Diffeomorphic Invariance implies Poincare Invariance?

[FedEx]

However, this is not an isometry of the metric, so this is an active transformation that actually changes the physics (in this case the potential energy between the particles), unless we also take along the metric:

In GR it makes less sense to make this distinction, because you cannot transform the content and geometry separately: every active transformation necessarily also takes along the metric, getting rid of the problem altogether.
I still think we're saying more or less the same thing.

The entire mechanism goes through only cause the metric is taken along as well. Which as someone mentioned is equivalent to saying that there is no background. The metric along with mass etc form the "foreground". Great. :-D

Now in this context, can one say that there might be solutions to EFE, say two black holes having different masses, i.e. two inequivalent physical situations, which are related by a diffeomorphism?

And if this is true, that would mean two different observers would see different curvature. But we know that co ordinate transformations cannot change the curvature. Does this imply that Active != Passive? And how would this affect the statement "Active and Passive are the same mathematically"

Sorry for all the far fetched (confused) implications.

 

[WannabeNewton]

No, if \left \{ T^{(i)} \right \}_{i} are the collection of tensor fields on the space - time M that represent physically measurable quantities of interest (which we for simplicity take to be a finite collection) and similarly we have \left \{ T'^{(i)} \right \}_{i} for the space - time N and N,M are not related by a diffeomorphism, then (M,T^{(i)}) will be physically distinguishable from (N,T'^{(i)}) when experiments are carried out and measurable quantities are obtained with regards to the tensor fields.

 

[FedEx]

No, if \left \{ T^{(i)} \right \}_{i} are the collection of tensor fields on the space - time M that represent physically measurable quantities of interest (which we for simplicity take to be a finite collection) and similarly we have \left \{ T'^{(i)} \right \}_{i} for the space - time N and N,M are not related by a diffeomorphism, then (M,T^{(i)}) will be physically distinguishable from (N,T'^{(i)}) when experiments are carried out and measurable quantities are obtained with regards to the tensor fields.

True. There are ofcourse different physically inequivalent situations which are not related by a diffeo.

But I was looking for : Two descriptions related by a diffeo can they correspond to different physically inequivalent situations? They should, shouldn't they? Since, active diffeos will change the metric(since an arbitrary diffeo need not be an isometry, as you rightly mentioned in the earlier posts) which might correspond to a different mass distribution.

 

[PeterDonis]

No, if \left \{ T^{(i)} \right \}_{i} are the collection of tensor fields on the space - time M that represent physically measurable quantities of interest (which we for simplicity take to be a finite collection) and similarly we have \left \{ T'^{(i)} \right \}_{i} for the space - time N and N,M are not related by a diffeomorphism, then (M,T^{(i)}) will be physically distinguishable from (N,T'^{(i)}) when experiments are carried out and measurable quantities are obtained with regards to the tensor fields.

Just to inject some more handwaving here , I still think there's a "gap", so to speak, when we're talking about active diffeomorphisms. The above seems to imply that no transformation that changes physically measurable quantities can be a diffeomorphism; but is that true? It seems like an active diffeomorphism that is not an isometry could change physically measurable quantities.

For example, consider a simple case: we have a 2-sphere with a given radius, and we have an atlas of coordinate charts on the sphere--for concreteness, say the atlas consists of the stereographic projections from the North and South poles, along with an appropriate metric.

A passive diffeomorphism would be changing coordinate charts to, say, two "latitude-longitude" type charts that overlap in such a way as to cover the entire sphere (with an appropriate change in the metric).

An active diffeomorphism that was an isometry would be rotating the sphere without changing coordinate charts; the coordinate labels of points in each chart would change, but the metric would not change, and neither would any other tensor field at any point.

An active diffeomorphism that was *not* an isometry would be changing the sphere's radius, again without changing coordinate charts: the metric would have to change because the curvature changes (and probably any other tensor field would have to change too).

So the two spheres with two different radii would seem to be physically distinct manifolds related by a diffeomorphism.

