The Christoffel formula was derived from metric compatibility and zero torsion

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Relativity & Gravitation · Differential geometry of curved spacetime

The Geometry of Gravity, Ch The covariant derivative, parallel transport and geodesics

Problem

The Christoffel formula was derived from metric compatibility and zero torsion. Prove independently that these two conditions allow no other connection. Suppose Γλμν\Gamma^\lambda{}_{\mu\nu} and Γ~λμν\tilde\Gamma^\lambda{}_{\mu\nu} are two connections, both metric compatible for the same metric (∇αgμν=0\nabla_\alpha g_{\mu\nu}=0 and ∇~αgμν=0\tilde\nabla_\alpha g_{\mu\nu}=0) and both torsion-free, and set Cλμν≡Γ~λμν−ΓλμνC^\lambda{}_{\mu\nu}\equiv\tilde\Gamma^\lambda{}_{\mu\nu}-\Gamma^\lambda{}_{\mu\nu}. (a) Explain briefly why CλμνC^\lambda{}_{\mu\nu} is a tensor, and why Cλμν=CλνμC^\lambda{}_{\mu\nu}=C^\lambda{}_{\nu\mu}. (b) Using metric compatibility of both connections, and writing Cναμ≡gνλCλαμC_{\nu\alpha\mu}\equiv g_{\nu\lambda}C^\lambda{}_{\alpha\mu}, show that Cναμ+Cμαν=0C_{\nu\alpha\mu}+C_{\mu\alpha\nu}=0 for every choice of α,μ,ν\alpha,\mu,\nu. (c) Write this relation for the three cyclic permutations of (α,μ,ν)(\alpha,\mu,\nu), combine them by the same add-two-subtract-one trick used to isolate the Christoffel formula itself, and show that every CναμC_{\nu\alpha\mu}, and hence every CλμνC^\lambda{}_{\mu\nu}, vanishes. Conclude that Γ~=Γ\tilde\Gamma=\Gamma.

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