# Symmetry group in GR

## Main Question or Discussion Point

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?).

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

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?).

haushofer
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.

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.

Fredrik
Staff Emeritus
Gold Member
The Poincaré group is the isometry group of Minkowski spacetime. Most spacetimes that are solutions of Einstein's equation don't have any non-trivial 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 n-dimensional smooth manifolds, and consider a diffeomorphism

$$\phi:M\rightarrow N$$.

We can use $\phi$ to define a new function

$$\phi_*:T_pM\rightarrow T_{\phi(p)}N$$

for each $p\in M$, by

$$\phi_*v(f)=v(f\circ\phi)$$

for all smooth functions $f:N\rightarrow\mathbb R$. $\phi_*v$ is said to be the "pushforward" of v. We can also use $\phi$ to define a "pullback" of covectors at each point in the manifold

$$\phi^*:T_{\phi(p)}^*N\rightarrow T_p^*M$$

$$\phi^*\omega(v)=\omega(\phi_*v)$$

for all $v\in T_pM$. The generalization to other types of tensors is straightforward, and so is the generalization to tensor fields. For example, if $\omega$ is a covector field, we define $\phi^*\omega$ by

$$(\phi^*\omega)_p(v)=\phi^*\omega_{\phi(p)}v$$

for all $v\in T_pM$. Now consider the special case M=N. The diffeomorphism $\phi:M\rightarrow M$ defines a pullback $g^*$ of the metric g:

$$(\phi^*g)_p(u,v)=\phi^*g_{\phi(p)}(u,v)=g_{\phi(p)}(\phi_*u,\phi_*v)$$

$\phi$ is said to be an isometry if $\phi^*g=g$. 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 $\mathbb R^4$ with the Minkowski metric, there's a ten-parameter 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 non-trivial isometries.

If $x:U\rightarrow\mathbb R^4$ is a coordinate system, then so is $x\circ\phi$, 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 $GL(\mathbb R^4)$ or $GL(T_pM)$, but the pushforward maps it induces (a different one at each point) can be thought of as members of $GL(\mathbb R^4)$ if we identify the tangent space at each point with $\mathbb R^4$. (But I don't know if that's useful in any way).

Ben Niehoff
Gold Member
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.

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.

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?

Locally and approximately.

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

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

Ben Niehoff
Gold Member
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?

George Jones
Staff Emeritus