
#1
Mar710, 10:43 AM

P: 256

What is the symmetry group of manifold which models our world in general relativity.
In special relativity this group is Poincare group. Its elements preserve standard lorentz inner product. What structure is preserved by elements of symmetry group in GR (sygnature of metric, maybe sth else?). 



#2
Mar710, 10:50 AM

Sci Advisor
P: 869

Is it not the group of general coordinate transformations, GL(4,R)?




#3
Mar710, 10:59 AM

P: 256

GL menas general linear  if the space is curved it doesn't contain linear structure
so we cannot define linear maping. Symmetry group in GR contains probably all diffeomorphic coordinate transformations (but all of them preserve signature?). 



#4
Mar710, 04:12 PM

Sci Advisor
P: 869

Symmetry group in GR
What do you mean, "a curved space doesn't contain linear structure"? A tangent space is a vector space and as such linear. The covariant derivative and Lie derivative are both linear operators. Tensors are multilinear mappings from products of tangent spaces and cotangent spaces to the real line.




#5
Mar810, 01:46 AM

P: 256

Tangent bundle to any manifold is linear space but manifold itself need not to have linear structure.
For example Minkowski spacetime itself is linear space but spacetime near a star doesn't have this structure, although of course the tangent bundle in each point is linear space. I think that the group I look for should act on the whole spacetime. 



#6
Mar810, 03:07 AM

Emeritus
Sci Advisor
PF Gold
P: 8,990

The Poincaré group is the isometry group of Minkowski spacetime. Most spacetimes that are solutions of Einstein's equation don't have any nontrivial isometries. So there's nothing that corresponds to the Poincaré group in general spacetimes. But isometries are diffeomorphisms, and you could consider the group of all diffeomorphisms a symmetry group.
Suppose M and N are both ndimensional smooth manifolds, and consider a diffeomorphism [tex]\phi:M\rightarrow N[/tex]. We can use [itex]\phi[/itex] to define a new function [tex]\phi_*:T_pM\rightarrow T_{\phi(p)}N[/tex] for each [itex]p\in M[/itex], by [tex]\phi_*v(f)=v(f\circ\phi)[/tex] for all smooth functions [itex]f:N\rightarrow\mathbb R[/itex]. [itex]\phi_*v[/itex] is said to be the "pushforward" of v. We can also use [itex]\phi[/itex] to define a "pullback" of covectors at each point in the manifold [tex]\phi^*:T_{\phi(p)}^*N\rightarrow T_p^*M[/tex] [tex]\phi^*\omega(v)=\omega(\phi_*v)[/tex] for all [itex]v\in T_pM[/itex]. The generalization to other types of tensors is straightforward, and so is the generalization to tensor fields. For example, if [itex]\omega[/itex] is a covector field, we define [itex]\phi^*\omega[/itex] by [tex](\phi^*\omega)_p(v)=\phi^*\omega_{\phi(p)}v[/tex] for all [itex]v\in T_pM[/itex]. Now consider the special case M=N. The diffeomorphism [itex]\phi:M\rightarrow M[/itex] defines a pullback [itex]g^*[/itex] of the metric g: [tex](\phi^*g)_p(u,v)=\phi^*g_{\phi(p)}(u,v)=g_{\phi(p)}(\phi_*u,\phi_*v)[/tex] [itex]\phi[/itex] is said to be an isometry if [itex]\phi^*g=g[/itex]. The identity map I defined by I(p)=p for all p is of course an isometry. It's not a very interesting isometry, but sometimes it's the only one. When M is [itex]\mathbb R^4[/itex] with the Minkowski metric, there's a tenparameter group of isometries called the Poincaré group. The other "simple" solutions of Einstein's equation also have isometry groups, but as far as I know, there's no guarantee that an arbitrary solution has any nontrivial isometries. If [itex]x:U\rightarrow\mathbb R^4[/itex] is a coordinate system, then so is [itex]x\circ\phi[/itex], so the group of "general coordinate transformations" that Haushofer mentions can be identified with the group of diffemorphisms. It's certainly isn't isomorphic to [itex]GL(\mathbb R^4)[/itex] or [itex]GL(T_pM)[/itex], but the pushforward maps it induces (a different one at each point) can be thought of as members of [itex]GL(\mathbb R^4)[/itex] if we identify the tangent space at each point with [itex]\mathbb R^4[/itex]. (But I don't know if that's useful in any way). 



#7
Mar810, 03:41 AM

Sci Advisor
P: 1,562

It should be noted that the group of diffeomorphisms is a local gauge symmetry, so it is not the sort of symmetry one can use to construct Noether charges. Noether charges result from continuous, global symmetries. For example, the continuous global symmetries of Minkowski spacetime lead to conservation of momentum, energy, and angular momentum. And in electrodynamics, global gauge symmetry leads to the conservation of electric charge.
The GR analog of a global gauge symmetry is a Killing vector field. These are vector fields whose flows preserve the metric; that is, they are generators of isometries. An isometry is a rigid motion of the manifold that carries it into itself (cf. rotations of a sphere). As you can imagine, generic curved manifolds don't generally have many isometries, and might not have any at all. When you do have an isometry, there is a conservation law associated with that isometry. Timelike Killing vectors yield conservation of energy; spacelike ones give conservation of momentum in the direction of the Killing vector. 



#8
Mar810, 03:54 AM

P: 1,743

For example, in quantum mechanics states and observables are represented in the Hilbert space as vectors and Hermitian operators, respectively. So, in order to find how states and/or observables are affected by a change of the observer one just needs to construct a (unitary) representation of the Poincare group in the Hilbert space. If this is done, then we can find, for example, how results of observations change with time, i.e., dynamics. It is important that we can also find how results of observations are affected by boosts of the observer. Thus we should be able to derive Lorentz transformation formulas without ever mentioning the Minkowski spacetime. The same approach should be valid for all systems, interacting or not. Including those systems, which are governed by the gravitational interaction. Eugene. 



#9
Mar810, 05:21 AM

P: 256

So how one can imagine particle in GR.
According to Quantum Filed Theory particle is irreduciable representation of the symmetry group. In flat spacetime this group is Poincare group and this fact has lots of implication (spin, helicty, ...). How one can formulate QFT in curved spacetime? 



#10
Mar810, 05:29 AM

P: 986

Locally and approximately.
You can still think about "particles" as long as their wavelengths are much smaller than curvature of space. 



#11
Mar810, 06:11 AM

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P: 869





#12
Mar810, 02:24 PM

Sci Advisor
P: 1,562





#13
Mar910, 06:20 AM

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P: 869




#14
Mar910, 07:37 AM

Mentor
P: 6,037

I am going to try and tie this stuff together, and I hope that I don't get it too wrong.
A Noether symmety of a system is a local oneparmeter group of of diffeomprphisms that leaves the Lagrangian of the system invariant, and a local oneparmeter group of of diffeomprphisms is associated with a (local) vector field (see, e.g., the last paragraph that starts on page 473 of A Course in Modern Mathematical Physics by Peter Szekeres). Wald considers "a diffeomorphism invariant Lagrangian," so, for the systems that Wald considers, all vector fields generate Noether symmetries. Ben considers Killing vector fields that generate symmetries that leave the metric invariant. In general relativity, the Lagrangian for motion of a particle is constructed from the metric, so Killing vectors generate symmetries of this Lagrangian. 


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