Geodesic equation for the motion of a particle

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

$$$$m\frac{dp_\mu}{d\tau}=\frac{1}{2}p^\sigma p^\rho\partial_\mu g_{\rho \sigma}$$$$

i expanded the covariant derivative out and lowered some of the indices using the metric to obtain that

$$$$g^{\mu\rho}p^\nu[\partial_\nu p_\rho-\Gamma^\alpha_{\nu \rho}p_\alpha]=0$$$$

the first term in that expression can be written as $$g^{\mu \rho}m\frac{\partial p_\rho}{\partial \tau}$$ 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.

I feel like im so close. Any help is much appreciated.

Consider the term

$$p^\nu \Gamma^\alpha_{\nu \rho}p_\alpha = p^\nu p_\alpha \frac{g^{\alpha \beta}}{2} \left( \partial_\nu g_{\rho \beta} + \partial_\rho g_{\nu\beta} -\partial_\beta g_{\nu\rho}\right)$$

$$=\frac{1}{2}\left[ p^\nu p^\beta \partial_\nu g_{\rho \beta} + p^\nu p^\beta \partial_\rho g_{\nu\beta} - p^\nu p^\beta \partial_\beta g_{\nu\rho} \right]$$

Relabel the indices $$\nu \leftrightarrow \beta$$ in the last term

$$=\frac{1}{2}\left[ p^\nu p^\beta \partial_\nu g_{\rho \beta} + p^\nu p^\beta \partial_\rho g_{\nu\beta} - p^\nu p^\beta \partial_\nu g_{\beta\rho} \right]$$

The first and last terms cancel, leaving the term you want

$$=\frac{1}{2}p^\nu p^\beta \partial_\rho g_{\nu\beta}$$

• Then i can just time each side by the inverse right. Your a saint thanks a bunch i keep getting bogged down in these subscript notation questions. – Yep Jan 21 at 23:06