only for one coordinate patch, not for all of $M$.)

My opinion is that Einstein's understanding of the principle of general covariance had a physical part and a mathematical part. The physical part was the principle of equivalence. The mathematical part was that a physical law should be written in terms of quantities with the same transformation properties under local changes of coordinates. The fact is that there does not seem to be any clear-cut way to say this, and Einstein did not give any completely abstract definition of the principle of covariance, so my opinion is based on looking at what he did. Physicists even in his day misunderstood the principle of equivalence and the requirement of general covariance, one of them even accusing Einstein of betraying it, see pp. 237-239 of Volume 6, English translation supplement, of Einstein's Collected Papers.

I may have found the reference that you would like to see, although it is very in-depth. http://www.pitt.edu/~jdnorton/papers/decades.pdf by prof. Norton of Pittsburgh talks about many of the misunderstandings and disagreements about this principle.

Answer 3

Related questions Why can't General Relativity be written in terms of physical variables ? and Diff(M) and requirements on GR observables

I think the "controversy" being referred to is the invariance of GR under active diffeomorphisms and its corresponding interpretation as a gauge theory. An active diffeomorphism is to be thought of as moving the points of the spacetime manifold around, not just as a relabelling of points with new coordinates. GR has the property that, if you do this consistently, you map solutions of Einstein's equations to new solutions of Einstein's equations (see some discussions on Einstein's Hole argument to get a feeling for this). Thinking of this as a gauge freedom, you end up with with the space of solutions of Einstein's equations being partitioned into equivalence classes.

Traditionally, in gauge theories, the physical observables are gauge invariant quantities - in this case, these would be quantities which are preserved under active diffeomorphisms. I don't think a complete set of such quantities is known, but they would include things like integrals of curvature tensor invariants over all spacetime. These observables are generally non-local, and haven't proved too useful for explicit calculations (as far as I know).

I don't think this so-called controversy is anything significant. It seems to just revolve around whether or not GR should have the term "gauge theory" applied to it, since the character of the theory is different to other gauge theories in which the gauge symmetries apply fibre-wise in the relevant bundles.

Answer 4

EDIT: I misunderstood the question, so my answer deals with 'controversies concerning equivalence principle'. Sorry for the mess.

Well... that is the point. As you may know, Einstein equations can be derived from Hilbert action principle. Now, if you want to add matter, you add its lagrangian to the action. The strong equivalence principle tells you, that you can not use any explicit geometrical entities in lagrangian of matter fields. This is to be understood as follows. Let's say that you have a lagrangian for a scalar field in Minkowski spacetime $$ L = \partial_\mu \, \phi \, \partial_\nu \phi \, \, \eta^{\mu \nu} - m^2 \phi^2 $$ Then, you can generalize it to curved spacetime like this $$ L = \sqrt{-g} (\nabla_\mu \, \phi \, \nabla_\nu \phi \, \, g^{\mu \nu} - m^2 \phi^2) $$ but not like this $$ L = \sqrt{-g}(\nabla_\mu \, \phi \, \nabla_\nu \phi \, \, g^{\mu \nu} - m^2 \phi^2 + R\phi^2) $$

Since in vacuum scalar curvature R is zero one might be tempted to include it in the lagrangian and in fact people do it - for many reasons. Even I do it sometimes to study some interesting mathematical implications of such a choice. This, however, strong equivalence principle, as laws of physics in inertial coordinate systems are no longer equivalent! So, in short, people are tempted to add other physical theories to GR without any respect to general covariance because either they do not believe in it, or because such couplings are sometimes more interesting than 'normal' ones.