# Active diffeomorphism

1. Nov 23, 2009

### exponent137

I read about Einstein's hole argument and about diffeomorphism. Pasive diffeomorphism is changing of coordinate system. I do not understand, what is active diffeomorphism. I hear an example with rubber, which is streched. But I imagine this as passive diffeomorphism.

Regards

2. Nov 23, 2009

### atyy

Pick a manifold, choose a coordinate system, find a metric that is a solution of the Einstein equations.

If you use a diffeomorphism to map the manifold to another manifold, and map the metric to a metric on the other manifold, then the metric on the other manifold is also a solution of the Einstein equations THAT represents the same physical situation.

As you can see, there is no big deal between an active diffeomorphism and a coordinate change, since you can use the active diffeomorphism to make a coordinate change. Or at least this is the point of view found in Wald.

The hole problem isn't very deep unless you are Carlo Rovelli. The metric is a geometric structure, and if we take it seriously, then isometric manifolds should of course represent the same physical situation. However, it does mean that the initial conditions are not enough to determine a unique solution, and one has to gauge fix, see eg. section 1 of http://relativity.livingreviews.org/Articles/lrr-2005-6/ [Broken].

Also useful is http://arxiv.org/abs/gr-qc/0603087. Some people use a different definition of passive and active diffeomorphisms which correspond to diffeomorphism covariant and diffeomorphism invariant in that article.

Last edited by a moderator: May 4, 2017
3. Nov 24, 2009

### haushofer

I opened

a while ago, where I was struggling with the same question. And my experience is that there are a very few people who really understand the relation between passive and active diffeomorphisms. Maybe it's just laziness of them and I'm making things way too complicated, but it's a very subtle issue if you ask me.

4. Nov 25, 2009

### exponent137

I wish to know this, because I read
http://www.iop.org/EJ/abstract/0034-4885/64/8/301
(To my sorrow it is only payable)
There it is written, what the problems are at uniting of quantum mechanics and general relativity.
It is also written in this link: "on a fixed background spacetime M, the points in M are physically meaningful. It makes sense, for example, speak of the value of the electromagnetic field at the point x. ...GR is invariant under diffeomorphism, active coordinate transformations that move points in M, and points no longer have any independent meaning. "
I found also an article
http://en.wikipedia.org/wiki/Background_independence
But this problem (active diffeomorphism) I imagine as connected that distance between two events in GR cannot be defined on only one way and with only one number.
But obvious difference between active and passive diffeomorphism is that pasive diffeomorphism does not mean automatically background independence.

Regards

5. Nov 25, 2009

### George Jones

Staff Emeritus
The article is freely available on the arXiv,

http://arxiv.org/abs/gr-qc/0108040.

The relationship between (active) diffeomorphisms and (passive) coordinate transformations is given Chapter 5 in Sean Carroll's Lecture Notes on General Relativity,

http://arxiv.org/abs/gr-qc/9712019.

This material is covered in Appendices A and B in the book, An Introduction to General Relativity: Spacetime and Geometry, into which Carroll's notes evolved.

6. Nov 25, 2009

### xepma

I wrote the following in another topic, so I'll copy/paste it here. I'm happy to hear any remarks:

The notion of gauge invariance in General Relativity has to do with the principle of diffeomorphism invariance. The "action" of a diffeomorphism results in the same mathematics as a coordinate transformation, yet the two concepts have different origins.

As Carroll in his book puts it: diffeomorphisms are "active" (coordinate) transformations, while traditional [sic] coordinate transformations are "passive".

To be more precise: a passive transformations corresponds to a new choice of coordinates. You have some manifold $$M$$, and some coordinate system $$x^{\mu}: M\rightarrow\mathbf{R}^n$$. We can choose some other coordinate system, $$y^{\mu}: M\rightarrow\mathbf{R}^n$$. Standard rules tell you how components of tensors change according to this coordinate transformation. This is the passive point of view, since we only change the way we choose to describe the system.

In the active point of view we make use of diffeomorphisms, pullbacks and pushforwards. We choose one particular (global) coordinate system, $$x^{\mu}: M\rightarrow\mathbf{R}^n$$. Suppose we now have a diffeomorphism of the manifold M to itself: $$\phi: M\rightarrow M$$, i.e. some smooth mapping of the manifold to itself which is invertible. This mapping is an active mapping: it maps point on the manifold M to other points on the manifold. We can use this diffeomorphism to construct pushforwards, which map tensors, specifally, the metric $$g_{\mu\nu}$$, from the manifold M to itself.

