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fr.jurain

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One thing I found puzzling about several intros to GR is that, after a summary exposition of absolute diff. calculus, without further ado they posit a Levi-Civita connection, then derive the Einstein field eq. As if all manifolds had to be pseudo-Riemann, then.

Do they really have to, though? I tried to look a bit deeper into the matter and my provisional, & somewhat dismaying, conclusion is that yes, by a weird spin-off of the Equivalence principle they sort of have to. At least those posing as space-time models, that is.

That is, it could pretty well be that in practice we can't test whether our space-time continuum is pseudo-Riemann like it says in the papers, or not.

Did I miss s/thing? If so, what would be a practical test of pseudo-Riemannian-ness?

Here's how I understand the modelling of space-time in GR:

1) we need it 4-dim because of how we experience space & time,

2) we want a smooth manifold because the laws of Nature come out to us as differential eq's,

3) we want a tangent (1, 3)-metric tensor at every 4-point because again, it's our experience that lengths & velocity of light can be measured locally, and that SR holds locally,

4) we need a connection because the associated covariant derivative (CD) is what'll make the laws of Nature "generally covariant".

However, the only motivation for, in addition, requiring the CD of the metric to vanish everywhere, i. e. the manifold to be pseudo-Riemann, seems to be mathematical coziness, not physical necessity.

Now, imagine a (classical, i.e. non quantum) universe, whose physicists were lucky enough that their home world verified (1)-(4) exactly; only they all, like one man, picked the same wrong connection when putting it into theoretical shape.

I think their resulting GR will be mostly OK; they'll have inertia/fictitious forces, curvature-based gravitation, EP, local SR, general covariance, all there & accounted for. It's not even certain their EFE would look awkward.

Of course, they can't fail to notice discrepancies; e. g., test particles which we (who watch their universe from outside) know to be free-falling, will not follow their notion of a geodesic. Not enough to let them revamp their connection though, as they're still free to blame these discrepancies on a physical cause: one of the as yet unmodelled, but not fictitious, "fundamental forces of Nature"; because, as the difference of 2 CD's, the cause in question has the tensor nature.

In other words: there is no (practical) way to sort out "fundamental forces of Nature not described by GR" from "bits we dispensed with adding to the expression of the connection"; every bit of what we describe one way we could just as consistently describe the other way.

This is true as long as the theory comes out with the disclaimer "not completely unified yet".

Hence the odd conclusion: as long as we refrain from unification, i. e. just trying to account for inertia, the EP, and gravitation, then "everything goes"; that is, every connection is roughly as good as any other.

Or is it? Is there a physical necessity to have the CD of the metric vanish? Any light appreciated.

François Jurain.