# How to show pt=-E

1. Nov 27, 2017

### Vrbic

1. The problem statement, all variables and given/known data
It is not a homework but I'm interested in. How to show, very often used, $p_t=-E$

2. Relevant equations
$p^{\mu}p_{\mu}=-1$
$H=\sum{p_i \dot{q}^i}-L$
$2L=u^{\mu}u_{\mu}$
$p_{\mu}=g_{\mu\nu}\dot{q}^{\nu}$

3. The attempt at a solution
$H=1/2(g^{\mu\nu}p_{\nu})=1/2(g^{\mu t}p_t^2+g^{\mu r}p_r^2+g^{\mu \theta}p_\theta^2+g^{\mu \phi}p_\phi^2)$. I suppose that $H=E$ and $p_\theta=p_\phi=0$ when $r\rightarrow \infty$. But it is probably wrong way...there are squares.