 

[NanakiXIII]

True. There are ofcourse different physically inequivalent situations which are not related by a diffeo.

But I was looking for : Two descriptions related by a diffeo can they correspond to different physically inequivalent situations? They should, shouldn't they? Since, active diffeos will change the metric(since an arbitrary diffeo need not be an isometry, as you rightly mentioned in the earlier posts) which might correspond to a different mass distribution.

They change the metric, but they also change anything that lives in your manifold that might care about the metric to compensate.

Just to inject some more handwaving here , I still think there's a "gap", so to speak, when we're talking about active diffeomorphisms. The above seems to imply that no transformation that changes physically measurable quantities can be a diffeomorphism; but is that true? It seems like an active diffeomorphism that is not an isometry could change physically measurable quantities.

For example, consider a simple case: we have a 2-sphere with a given radius, and we have an atlas of coordinate charts on the sphere--for concreteness, say the atlas consists of the stereographic projections from the North and South poles, along with an appropriate metric.

A passive diffeomorphism would be changing coordinate charts to, say, two "latitude-longitude" type charts that overlap in such a way as to cover the entire sphere (with an appropriate change in the metric).

An active diffeomorphism that was an isometry would be rotating the sphere without changing coordinate charts; the coordinate labels of points in each chart would change, but the metric would not change, and neither would any other tensor field at any point.

An active diffeomorphism that was *not* an isometry would be changing the sphere's radius, again without changing coordinate charts: the metric would have to change because the curvature changes (and probably any other tensor field would have to change too).

So the two spheres with two different radii would seem to be physically distinct manifolds related by a diffeomorphism.

There's no such thing as a diffeomorphism that changes the radius of the sphere, at least not in any meaningful sense. There are two ways to look at the diffeomorphism, either they

1) act on your manifold only, in which case the "radius" of your sphere is not defined at all, because you need to impose a geometry first;

2) act on your geometry and your other content, in which case the fact that you're changing the radius is unimportant because you're also scaling along anything that might care about this radius.

 

[WannabeNewton]

Take the following example: Consider an observer S and another observer S' who set up respective coordinates x^{\mu },x'^{\mu} on some open subset U\subseteq M where (M,g_{ab}) is a space - time. For each p\in U, the observers use their respective coordinate bases \left \{ \frac{\partial }{\partial x^{i}}|p \right \}_{i},\left \{ \frac{\partial }{\partial x'^{i}}|p \right \}_{i} for T_{p}(M). Say we have a rank 2 tensor T\in T_p(M)\otimes T_p(M) representing a physically measurable quantity. In this passive point of view, what general covariance tells us is that if S makes a measurement, using the measuring apparatus used to define his local coordinates, of the physical quantity represented by T and S' does the same then the values T^{\mu \nu } obtained in measurement by S will relate to the values T'^{\mu \nu } measured by S' by T'^{\mu \nu } = \frac{\partial x'^{\mu }}{\partial x^{\alpha}}\frac{\partial x'^{\nu}}{\partial x^{\beta}}T^{\alpha\beta}. If in particular, we instead talk about the value of a scalar field at a point then such observers, S,S' will measure the same exact value for the value of the scalar field at that point.

The active point of view, instead of looking at coordinates and coordinate transformations on a given space - time, looks at the image of the space - time under the diffeomorphism and the pull back and pushforward, under this diffeomorphism, of tensor fields representing physically measurable quantities and again asserts general covariance.

 

[FedEx]

They change the metric, but they also change anything that lives in your manifold that might care about the metric to compensate.

The mass distribution for instance.

Hence the statement " Two descriptions related by a diffeo can correspond to different physically inequivalent situations " should hold good.

 

[NanakiXIII]

The mass distribution for instance.

Hence the statement " Two descriptions related by a diffeo can they correspond to different physically inequivalent situations? " should hold good.

