Question

∇ is said to be a symmetric or torsion-free covariant derivative when ∇uv−∇vu=[u,v]

. Other types of covariant derivatives, as studied by mathematicians, have no relevance for any gravitation theory based on the equivalence principle.
A)
∇
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−
∇
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=
[
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,
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]
∇uv−∇vu=[u,v]
B)
∇
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+
∇
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=
[
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,
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]
∇uv+∇vu=[u,v]
C)
∇
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−
∇
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=
0
∇uv−∇vu=0
D)
∇
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+
∇
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=
0
∇uv+∇vu=0

Answer 1

A covariant derivative ∇ is said to be symmetric or torsion-free when ∇(uv) - ∇(vu) = [u,v]. Therefore, the correct option is C) ∇uv - ∇vu = 0.

Step-by-step explanation:

A covariant derivative ∇ is said to be symmetric or torsion-free when ∇(uv) - ∇(vu) = [u,v]. This means that the derivative is the same regardless of the order in which the vectors u and v are differentiated. In the context of gravitational theory based on the equivalence principle, this type of covariant derivative is relevant.

Therefore, the correct option is C) ∇uv - ∇vu = 0, indicating that the covariant derivative is symmetric or torsion-free.