The gauge of general Relativity

[tunafish]

Hi guys!

I'll go straight to my question.. how come the lorentz group is not the gauge group of general relativity but it is instead the two sheet $SL(2,\mathbb{C})$ covering of it??

Thanks!

 

[TrickyDicky]

Hi guys!

I'll go straight to my question.. how come the lorentz group is not the gauge group of general relativity but it is instead the two sheet $SL(2,\mathbb{C})$ covering of it??

Thanks!

Do you have some reference where that is stated or explained?

 

[Matterwave]

General relativity expands on SR. Whereas SR has vectors which transform properly under the Poincare group operations, GR has vectors which transform properly under general coordinate transformations (as long as the transformation has a non-degenerate Jacobian I believe, although I'm a little fuzzy on the restrictions). As such, GR is generally covariant under all these sets of transformations.

I'm not familiar with your terminology of the two sheet SL(2, C). Perhaps someone equipped with more sophisticated terminology can chime in.

 

[Mentz114]

Hi guys!

I'll go straight to my question.. how come the lorentz group is not the gauge group of general relativity but it is instead the two sheet $SL(2,\mathbb{C})$ covering of it??

Thanks!

Surely if $SL(2,\mathbb{C})$ gives double coverage of the Lorentz group, and is the gauge field, then so is the Lorentz group ? Using 2 $SL(2,\mathbb{C})$ generators instead of one Lorentz generator is actually the same thing.

I don't know much about $SL(2,\mathbb{C})$, but in analogy, $SU(2,\mathbb{R})$ covers $SO(3)$ in the same way, and invariance under one implies invariance under the other.

 

[dextercioby]

Lorentz gauge theory or Poincare gauge theory, two ideas, 3 names which come to my mind: Utiyama, Kibble, Hehl.

The purpose of gauging the local SR symmetry is finding a method to describe in a classical (usually Lagrangian) theory the relevant couplings between general relativity (the metric field is replaced by vierbein) and all the possible matter fields, including or rather specializing on spinorial matter fields, like the Dirac field.

To the reach the latter, one knows from the QFT in flat Minkowski space-time that he needs the double-cover of the symmetry group (Poincare or Lorentz) which furnishes the spinorial representations, one of them being the Dirac spinor. The double-cover in 4D is/includes SL(2,C).

More on this in a relatively recent book by Milutin Blagojevic "Gravitation and gauge symmetries" which has the merit of not using the 'hardcore' language of fiber bundles.

@Mentz114: it's SU(2,C) not SU(2,R). The apparently unanimous convention is to drop the 'C' => SU(2), notation which, however, can create a confusion.

 

[Mentz114]

@Mentz114: it's SU(2,C) not SU(2,R). The apparently unanimous convention is to drop the 'C' => SU(2), notation which, however, can create a confusion.

Thanks for the correction.

 

[samalkhaiat]

Hi guys!

I'll go straight to my question.. how come the lorentz group is not the gauge group of general relativity but it is instead the two sheet $SL(2,\mathbb{C})$ covering of it??

Thanks!

In QFT, one needs to know representations of the Lorentz algebra so(1,3) associated with Lorentz group SO(1,3) rather than representations of SO(1,3) itself. Lorentz group is connected but not simply connected. The main property of simply connected groups is a one-to-one correspondence between representations of the group and the corresponding Lie algebra; any representation of the Lie algebra $\mathcal{L}$ of a simply connected Lie group G is the differential of some representation of G. This, however, is not true for non-simply connected groups. Therefore, it is not true for the Lorentz group SO(1,3). However, for any connected Lie group (such as SO(1,3)) one can find a (unique) simply connected covering group ( SL(2,C) for SO(1,3)). So, to construct representations of the Lorentz algebra so(1,3), it is sufficient to find a universal covering group for SO(1,3), denoted by Spin(1,3)=SL(2,C), and to determine its representations. So, the short answer to your question is :
SL(2,C) is simply connected whereas SO(1,3) is non-simply connected.

sam

 

[martinbn]

It actually is SU(2,R) not SU(2,C).

 

[Ben Niehoff]

$\mathfrak{su}(2)$ is a real Lie algebra. Its complexification is $\mathfrak{sl}(2, \mathbb{C})$.

For the full groups, $SU(2, \mathbb{C})$ is the correct notation. However, the $\mathbb{C}$ is usually left out, because the 'U' stands for "unitary" which already implies that the matrices have complex entries (otherwise we would see an 'O' for "orthogonal").

Specifically, $SL(2, \mathbb{C})$ is the double cover of $SO^+(1,3)$, which are the proper, orthochronous Lorentz transformations. Remember that $SO(1,3)$ has four disconnected components, differing by parity, time reversal, and both parity AND time reversal. $SO^+(1,3)$ is the component connected to the identity.

 

[dextercioby]

[...]. Remember that $SO(1,3)$ has four disconnected components, differing by parity, time reversal, and both parity AND time reversal. [...]

Small correction, SO(1,3) has only 2 disconnected components, while O(1,3) (the full/complete Lorentz group) has 4.

 

[Ben Niehoff]

Small correction, SO(1,3) has only 2 disconnected components, while O(1,3) (the full/complete Lorentz group) has 4.

Yes, you're right.

 

[TrickyDicky]

Having no previous knowledge of group theory I searched for SL(2,C) in Wikipedia, and redirected me to the Moebius transformations that I remotely recalled from complex analysis.
So how are these transformations related to GR? can someone explain a bit in not very technical terms how these transformations can give GR a gauge symmetry?

 

[dextercioby]

Having no previous knowledge of group theory I searched for SL(2,C) in Wikipedia, and redirected me to the Moebius transformations that I remotely recalled from complex analysis.
So how are these transformations related to GR? can someone explain a bit in not very technical terms how these transformations can give GR a gauge symmetry?

From my perspective, the occurrence of SL(2,C) in the context of gravitational theories is explained above in post #5. This group has therefore a proeminent role in the physics which is/was built on the principles of quantum mechanics.

Actually, gauging a global/rigid symmetry like the Lorentz/Poincare one can only be correctly interpreted if the geometrical assumptions which come with GR are left aside and more general manifolds are accepted. But one can do GR in the absence of any of these (frame fields, tetrads, spin, ...) and include only classical matter (pointlike particles, relativistic fluids and electromagnetic fields). This is done in any standart GR source which can be used as a textbook for a university class on GR, because this is really the theory.

Going to gauge groups means already to put at least a step outside GR and venture into theories (supergravity for example) which have yet to have the same success and worldwide recognition like Einstein's one.

As for a non-technical discussion for such matters like SL(2,C) and GR, I guess one can make an attept which would probably confuse and leave many unanswered questions. It's hard even to explain in laymen terms what a pull-back is...

Technical discussions: Well, there's the famous chapter 13 of Wald, any of the known group theory & GR books by Moshe Carmeli, the 2 volumes of Penrose and Rindler (lots of Moebius there) and even some pages in the <Phone book>, but the latter was written before SuGra and PGT as shaped by Hehl.