They're not physically inequivalent. Suppose my mass distribution is two point particles. If I change my metric so that the particles are now twice as far apart, but I also act on the particles, moving them closer together, I end up with the same situation. See my examples in post #21.

Yes, the metric changes and yes, the mass distribution changes, but they change in such a way that the physics is identical.

 

[FedEx]

They're not physically inequivalent. Suppose my mass distribution is two point particles. If I change my metric so that the particles are now twice as far apart, but I also act on the particles, moving them closer together, I end up with the same situation. See my examples in post #21.

Yes, the metric changes and yes, the mass distribution changes, but they change in such a way that the physics is identical.

That helped. :-)

Thanks a lot.

 

[WannabeNewton]

So the two spheres with two different radii would seem to be physically distinct manifolds related by a diffeomorphism.

I don't disagree with you at all here but I think the issue is in the great ambiguity of the phrase "represents the same physical space - time" and what "physically the same" really means. I was thinking of it in terms of what I stated in my above post but I have not found like a standard definition for this notion of "physically equivalent".

 

[atyy]

Just to inject some more handwaving here , I still think there's a "gap", so to speak, when we're talking about active diffeomorphisms.

Yes, a diffeomorphism that is not an isometry will change physical quantities (Hawking & Ellis). IIRC, Wald's definition of "active diffeomorphism" is the same as Hawking & Ellis's "isometry".

 

[TrickyDicky]

Perhaps it would be useful to try and agree about a few concepts in order to avoid the different degrees of confusion that discussions about "diffeomorphism invariance" often stir up.

-First I think it is important to acknowledge that there is differences between what the different terms used mean in differential geometry texts like Lee's to mention a well known here good reference about smooth and Riemannian manifolds, versus what they convey in GR and physics textbooks. This comes from the fact that most differential geometry books center much more in Riemannian rather than pseudoRiemannian manifolds/dynamical spacetimes but not only from this. This can lead to confusion IMO.

-In purely mathematical terms a tensor equation like the EFE is completely indifferent to coordinate transformations/diffeomorphisms so it has diffeomorphism invariance (a.k.a. general covariance) by definition. The problem is that GR as a theory is a bit more than just the EFE, basically it adds that the context in which the EFE must be applied is that of a Lorentzian manifold, a dynamical spacetime, and this basic tenet of GR can make the general covariance ambiguous at the least because it is either trivial as a differential topology statement at the smooth manifold level, or not true when taken to the manifolds with metric differential geometry level of (global)isometry invariance.

-GR is concerned only with the local geometry of the manifold not with the global topology, so I always understood that when GR books talk about both both diffeomorphisms and isometries they refer to local diffeomorphisms and local isometries, but this distinction is never addressed in GR books. It is however perfectly explained in diff. geometry texts like Lee(WN I'm sure you know what I'm referring to since you mentioned Lee's books).

 

[PeterDonis]

There's no such thing as a diffeomorphism that changes the radius of the sphere, at least not in any meaningful sense.

This might be another issue of terminology. Let me give a concrete example. Suppose we start out with a sphere of radius 1, a coordinate chart ( \theta, \varphi ) on it, similar to the standard "latitude, longitude" coordinates, and a curve going from (0, 0) to (1, 0) in coordinate values. (We'll gloss over the fact that we need two such charts to cover the whole sphere; everything I say will apply to both charts and won't affect the transition map between them.) Then, using your ( M, g_{\alpha \beta}, \psi ) terminology, we have M = S^2, g_{\alpha \beta} = diag ( 1, sin^2 \theta ), and \psi = [ (0, 0), (1, 0) ].

Now we transform to (M', g'_{\alpha \beta}, \psi'), where M' = S^2, g'_{\alpha \beta} = diag ( 2, 2 sin^2 \theta ), and \psi = [ (0, 0), (1, 0) ]. We haven't changed M or \psi, but we have changed the metric to that of a sphere with radius 2. (This will also change the arc length of \psi to 2 instead of 1; if we treat the arc length as part of \psi, then \psi does change as a result of the transform. Or we could treat the arc length as a function of the metric and \psi.)