The similarity with the passive point of view immediatly comes into play: the pushforward of the coordinate mapping defines a new coordinate system! By definition the pushforward is $$(\phi^\star x)^\mu:M\rightarrow \mathbf{R}^n$$. Hence, by starting with a coordinate system and a diffeomorphism we construct a new coordinate system, using the concept of a pushforward. The mathematics give precisely what you expect of you were 'simply' performing a coordinate transformation. This is why these two concepts are so similar.

But back to diffeomorphism: what does it mean in the context of general relativity? Suppose we have a manifold $$M$$, with some metric $$g_{\mu\nu}$$ caused by some matter distributions represented by the fields $$\psi$$, i.e. the set $$(M,g_{\mu\nu},\psi)$$. Implicitly we have chosen a coordinate system, since we assume to know all the components of the metric.

We can use some diffeomorphism $$\phi: M\rightarrow M$$ and construct a new physical system on M, defined by the triple $$(M,\phi^{\star}g_{\mu\nu},\phi^{\star}\psi)$$. Diffeomorphism invariance means that these two systems represent the same physical situation, since with respect to the diffeormorphisms Einstein's equation do not change. We still deal with the same Riemann curvature.

Is this a powerful statement? Is this an eye-opener? As Carroll puts it: it actually conveys very little information. To quote him: the theory is free of "prior geometry", and there is no preferred coordinate system for spacetime. What's the consequence of this? Well, it's the idea of some "hidden" gauge structure - a redundancy present in the way we choose to describe the physics. In this case we choose some geometric structure, with a metric, matter fields and a coordinate system. But another configuration, with a different metric, matter fields, etc, might describe the same physical system - because of diffeomorphism invariance. So these two systems are not physically distinguishable.

One situation where this comes into play is with the treatment of perturbed spacetimes. We might have some background metric $$g_{\mu\nu}$$ with some small perturbation on this metric, $$h_{\mu\nu}$$. The existence of diffeomorphism invariance allows us to construct a whole family of perturbations, $$h_{\mu\nu}^{\epsilon}$$, all small and all with respect to the same background metric. They are related through (infinitesimal) diffeomorphisms and we can switch from one to the other via a gauge transformation. Under such tranformations the perturbation $$h_{\mu\nu}$$ changes, but the Riemann tensor does not. (Note that there also exist diffeomorphisms which blow up the perturbation, such that it isnt small anymore). The presence of this gauge reduces, for instance, the number of physical degrees of freedom present in the perturbation then you would perhaps expect from simply counting the number of components.

But this comes down to the same idea that we can choose a different coordinate system to describe the perturbation! Yet the origin of the two concepts is not completely the same.

7. Nov 25, 2009

### atyy

Yes, this has no information because it's true even of special relativity if we do it for (manifold, metric, matter) - it's just using the "active" diffeomorphism to do a "passive" coordinate change.

This is where GR differs from SR. If instead of the defining "active/passive diffeomorphism" so that they are just coordinate changes, let's take a new definition and say a passive diffeomorphism acts on everything (manifold, metric, matter), while an active diffeomorphism acts only on the dynamical fields. Then SR differs from GR. In SR, the metric is not dynamical, so the active diffeomorphism will act on (manifold, matter) which will not produce equivalent physical situations; in GR the metric is dynamical, so the active diffeomorphism will act on (manifold, metric, matter) which will produce equivalent physical situations. So in this definition, GR is distinguished by invariance under active diffeomorphism which is equivalent to the metric being dynamical which is no prior geometry. This is discussed as "diffeomorphism covariance" versus "diffeomorphism invariance" in http://arxiv.org/abs/gr-qc/0603087 .

So the notion of "no prior geometry" or of diffeomorphism invariance applied to the dynamical elements only or presumably equivalently at the level of the Lagrangian is interesting, and Carroll applies it to the Lagrangian with matter to get local energy conservation which is related to the equivalence principle. This is discussed in http://arxiv.org/abs/0707.2748 "As discussed in section 3, following Will’s book one can argue that the EEP can only be satisfied if there exists some metric and the matter fields are coupled to it not necessarily minimally but through a non-constant scalar"

At this point we should note that GR does have elements that are not dynamical - the topology of the manifold, and the signature of the metric.

We should also note that combining special relativity and the weak equivalence principle does not require no prior geometry as shown by Nordstrom's second theory.

8. Nov 27, 2009

### exponent137

http://arxiv.org/abs/gr-qc/0108040.
In page 138 Carroll write: "The theory is free of "prior geometry"".
In principle, this is what I searced.
I have some questions.
1. Is this sentence solid?
2. Is this sentence anything connected with active diffeomorphism?

