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Meaning of diffeomorphism invariance

  1. Mar 14, 2008 #1


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    I was watching one of Smolin's online lecture (the link was provided by Marcus in the thread "What's new that's happening in quantum gravity" or something to that effect) and Smolin makes a big deal on the difference between diffeomorphism invariance and invariance under general coordinate transformation.

    Can someone explain what is and why there is diffeomorphism invariance (I knoe that it's a GR question but it is so central to the whole loop quantum gravity approach that I thought this would be a good place to ask)?

    Diffeomorphism invariance is an invariance under a mapping (obeying certain conditions) of the points of the manifold into different points on the manifold (as ooposed to a simple relabelling of the points in which case there is no real change of the manifold). This sounds crazy at first. I mean, if there an observer falling into a black hole, let's say, clearly the physics is changed is the spcetime point of the observer is suddenly mapped to another point very far away from the black hole. The curvature near the observer can't stay the same (since the metric would no longer be a Schwarschild materic). So what is going on?

    Thanks in advance
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  3. Mar 14, 2008 #2


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    Hi kdv,

    Diffeomorphism invariance is something you have already in classic 1915 General Relativity. Einstein has quite a lot to say about it under the heading of "general covariance".

    One thing to note about your example is that in the Schwarzschild geometry the singularity is not a point of the manifold because the metric is not defined there and all the other stuff is not defined there.

    So you can't have a diffeomorphism that moves the singularity around. This kind of nails things down some.

    The simplest way to get some understanding is to think of a simple continuum with no singularities. Everything is smooth uniform and no holes.

    And then you have some matter and you have a metric which is a solution to the Einstein equation and you come in with a smooth map (a diffeo.)

    The map moves the matter around and it moves the metric around and what invariance says is that you STILL have a solution to the Einstein equation.

    After applying the mapping you have a new distribution of matter and you have a new metric and it is still a solution.
    Solutions are invariant under diffeo.

    The new distribution of matter and the new metric do, in a sense, describe the SAME situation as before. They just use the manifold differently as a means of describing it.
    The manifold, in a sense, becomes unimportant, just an accessory. A formal convenience for defining physical events and the gravitational field.

    It doesnt matter at what points of the manifold the events get mapped to because everything is covariant and gets mapped along with them.

    Einstein had some nice words about this like "the principle of general covariance deprives the last remnant of physical reality from points in spacetime." (1916)

    You might want to take this up in the General Relativity forum. It is really a classical thing. It has been around for almost 100 years but a lot of people including physicists still think of points of space as having some absolute reality and they have not assimilated this diffeo invariance thing. they might like to have a discussion about it in the GR forum, since it is a classical thing.

    Loop QG just tries to carry over diffeo invariance from classical GR, as one of the main lessons.
  4. Mar 15, 2008 #3


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    Thank you Marcus. I will bring it up in the GR forum.

    It is hard to grasp because it seems as if under such a mapping one could actually chnage the curvature of spacetime and therefore get a manifold that woul dnot respect Einstein's equation. Let's say that we take a region very far from any star. There is no matter there and the curvature is very small. Now I do a mapping of the manifold such that this region of space becomes curved. Clearly this would no longer obey Einstein's equations! I guess that there must be some very clear restriction on the type of mappings that are possible!

    Anyway, I will go to the GR forum.

    Thanks again
  5. Jul 28, 2008 #4
    Could you specify on the meaning of diffeo? Thanks.
  6. Jul 28, 2008 #5


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    Some people mean different things when they say Diffeomorphism invariance.

    General relativists usually mean "active" diffeomorphisms, eg the gauge transformations of General relativity or really the active part of a coordinate transformation.

    Meanwhile there is also a 'passive' coordinate transformations (simply changing coordinates).
    The two can sometimes be confused, because they look similar in some contexts.

    The hole argument (as described above) gives a way to see whats going on physically.
  7. Jul 28, 2008 #6


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    I was talking in shorthand to kvd, who asked the question. What I meant by "diffeo invariance" was diffeomorphism invariance. As students we used to do this--shorten the word, like calling someone Chris instead of Christopher. because you get tired of saying so many syllables.

    You can look up diffeomorphism in Wikipedia. A diffeomorphism is a smooth one-to-one onto function from a manifold U to a manifold V which has a smooth inverse.

    In the informal shorthand talk of mathematicians it can be described as a "smooth mapping" from U to V, or a "smooth map". Smooth means the derivatives exist---with smooth mappings you are relieved from bother about whether or not you can take the derivative. No sharp creases or kinks.
    There are degrees of this---I am simplifying. We wont worry about degrees of smoothness. Smooth is an intuitive word that gets the idea across.

    As a special case, U and V can be the same manifold. So a diffeo in that case is a smooth map from the manifold to itself.

    As I said in post #2, a diffeomorphism carries the matter along. So what concerned KdV in post #3 does not actually lead to an inconsistency. The mapping moves points in the manifold around and, simultaneously, moves the matter around----so you end up with a new situation which nevertheless still satisfies the Einstein equation. The matter is in new places and the curvature is in new places, but they remain associated so that one still causes the other!
    Last edited: Jul 28, 2008
  8. Jan 1, 2011 #7
    At the risk of being labelled a necroposter, I'll revive this thread.

    The topic came up while reading a paper suggested in another thread, and I wanted to make sure my understanding was precise. So I searched google for "diffeomorphism invariance."

    This thread comes up as one of the top google searches so I thought it would be good to add this very helpful reference here too.

    I found Section 4.1 of the paper by Gaul and Rovelli, Loop Quantum Gravity and the Meaning of Diffeomorphism Invariance, the section entitled: "Passive and Active Diffeomorphism Invariance," to be quite easy to comprehend and a very clear description of the difference.
  9. Jan 1, 2011 #8


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    That's a good paper! It gives one of the best available elementary introductions to LQG as it was at that time, primarily the canonical formulation, mainly the kinematics. I remember finding Gaul Rovelli very useful, also Upadhya Rovelli, an even shorter introduction, which came out about the same time
  10. Jan 2, 2011 #9
    I found Gaul Rovelli to be much easier than Rovelli Upadhya. The latter seems to assume much more math proficiency than I currently have. Gaul Rovelli explains much more as it goes.
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