Is this transformation a diffeomorphism? I don't see why not. The transformation of the manifold itself, S^2, is just the identity, so the differentiable structure of the manifold itself is certainly preserved. The metric changes, but I don't see how that would have any impact on differentiability. You could say that the diffeomorphism is trivial because the manifold doesn't change at all, but that just means it's trivial; it doesn't mean it isn't a diffeomorphism. And it does change the metric, without changing anything else, so it does change things that are "physically measurable", like the arc length of the curve \psi.

 

[PeterDonis]

GR is concerned only with the local geometry of the manifold not with the global topology

I'm not sure I agree with this; I wouldn't say that GR is not concerned with global topology, just that solving the EFE, by itself, doesn't tell you the global topology, since you can have solutions that are locally identical but have different global topology. But I would agree that GR texts don't stress the fact that the EFE is local, so all the talk about invariance under coordinate transformations is also local.

 

[WannabeNewton]

This might be another issue of terminology.

It isn't Peter. You are correct and he/she isn't if what was typed is what was intended. Note first that closed balls in \mathbb{R}^{n} of all radii are diffeomorphic to one another therefore so are their manifold boundaries, which happen to be n-1 - spheres.

 

[atyy]

I suppose while we hairsplitting, even two manifolds related by an isometry need not be physically equivalent - as NanakiXIII points out one has to move everything so that nothing moves - an isometry only moves the metric, and not matter, so it should be iso-everything:)

 

[atyy]

I don't disagree with you at all here but I think the issue is in the great ambiguity of the phrase "represents the same physical space - time" and what "physically the same" really means. I was thinking of it in terms of what I stated in my above post but I have not found like a standard definition for this notion of "physically equivalent".

It means all physical observables are the same:)

As for what is physically observable, that's defined by the theory. If your theory doesn't say what is physically observable, it's not a theory of physics:)

Ok, joking aside, within general relativity, my guess is that it's something like all quantities which remain the same after arbitrary changes of coordinates.

But there may be more than that. For example, sticking to SR, if one formulates electromagnetism in terms of the scalar and vector potential, then one would also have to add, and under arbitrary changes of gauge. But one wouldn't have to specify the additional condition if electromegnetism had been formulated in terms of electric and magnetic fields.

I think the main problem in GR is that in pure gravity there are no local observables. So in a vacuum solution, one usually puts distinguishable test particles all over and the intersection of their worldlines are then events. But since in real GR, there are no test particles (ie. particles that don't contribute to spacetime curvature), then one has to add matter, something like the discussion in the introduction of http://arxiv.org/abs/gr-qc/9404053. Another interesting discussion is http://arxiv.org/abs/gr-qc/0110003.

Another question then is why do we ever introduce gauge descriptions - why can't we work with gauge invariant objects only? I think the answer is that in many cases, the gauge invariant objects are nonlocal - things like Wilson loops. So if we want to describe physics with local equations, we use a description with gauge redundancy.

 

[NanakiXIII]

It isn't Peter. You are correct and he/she isn't if what was typed is what was intended. Note first that closed balls in \mathbb{R}^{n} of all radii are diffeomorphic to one another therefore so are their manifold boundaries, which happen to be n-1 - spheres.

This might be another issue of terminology. Let me give a concrete example. Suppose we start out with a sphere of radius 1, a coordinate chart ( \theta, \varphi ) on it, similar to the standard "latitude, longitude" coordinates, and a curve going from (0, 0) to (1, 0) in coordinate values. (We'll gloss over the fact that we need two such charts to cover the whole sphere; everything I say will apply to both charts and won't affect the transition map between them.) Then, using your ( M, g_{\alpha \beta}, \psi ) terminology, we have M = S^2, g_{\alpha \beta} = diag ( 1, sin^2 \theta ), and \psi = [ (0, 0), (1, 0) ].