We should also note that combining special relativity and the weak equivalence principle does not require no prior geometry as shown by Nordstrom's second theory.

But, it is not this already very close to GR?

Last edited: Nov 27, 2009
9. Nov 27, 2009

### haushofer

Yes. It means that the metric is not a fixed background as in Newton or special relativity, but has to be solved from the field equations; the metric has become a dynamical field.

Shifting all the dynamical fields in a theory with a fixed background changes something, but without such a fixed background nothing happens because you "shift the geometry along" (because the metric is also shifted).

10. Nov 28, 2009

### exponent137

So, even Carroll gives that "The theory is free of "prior geometry"
This is an essential thing. If it is not really connected with active diffeomorphism, it is not the most important.

However about diffeomorphism - In article
http://arxiv.org/abs/gr-qc/0108040
it is also written:
"essentially because active coordinate transformations “move points” and cannot preserve a quantity defined by its value at individual points."
I never understood this sentence - how it is possible that magnetic field at point P1 is a1, how it is possible, that in a new point its value is zero?

11. Jan 31, 2010

### jdstokes

I do not quite agree with you on this point. Consider as an example the action for a massless $U(1)$ gauge field on a fixed, non-dynamical background $g_{\mu\nu}$

$S = \int d^4 x \sqrt{-g}\left(-\frac{1}{2} g^{\mu\sigma}g^{\nu \rho}F_{\mu\nu}F_{\sigma\rho} \right)$

Under a general coordinate transformation, the following things change: the manifold (represented by the coordinate differential $d^4x$), the metric $g_{\mu\nu}$ (and thus $\sqrt{-g}$) and the matter fields $F_{\mu\nu}$.

Under an active diffeomorphism, only $F_{\mu\nu}$ changes ($d^4x$ and $g_{\mu\nu}$ are both non-dynamical) so we clearly don't have diffeomorphism invariance.

If we go the GR, then $g_{\mu\nu}$ becomes dynamical, so that under active diffeomorphism, the quantity in brackets does not change. Nor does $d^4x$. The quantity $\sqrt{-g}$ does change, however. It can be shown (see Bertschinger's notes) that if by active diffeo you mean pull-back induced by infinitesimal diffeomorphism, then the whole action doesn't change. Imposing diffeomorphism invariance for the above action implies the conservation of stress-energy for the Maxewell field $\nabla_\mu T^{\mu\nu} = 0$.

So I would summarize by saying that coordinate transformations act of everything (manifold, metric, matter) but active diffeomorphisms act on the dynamical fields matter and perhaps metric, but not the manifold itself.

This is confusing because by diffeomorphism, a mathematician means a map $\phi : \mathrm{manifold} \to \mathrm{manifold}$ which does change the manifold.

I attribute this to bad terminology on the part of physicists :)

12. Dec 22, 2011

### exponent137

It is not yet clear to me, what is the active diffeomorphism. It changes vectors, not a coordinate system. How vectors change in general relativity (that we understand that space-time is background free)?

In
http://plato.stanford.edu/entries/spacetime-holearg/notes.html
is written:
In brief, the distinction between the passive and active general covariance is as follows. Passive general covariance allows use of all coordinate charts of the differential manifold and is conferred automatically on theories formulated by modern methods. Active general covariance considers the dual point transformations induced by coordinate transformations. These amount to diffeomorphisms on the manifold M and the transformations of the fields correspond to maps that associate an object field O with its carry along h*O under diffeomorphism h.

I think that dual point transformations are explained here enough clear:

Why dual transformations in general relativity? Is this connected with dual vectors:
http://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors

What those dual point transformations have with general relativity? Is this connected with dual vectors?

