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Metric tensor transformations

  1. Mar 7, 2012 #1
    Hi, i was thinking about the metric tensor transformation law:
    [tex]g_{cd}(x) = \frac{{dx'}^a}{{dx}^c} \frac{{dx'}^b}{{dx}^d} g'_{ab}(x')[/tex]
    and, in view of this definition, the differences between Poincare transformations and reparametrization-like transformation (f.e. various conformal transformations of Schwarzschild metric to obtain Penrose diagrams). Maybe someone could point out the differences between them? I think there should be some because Poincare represent some physical situation whereas reparametrization is just change of coordinate charts. I've read about Poincare transformations being an isometry which suppose to mean that they preserve the manifold structure but I am not sure about this. Hope someone will clarify this.

    Thanks for reply in advance.
    Last edited: Mar 7, 2012
  2. jcsd
  3. Mar 7, 2012 #2
    You are intechanging different concepts.

    One thing is a general coordinate transformation in general relativity. In general relativity we want tensors to represent 'real' physical entities, and therefore we need them to transform accordingly. The law you are reffering to is just an requirement that metric tensor transforms like tensor of type (0,2). In that case, when the metric tensor acts on two (arbitrary) vectors, the result is a scalar, i.e. it is invariant under coordinate transformations, which is the desired property of general relativity. Specially, as a consequence, the lenghts are invariant.

    Poincare transformation is a very special transformation on very special manifold: it is a coordinate transformation on Minkowski space that does preserve *the components* of metric tensor: g'=g. (And here g is not a general metric tensor, it assumed to be g=diag(+1,-1,-1,-1), or diag(-+++) based on convention.) Maybe it should be added that these transformations are just transformations between different inertial frames in special relativity.

    And conformal transformation is yet another type of transformation, since it does not amount only to an coordinate transformation (and appropriate transformation of the components of tensors), but also to an multiplication of metric tensor by some scalar. Therefore the distances are no longer preserved.
    Last edited: Mar 7, 2012
  4. Mar 7, 2012 #3
    Thanks for response. Do you mean that Poincare transformation applies only to special relativity? I thought it is a special type (given interpretation in terms of observer movement) of general coordinate transformation you mentioned in the first paragraph and can be also used in GR. And in the case of special relativity it indeed preserves the metric.
  5. Mar 7, 2012 #4
    I haven't encountered Poincare transformation outside of special relativistic settings so far,
    but I'm no expert either. (Far from it)
    Also everything that applies to special relativity does in a sense apply to general relativity (equivalence principle), so you maybe could define kind of (local) Poincare transformations on general manifold.

    And yes, you are right, Poincare tran. is a special type of coordinate transformation, but I don't see right now how you would naturally generalize it to less special settings. (And why would you wish to do so.)
  6. Mar 8, 2012 #5
    I was thinking only about various transformations and how they change the metric. In particular, is it possible to think of transformation changing not only the coordinate chart on a manifold but also defining a new manifold. In this respect, every "physical" transformations shouldn't have this ability(change manifolds by boosting ourselves) whereas others could maybe change the manifold. I am not very fluent in differential geometry but i believe this is more a mathematical question.

    As for Poincare in GR, I would plug in the same as in SR... Or almost the same ;). It would give me GR metric in some boosted ref. of frame. For example, Schwarzschild metric is the one in which mass in the center is at rest and observers(frames) naturally move towards the center. In order to get myself hoovering over the BH at fixed distance i need to boost myself properly (it should even contain some accelerations) and this would be my Poincare transformation. You are probably right that not simple Poincare with constant velocity but some local version of it.

    Hopefully someone else will join the discussion ;).
  7. Mar 8, 2012 #6
    In differential geometry the idea is that you have a manifold and some other quantities or geometrical objects on it that (both the manifold and those quantities) are quite independent of coordinates we use in their description. Therefore by definition, the coordinate transformation do not change the geometry.

    But of course, you can define a transformation that changes the manifold (or fields).
    Knowledge of both the metric tensor and the tensor of torsion is enough to determine whole geometry of manifold.

    The most general transformation will transform coordinates, the metric tensor, the torsion tensor and also all other fields defined on the manifold. In case when metric, torsion and other tensor fields transform accordingly to coordinates the way tensor fields should transform, we have an coordinate transformation and manifold and all ''geometry'' (and physics) is preserved. In case metric and torsion transforms accordingly but other physical quantities does not, you are probably changing physics but preserving manifold. In case torsion or metric transformed differently than they should based on tensor transformation laws, you have defined new manifold.

    As far as Poincare transformation goes, to me it looks just like semantics problem. You surely can define something that resembles Poincare transformations on general manifold.
    And of course you can define transformations that amounts to boosts and rotations and translations on general manifold, but remember that these kind of ''euclidean'' concepts have some sense, as far as we are concerned in coordinate transformations, only locally.
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