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lalbatros
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That's a crucial point of GR !
And I have always problems with that.
Back to the basics, with your help.
Thanks
Michel
And I have always problems with that.
Back to the basics, with your help.
Thanks
Michel
pervect said:I've never quite understood torsion, though, or torsion-based theories of gravity.
The covariant derivative of the metric tensor being 0 implies that the metric tensor is constant throughout the space or spacetime. This means that the distances and angles between any two points in the space or spacetime remain the same, regardless of the coordinates used to measure them.
The covariant derivative of the metric tensor being 0 is a necessary condition for a space or spacetime to be flat or have zero curvature. This means that the space or spacetime is Euclidean, and the rules of Euclidean geometry apply.
Yes, the covariant derivative of the metric tensor can be non-zero in cases where the space or spacetime is curved. This means that the distances and angles between two points will vary depending on the coordinates used to measure them, and the rules of non-Euclidean geometry will apply.
The covariant derivative of the metric tensor is defined as the partial derivative of the metric tensor with respect to one of its indices, plus the Christoffel symbols multiplied by the metric tensor. This can be written as Dg^{αβ}/dx^{γ} = ∂g^{αβ}/∂x^{γ} + Γ^{α}_{γρ}g^{ρβ}, where Γ^{α}_{γρ} are the Christoffel symbols.
In general relativity, the covariant derivative of the metric tensor is used to define the curvature tensor, which is a measure of the curvature of space or spacetime. It is also used in the Einstein field equations, which describe how matter and energy affect the curvature of space or spacetime.