13. Dec 25, 2011

### exponent137

Once again about differomorphism from Carroll:
In article http://arxiv.org/abs/gr-qc/0108040
it is also written:
You will often hear it proclaimed that GR is a “diffeomorphism invariant” theory. What this means is that, if the universe is represented by a manifold M with metric gμν and matter fields ψ, and φ : M →M is a diffeomorphism, then the sets (M, gμν , ψ) and (M, φ∗gμν, φ∗ψ) represent the same physical situation. Since diffeomorphisms are just active coordinate transformations, this is a highbrow way of saying that the theory is coordinate invariant. Although such a statement is true, it is a source of great misunderstanding, for the simple fact that it conveys very little information. Any semi-respectable theory of physics is coordinate invariant, including those based on special relativity or Newtonian mechanics; GR is not unique in this regard. When people say that GR is diffeomorphism invariant, more likely than not they have one of two (closely related) concepts in mind: the theory is free of “prior geometry”, and there is no preferred coordinate system for spacetime. The first of these stems from the fact that the metric is a dynamical variable, and along with it the connection and volume element and so forth. Nothing is given to us ahead of time, unlike in classical mechanics or SR. As a consequence, there is no way to simplify life by sticking to a specific coordinate system adapted to some absolute elements of the geometry. This state of affairs forces us to be very careful; it is possible that two purportedly distinct configurations (of matter and metric) in GR are actually “the same”, related by a diffeomorphism. In a path integral approach to quantum gravity, where we would like to sum over all possible configurations, special care must be taken not to overcount by allowing physically indistinguishable configurations to contribute more than once. In SR or Newtonian mechanics, meanwhile, the existence of a preferred set of coordinates saves us from such ambiguities. The fact that GR has no preferred coordinate system is often garbled into the statement that it is coordinate invariant (or “generally covariant”); both things are true, but one has more content than the other.
As it seems now to me, Carroll explained connection between diffeomorphism and general covariance. But a new question appears - is Norton wrong?
or
http://plato.stanford.edu/entries/spacetime-holearg/notes.html
??

