The discussion centers on the covariant derivative of the metric tensor, specifically the assertion that in flat space, the covariant derivative of the metric tensor, denoted as \(\nabla g = 0\), is valid. Bernard F. Schutz's "A First Course In General Relativity" suggests this holds true in Riemannian manifolds due to their local flatness. However, participants debate whether \(\nabla g = 0\) should be treated as an axiom or can be derived from the properties of the Levi-Civita connection, which is the unique torsion-free and metric-compatible connection. The conversation highlights the importance of specifying the connection when discussing the covariant derivative in various coordinate systems.
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You could try Sean Carroll's GR lecture notes.snoopies622 said:Actually this thread has been very helpful to me. I'm especially interested in following up on the method outlined in the Einstein paper, where one arrives at Christoffel symbols via the geodesic approach. Would anyone like to recommend a good source? I've never taken a course in the calculus of variations.
Thanks, Michael!Michael Price said:You could try Sean Carroll's GR lecture notes.
https://arxiv.org/abs/gr-qc/9712019
Exactly Schutz's approach!Michael Price said:I've come round to thinking that the best demonstration of ∇g=0 - in GR - is to note that it is trivially true in flat Minkowski space time, and there is always a locally Lorentzian frame in GR, so it has to hold everywhere in GR, as well.
snoopies622 said:Thanks, Michael!
Exactly Schutz's approach!
Sure. The usual assumption is that you have the Levi-Civita connection. You can also do something somewhat different and assume the connection and metric to be independent. In that case the Levi-Civita connection drops out from the stationary action principle (with some assumptions).lavinia said:The local Lorentz frame seems to be interpreted to say that the Space-Time metric can be represented in normal coordinates as diagonal ±1 with first partial derivatives equal to zero at a central point. But doesn't this assume that the connection and the metric are compatible? Why does one do a proof?
It also seems that the Christoffel symbols are assumed to to be zero at the central point. Doesn't this require the affine connection to be torsion free? If not, I don't see how all of the Christoffel symbols can all vanish.
As a counter example: in spherical polars in a flat space and flat spacetime, the Christoffel symbols are not zero.lavinia said:I don't know the Physics so this is a naive question.
The local Lorentz frame seems to be interpreted to say that the Space-Time metric can be represented in normal coordinates as diagonal ±1 with first partial derivatives equal to zero at a central point. But doesn't this assume that the connection and the metric are compatible? Why does one do a proof?
It also seems that the Christoffel symbols are assumed to to be zero at the central point. Doesn't this require the affine connection to be torsion free? If not, I don't see how the Christoffel symbols can all vanish.
This has nothing to do with the central point of normal coordinates. The Christoffel symbols of normal coordinates do indeed vanish at the central point and this does require the connection to be torsion free (at least at the central point).Michael Price said:As a counter example: in spherical polars in a flat space and flat spacetime, the Christoffel symbols are not zero.
You're right, I wasn't thinking of normal coordinates which I should have been.Orodruin said:This has nothing to do with the central point of normal coordinates. The Christoffel symbols of normal coordinates do indeed vanish at the central point and this does require the connection to be torsion free (at least at the central point).