Metric tensor transformations

In summary: It preserves the geometry in special relativity, but in general relativity it does not preserve the field components.
  • #1
grelade
5
0
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.
 
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  • #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.
 
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  • #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.
 
  • #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.)
 
  • #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 ;).
 
  • #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.
 

1. What is a metric tensor in general relativity?

A metric tensor is a mathematical object used in general relativity to describe the geometry of spacetime. It is a symmetric tensor that assigns a distance between any two points in spacetime, and allows for the measurement of lengths, angles, and volumes.

2. How does the metric tensor transform under a change of coordinates?

The metric tensor transforms according to the rules of tensor transformation, which involves multiplying the tensor by the Jacobian of the coordinate transformation. This ensures that the metric tensor remains a symmetric tensor and that the distance between points in spacetime remains unchanged.

3. What is the significance of the metric tensor in general relativity?

The metric tensor is a fundamental concept in general relativity as it describes the curvature of spacetime and is used to calculate the motion of particles and the behavior of light. It allows for the formulation of Einstein's field equations, which describe the relationship between matter and the curvature of spacetime.

4. How does the metric tensor relate to the concept of spacetime curvature?

The metric tensor describes the curvature of spacetime by assigning a distance between points. In flat spacetime, the metric tensor is constant, and the distance between points is always the same. In curved spacetime, the metric tensor varies, and the distance between points changes depending on the curvature at that point.

5. What is the difference between contravariant and covariant metric tensors?

The contravariant metric tensor is used to raise indices in tensor equations, while the covariant metric tensor is used to lower indices. They are related by the metric tensor's inverse, and both are necessary for the formulation of tensor equations in general relativity.

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