# What is the physics behind GR's diffeomorphism invariance?

1. Mar 8, 2015

### itssilva

Hi. This is my first post here in PF ( :) ). I've been reading some threads on "passive" versus "active" diffeomorphisms, and I wondered: what is the physical motivation for having GR be diffeomorphic invariant? Sure, this allows us to have solutions to Einstein's equations (EFE) up to diffeomorphism (pick one and obtain another by "pushing forward" along the morphism), but, as far as I know (and may be incorrect, please feel free to comment on inadequacies), relativistic equations are only required to be invariant under some representation of the Poincaré group P, which is, itself, a subset of GL(4, R), the set of matrix representations of all possible real linear transformations in 4-space; why not simply restrict ourselves to morphisms associated to P?

2. Mar 8, 2015

### Staff: Mentor

Hi itssilva, and welcome to PF!

This is true in special relativity, where spacetime is flat; but in GR, spacetime is curved, and this restriction doesn't apply. Basically, this is because in flat spacetime, the Poincare group is always the isometry group of the spacetime; but a general curved spacetime may not have an isometry group at all. So the only way to write laws of physics that are invariant in a general curved spacetime is to make them invariant under general diffeomorphisms.

3. Mar 8, 2015

### Roy_1981

"physical motivation for having GR be diffeomorphic invariant"

This comes from the physical picture that "laws" of physics should be written in a form that all observers agree on it. Now observers or reference frames in physics is defined by a set of coordinate systems and clocks. GR first assumes that spacetime (or curved spacetime but I think the adjective curved is redundant) is a (pseudo)Riemannian manifold. As you know very well, changing reference frames or observers then means simply changing the labels of spacetime coordinates i.e. relabeling of the points on the manifold (or in math lingo going from one coordinate patch to another one which overlap at the points being relabelled). Einstein argued that since under the relabelings (aka diffeomorphisms) the manifold does not change, so should the laws of physics as well. This is how he promoted diffeomorphism covariance or general covariance to a principle. However this does not mean anything more than the fact the laws of physics should be of the form, "Some tensor field =0" and nothing more. How gravity comes into the picture is a whole different and interesting part of the story and there we need something called the equivalence principle.

4. Mar 8, 2015

### WannabeNewton

In fact the real content of diffeomorphism invariance comes about when one makes the metric tensor dynamical. This makes it more than just a mere statement about general covariance which is a rather trivial statement by itself.

5. Mar 8, 2015

### Roy_1981

"In fact the real content of diffeomorphism invariance comes about when one makes the metric tensor dynamical."

This statement is akin to putting horse before the cart. A pseudo-Riemannian manifold in general would have a metric (tensor) components to be time-dependent. Stationary spacetimes are special cases where metric is without time-dependence, aka non-dynamical. General covariance is content free when it comes to linking metric tensor to gravitation. In fact, one can write down a general covariant version of Newton's scalar theory of gravitation as was done by Cartan which is unfortuantely not covered in a GR course or most texts. But the encyclodepic treatise of Misner, Thorne & Wheeler has a nice description of Cartan's theory. Bottomline is one needs the principle of equivalence to link the metric tensor to gravitation. General covariance alone gives you nothing, I won't use the belittling adjective "trivial" but instead would use the adjective "natural or expected".

6. Mar 8, 2015

### dx

The following is approximately how Einstein argues in his original paper:

In pre-general relativistic physics, the differences in coordinates had a natural interpretation in terms of measuring rods and clocks. To apply the equivalence principle, we assume that the freely falling frame, in which there is no gravitational field, is the truly inertial one, and special relativity applies only in the freely falling frame. This implies that the 'stationary frame' in which there is a gravitational field, has a metric tensor that is different from the Minkowski one, and for example the rate of clocks depend on their position, and light rays do not move in straight lines, so this coordinate system cannot be interpreted in the customary way as we did in special relativity.

If we want to base a theory of gravitation on these ideas, we must therefore regard any system of coordinates as equally good from the point of view of the laws, i.e. the equations must be generally covariant.

There is more to this issue, which ultimately boils down to what exactly the coordinates mean in physics. Einstein calls it his "struggle with the meaning of the coordinates." Check out the 'hole argument' for example.

