# General covariance

1. Oct 2, 2012

### zonde

I am trying to understand what exactly general covariance states. As I understand general covariance appeared as generalization of relativity principle so I will try to state relativity principle in a manner that I consider more convenient for my purpose.

So let's say we have inertial coordinate system K and in that coordinate system we have coordinate dependent formulation of physical law A. Now in a certain way we transform inertial coordinate system K into inertial coordinate system K' and in that new coordinate system we have coordinate dependent formulation of physical law A' that takes the same mathematical form as law A in K. Relativity principle states that if law A in K holds then law A' in K' holds as well.
And we can experimentally test this statement. We take well tested physical law A in K then find K', formulate A' and then translate K' back into K along with physical law A' so that we get physically identical coordinate dependent law B as law A' but in different mathematical form as law A.
Within coordinate system K we test law B and if it holds we say that relativity principle holds.

So we can use relativity principle to formulate new coordinate dependent physical law B in K if we have coordinate dependent physical law A in K. This might not be very popular formulation of relativity principle but nonetheless just as valid. And the point of this formulation is that relativity principle leads to a new physical laws within single coordinate system.

Now the question about generalization of relativity principle to general covariance. In what sense relativity principle is generalized to arrive at general covariance?
I would imagine that general covariance applies to coordinate dependent laws in any coordinate system (with primary interest in accelerated coordinate systems) if we have such laws (that are most conveniently formulated in accelerated coordinate system). And then transforming this coordinate system in a certain way we arrive at new laws (with the same mathematical form as primary law) that we think will hold in this new coordinate system. And of course we can translate it back into original coordinate system and get new laws in the same coordinate system (but expressed in different mathematical form as primary law).

Does this seem correct?

2. Oct 2, 2012

### Ben Niehoff

In some sense, the notion of general covariance is trivial. All it says is that we should be able to write physical laws in a way that does not depend on the coordinates used; i.e., that physical laws should take the same form in every coordinate system.

However, ANY physical theory can be written in such a way. So it is not really any restriction on the kinds of physics we can write down; it just tells us we should be clever enough to be aware that coordinates are just a way to label points, and hence arbitrary.

3. Oct 2, 2012

### PAllen

There are a number of subtleties here. At this moment, I only have time to make a few brief observations:

1) Einstein originally hoped that general covariance, or the principle of general relativity, would be analogous to the principle of relativity in SR. The most fundamental feature of the relativity principle of SR is that 'inside a box' you truly cannot distinguish one state of inertial motion from another. Obviously, within a box, you can tell if you are accelerating. The hope was that at least you could say that you can't distinguish acceleration from gravity: thus, even if you feel an inertial force inside a box, you still can't tell your actual state of motion. However, you certainly can tell you are not 'inertial'. Further, the equivalence is between acceleration and gravity is only true to a good approximation, not exact as for relativity in SR. This came to be called the principle of equivalence (in Einstein's formulation; some modern formulations are quite different).

2) Then, general covariance was initially thought by Einstein to be able to serve a filter of admissible theories. A possibly correct theory would have to be expressed in terms of geometrical objects that are covariant with respect to coordinate transformations (tensors, vectors, invariant scalars). In modern terms, a law should have a coordinate free expression. Unfortunately, it was almost immediately shown that any law could be written in such a form by introducing 'absolute' geometric objects. So this formulation ended up having no physical content. Note, that once a law is expressed in the appropriate way, the fact that it makes identical predictions in every possible coordinate system is mathematical fact. To disbelieve it is to claim mathematical definitions and proofs of differential geometry are wrong.

3) To restore meaning to what Einstein intended, various alternative principles have been proposed:
- A 'good' theory has no 'absolute' geometric objects. To my knowledge, this idea has never been rigorously formulated.
- The principle of minimal coupling states how matter and non-gravitational laws should couple to gravity. However this really functions more as a formalization of the principle of equivalence rather than a relativity principle.

The bottom line is that, for the most part, general covariance ends up being used today in sense (2) - a rule for formulating theories so they are coordinate independent by construction. When someone then comes along saying they doubt general covariance, it looks quite silly: "I don't believe the definitions and proofs of differential geometry are correct."

4. Oct 2, 2012

### PAllen

Here is the quip version of general covariance:

physics is unaffected by whether you choose to use polar coordinates or rectilinear coordinates - or any other coordinates. Physics does not operate in coordinates.

