E.O.M of a free- particle with "dynamical mass" from action principle in General Relativity If one tries to obtain the E.O.M of a (massive) free-particle, one should extremize the action:
$$ S_{m} = \int ds = - \int  m  \sqrt{-g_{\mu \nu}\frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}} d\lambda $$
In general $m$ does not depend on $x^{\mu}$, so we can put it outside the integral and by least action principle one does find the geodesic equation.
My question is: in a scenario where $m \equiv m( \phi ) $ but $\phi$ itself depends on the spacetime coordinates, particles (now) should not follow geodesics. Unfortunately my result does not seem correct for the E.O.M, and I need some direction.
In the desired expression, as obtained in equation (3.2) of reference https://doi.org/10.1098/rsta.2011.0293 the RHS of the geodesic equation sould appear the following expression:
$$ \frac{\partial \,\mbox{ln}  \, m}{\partial \phi} \frac{\partial \phi}{\partial x^{\mu} }\left( g^{\beta \mu} + u^{\beta}u^{\mu}\right) $$
I can't get the seccond term on inside the brackets.
 A: It appears that $\frac{\partial \alpha_i}{\partial x^{\beta}}$ in the original text refers to derivative wrt spatial coordinates and not the space-time coordinates (I'm not sure about the notations used in that paper). The spatial coordinates are defined using the time-like vector field $\textbf{u}$. The construction is as follows:
Consider our space-time ($\mathcal{M}$, $g$) with family of space-like hypersurface $S(x)$=const. We can define time-like vector field $\textbf{u}$ whose components :
$$u^{\mu}:=f(x)g^{\mu\nu}\partial_{\nu}S(x)$$where $f(x)$ is a normalizing function defined such that $u^{\mu}u_{\mu}=-1$ (considering signature (-,+,+,+)). The induced metric on the space-like hypersurface $S(x)=const$ is then given by the projection tensor $$h_{\mu\nu}=g_{\mu\nu}+u_{\mu}u_{\nu}$$ Example, for Minkowski space-time, if we consider spacelike hypersurface defined as $t=const$, then the vector field $u^{\mu}=(1,0,0,0)$ and induced metric $h_{\mu\nu}=$diag($-1,1,1,1$)+diag($1,0,0,0$)$=$diag($0,1,1,1$) is the usual Euclidean 3D metric. We can use this time-like vector field to define projection of vectors : (1) along $u^{\mu}$ (2) on hypersurface $S(x)=const$
Spatial coordinates (here denoted as $s^a$)  refers to the orthogonal projection of space-time coordinates ($x^a$): $$ds^a=h^{ab}dx_b=(g_{ab}+u_au_b)dx_b$$ and so we have $$\frac{\partial \phi}{\partial x_b}=\frac{\partial \phi}{\partial s^a}(g_{ab}+u_au_b)$$
