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. :)

  • $\begingroup$ Although @RonMaimon's answer is good, general covariance principle is not that physics can simply use any coordinate system (which is trivially true), but that the form of the equations should be invariant, under any coordinate choice. This is a postulate of some symmetry, but is there indeed such symmetry? This is the major controversy IMO $\endgroup$
    – Nikos M.
    Nov 10, 2021 at 16:18
  • 1
    $\begingroup$ You may be interested in this article $\endgroup$
    – Nikos M.
    Jul 6, 2022 at 20:23

4 Answers 4


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.

  • $\begingroup$ based on the recent answer by Mr. Josephson, I feel obliged to withdraw my acceptance of your answer. Sorry for the flickering behavior. $\endgroup$ Feb 1, 2012 at 11:21
  • $\begingroup$ @Joar Bolstad: Joseph f. Johnson (not to be confused with Josephson) is free to say what he wants, but my answer is really the correct answer. There is no real controversy about what General Covariance means, it means what I said. $\endgroup$
    – Ron Maimon
    Feb 1, 2012 at 11:31
  • $\begingroup$ I stand corrected. Mr. Johnson it is, of course. I agree that there probably isn't a great deal of controversy surrounding the subject, but in honor of Einstein's philosophical legacy I feel I need to consider even the philosophers objections to the matter at hand. Even so: I am reassured by your answer regarding the operational aspects of GR, and my previous upvote still stands. Thanks! :) $\endgroup$ Feb 1, 2012 at 12:16
  • $\begingroup$ Just a slight remark: general covariance starts with equivalence principle, but, in fact, it is much weaker than eq. pr. $\endgroup$
    – Terminus
    Feb 1, 2012 at 13:21
  • $\begingroup$ @Terminus: it depends on the interpretation of General Covariance. The way Einstein intended it, it means no fixed background, with the variations in the background metric to be interpreted as the local gravitational field. In this sense, it includes the EP and is stronger than the EP. But the principle, if formulated as "you can change coordinates" is vacuous and trivial, and is much weaker than anything, because it is the weakest statement you can make--- it is trivially true of any theory. $\endgroup$
    – Ron Maimon
    Feb 1, 2012 at 14:41

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.

  • $\begingroup$ thank you very much for a sobering and thoughtful answer. First of all, I would like to address my acceptance of Mr. Maimon's answer. I have no a priori reason to value Mr. Maimon's opinion over any of the other posters, and I'd like to stress that me accepting his answer over twistor59's is more a matter of being limited to only one accepted answer per question than anything else. My primary motivation for asking this question in the first place, was to uncover any largely debated controversy I didn't know about regarding one of the corner stones of GR. $\endgroup$ Feb 1, 2012 at 11:17
  • $\begingroup$ Since the answers seemed to diverge in opinion, I (hastily) assumed that no such controversy was known. I truly appreciate your effort, and in particular the reference to prof. Norton's article. Based on this new information I will withdraw my acceptance and, if time allows me, study prof. Norton's work and subsequently review all incoming answers to my original question. $\endgroup$ Feb 1, 2012 at 11:17
  • $\begingroup$ @Joar Bolstad: You can do what you want, but there is no controversy among literate people. The controversy is really only among philosophers, who you can't take too seriously because they are by and large mathematically illiterate. The meaning of the principle of general covariance is that you shouldn't use external background tensors in your equations, even if they are relativistically invariant. As for the Geroch article cited here, it is a total red herring. $\endgroup$
    – Ron Maimon
    Feb 1, 2012 at 11:35
  • $\begingroup$ I don't know why you go about discussing my answer--- you seem to think there is something wrong with it. There certainly isn't, but I'll let the OP figure that out for himself. $\endgroup$
    – Ron Maimon
    Feb 1, 2012 at 11:36
  • $\begingroup$ very good summary, +1 $\endgroup$
    – lurscher
    Feb 1, 2012 at 14:38

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.

  • $\begingroup$ I don't believe there is any valid difference between the "active" and "passive" point of view regarding diffeomorphisms. What's the point in talking about this difference when it doesn't exist? $\endgroup$
    – Ron Maimon
    Jan 31, 2012 at 8:01
  • 2
    $\begingroup$ Indeed you can represent an active diffeomorphism by coordinate changes, but the reason I chose to emphasize the active view is to make it explicit that the Diff(M) action can be interpreted as relating different metrics, not just different representations of a given metric tensor object by coordinate functions. This active view relates nicely to the hole argument. The controversy that I'm thinking of is not active vs passive, but the question of whether Diff(M) invariance is really a gauge theory(Weinstein for example. IMO it is a vacuous discussion. $\endgroup$
    – twistor59
    Jan 31, 2012 at 9:50
  • $\begingroup$ coordinates don't have to be global. If the Laws of Nature can be expressed by local diff. eqs, then examining their transformation properties inside one coordinate patch with respect to a change of (local) coordinates is one thing, which I think is what Einstein intended (using the example of the lift), and talking about a diffeomorphism is quite another. Especially if the Universe is compact. $\endgroup$ Feb 1, 2012 at 15:45

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.

  • $\begingroup$ Why are they no longer equivalent? Which transoformations in these (I guess local) intertial systems do mess what up? $\endgroup$
    – Nikolaj-K
    Jan 30, 2012 at 23:32
  • $\begingroup$ This answer is completely wrong. Both expressions are generally covariant. If you use $\eta_{\mu\nu}$ instead of $g_{\mu\nu}$, that's not covariant. $\endgroup$
    – Ron Maimon
    Jan 31, 2012 at 8:00
  • $\begingroup$ Terminus: So basically it reduces the overall mass by $2R$? $\endgroup$
    – Nikolaj-K
    Jan 31, 2012 at 8:11
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    $\begingroup$ @Terminus: Your principle is called "minimal coupling", not "General Covariance". Nonminimal terms can be used when you have a nonstandard coupling of scalars to gravity, as in the Coleman et. al. improved energy tensor coupling, which used an R term. This does not violate General Covariance. $\endgroup$
    – Ron Maimon
    Jan 31, 2012 at 8:23
  • $\begingroup$ Ok, i misunderstood the question - I thought it is concerned with strong equivalence principle - which is of course violated by nonminimal terms as it states less-more 'all identical non-gravitational experiments should give the same results in all inertial frames' $\endgroup$
    – Terminus
    Feb 1, 2012 at 13:07

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