5. Oct 2, 2012

### dextercioby

AFAIK general covariance is pretty straightforward, the equivalence principle is the sensitive point in classical relativity. What I'm saying it that your worries needn't be here.

6. Oct 2, 2012

### zonde

Tanks for replays!

From what you (and Ben Niehoff) say I understand that your understanding of general covariance is that it lacks physical content.
But in that case general covariance is not generalization of relativity principle, right? Because relativity principle has physical content (and I am ready to defend that part).

7. Oct 2, 2012

### PAllen

Correct. Where it comes into play is when someone says: Kruskal coordinates represent different physics than SC coordinates. That is analogous to saying the geometry of a Euclidean plane is different if you draw polar coordinates on it rather than rectilinear coordinates.

8. Oct 3, 2012

### haushofer

Of course, but here we talk about general coordinate transformations. In GR, both the field equations AND the solutions are invariant under gct's. That's non-trivial. E.g., in Newton-Cartan theory one does not have this feature; in order to find solutions of the gravitational theory, one has to gauge-fix, and of course in this process the gct's break down to Galilei-transformations plus linear accelerations. This is obvious; one is just describing Newtonian gravity, and the flat spatial background is not invariant under gct's, but only under the Galilei+accelerations group (Milne group).

The modern point of view is that one can always introduce Stuckelberg fields to make an action or field equation invariant under certain gauge symmetries. An example is the massive vector field, in which one can introduce a U(1) gauge symmetry by adding a scalar field (see e.g. Hinterbilcher's notes on massive gravity). As such one can wonder what it means to write down gauge-invariant equations of motion.

From a field theory point of view general covariance is a consequence of the fact that one is describing self-interacting massless spin-2 fields. At lowest order this is just Fierz-Pauli theory, and one needs linearized gct's to make sense of the theory. So general covariance is not a defining property of the theory, but a consequence. There is a procedure to obtain the full non-linear theory, namely GR, of this Fierz-Pauli theory.

To come back to your statement: if both the gravitational and inertial mass of a particle would be different, would one then still be able to write down a theory of gravity of which the solutions of the gravitational field are gct-invariant?

Last edited: Oct 3, 2012
9. Oct 3, 2012

### bcrowell

Staff Emeritus
This may be helpful: http://arxiv.org/abs/gr-qc/0603087 , Some remarks on the notions of general covariance and background independence, Domenico Giulini.

From a modern point of view, what may be more interesting than general covariance is background independence.

10. Oct 3, 2012

### PAllen

I am not familiar enough with QFT to answer any your field theory questions/issues.

As for this distinction: " In GR, both the field equations AND the solutions are invariant under gct's", can you clarify what you mean? What I've read are completely coordinate free descriptions of Newton-Cartan gravity. The geometry is not pseudo-riemannian (there is no spacetime metric, only a spatial metric). In a coordinate free formulation, I don't understand how to make the distinction you allude to.

For concreteness, we can refer to Newton-Cartan as presented in Box 12.4 (p.300 of my copy) of MTW. See also section 12.5.

11. Oct 3, 2012

### PAllen

Right, I prefer to keep separate the trivial 'general covariance' from efforts to distinguish GR from other geometrically formulated theories based on some physical principle. In this sense, general covariance is nothing more than: how to formulate a theory so it is clear how to work with it in any coordinate system (to obtain the same physics).

12. Oct 3, 2012

### haushofer

Well, that's one reason why these notes of Hinterbilcher on massive gravity are so interesting: he shows how one can extend massless Fierz Pauli, which is a spin-2 theory on a flat background, to GR, which is background independent (BI).

In that sense I have sometimes the feeling that some physicists dweep with BI, for instance when they critize string theory. It's obvious that doing perturbation theory on a theory of gravity is not background independent, but the example above shows that a non-perturbative extension can still be BI.

13. Oct 3, 2012

### haushofer

Yes, the field equations are general covariant (gc). Let me compare with GR.

In gr, the einstein equations are gc. Now i can solve for the metric, giving e.g. the Schwarzschild solution. However, i can still perform a gct on this solution, and the result is just the same metric in another coordinate system. It is again a solution of the einstein equations.

In newton- cartan the " einstein equation" as shown in MTW is also gc. I can again solve for the metric. This gives me that space is flat, and all the other metric components gather into a Galilei-scalar which is the newtonian potential. But now this solution is not gct invariant, but only under a subgroup of them: the galilei group plus linear accelerations.

