Diffeomorphism Invariance, Passive/Active Interpretations GR Insight

[haushofer]

Dear all,

in my current week of holidays, where all the Corona-dust settles down a bit, I came across some personal notes I made a while ago about the meaning of diffeomorphism invariance, the difference between passive and active coordinate transformations, and the notion of background independence and the hole argument in GR. Right now I'm too lazy to turn them into an Insight, and they're probably too long anyway for that, but maybe some other people find these notes useful. It could contain some typo's and mistakes, but otherwise these notes would just sit down unnoticed on my hard disk. Since these topics come along every now and then, they could be useful for some people here. All the best,

Haushofer

 

[jbergman]

Interesting document. I still working through it but it is thought provoking.

Some of it bothered me a bit, maybe because I didn't fully understand it or read it super carefully. You talk about diffeomorphisms but when I think of coordinate transformations, I usually think of isometries. For instance, in SR you want to preserve the Minkowski metric.

The second thing that bothered me was the idea that active transformations happen on the Manifold and not in $\mathbb{R}^n$. Not all manifolds are isotropic, so moving something on the manifold could place somewhere where things are not the same which, I think would be different than a passive transformation, i.e., a relabeling.

 

[haushofer]

Isometries are coordinate transformations, not necessarily the other way around :)

I have to think about peculiar cases where the passive-active duality doesn't hold. Thanks!

 

[vanhees71]

I never understood this obsession to distinguish between "active and passive transformations". At the end it all boils down to applying transformations from one coordinate system to another in a specific context appropriate for the theory under consideration.

 

[jbergman]

I never understood this obsession to distinguish between "active and passive transformations". At the end it all boils down to applying transformations from one coordinate system to another in a specific context appropriate for the theory under consideration.

Personally, the passive POV always felt more natural to me and I never thought about it that much. But Ballentine derives everything using an active POV which still is a bit awkward for me.

 

[haushofer]

I never understood this obsession to distinguish between "active and passive transformations". At the end it all boils down to applying transformations from one coordinate system to another in a specific context appropriate for the theory under consideration.

Well, personally, I like to understand conceptually what's going on if I work with tensors. It also entails one of the deepest and most profound aspects of GR, culminating in the hole argument. And I know from personal experience that you can confuse a lot of physicists with that argument.

Paraphrasing Feynman: it's good practise to understand physics in different ways. And the passive-active duality is by no means trivial. Either that or Einstein and I are complete morons :P

 

[vanhees71]

Well, from a mathematical point of view there's not so much difference between the active and passive point of view. From a physical perspective of course there is.

The most intriguing example is time-reversal symmetry. The passive point of view, i.e., considering a change of coordinates/reference frames, is simply to make $t \rightarrow -t$ and $\vec{x} \rightarrow \vec{x}$ [EDIT: corrected in view of #8] and derive how the observables transform (with the subtle point that from the behavior of $\hat{H}$ and the demand that it must be bounded from below you conclude that time-reversal must be implemented as an anti-unitary transformation rather than a unitary one).

The active point of view, i.e., to consider a physical situation and its time-reversal of course you cannot make in any sensible way $t \rightarrow -t$. What's meant by time-reversal is of course to look at a physical situation by giving the corresponding initial conditions and consider its time evolution (of course always "forward in time") and then you look at the situation, where initially you prepare the time-reversed final state of the former situation and then consider the time evolution (of course always "forward in time" again). There is time-reversal symmetry if you end up with the time-reversal of the initial conditions of the first observation (after the corresponding time).

 

[PeterDonis]

The most intriguing example is time-reversal symmetry. The passive point of view, i.e., considering a change of coordinates/reference frames, is simply to make $t \rightarrow -t$ and $\vec{x} \rightarrow -\vec{x}$

This isn't just time reversal, it's time reversal plus space reversal (parity inversion).

 

[vanhees71]

Argh. That's a typo of course. $\mathcal{T}$ means of course $t \rightarrow -t$ and $\vec{x} \rightarrow \vec{x}$. What I wrote is the PT transformation.

 

[dx]

We were discussing this a while ago in a thread I started, and I have spent some time going through the historical literature on this, trying to read all the comments made by Einstein about this issue. I realize that not everyone may have done this, and since this issue is not really discussed by people other than philosophers, let me share what I have gathered from Einstein's comments. I am quite certain that what I am about to share is really what Einstein thinks, and I would be happy to quote to you his comments/writings directly if anyone is interested.

The conclusion that Einstein draws from the diffeomorphism gauge invariance of GR is the following. Before general relativity, there were two independent conceptual elements in physics, a 'container' called space, and objects/fields which move in that space. Genereal relativity modifies this by getting rid of the 'container.' There is only one conceptual element, which is the field. The underlying manifold is an artifact of the diffeomorphism invariance, i.e. it is a 'gauge artifact.'

Take a piece of paper, and draw physical events on this piece of paper. For instance, draw two world lines. The only observable thing is the coincidence of the two world-lines at a paticular point. If you deform or move everything on the paper (diffeomorphism), nothing physical would change since a ruler which measures the height of a man will coincide with the head of the man at the same point, since both are streched by the same amount. So the location on the piece of paper may change, but the measured value of the height would be the same. Therefore the underlying piece of paper is an artifact of this freedom to perform diffeomorphisms, and its purpose is merely to facilitate the description of such 'point concidences' of independent events.

I should mention that loop quantum gravity advocates are rather obsessed with this idea, at least more so than other groups. The word 'gauge artifact' I borrowed from Carlo Rovelli. Some philosophers talk about 'spacetime substantivalism', referring to the idea of whether space is an independent conceptual element from the idea of a field. This issue of distinguishing between active and passive viewpoint is not a trivial issue. It is central to Einstein's whole philosophy about GR. An coordinate change is not a transformation of the fields. Gauge symmetries are transformations of the fields themselves, and indeed the Einstein equations have this property that active diffeomorphisms generate new solutions of these equations, which cannot be ascribed merely to their coordinate invariance.

 

[dx]

The above comments may seem rather irrelevant for practical purposes, but actually this kind of idea does have some specific concrete implications like the claim in this video at 3:25