Now we transform to (M', g'_{\alpha \beta}, \psi'), where M' = S^2, g'_{\alpha \beta} = diag ( 2, 2 sin^2 \theta ), and \psi = [ (0, 0), (1, 0) ]. We haven't changed M or \psi, but we have changed the metric to that of a sphere with radius 2. (This will also change the arc length of \psi to 2 instead of 1; if we treat the arc length as part of \psi, then \psi does change as a result of the transform. Or we could treat the arc length as a function of the metric and \psi.)

Is this transformation a diffeomorphism? I don't see why not. The transformation of the manifold itself, S^2, is just the identity, so the differentiable structure of the manifold itself is certainly preserved. The metric changes, but I don't see how that would have any impact on differentiability. You could say that the diffeomorphism is trivial because the manifold doesn't change at all, but that just means it's trivial; it doesn't mean it isn't a diffeomorphism. And it does change the metric, without changing anything else, so it does change things that are "physically measurable", like the arc length of the curve \psi.

I was under the impression a diffeomorphism is only defined to act on manifolds, not on their geometry. If that is a mistake, then what you're saying is correct.

However, it's not very meaningful to me. Of course you're going to change your system if you suddenly impose a different metric. But what is that kind of transformation supposed to signify? It's certainly not a change of coordinates, so I'm not comfortable calling it either an active or a passive coordinate transformation.

P.S.: I'd also like to point out that under your terminology, GR is not diffeomorphism invariant at all, so your definition does not seem to be the one used in the literature.

P.P.S.: Also, under my definition, spheres of all radii are also diffeomorphic to one another, precisely because in my terminology diffeomorphisms don't care about the geometry and hence they're all trivially identical.

P.P.P.S.: I had a look in Spivak and he clearly defines diffeomorphisms without making any mention of higher structure like geometry. His diffeomorphisms act on a manifold equipped with an atlas. It doesn't touch anything that lives on the manifold. So your transformation, Peter, consists of a (trivial) diffeomorphism on the manifold, but you added something separate to act on the metric.

 

[WannabeNewton]

Yes diffeomorphisms make no mention of geometry which is what I was trying to stress in many of my posts on the first page =D

 

[NanakiXIII]

Yes diffeomorphisms make no mention of geometry which is what I was trying to stress in many of my posts on the first page =D

Then Peter's concrete example is not a diffeomorphism, but something more, since there is something specifically acting on the geometry.

 

[haushofer]

True. There are ofcourse different physically inequivalent situations which are not related by a diffeo.

But I was looking for : Two descriptions related by a diffeo can they correspond to different physically inequivalent situations? They should, shouldn't they? Since, active diffeos will change the metric(since an arbitrary diffeo need not be an isometry, as you rightly mentioned in the earlier posts) which might correspond to a different mass distribution.

No. That's the crux of the hole argument: spacetime points, or events, only get their physical meaning after you've introduced the metric. That's another way of saying that diffeo's are gauge transformations in GR, and is unrelated to any possible isometries of the metric. Isometries are investigated after you introduced a metric.

 

[haushofer]

In a cruel way it's funny to see how these discussions often end in semantics :P

 

[PeterDonis]

Yes diffeomorphisms make no mention of geometry which is what I was trying to stress in many of my posts on the first page =D

If that's true, then the statement that GR is diffeomorphism invariant is either trivial or meaningless. The whole point of a coordinate transformation is to leave the underlying geometry invariant, not just the underlying topological manifold. So it seems to me that GR physicists, at least, must mean something more by "diffeomorphism" than just a transformation of the underlying manifold; by that definition all GR coordinate transformations are just the identity diffeomorphism.

 

[PeterDonis]

P.S.: I'd also like to point out that under your terminology, GR is not diffeomorphism invariant at all, so your definition does not seem to be the one used in the literature.

