# Short question about diffeomorphism invariance

#### marcus

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This whole topic of coordinate invariance versus diffeomorphism invariance is notoriously poorly explained in the literature. In fact, I am not aware of a single reference which explains it to my satisfaction.
JD, I'm interested in how you characterize the unique role played by the gravitational field in all of this.
The intuitive idea that the other fields rely on the metric tensor for their diffeomorphism invariance. What you say sounds like it might be a way of expressing what I have in mind.

Diffeomorphism invariance follows when the theory is devoid of background geometrical data...
I agree that if we include the gravitational field and transform everything together then the other fields typically acquire diffeomorphism invariance. And because then the geometry is represented by metric tensor the situation seems, as you put it, "devoid of background geometrical data." But suppose we consider the invariance question for some of these fields without putting gμν into the picture. That may seem a strange, unmotivated thing to do---leave out the gravitational field. It seems to me that typically (perhaps with some exceptions which you can point out) the other fields fail to have a satisfactory formulation.

Maybe this is even obvious. Imagine using a fixed background metric, one that does not transform under diffeomorphisms, or no metric at all. How to put the other fields into covariant form?

It seems to me that the criterion you offer---"devoid of background geometric data"---could be interpreted as saying that the gravitational field is essential to the proper formulation of the other fields. That in order to transform properly the other fields must "ride" on the gravitational field. Does this make any sense to you?

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#### jdstokes

Maybe this is even obvious. Imagine using a fixed background metric, one that does not transform under diffeomorphisms, or no metric at all. How to put the other fields into covariant form?
The answer is that you simply can't make the theory generally covariant. The theory will at most be invariant under the symmetries of your (fixed) background metric. Note that this is exactly the situation in Poincare-invariant QFTs, which are studied by particle physicists all the time.

To a particle physicist $\eta_{\mu\nu}$ is considered as a fixed matrix of numbers, not as a tensor field which transforms. The Noether currents associated with Poincare invariance are derived using this fact.

It seems to me that the criterion you offer---"devoid of background geometric data"---could be interpreted as saying that the gravitational field is essential to the proper formulation of the other fields.
I agree with this. Demanding general covariance forces one to couple matter to the metric (this is the definition of general relativity). Einstein's theory of gravity is obtained from general relativity by endowing the metric with its own (diffeomorphism invariant) kinetic term.

#### marcus

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The gravitational field is the one diff-invariant one--morally :-D

...It seems to me that the criterion you offer---"devoid of background geometric data"---could be interpreted as saying that the gravitational field is essential to the proper formulation of the other fields. That in order to transform properly the other fields must "ride" on the gravitational field. Does this make any sense to you?
...I agree with this. Demanding general covariance forces one to couple matter to the metric (this is the definition of general relativity). Einstein's theory of gravity is obtained from general relativity by endowing the metric with its own (diffeomorphism invariant) kinetic term.
One way to put this--would you agree?--is to say that the other fields acquire their diff-invariance from the gravitational field.

Morally, the metric tensor is the source of diff-invariance, which the other fields get by "riding" on it or being formulated using it. This is just interpretive language, but I think it is in line with the idea of "no prior geometry" (phrase used in the MTW text), independence of geometric background, and what you mean by "devoid of background geometric data."

That may be why many of us, including myself, think of the gravitational field (the spacetime geometry) as playing a unique role.

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#### Finbar

I'd like to check my understanding in general, and in particular, whether I understand the terms being used in the same way as others.

In general, this is false. All that's required from a diffeomorphism is that it be smooth. There's nothing in the definition that requires that the proper distance between two points be preserved.

It's true that manifolds that are diffeomorphic are (essentially) the same manifold. But that's because a manifold has no metric structure defined on it. So understood, a manifold isn't yet anything like a space-time with a `shape'.

Is this Right? Wrong? Not even wrong?
I think your right. But once you have a manifold M with a metric defined on it you can push this metric forward onto a diffeomorphic manifold N and then the proper distances are preserved. You could of coarse define another metric on the N if you liked.

#### yossell

I think your right. But once you have a manifold M with a metric defined on it you can push this metric forward onto a diffeomorphic manifold N and then the proper distances are preserved. You could of coarse define another metric on the N if you liked.
Thanks, Finbar - I see what you mean. As atyy pointed out, there appear to be a lot of different ideas about how exactly the hole construction is to be done, and this may be what's behind so much apparent disagreement and confusion.

#### nrqed

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Well, I thought I would be straightened out within a couple of posts. I did not realize that
this is such a confusing topic.

I want to thank again everybody for their input and explanations.
I am still sorting some details out but I think that I now am starting to understand much better.

