# From local symmetry to General Relativity

1. Jul 5, 2014

### quangtu123

First I want to consider an example of 1D motion. Lagrange equation:

$$\frac{d}{dt} \frac{\partial L}{\partial \dot x} - \frac{\partial L}{\partial x} = 0$$

If we transform $$L \rightarrow L+a$$ with a is constant, the equation of motion remains unchanged. This is global symmetry.

To obtain local symmetry we want when transforming $$L \rightarrow L+a(x)$$ we still have the same equation. To obtain that we introduce the "total derivative":

$$\frac{Df}{dt} = \frac{df}{dt} + \frac{\partial a}{\partial x}$$

Then the equantion of motion would be unchanged under any local transformation:

$$\frac{D}{dt} \frac{\partial L}{\partial \dot x} - \frac{\partial L}{\partial x} = 0$$

The quantity $$\frac{\partial a}{\partial x}$$ is similar to the Christoffel symbols in general relativity.

Is there anyway to construct General Relativity by demanding local symmetry like this?

2. Jul 5, 2014

3. Jul 5, 2014

4. Jul 5, 2014

### WannabeNewton

Yes certainly. But bear in mind there is a difference between the consideration of classical GR as a gauge theory under diffeomorphisms and the usual considerations of Yang-Mills gauge theories.

The diffeomorphism invariance of the Einstein-Hilbert action is certainly not the same thing even in spirit as the invariance of the minimally coupled Dirac action under a local $U(1)$ gauge transformation. This is explained quite well in the Weinstein paper above.

5. Jul 5, 2014

### quangtu123

As the way I see all general relativity and non-gravitation quantum physics can be constructed from the groups of symmetry (in broad sense). It's somewhat really awesome.

Is there any idea like this to pursue a quantum gravity theory?

6. Jul 5, 2014

### WannabeNewton

I have less than zero knowledge about quantum gravity, sorry!

7. Jul 7, 2014

### Mentz114

Effects of curvature and gravity from flat spacetime

EFFECTS OF CURVATURE AND GRAVITY FROM FLAT SPACETIME

This is the title of a thesis which shows (amongst other things) that the Poincare gauge invariance is the same as the diffeomorphism invariance. Worth a read with lots of local gauge invariant stuff.

arXiv:1406.4303v1 [gr-qc] 17 Jun 2014 (http://arxiv.org/abs/1406.4303

One thing that arises from gauging the Poincare group is two gauge covariant derivatives (p39) $\nabla_\mu$ and $\nabla_i$, the first has field strength

$[\nabla_\mu, \nabla_\nu]\phi = (1/2){R^{ij}}_{\mu\nu}\Sigma_{ij}\phi$

where ${R^{ij}}_{\mu\nu}$ can be written in terms of the spin connections (gauge fields) ${\omega^{ij}}_{\mu}$.

The field strength seems to be a gauge based version of the geometric definition of the curvature tensor in GR (Wald p37~)
$(\nabla_{a}\nabla_{b}-\nabla_b\nabla_a)\omega_c={R_{abc}}^d \omega_d$

This is either blindingly obvious, or an interesting tie-up.

He also gets

${R^{ij}}_{\mu\nu}=\partial_\mu {\omega^{ij}}_\nu-\partial_\nu {\omega^{ij}}_\mu + {\omega^i}_{l\mu}{\omega^l}_{j\nu}-{\omega^i}_{l\nu}{\omega^l}_{j\mu}$

which has the same structure as the Riemann tensor defined in terms of the Levi-Civita connection ${\Gamma^a}_{bc}$.

Last edited: Jul 7, 2014
8. Jul 8, 2014

### haushofer

In gauging the poincare algebra, you can show that by putting the translational curvature to zero eliminates the local translations. This leaves you with a soft algebra, consisting of LLT's and gct's, the symmetries of GR. The same can be done for other algebras, like n=1 superpoincare, (n=2) superbargmann and the (super)conformal algebra.