Well, under your and WannabeNewton's definition, as I just posted in response to him, the statement that GR is diffeomorphism invariant is either trivial or meaningless. So there's got to be something more to it than just a transformation on the manifold.

Here's what I get from the physicists', as opposed to the mathematicians', discussion of this:

(1) A "solution" in GR includes a topological manifold, a coordinate chart on that manifold, and an expression for the metric and the stress-energy tensor in that coordinate chart, such that the Einstein tensor derived from the metric is a formal solution of the Einstein Field Equation with that stress-energy tensor as source. The "geometry" of the solution is the set of all scalar invariants contained in it: for example, the arc lengths of all curves, the values of all Lorentz scalars at each point, etc.

(2) A passive diffeomorphism is a coordinate transformation that leaves the underlying geometry invariant. Thus, such a transformation changes the coordinate chart and the expressions for the metric and the stress-energy tensor, but not the topological manifold. The statement that GR is invariant under passive diffeomorphisms means that: (a) the transformed metric and stress-energy tensor will still give a formal solution of the Einstein Field Equation; and (b) all of the scalar invariants will be unchanged.

Example: switching from Schwarzschild to Painleve coordinates on Schwarzschild spacetime with a specific mass M. Formally, the metric looks different, and the components of the Einstein Field Equation look different; but both metrics express a formally valid vacuum solution to the EFE. And all scalar invariants are unchanged by the transformation.

(3) An active diffeomorphism is a transformation that may or may not change the coordinate chart, but it does change the expressions for the metric and the stress-energy tensor, and it does change the underlying geometry; it does not change the topological manifold. (There seems to be less agreement about this in the literature, so what I'm giving here is just the version that I feel I understand the best.) The statement that GR is invariant under active diffeomorphisms means that: (a) the transformed metric and stress-energy tensor will still give a formal solution of the Einstein Field Equation; but (b) scalar invariants may be changed.

Example: taking Schwarzschild spacetime with mass M1, to Schwarzschild spacetime with a different mass M2, using the same coordinate chart, say Schwarzschild coordinates. Formally, the expression for the metric changes, and so do the components of the Einstein Field Equation (though not much, since only one parameter changes). But again, both metrics express a formally valid vacuum solution to the EFE. However, now scalar invariants are changed.

It may well be that "diffeomorphism" is not a good term for these transformations, mathematically speaking. But it seems to be often used by physicists in the way I've used it here.

 

[sheaf]

If that's true, then the statement that GR is diffeomorphism invariant is either trivial or meaningless. The whole point of a coordinate transformation is to leave the underlying geometry invariant, not just the underlying topological manifold. So it seems to me that GR physicists, at least, must mean something more by "diffeomorphism" than just a transformation of the underlying manifold; by that definition all GR coordinate transformations are just the identity diffeomorphism.

I think active diffeomorphisms are definable independent of the geometry, but if you *do* a diffeomorphism, then you surely do change the geometry. If you have a pair of points and a metric-defined distance, then you drag those points off somewhere then the distance between them in the dragged-along metric is different from the original distance.

 

[WannabeNewton]

If that's true, then the statement that GR is diffeomorphism invariant is either trivial or meaningless.

You might find the discussion starting on page 434 in appendix B of Carroll interesting with regards to this.

 

[PeterDonis]

You might find the discussion starting on page 434 in appendix B of Carroll interesting with regards to this.

Which Carroll do you mean? I'm familiar with his online lecture notes on GR, but they don't have an Appendix B (and they only have 231 pages).

 

[atyy]

If that's true, then the statement that GR is diffeomorphism invariant is either trivial or meaningless. The whole point of a coordinate transformation is to leave the underlying geometry invariant, not just the underlying topological manifold. So it seems to me that GR physicists, at least, must mean something more by "diffeomorphism" than just a transformation of the underlying manifold; by that definition all GR coordinate transformations are just the identity diffeomorphism.

Yes, only manifolds that are isometric are physically equivalent.

