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From local symmetry to General Relativity

  1. Jul 5, 2014 #1
    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. jcsd
  3. Jul 5, 2014 #2


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  4. Jul 5, 2014 #3
  5. Jul 5, 2014 #4


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    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.
  6. Jul 5, 2014 #5
    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?
  7. Jul 5, 2014 #6


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    I have less than zero knowledge about quantum gravity, sorry!
  8. Jul 7, 2014 #7


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    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
  9. Jul 8, 2014 #8


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    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.
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