I am working on a simple and popular GR textbook exercise. In Dodelson's Modern Cosmology (p. 54), it is stated thus:
The metric for a particle traveling in the presence of a gravitational field is $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$ where $h_{00} = -2\phi$ where $\phi$ is the Newtonian gravitational potential; $h_{i0}=0$; and $h_{ij} = -2\phi\delta_{ij}$. Show that the time component of the geodesic equation implies that energy $p^0+m\phi$ is conserved.
I have determined that the nonzero Christoffel symbols are $$\Gamma^0_{00}=\frac{\partial\phi}{\partial t},\quad\Gamma^i_{00}=c^2\frac{\partial\phi}{\partial x^i},\quad\Gamma^0_{ii}=\frac{-1}{c^2}\frac{\partial\phi}{\partial t},$$ by assuming that $\phi \ll 1$. From there I use the time component of the geodesic equation $$\frac{d^2t}{d\tau^2} = \Gamma^0_{\alpha\beta}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau},$$ which simplifies considerably when written in terms of $\gamma=(1-v^2/c^2)^{-1/2}$: $$\gamma\frac{d\gamma}{dt} = -\frac{\partial\phi}{\partial t} - 2\gamma^2\left(\frac{\partial\phi}{\partial x}\frac{dx}{dt}+\frac{\partial\phi}{\partial y}\frac{dy}{dt} + \frac{\partial\phi}{\partial z}\frac{dz}{dt}\right).$$
According to the problem statement, however, I should be ending up with something like $$\frac{\partial}{\partial t}\left(p^0 +m\phi\right)=0$$ (in natural units with $c=1$). Does my result somehow reduce to this one? Thus far, I can't even see how the term $m\phi$ would arise.