Last edited: Dec 25, 2011
14. Mar 25, 2012

### momchil

Here are some thoughts about the (active) diffeomorphism invariance in GR. Some of my notations are slightly sloppy and there might be some inaccuracies, but I think I have clarified the issue for myself. I'll be glad to hear criticism of these notes. The above issue is somewhat controversial and obscure, but it originated with the famous Einstein's hole argument and lingered for a hundred years. On the other hand, diffeomorphism invariance is one of the key concept in String Theory and Quantum Loop Gravity, so it is very popular whenever GR is addressed. I am going to argue that in classical GR, diffeomorphism invariance is unimportant and somewhat misleading. Einstein's hole argument is also misleading. In fact from Category Theory point of view, while diffeomorphisms are isomorphisms in the category of smooth manifolds, the isomorphisms in the category of Riemann manifolds are the diffeomorphisms that leave the metric invariant. Here I have in mind the metric not just as a tensor product of the cotangent bundle over the manifold, but the equivalent requirement that all distances from a given point to any other point are preserved by the morphism. On the other hand, in the quest for quantum gravity, often the scene of the action is a smooth manifold, but the metric is something that has to emerge later as a the expectation value of a quantum object. The smooth manifolds without metric are naturally diffeomorphism invariant.
Let me introduce briefly the active diffeomorphisms. A smooth manifold $\cal{M}$ is a topological space that is locally homeomorphic to $ℝ^{n}$ . If $U_i \subset \cal{M}$ is a collection of open sets, the images of the homeomorphisms $h_i(U_i)=E_i \subsetℝ^n$ are called charts and whenever two charts overlap, $h_j \circ h_i^{-1}: E_i\cap E_j\rightarrow E_i\cap E_j$are smooth (infinitely differentiable)$ℝ^{n}\rightarrow ℝ^{n}$ functions. They are called transition functions. (Active) Diffeomorphisms are defined in a similar way. If $d: \cal{M}\rightarrow \cal{N}$, it is a diffeomorphism iff $h_j \circ d \circ h_i^{-1}$ is a smooth $ℝ^{n}\rightarrow ℝ^{n}$ function. Here $g_j (V_j) \rightarrow ℝ^{n}$ is a chart of $\cal{N}$ . Sometimes the transition functions, which can be thought as coordinate changes are called passive diffeomorphisms. Whenever we have a Riemann manifold, i.e. a smooth manifold with metric, we can extend the diffeomorphism to the whole bundle and transform the metric at point $P\in \cal{M}$to the metric at the point $d(P)\in \cal{N}$. Note however, that the isomorphisms of the category of smooth manifolds are the one to one diffeomorphisms, while the isomorphisms in the category of Riemann manifolds are those one to one diffeomorphisms for which the metric at point $P$ is the same at the one at $d(P)$. Diffeomorphisms $d: \cal{M}\rightarrow \cal{M}$ for which the above is not satisfied can be thought as simple relabeling of the points of the manifold.
A specific application of active diffeomorphisms in GR is Einstein's hole argument. Let's have a Riemann manifold, with a metric that is solution of Einstein's equations (with proper boundary and initial conditions): a Cauchy problem. Let's take an open set (the hole) in the region where we have found the solution and change the coordinates there. We have switched to another chart, and we can find the solution of the Einstein's equations in the new coordinates. As a function of the coordinates, the solution will be everywhere the same, except in the hole. Let's re-interpret the coordinate change as a new solution of Einstein's equations in the region of the hole. This is possible, because of the general covariant form of Einstein's equations. If we take the coordinates of point P in the old solution, we re-interpret the coordinate transformation as a map of the same chart onto itself, which produces an active diffeomorphism of the hole onto itself. In this transformation, also all matter fields are moved around (i.e. their values at the original points are moved to the images of these points). An important and confusing question is whether the new solution has the same metric, i.e. if it is the same Riemann manifold. It is not hard to see that the metric at the point P, labeled by the same coordinates in the old and in the new solution is different (as well as all material fields at that point). But does the solution correspond to a new Riemann manifold? Einstein was able to show that if we take physical signals that label the events (points), the distance between any pair of such physical events is the same in the new and in the old solution. Einstein claimed that the points of the spacetime by themselves do not have physical meaning, and only the intervals between the physical events are real. But one can alternatively say, that the active diffeomorphism in the hole was just relabeling of the space-time points, which was meaningless and unnecessary. The manifold remained the same, because the points labeled by physical events have the same metric in the two solutions. It is interesting that some philosophers take Einstein's remark as a confirmation of Aristotle's, Descartes' and Mach's viewpoint that space(time) is not real and it is just a tool to describe the ordering of the physical objects (events).
Another controversial issue is the treatment of active diffeomorphisms as gauge transformations. Gauge transformations of a dynamic variable (field) correspond to the same physical description (state), both in classical and quantum mechanics. The dynamic variables are redundant, the real physical object is the equivalence class defined by the gauge transformation. The reason that we use the particular redundant dynamic variables is that they are described by a nice linear objects, while the equivalence classes are hard to work with (they are not linear objects). Both in quantum and in classical mechanics, when we need to remove the redundancy, we choose a particular representative from each equivalence class (called also a gauge group orbit), by imposing a gauge-fixing condition. It is a surface that intersects each gauge orbit in a single point. In General Relativity, when described by the metric, there is no redundancy and no need to perform diffeomorphisms, which either bring us to a different point (with different metric) on the same Riemann manifold or are trivial isomorphisms of the Riemann manifold. There are two subtleties: 1. If the Riemann manifold possess a symmetry, described by a Killing field, there are non-trivial isomorphisms, but they correspond to true symmetries, which map one physical state to another physical state and not to redundancies (gauge symmetries). 2. Sometimes in GR the description uses not the metric tensor but another quantity, e.g. the tetrads. They provide a redundant description of the Riemann manifold, but it makes manifest the local Poincare symmetry, which is a gauge symmetry for the tetrads. However it is unrelated to the diffeomorphisms.
One case where people use gauge fixing is the description of gravitational waves and in general any solution of the Einstein's equations, which is written with respect to some background metric. Now we have two different Riemann manifolds, both satisfying Einstein's equations. Let's treat the second manifold as a produced by infinitesimal diffeomorphism of the first one. Lets write the metric of the second manifold as $g_{αβ}+h_{αβ}$ , where $g_{αβ}$ is the metric of the background manifold and $h_{αβ}$ is an infinitesimal change. Now we treat $h_{αβ}$ as a tensor field on the old manifold. This is not possible in general and only the infinitesimal nature of $h_{αβ}$ allows us to do this. It is a function of 20 arbitrary parameters in 4D: an infinitesimal 4x4 matrix (16 parameters), that transforms gij and the 4 infinitesimal coordinate shifts that move the point P to its new position. Requiring that the new metric is torsion free, imposes 6 conditions and that it is solution of Einstein's equations imposes another 10 conditions on the parameters. We are left with 4 parametric redundant description of $h_{αβ}$ , but this is the price to pay for treating it as a tensor field on the old manifold. To remove the redundancy, we have to impose a 4 parametric gauge-fixing condition.
Finally most of the attempts to build quantum gravity desire to have a background-free description of the theory. This means that you don't start with a particular Riemann manifold, the metric has to emerge as the expectation value of some quantum operator in some physical state. But even in these descriptions of quantum gravity there is a “scene” at which the action happens: a smooth manifold. It is a hope only, that in an even more fundamental approach, the space-time itself will emerge from some other mathematical object, like spin-foam, twistor space or something else.
The “scene” of the current quantum gravity attempts is diffeomorphism invariant; two smooth manifolds, related by one to one diffeomorphism and without any additional structure are indistinguishable . That is why, diffeomorphism invariance is so essential in String Theory and Loop Quantum Gravity. However as we saw, it is not important to classical GR and also it is not clear that the true quantum gravity theory, necessarily will have to start from or go at an intermediate stage through the the diffeomorphism invariant smooth manifold without metric.