Momentum conservation for a free-falling body in GR

[complexconjugate]

Homework Statement

The conservation law for the energy-momentum vector of a free-falling massive particle is: $$m\frac{\mathrm{d}p^\alpha}{\mathrm{d}t} = -\Gamma^\alpha_{\beta\mu}p^\beta p^\mu$$. Show the conservation law for the covector reads: $$m\frac{\mathrm{d}p_\beta}{\mathrm{d}t} = \frac{1}{2}g_{\mu\nu,\beta}p^\beta p^\mu$$

Relevant Equations

I: $$m\frac{\mathrm{d}p^\alpha}{\mathrm{d}t} = -\Gamma^\alpha_{\beta\mu}p^\beta p^\mu$$
II: $$m\frac{\mathrm{d}p_\beta}{\mathrm{d}t} = \frac{1}{2}g_{\mu\nu,\beta}p^\beta p^\mu$$

Hello everyone!
It seems I can't solve this exercise and I don't know where I fail.
By inserting the metric on the lefthand side of I. and employing the chain rule, the equation eventually reads (confirmed by my notes from the tutorial):
$$m\frac{\mathrm{d}p_\delta}{\mathrm{d}t} = \Gamma^\gamma_{\beta\delta}g_{\mu\gamma}p^\beta p^\mu$$
Now contracting the metric with the Christoffel symbol and renaming indices gives $$m\frac{\mathrm{d}p_\beta}{\mathrm{d}t} = \frac{1}{2}\left[g_{\mu\beta,\nu}+g_{\nu\mu,\beta} - g_{\nu\beta,\mu} \right]p^\nu p^\mu$$
Now I don't understand why two terms in the brackets vanish. Is there some symmetry in the indices I'm missing?
Thanks for any hints.

 

[TSny]

Welcome to PF!

Homework Statement:: Show the conservation law for the covector reads: $$m\frac{\mathrm{d}p_\beta}{\mathrm{d}t} = \frac{1}{2}g_{\mu\nu,\beta}p^\beta p^\mu$$

Something's wrong here. On the left side, $\beta$ is a lone index. But on the right side, $\beta$ is a summation index.

$$m\frac{\mathrm{d}p_\beta}{\mathrm{d}t} = \frac{1}{2}\left[g_{\mu\beta,\nu}+g_{\nu\mu,\beta} - g_{\nu\beta,\mu} \right]p^\nu p^\mu$$
Now I don't understand why two terms in the brackets vanish. Is there some symmetry in the indices I'm missing?
Thanks for any hints.

In general, none of the terms within the bracket vanish or cancel. However, after contracting the terms in the bracket with $p^\nu p^\mu$, you will see that there will be some cancellation.

 

[complexconjugate]

Something's wrong here.

I noticed, I copied the Latex code and forgot to swap out the beta... the upper beta should be a nu.

you will see that there will be some cancellation

I think I see what you mean, can you tell me how to justify it properly? Something like symmetry between nu and mu because it's both times contracted with the momentum?

Thank you a lot!

 

[TSny]

I think I see what you mean, can you tell me how to justify it properly? Something like symmetry between nu and mu because it's both times contracted with the momentum?

Yes

$$m\frac{\mathrm{d}p_\beta}{\mathrm{d}t} = \frac{1}{2}\left[g_{\mu\beta,\nu}+g_{\nu\mu,\beta} - g_{\nu\beta,\mu} \right]p^\nu p^\mu$$

To see what happens, consider the first and last terms in the bracket of the right-hand side and contract them with $p^\nu p^\mu$. Write out these terms explicitly; i.e., carry out the sum over $\mu$ and $\nu$. You'll probably never need to do that again

The idea is that $g_{\mu\beta,\nu} - g_{\nu\beta,\mu}$ is antisymmetric with respect to $\mu$ and $\nu$ and it's being contracted with $p^\nu p^\mu$ which is symmetric with respect to $\mu$ and $\nu$