7. Mar 8, 2015

8. Mar 8, 2015

### itssilva

I agree; but my question is kinda indirectly tied to the equivalence principle, which I understand physically but haven't been able to translate to math, because, roughly speaking, I imagine GR ought to 'localize' SR, in possibly one of two ways:
1) Suppose you have a tensor field T and a diffeomorphism f, that induces a change in the original T (let's call it *T); then, for every point of spacetime, we could say that the theory is 'diffeomorphic invariant' but restrict ourselves to those f such that the pullback of *T is T transformed by a Poincaré rep; or
2) In a Yang-Mills-like way; in QED, you have global U(1) symmetry, and, if you want that to be local, you add a gauge field A with field strength (curvature) F (all of these geometrical objects defined in a principal bundle), and, for kicks, that introduces interactions in the Lagrangian; I think it suggestive to make the analogy P=U(1), affine connection=A, Riemmann tensor=F and spacetime=some global section after gauge fixing. In fact, there's a line of research, gauge theory gravity, that seems to do something like this, although they don't use the P, for some reason.
As of 'general covariance', I see people using this term in different contexts, but what I understand by that is that means simply writing equations in tensor form, so they are coordinate-independent (and also spacetime-independent, if you use the minimal-prescription of derivatives go to covariant derivatives, etc); thus, even if you Poincaré-transform a coordinate patch, your equations remain the same.

9. Mar 8, 2015

### Staff: Mentor

This amounts to limiting consideration to local inertial coordinates. The problem with that is that there are a lot of things we would like to do in GR that are a lot easier if we aren't limited to local inertial coordinates.

But a Poincare transformation only works in local inertial coordinates on a sufficiently small patch of spacetime, because it assumes that the region of spacetime on which it is operating is flat. Again, if we want to allow general coordinate charts on larger regions of spacetime, we can't limit ourselves this way.

10. Mar 8, 2015

### atyy

The "equivalence principle" is related to the "gauge principle" of field theory. Both are "minimal coupling" principles.

As WannabeNewton said above, general covariance is trivial. Special relativity is generally covariant also, since it just means that the physics is formulated on a manifold, and we can use any coordinates we want to describe the same physics. The physical content of general relativity is that matter curves spacetime, and spacetime tells matter how to move. (Alternatively, it can be argued to be spin 2, but whatever it is, there is hardly any physics in general covariance. However, terminology varies, and Weinberg's textbook uses "General Covariance" to mean the equivalence principle, which is a non-trivial principle.)

Since general covariance is a kind of gauge equivalence (more than one description of the same physics), one sees that the "gauge principle" is also badly named since it is possible to use non-minimal couplings that also (at least classically) retain gauge invariance.

Last edited: Mar 8, 2015
11. Mar 8, 2015

### dx

It is true that general covariance interpreted as a requirement that the laws should be covariant with respect to arbitrary changes of coordinates is in a sense 'trivial.' However the idea of general convariance in Einstein's mind was a little different, and is not exactly a mathematical requirement, and is in his words also a kind of 'heuristic' principle.

Newton's laws for example, prefer particular coordinate systems in the sense that they are simplest when they refer to inertial frames. The law of inertia that a body free from forces moves in a straight line for example is not generally covariant, although a more complicated formulation (Cartan) can be generally covariant.

Einstein's idea was that the the simplest and most natural formulation of gravity theory must not prefer any particular coordinate systems, i.e. the simplest description must be applicable in arbitrary coordinates without change.

The preference for particular coordinate choices arises as a description of particular situations, and cannot therefore be part of the most natural formulation. Einstein compares this to the following situation: on the Earth's surface, there is a preferred direction, the vertical, connected to the uniform gravitational field near the surface. If we were to formulate the theory in such a way that this direction was embedded in the foundation of the theory, it would make things very difficult when we consider more general situations, because the preferred direction which has been embedded into the foundations completely complicates things when we are no longer on the surface of the earth. The preference given to inertial frames is kind of analogous to this, and the the most natural description is one where such preferred systems are not embedded in the foundations.

Last edited: Mar 8, 2015
12. Mar 8, 2015

### atyy

Is the argument here simply that there is no global inertial frame, because spacetime is curved? If so, does it require a cosmological solution, or some circumstances like a non-zero cosmological constant under which the gravity cannot be written as spin 2 on flat spacetime? If that is the case, it would be funny that Einstein first tried to make the universe static - but at least he had to the good sense to use the cosmological constant to do that?

13. Mar 9, 2015

### dx

Yes, I think so. Not only is there no global inertial frame, but also that as soon as we admit non-linear transformations in order to use the equivalence principle, we lose the interpretation of differences in coordinates as measurable lengths and times using rods and clocks. However we cannot avoid admitting these non-linear transformations if we want to base the theory on the equivalence principle, and there seems to be no natural way to restrict these non-linear coordinates for the reason given before (i.e. their lack of direct connection to metric measurements), so we must admit all possible coordinates on equal footing.

