i know i can write the geodesic equation for a massive particle as: dot x^{nu}nabla_nu dot x^(mu)=0 and then we can express this using the 4 momentum, p^(mu) = mu^(mu)=m dot(x)^(mu) p^{nu}nabla_nu p^{mu}=0 I want to show that this can be written as m (dp_mu)/(d tau)=(1)/(2)p^sigma p^rho del_mu g_{rho sigma}

Jacoby Erickson 2022-10-22 Answered
i know i can write the geodesic equation for a massive particle as:
x ˙ ν ν x ˙ μ = 0
and then we can express this using the 4 momentum, p μ = m u μ = m x ˙ μ
p ν ν p μ = 0
I want to show that this can be written as
m d p μ d τ = 1 2 p σ p ρ μ g ρ σ
i expanded the covariant derivative out and lowered some of the indices using the metric to obtain that
g μ ρ p ν [ ν p ρ Γ ν ρ α p α ] = 0
the first term in that expression can be written as
g μ ρ m p ρ τ
and i know i can get a 1/2 out of the christofell symbol but everything i try ends in a mess of metrics derivatives and indices.
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Answers (1)

Amadek6
Answered 2022-10-23 Author has 21 answers
Consider the term
p ν Γ ν ρ α p α = p ν p α g α β 2 ( ν g ρ β + ρ g ν β β g ν ρ )
= 1 2 [ p ν p β ν g ρ β + p ν p β ρ g ν β p ν p β β g ν ρ ]
Relabel the indices ν β in the last term
= 1 2 [ p ν p β ν g ρ β + p ν p β ρ g ν β p ν p β ν g β ρ ]
The first and last terms cancel, leaving the term you want
= 1 2 p ν p β ρ g ν β
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