The key thing that I realized (and that Atyy emphasized) is that there is more than one definition
of diffeomorphism being used in the literature. Unfortunately, most people only use words in such discussions
and that leads to a lot of confusion when people use the same words to mean different things. Discussions
would be much clearer (and there would be much less arguing) if people would start by defining (mathematically!)
what they mean by a diffeomorphism (not only how the manifold is transformed but also how tensors are transformed), by general covariance, etc.

I think that Rovelli and Carroll are both very bright guys and I think that they understand what they
are talking about when they make statements about diffeomorphisms. They just have different definitions
of diffeomorphisms.

My understanding now is that according to a certain definition (adopted by Carroll among others),
invariance under diffeomorphisms is truly trivial and completely equivalent to invariance under a change
of coordinates. On the other hand, under Rovelli's definition of active diffeomorphisms, it is a non-trivial
statement to say that a theory is diffeomorphism invariant. GR is but QED, QCD, etc, are not.

As for general covariance, I think there are (at least) two ways to define that too.
In one definition, all theories can be made generally covariant and it does not tell us anything to
say that a theory is generally covariant. On the other hand, in understanding the resolution of the
hole argument, I have seen a definition of general covariance that is *not* trivial and that applies to GR
but not, say, to Newtonian gravity. And this is this definition that Rovelly has in mind in his book.

As for ''prior geometry", I am still not sure if this is used to refer simply to the fact that
spacetime intervals between points in spacetime are not defined a priori, i.e. that spacetime distances
are determined dynamically through Einstein's equations. Or if it is meant to also include what Rovelli
discusses in his book, which is the stronger (it seems to me) implication that not spacetime intervals
between points in spacetime are not defined independently of dynamics, but that the actual points in spacetime
have no physical signicance in the first place.

Anyway, these are just some thoughts. I am still trying to understand the details and I will surely come back with more
specific questions.

Thanks

#### atyy

As for ''prior geometry", I am still not sure if this is used to refer simply to the fact that spacetime intervals between points in spacetime are not defined a priori, i.e. that spacetime distances are determined dynamically through Einstein's equations. Or if it is meant to also include what Rovelli discusses in his book, which is the stronger (it seems to me) implication that not spacetime intervals between points in spacetime are not defined independently of dynamics, but that the actual points in spacetime have no physical signicance in the first place.
Actually, the individuation of spacetime points is already a problem in special relativity. Again, Giulini has interesting comments in http://arxiv.org/abs/0802.4345 , the part "From a General-Relativistic point of view, Minkowski space just models an empty spacetime, that is, a spacetime devoid of any material content. It is worth keeping in mind, that this was not Minkowski’s view. ..... Even if the need to incorporate gravity by a variable and matter-dependent spacetime geometry did not exist would the concept of a rigid background spacetime be of approximate nature, provided we think of spacetime points as individuated by actual physical events."

I agree particularly with Giulini's comment "If we mentally individuate the points (elements) of spacetime, we—as physicists—have no other means to do so than to fill up spacetime with actual matter, hoping that this could be done in such a diluted fashion that this matter will not dynamically affect the processes that we are going to describe." - except that I would say "experimentally individuate" rather than "mentally individuate".

In other words, the metric in special relativity corresponds to matter. In Maxwell's equations on flat spacetime, the metric corresponds to electrically neutral measuring rods. Although real measuring rods are composed of electrically charged particles, those clump together so that on the scale on which Maxwell's equations in flat spacetime are true, the rods are electrically neutral. In general relativity, there is no such thing as a measuring rod that does not interact with other matter, since mass couples universally via gravity, and the metric becomes dynamical.

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#### nrqed

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Gold Member
Actually, the individuation of spacetime points is already a problem in special relativity. Again, Giulini has interesting comments in http://arxiv.org/abs/0802.4345 , the part "From a General-Relativistic point of view, Minkowski space just models an empty spacetime, that is, a spacetime devoid of any material content. It is worth keeping in mind, that this was not Minkowski’s view. ..... Even if the need to incorporate gravity by a variable and matter-dependent spacetime geometry did not exist would the concept of a rigid background spacetime be of approximate nature, provided we think of spacetime points as individuated by actual physical events."

I agree particularly with Giulini's comment "If we mentally individuate the points (elements) of spacetime, we—as physicists—have no other means to do so than to fill up spacetime with actual matter, hoping that this could be done in such a diluted fashion that this matter will not dynamically affect the processes that we are going to describe." - except that I would say "experimentally individuate" rather than "mentally individuate".

