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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?).

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- Thread starter paweld
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- #1

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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

haushofer

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

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so we cannot define linear maping.

Symmetry group in GR contains probably all diffeomorphic coordinate transformations

(but all of them preserve signature?).

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haushofer

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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

Fredrik

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Suppose M and N are both n-dimensional 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

[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

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

Ben Niehoff

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The GR analog of a

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

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The Poincaré group is the isometry group of Minkowski spacetime.

This is true. But in my opinion this is not why the Poincare group is useful in physics. The true significance of the Poincare group is that it is the group of transformations in the set of inertial observers. This means that applying a translation, rotation or boost to any inertial observer we will obtain another equivalent inertial observer. The composition law of such transformation is exactly the same as in the Poincare group. So, relativistic theories can be build without any reference to the 4D Minkowski "spacetime".

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.

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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?

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

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haushofer

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But what about the Wald prescription of black hole entropy, in which the entropy is constructed with the Noether charge associated to diffeomorphism invariance?It should be noted that the group of diffeomorphisms is alocal gauge symmetry, so it is not the sort of symmetry one can use to construct Noether charges. Noether charges result from continuous,globalsymmetries. 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.

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Ben Niehoff

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But what about the Wald prescription of black hole entropy, in which the entropy is constructed with the Noether charge associated to diffeomorphism invariance?

I don't know, I'm not familiar with that. Do you have a link?

- #13

haushofer

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I don't know, I'm not familiar with that. Do you have a link?

http://arxiv.org/abs/gr-qc/9307038

Black hole entropy is Noether charge, by Robert Wald. :)

- #14

George Jones

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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 one-parmeter group of of diffeomprphisms that leaves the Lagrangian of the system invariant, and a local one-parmeter 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.

A Noether symmety of a system is a local one-parmeter group of of diffeomprphisms that leaves the Lagrangian of the system invariant, and a local one-parmeter 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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