i know i can write the geodesic equation for a massive particle as: x ˙ ν ∇ ν x ˙ μ = 0and then we can express this using the 4 momentum, p μ = m u μ = m x ˙ μ p ν ∇ ν p μ = 0I 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 ] = 0the 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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i know i can write the geodesic equation for a massive particle as:                    x        ˙                    ν            ∇    ν                      x        ˙                    μ        =  0and then we can express this using the 4 momentum,       p    μ    =  m      u    μ    =  m                    x        ˙                    μ            p          ν            ∇    ν        p          μ        =  0I 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    ]  =  0the 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.

Amadek6 · Accepted Answer

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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