Of course, in solving for the metrics, you need to choose coordinates. Perhaps that is where the confusion comes from.

14. Oct 3, 2012

### PAllen

I still don't see this. It seems to be a matter of what objects you use to represent the solution. If you express the specific solution in terms of non-covariant objects ... it is not covariant. If you express the solution in terms of the objects used in the covariant equations, the solution is covariant.

15. Oct 4, 2012

### haushofer

What do you mean exactly with that last phrase? Can you give the solution of NC as you describe?

To put it differently: what exactly then is the difference between GR and NC in terms of covariance, according to you? That in NC the solution is not gct-covariant upon using coordinates, while in GR it is?

Last edited: Oct 4, 2012
16. Oct 4, 2012

### PAllen

I think the difference for GR is additional criteria like 'no absolute geometric objects' or 'background independence', nothing to do with gct covariance.

If you express the solution as a Newtonian potential on Cartesian coordinates, it is not gct-covariant. If you express the solution as Ricci tensor plus the various absolute geometric objects, it is gct covariant.

17. Oct 4, 2012

### haushofer

I agree, that's straightforward.

I wouldn't call that a solution; I'd call that the field equations:

$$R_{\mu\nu}(\Gamma) \sim \tau_{\mu\nu} \ \ \ \ \ \ \ \ (1)$$

The Riemann tensor can be written in terms of a connection Gamma, which on its turn can be deduced from two metrics (h and tau) via metric compatibility. However, one then needs an additional constraint (called the Trautman condition) because metric compatibility doesn't give Gamma uniquely, and there appears an extra "Coriolis term". Both (1) and the Trautman condition are gct-covariant.

But that's deceiving, because these equations are "Stuckelberged", in the sense that they are just Newtonian gravity supplemented by a lot of extra gauge degrees of freedom. That allows you to rephrase Newtonian gravity in a geometric way. In that sense it would be best if one looks at the symmetries of the solutions, not of the EOM.

Another simple example is given by a massless scalar field in Minkowski spacetime. I can rewrite the Klein-Gorden equation
$$\eta^{\mu\nu}\partial_{\mu} \partial_{\nu} \phi = 0$$
as
$$g^{\mu\nu}\nabla_{\mu}\partial_{\nu} \phi = 0 \ \ \ \ \ \ \ \ \ (2)$$
and
$$R^{\rho}_{\ \ \mu\nu\sigma} (\Gamma) = 0 \ \ \ \ \ \ \ \ (3)$$
These equations are gct-invariant. I have done nothing fancy; I just rewrote the Minkowski metric as a general metric g plus a flat-spacetime condition. Newton-Cartan theory does exactly the same: one introduces two metrics and a vector field, which in the end are all gauge-fixed such that one ends up with a Newton potential and (1) becomes the Poisson equation.

I hope I'm clear, because this can be a bit confusing.

Last edited: Oct 4, 2012
18. Oct 4, 2012

### PAllen

This is clear enough, but I disagree with your terminology. To my mind, a solution of geometrically expressed theory is particular manifold with an associated geometry (which can take many forms; for Newton-Cartan it is not pseudo-riemannian, for example). The gct covariance of the solution (IMO) means the trivial fact that any mapping of the coordinate charts defining the manifold to another set of coordinate charts, pulling the geometric structure with it, is the same manifold and the same geometry. In Newton-Cartan, the Ricci tensor is the specific piece of the geometry that encodes the matter density across space and time. This is what distinguishes one solution (= hypothetical universe) from another.

I think it is better to separate this triviality in formulating a theory from other characteristics of a theory: symmetry groups; the nature of the geometric structures, etc.

19. Oct 4, 2012

### PAllen

Maybe it would be useful to express your disagreement with the paper Bcrowell posted. It is using covariance and invariance in the sense I mean (trivial in the sense of not acting as a theory filter; can be made true for any theory). This is contrasted with other principles that aim to distinguish GR from most other theories. This paper nicely sums up the thrust of history of thought on this that I've read.

Or perhaps you can point to a paper clarifying your concept of gct-of solution as a theory filter principle. In a dozen or so papers I've read on the history of the general covariance concept, I haven't seen this one presented.

20. Oct 4, 2012

### haushofer

Ok, then it's a matter of terminology. I'll try to read the paper; I was already pointed to it before. :)

The paper I mentioned was http://arxiv.org/abs/1105.3735. It has a very nice introduction. Also, from page 30 on the author explains the Stuckelberg trick very nicely.