Basically, I want to know the number of independent components of the Christoffel symbols in a 3+1 spacetime.

I've recently being doing a research review on classical gauge theories and, in doing so, been learning a bunch of maths I'd never learnt before, in particular about differential forms and fibre bundles. In gauge theories, the connection is a Lie algebra valued 1-form defined on a principle bundle. From this, analogy to the connection I'm familiar with is sufficient to predict the expressions for the gauge covariant derivative and the curvature form which is all great.

The issue is, whilst I know that, geometrically, the Levi-Civita connection and the gauge connection are "the same", I'm nervous about how the fact that the former is defined over a tangent bundle and the latter over a principle bundle affects things.

Assuming the analogy is solid, the Christoffel symbols should also be Lie algebra valued 1-forms and, after some thinking, it seems clear that this should be the Lorentz group which would imply that the Christoffel symbols have 24 free components (4 horizontal components each of which lies in the Lorentz algebra which has 6 dimensions).

To convince myself of this analogy further, I decided to try and derive that this was the number of independent components of the Christoffel symbols from the expressions I knew from GR and this proved difficult.

A general 3-index object has 64 independent components but torsion-free-ness gives us 24 conditions (in terms of the structure group of the tangent bundle, it removes shears from the initial general linear group).

I then thought it should be easy to show how many independent conditions we get from metric compatability but this proved much harder than I anticipated.

Ultimately, the best approach we were able to get was to consider the formula for Γ

_{abc}, use the fact that (from torsion-free-ness) Γ

_{abc}=Γ

_{acb}and consider the parts symmetrised and antisymmetrised in the first two indices for both sides of this equals sign.

This gave us 10 free components of the antisymmetrised version (using torsion-free-ness) and 20 from the symmetrised version.

Normally these would have to be independent (suggesting 30 independent components of the Christoffel symbols which would be incompatible with it being a Lie algebra valued 1-form) but I think that the torsion-free-ness actually enables us to say there might be redundancies between the expression for the symmetrised and antisymmetrised parts derived from opposite sides of the torsion-free-ness equation and thus reduce this number of free components further.

This showed us that it had to have fewer than 30 independent components but, if it's at all possible to consider the connection as a 1-form taking values in some other space, its number of free components should be a multiple of 4 and, by analogy to gauge theories, we'd expect this multiple to be the dimension of the structure group. On physical grounds, this structure group should probably be the Lorentz group (giving the 24 free components I'm expecting) but I'm not entirely sure I can rule out a "dilation", either spatially isotropic, a multiple of a dirac delta or a multiple of η giving 28 components.

So, I was wondering if anyone knew how to show the number of independent components there are to the Christoffel symbols. Or, if not quite that much, provide an argument as to why the dilation can be disregarded.