Geodesic equations with varying mass and the variational principle Consider the action,
$$
S 
=
\int d\lambda\ 
\phi(x)
\left(
-g_{\mu\nu}\frac{d x^\mu}{d \lambda}\frac{dx^\nu}{d\lambda}
\right)^{1/2}.
$$
Using the variation principle we obtain,
$$
\delta S
=
\int d\lambda\ 
\left\{
\delta\phi(x)
\left(
-g_{\mu\nu}\frac{d x^\mu}{d \lambda}\frac{dx^\nu}{d\lambda}
\right)^{1/2}
+
\phi(x)
\delta
\left(
-g_{\mu\nu}\frac{d x^\mu}{d \lambda}\frac{d x^\nu}{d \lambda}
\right)^{1/2}
\right\}.
$$
The second term gives the usual result. See here for example. The variation $x^\mu\to x^\mu + \delta x^\mu$ would give
$$
\delta\phi(x)
=
\frac{\partial\phi(x)}{\partial x^\sigma}
\delta x^\sigma.
$$
However, plugging this in and performing similar simplifications as in the usual case, I don't arrive at the expected article given in this article (eq. 20), i.e.
$$
\frac{d^2x^\mu}{d\tau^2}
+
\Gamma_{\rho\sigma}^\mu
\frac{d x^\rho}{d\tau}
\frac{d x^\sigma}{d\tau}
=
-
\left(
g^{\mu\nu}
+
\frac{d x^\mu}{d\tau}
\frac{d x^\nu}{d\tau}
\right)
\partial_\nu(\ln\phi). \tag{20}
$$
In particular, I am having trouble getting the second term on the right-hand side. Any help will be appreciated!
 A: Hints:

*

*The variable mass $$m~\propto~\phi\tag{A}$$  gives rise to an effective metric
$$\bar{g}_{\mu\nu}~=~\phi^2g_{\mu\nu}.\tag{B}$$


*The relationship between the corresponding proper times is
$$-\bar{g}_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}~=~ (c\mathrm{d}\bar{\tau})^2~=~(\phi c\mathrm{d}\tau)^2~=~ -\phi^2g_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}.\tag{C} $$


*Eq. (20) is a matter of transforming the geodesic equation
$$ \frac{d^2 x^{\mu}}{d\bar{\tau}^2} + \bar{\Gamma}^{\mu}_{\alpha\beta} \frac{dx^{\alpha}}{d\bar{\tau}} \frac{dx^{\beta}}{d\bar{\tau}} ~=~ 0\tag{D} $$
into the unbarred variables. There are 2 effects:

*

*Non-affine parametrization
$$ \frac{d^2 x^{\mu}}{d\tau^2} + \bar{\Gamma}^{\mu}_{\alpha\beta} \frac{dx^{\alpha}}{d\tau} \frac{dx^{\beta}}{d\tau} ~=~ \frac{d\ln\phi}{d\tau}\frac{dx^{\mu}}{d\tau}, \tag{E} $$
cf. e.g. my Phys.SE answer here.


*Transformation of the Christoffel symbols
$$\bar{\Gamma}^{\mu}_{\alpha\beta}-\Gamma^{\mu}_{\alpha\beta}~\stackrel{(B)}{=}~\left(\delta^{\mu}_{\alpha}\partial_{\beta}+\delta^{\mu}_{\beta}\partial_{\alpha}-g_{\alpha\beta}g^{\mu\nu}\partial_{\nu}\right)\ln\phi. \tag{F} $$
