# 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}

i know i can write the geodesic equation for a massive particle as:
${\stackrel{˙}{x}}^{\nu }{\mathrm{\nabla }}_{\nu }{\stackrel{˙}{x}}^{\mu }=0$
and then we can express this using the 4 momentum, ${p}^{\mu }=m{u}^{\mu }=m{\stackrel{˙}{x}}^{\mu }$
${p}^{\nu }{\mathrm{\nabla }}_{\nu }{p}^{\mu }=0$
I want to show that this can be written as
$m\frac{d{p}_{\mu }}{d\tau }=\frac{1}{2}{p}^{\sigma }{p}^{\rho }{\mathrm{\partial }}_{\mu }{g}_{\rho \sigma }$
i expanded the covariant derivative out and lowered some of the indices using the metric to obtain that
${g}^{\mu \rho }{p}^{\nu }\left[{\mathrm{\partial }}_{\nu }{p}_{\rho }-{\mathrm{\Gamma }}_{\nu \rho }^{\alpha }{p}_{\alpha }\right]=0$
the first term in that expression can be written as
${g}^{\mu \rho }m\frac{\mathrm{\partial }{p}_{\rho }}{\mathrm{\partial }\tau }$
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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Consider the term
${p}^{\nu }{\mathrm{\Gamma }}_{\nu \rho }^{\alpha }{p}_{\alpha }={p}^{\nu }{p}_{\alpha }\frac{{g}^{\alpha \beta }}{2}\left({\mathrm{\partial }}_{\nu }{g}_{\rho \beta }+{\mathrm{\partial }}_{\rho }{g}_{\nu \beta }-{\mathrm{\partial }}_{\beta }{g}_{\nu \rho }\right)$
$=\frac{1}{2}\left[{p}^{\nu }{p}^{\beta }{\mathrm{\partial }}_{\nu }{g}_{\rho \beta }+{p}^{\nu }{p}^{\beta }{\mathrm{\partial }}_{\rho }{g}_{\nu \beta }-{p}^{\nu }{p}^{\beta }{\mathrm{\partial }}_{\beta }{g}_{\nu \rho }\right]$
Relabel the indices $\nu ↔\beta$ in the last term
$=\frac{1}{2}\left[{p}^{\nu }{p}^{\beta }{\mathrm{\partial }}_{\nu }{g}_{\rho \beta }+{p}^{\nu }{p}^{\beta }{\mathrm{\partial }}_{\rho }{g}_{\nu \beta }-{p}^{\nu }{p}^{\beta }{\mathrm{\partial }}_{\nu }{g}_{\beta \rho }\right]$
The first and last terms cancel, leaving the term you want
$=\frac{1}{2}{p}^{\nu }{p}^{\beta }{\mathrm{\partial }}_{\rho }{g}_{\nu \beta }$