So when one says that GR is invariant under active diffeomorphisms, one always means active diffeomorphisms that are isometries - it's standard abuse of terminology.

Which Carroll do you mean? I'm familiar with his online lecture notes on GR, but they don't have an Appendix B (and they only have 231 pages).

In his lecture notes, it's probably the claim the diff invariance leads to covariant energy conservation - I'm not sure if that's true without the principle of equivalence, although Carroll says it is.

 

[PeterDonis]

when one says that GR is invariant under active diffeomorphisms, one always means active diffeomorphisms that are isometries - it's standard abuse of terminology.

But one could also do an active diffeomorphism that wasn't an isometry, but still yielded a valid solution of the EFE; see my example of a diffeomorphism between Schwarzschild spacetimes with different parameters M. I've seen discussions (IIRC in one of Rovelli's papers, for example) that appeared to say this would count as an active diffeomorphism.

 

[PeterDonis]

In his lecture notes, it's probably the claim the diff invariance leads to covariant energy conservation - I'm not sure if that's true without the principle of equivalence, although Carroll says it is.

I see several interesting statements in Carroll's discussion of diffeomorphisms in Chapter 5 of the lecture notes.

First, he says (p. 133, about halfway down the page): "If \phi is invertible...then it defines a diffeomorphism between M and N. In this case M and N are the same abstract manifold." (emphasis mine)

This seems to say that, as far as the topological manifold is concerned (which is what I think he means by "abstract manifold"), any diffeomorphism is trivial, since it's just the identity. The only thing a diffeomorphism can change, in this sense, is additional structures on the manifold.

Then, he says (p. 133, near the bottom): "If you like, diffeomorphisms are "active coordinate transformations", while traditional coordinate transformations are passive."

In the further discussion following this, he appears to view these "active coordinate transformations" as something like isometries. More precisely, he appears to view them as defining vector fields and families of integral curves on a constant underlying geometry; but there is no requirement that I can see for the vector field to be a Killing vector field, which is what would be required for the transformation to be an isometry, strictly speaking. But he is still holding the underlying geometry constant; so this notion of "active transformations" is less general than what I was calling "active diffeomorphisms" before, since those could change the underlying geometry.

For example, on Schwarzschild spacetime one could define an "active coordinate transformation" in Carroll's sense, it seems to me, using either of two vector fields: first, the vector field \partial / \partial t, which is a Killing vector field (note that it doesn't matter whether this is the t of Schwarzschild or Painleve coordinates, since it's the same vector field either way); second, the vector field \partial / \partial T - \sqrt{ 2M / r} \partial / \partial r in Painleve coordinates, which is the 4-velocity field of ingoing Painleve observers. An active coordinate transformation in Carroll's sense would "move points" along the integral curves of the vector field; in the first case, such a transformation would be an isometry, in the second it wouldn't. But the overall underlying geometry would remain the same either way.

Later (pp. 138-139), he discusses diffeomorphism invariance and covariant energy conservation, which looks like the passage you are referring to. His comment on p. 138 seems to confirm that what he is calling a diffeomorphism does not change the manifold itself, but only additional structures on the manifold. But he also makes this interesting comment: "it is possible that two purportedly distinct configurations (of matter and metric) in GR are actually "the same", related by a diffeomorphism". This is possible, he says, because GR has no preferred coordinate system; but if he just means that, for example, Schwarzschild and Painleve coordinates, each with their appropriate metric, both describe the same geometry, then "diffeomorphism" here should mean "passive" diffeomorphism, not active.

Then he derives covariant conservation of the SET from diffeomorphism invariance, basically by computing the variation of the matter Lagrangian and requiring that it be zero under arbitrary diffeomorphisms. This is interesting to me because all of the other texts I'm familiar with, such as MTW, say that covariant conservation of the SET is a consequence of covariant conservation of the Einstein tensor, which is due to the contracted Bianchi identities, plus the Einstein Field Equation. Carroll's argument here, however, would seem to apply even if the EFE were not valid.