The argument in my previous post about how the preference given to inertial systems is analogous to the preferred vertical direction near the surface of the Earth is given in Einstein's essay "Autobiographical Notes." He goes into these things much more deeply there.

14. Mar 9, 2015

### dx

Not only do the coordinates lack a direct connection to metric measurements, the coordinates in fact do not represent anything real. We cannot even say that a particular point (x1, x2, x3, x4) corresponds to a particular event. Such an identification becomes possible only once the gravitational field has been introduced (i.e. one of many possible fields which are related by gauge transformations.) The active diffeomorphisms of the fields defined on the coordinates also changes the location of events relative to the coordinates.

Last edited: Mar 9, 2015
15. Mar 10, 2015

### itssilva

Thanks everyone for your comments, I really appreciate your input; to more or less wrap things up, I'd like to point out that seeing gravity as the gauge theory of the Poincaré group is an old thing - just google it to see -, but I find it really odd that few people pursue this line of research, which is more on line with the modern approach to high-energy physics and it also seems the more parsimonious view; to illustrate this last point, let me use this modified version of the elevator thought experiment: suppose I'm at reference frame S and you are at S', accelerating away from me at acceleration a (S and S' are in a region where gravitational effects are negligible), and I shoot a light signal at angle θ from the direction of motion; you'd see light bend towards or away from you (depending on a), because c has to be constant; the weak PE asserts that gravitational phenomena, such as the bending of light, cannot be distinguished from the effects of acceleration as observed in inertial reference frames, so, if you didn't know about S, you might be inclined to say that there was some mass distribution nearby causing the signal to bend - conversely, you may swap to a non-inertial frame S'', which rotates slightly away from S' in such a way as to always see the signal at angle θ from your direction of motion. Both views, I believe, imply that spacetime is curved, and I only needed local P transformations to arrive at this conclusion; to me, it still feels that these are the only transformations (local or global) with real physical content; otherwise, how do you make sense of some arbitrary transformation of spacetime coordinates or spacetime-valued fields that is not a translation, a rotation or a boost?

16. Mar 10, 2015

### wabbit

This sounds a little strange to me, probably misreading what you say here - but if you want to exclude non-inertial frames, describing you standing in your lab becomes rather contrived as you would be using a (inertial) frame that accelerates away from the lab.

17. Mar 10, 2015

### Roy_1981

", I'd like to point out that seeing gravity as the gauge theory of the Poincaré group is an old thing - just google it to see -, but I find it really odd that few people pursue this line of research"

Actually this line of thought has been pursued since the late 60s thru 70s till mid-80s and followed to a unsuccessful conclusion apart from the case of 2+1 dimensions. The issue is that there is NO gauge theory action which corresponds to GR. Witten demonstrated in 1988 that this only possible in 2+1 dimensions where Einstein-Hilbert action/GR corresponds to a (sum of) Chern-Simons Gauge theory action. Witten's 1988 paper itself has a nice summary of this line of thought, called the Macdowell-Mansouri approach, and why it is ultimately not fruitful (https://inspirehep.net/record/264816?ln=en).

"which is more on line with the modern approach to high-energy physics and it also seems the more parsimonious view"

Actually the modern/high energy theorists approach to GR or Einstein-Hilbert is based on showing that for spin-2 massless particles, a consistent interacting theory can only be constructed without negative norm ghosts if one has linearized diffeomorphisms as a symmetry. Then adding (self)-interactions lead to an infinite series which when summed leads to Einstein field equations. You can find this quantum field theorists approach to GR in Deser (1970), republished as gr-qc/0411023.

18. Mar 10, 2015

### itssilva

The frames look non-inertial globally, but my point is that they can be made to look inertial locally by P transformations-and this is important in the example because the observer makes local observations. Dunno if that helped?

19. Mar 10, 2015

### itssilva

I don't want to be obtruse, but putting it that way feels a little like a circular argument: for, if you're trying to make a first-principles theory, you get the equations (be the H-E action or whatever) as a result of your physical considerations, as Einstein did starting from PE+general covariance, instead of considering them as a given-that is, any gravitation theory ought to be satisfactory if it satisfies our physical requirements of PE+general covariance plus agree with observation, which GR happens to do; sure, it might well be that it's mathematically impossible to find a gauge theory corresponding to the H-E action, but that doesn't necessarily exclude the possibility that you start with one and, by appropriate gauge fixing and/or other subtleties, you arrive at an action that at least approximates the H-E or its predictions (sorry if I've digressed too much here )

20. Mar 10, 2015

### wabbit

Hmm no but don't mind me I was just stopping by and this discussion is way over my head - sorry please carry on.