i know i can write the geodesic equation for a massive particle as:

\begin{equation} \dot{x}^{\nu}\nabla_\nu \dot{x}^{\mu}=0 \end{equation}

and then we can express this using the 4 momentum, $ p^\mu = mu^\mu=m\dot{x}^{\mu}$,

\begin{equation} p^{\nu}\nabla_\nu p^{\mu}=0 \end{equation}

I want to show that this can be written as

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

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

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

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

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

  • $\begingroup$ 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. $\endgroup$ – Yep Jan 21 at 23:06

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