In other words, the metric in special relativity corresponds to matter. In Maxwell's equations on flat spacetime, the metric corresponds to electrically neutral measuring rods. Although real measuring rods are composed of electrically charged particles, those clump together so that on the scale on which Maxwell's equations in flat spacetime are true, the rods are electrically neutral. In general relativity, there is no such thing as a measuring rod that does not interact with other matter, since mass couples universally via gravity, and the metric becomes dynamical.
Very interesting take. Thanks for sharing this view.

#### nrqed

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Gold Member
This is wrong.

The first point to understand is that any physical theory can be written in a Lorentz invariant form. This includes all of the theories you mention above. I should point out that Newton is slightly different from the other theories in that it is not manifestly Lorentz invariant, This problem can be overcome, however, simply by introducing a preferred timelike direction. The underlying theory is still Lorentz invariant.

As is well known, any Lorentz-invariant theory can be given a coordiante-invariant formulation using the minimal substitution prescription; that is, replace the fixed matrices $\eta_{\mu\nu}$ by the metric tensor field $g_{\mu\nu} = g_{\mu\nu}(x)$, partial derivatives by covariant derivatives etc...

Diffeomorphism invariance is a property of a coordinate-invariant theory which does not possess any background geometrical data. This is simply objects which do not obey field equations of motion (such as the preferred timelike direction in Newton). The metric does not fall into this category, however.

Therefore, Maxwell, QED, QCD are perfectly good diffeomorphism-invariant field theories, by virture of their Lorentz invariance, and absence of background geometrical data.

This whole topic of coordinate invariance versus diffeomorphism invariance is notoriously poorly explained in the literature. In fact, I am not aware of a single reference which explains it to my satisfaction.

I understand the point you are making, jdstokes, and I find your input very helpful.

One question: if I understand correctly from your posts, you use "general covariance" and "invariance under diffeomorphisms" to mean the same thing, right?

And one comment: want to point out that what most people mean by QED and QCD are the theories defined on a fixed spacetime background (usually Minkowski), *not* the theories coupled to gravity. And in that case, QED and QCD are inded not diffeomorphism invariant according to your definition (which would agree with Rovelli).

#### jdstokes

I understand the point you are making, jdstokes, and I find your input very helpful.

One question: if I understand correctly from your posts, you use "general covariance" and "invariance under diffeomorphisms" to mean the same thing, right?

And one comment: want to point out that what most people mean by QED and QCD are the theories defined on a fixed spacetime background (usually Minkowski), *not* the theories coupled to gravity. And in that case, QED and QCD are inded not diffeomorphism invariant according to your definition (which would agree with Rovelli).
Hi nrqed,

Firstly, I'm very surprised to hear of the plethora of definitions out there for coordinate invariance/diffeomorphism invariance. I have no idea what these authors are talking about because the only sensible definitions for these terms are the following:

Consider a field-theory action integral $S = \int d^4 x \sqrt{-g} \mathcal{L}$
where $\mathcal{L}$ is a scalar.
Under a coordinate transformation the following things change
$T_{\mu\nu}(x) \mapsto T_{\mu\nu}'(x') = \partial'_\mu x^\alpha \partial'_\nu x^\beta T_{\alpha \beta}(x)$ etc
$d^4 x \mapsto d^4x' = d^4 x J$
where T is any tensor field.

Under a diffeomorphism,
$T_{\mu\nu}(x) \mapsto T_{\mu\nu}'(x)$
$d^4 x \mapsto d^4 x$

As you can see, these transformations are unambiguously different, which is revealed by the fact that I have chosen to work local coordinates. If you write everything down in abstract notation, as some authors like to do, things are a lot more confusing than necessary.

Any physical theory whatsoever is coordinate invariant. This is not a big surprise if you consider that coordinate invariance is not a real symmetry (ever wondered what are the associated Noether currents?).

Diffeomorphism invariance is a true symmetry of the theory (the conserved current is the energy-momentum tensor). Therefore, not all theories are diffeomorphism invariant. As an exercise, try proving that the action for a coordinate-invariant scalar is diffeomorphism invariant. Then see if you can think up a coordinate-invariant field theory which is NOT diffeomorphism invariant.

P.S. QED and QCD formulated on non-dynamical flat space are diffeomorphism invariant. If you think about it, this is trivially true. Not convinced? Then just write down a coordinate-invariant formulation of QED in flat space.

#### haushofer

@jdstokes:

Would you say that invariance under Lie derivatives reveals a symmetry, while an invariance under general coordinate transformations reveals only something about how one describes the theory (in terms of tensors)?

I think people would call QED "not invariant under diffeomorphisms" because they put "invariant under diffeomorphisms" and "background independent" on equal footing.

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