The answer to this question Is spacetime symmetry a gauge symmetry? makes the following claim:

One may indeed view general relativity as a gauge theory whose gauge group is $GL(4,ℝ)$ and whose gauge field are the Christoffel symbols $Γ_𝜇$ viewed as a $GL(4,ℝ)$-valued field.

But does not provide a derivation of such. Can anyone provide a reference which gauges the general linear group, and derive general relativity from this approach? Or if possible even write the derivation as an answer.

All I can find in the literature are the Poincaré group $T(1,3) \times O(1,3)$ producing the Einstein-Cartan gauge theory, or the the $T(1,3) \times GL(4,ℝ)$ producing the metric-affine gauge theory, but nothing about the $GL(4,ℝ)$ group by itself?

Finally, the answer to this question To which extent is general relativity a gauge theory? appears to provide such an answer, but the author seems to backtrack in the comment section of his answer. Is the answer correct, or are there things missing to call it GR?

  • 2
    $\begingroup$ I would point out that in a gauge theory, the fundamental variable is the connection itself while in GR the connection is not fundamental, but rather derived from another field, the metric. The action of GR also does not take the form $F^2$ as in Yang-Mills because in GR there is a lower order invariant, namely $R$. The vielbein/spin connection formulation makes this clearer. $\endgroup$ May 15, 2021 at 2:28
  • $\begingroup$ @RichardMyers But metric-affine gravity is a gauge theory, and this is fine? Just not GR? $\endgroup$
    – Anon21
    May 15, 2021 at 2:58
  • $\begingroup$ There are indeed gauge transformations in the theory, if that's your criteria for calling it a gauge theory. $\endgroup$ May 15, 2021 at 4:20
  • $\begingroup$ Understanding Riemannian geometry from the perspective of principal Lie group bundle (more particularly Frame bundle) has been discussed in Schuller's Lecture notes on Geometrical Anatomy of Theoretical physics. $\endgroup$
    – KP99
    Feb 24 at 10:23


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