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Diffeomorphism and removal of all matter from the spacetime

  1. Oct 18, 2014 #1
    It is unclear to me (1) what precisely diffeomorphism means and (2) what happens when all matter is removed from the spacetime. Sean Carrol says that: "the theory is free of "prior geometry" and there is no prefered coordinate system for spacetime." http://arxiv.org/pdf/gr-qc/9712019v1.pdf page 138

    Including this I imagine that removal of all matter means, that even space-time does not remain if all matter is removed. I also understood isham similarly
    “However, the use of a fixed causal structure is an anathema to most general relativists and therefore, even if this approach to quantum gravity had worked (which it did not), there would still have been a strong compunction to reconstruct the theory in a way that does not employ any such background.”

    But, some physicists claim that spacetime of Minkowski remains after all matter is removed. Besides, Carroll says also that "In SR or Newtonian physics, meanwhile, the existence of a preffered set of coordinates saves us from such ambiguities."

    How it is with this?
  2. jcsd
  3. Oct 18, 2014 #2


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    1) A diffeomorphism is a one to one, onto, differentiable, mapping of one manifold to another and with a differentiable inverse. In other words, it's an isomorphism of manifolds with the added requirement of differentiability of the mappings. A lot of GR texts will use "diffeomorphism invariance" to mean invariance of physical laws under arbitrary differentiable coordinate transformations on the space-time manifold. Some authors find this to be an abuse of terminology and prefer "general covariance". Sadly, I don't know which convention is more popular these days, from the samplings of texts that I have read, I have seen both terminologies used.

    2) This seems to be more of a philosophical question. We can't really say much in physics about the literal "existence" of space-time or not, that's a topic for ontology, we can really only talk about the experimentally verifiable conclusions such a model gives us. Without any matter what so ever (no energy, no particles, nothing), then we are at a loss at how to define an "event" (not to mention we wouldn't be here to talk about this!), which are the fundamental building blocks (the points) of space-time. We would be adrift in an ocean without an anchor to pin ourselves down.

    Since I don't know off the top of my head, the contexts for your quotes by Carrol and Isham, I can't comment on them very well. However, the comments that "when all matter is removed Minkowski spacetime remains, is really just saying that Minkowski space-time is the limit for a space-time with no gravitational fields. Since any presence of matter (mass, energy, stress, pressure, momentum, etc.) will give rise to gravitational fields, technically Minkowski spacetime requires there to be "no matter" present. However, of course, we then run into the conundrum that "no matter = no events = no way to measure spacetime". The resolution is that practically speaking, gravity is so weak, that in most cases even in the presence of "some matter", the spacetime is approximately Minkowski. The existence of you or me (along with our experimental apparatus) would not create strong enough gravitational fields for us to measure any appreciable difference between the space-time we inhabit and Minkowski spacetime.

    In fact, if the "experimental apparatus" were you and me (our senses) and not some very sensitive instruments, then even the presence of the Earth and Sun and essentially anything other than a Neutron star/black hole (or perhaps a white dwarf, but I doubt it) would not affect the space-time enough for us to notice the difference between Minkowski spacetime. After all, we don't notice that people who travel to the top of Mount Everest and back are a fraction of a second older than we are since the time at the top of Mount Everest moves faster than it does down here. We also don't notice that people who travel a lot on airplanes are a fraction of a second younger because they are moving and we are not.
  4. Oct 19, 2014 #3
    I very agree with your opinon. But contradiction between diffeomorpism's "no prior geometry" and "Minkowski spacetime remains" annoys me. Is diffeomorpism something much different so that these two things are not in contradiction?

    As I understand Isham's quote, it seem that these things are in contradiction.
  5. Oct 19, 2014 #4


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    I would need the context of "no prior geometry" to discuss it well. I think what that quote is saying is that space-time is curved in the presence of matter, and that there is no "background" on top of which we define the curving of space-time. Either way, I don't think there is any conflict because the statement "Minkowski spacetime remains" is not a precise statement. It should read "Minkowski spacetime is the limit as the gravitational field goes to 0".
  6. Oct 19, 2014 #5
    "and that there is no "background" on top of which we define the curving of space-time."
    Can you please explain this a little differently. Is any example maybe close to black hole or in weak gravitational field?
  7. Oct 19, 2014 #6


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    No prior geometry simply means that the metric tensor is dynamical. This is the strong version of general covariance and is a much more profound restriction than diffeomorphism invariance which basically any theory of physics can be made to satisfy including Newtonian gravity. What makes GR more special (but not unique) is the dynamical evolution of the metric tensor according to some equation of motion i.e. Einstein's equation ("no prior geometry"). In other words we cannot a priori exactly fix the metric tensor (although we can certainly do so to various extents in approximation schemes by using e.g. reduction of order in the Landau-Lifshitz formulation).
  8. Oct 19, 2014 #7
    Does "no prior geometry" simply means that "equation of motion i.e. Einstein's equation" defines metric and metric defines the "equation of motion"? Thus, that this relationship means convergence toward correct metric?

    But in spacetime of Minkowski the metric is predefined (in the first aproximation). In the next approximations we need Einstens's equations.
  9. Oct 19, 2014 #8


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    "No prior geometry" is hard to define exactly, and should be taken as just a heuristic. Roughly it means that the geometry depends on matter, and matter depends on geometry. Einstein's equations with matter satisfy this idea.

    In general relativity, flat spacetime without matter does satisfy "no prior geometry" in the sense that when matter is added, we cannot have flat spacetime. In contrast, in special relativity, we put all sorts of matter in the theory, but the spacetime is always flat. This is not possible in general relativity where even having the same fields, but in a different configuration will change the spacetime geometry, hence we say "no prior geometry".

    In general relativity, flat spacetime is only one possibility when there is no matter. The Schwarzschild solution is another vacuum solution (but it has a singularity). However, the idea is the same - one cannot put matter into the Schwarzschild spacetime without affecting the geometry.
  10. Oct 19, 2014 #9


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    What I mean by that is that even if you can define gravity as "Minkowski space time+matter fields", you can't, even in principle, "discover" the Minkowski space-time background if you are in the presence of matter. In other words, even if there is a "background flat space-time", there's no way to conduct experiments to expose this background because in the presence of matter, all the particle trajectories that you would examine/experiment with, and even all the fields that you can measure, will act like they are in curved space time. I think this is what "no prior geometry" is getting at, but probably WBN's answer is more precise.
  11. Oct 19, 2014 #10
    Maybe you are now close to Isham quote "However, the use of a fixed causal structure is an anathema to most general relativists and therefore, even if this approach to quantum gravity had worked (which it did not), there would still have been a strong compunction to reconstruct the theory in a way that does not employ any such background." He tries to emphasis that General relativists do not like that "Minkowski metric" is causal background to quantized curved space time metric, as I understand.

    But, maybe this also means that removal of all matter do not give spacetime of Minkowski. We agree that "empty space time does not have any rest clock with which can define time". But now we can ignore this quoted sentence and even no prior space time gives the same conclusion.

    But some claim oppositely. For instance, Motl:
    "the only ''information'' that the vacuum carries at each point is the so-called ''metric tensor'' - a set of numbers that allow one to calculate the distance between any two nearby points. This is enough for the vacuum to be able to bend - much like any material. One doesn't need any atomic constituents to be able to talk about geometry of the space, and to guarantee that the environment is able to get curved (and to distinguish a flat region of the vacuum from a curved one."
    Last edited: Oct 19, 2014
  12. Jun 20, 2015 #11
    This is an interesting claim, but how it is with tidal forces. I think that tidal forces can always determine in background space time is not flat?
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