From local symmetry to General Relativity

[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?

 

[WannabeNewton]

http://web.mit.edu/edbert/GR/gr5.pdf
http://philsci-archive.pitt.edu/834/1/gr_gauge.pdf

 

[quangtu123]

http://physics.stackexchange.com/qu...h-extent-is-general-relativity-a-gauge-theory

It seems like there is some mathematical similarities between gauge theory and GR.

 

[WannabeNewton]

http://physics.stackexchange.com/qu...h-extent-is-general-relativity-a-gauge-theory

It seems like there is some mathematical similarities between gauge theory and GR.

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.

 

[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?

 

[WannabeNewton]

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

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

 

[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 ${\Gamma^a}_{bc}$.

 

[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.