Spherically symmetric solutions of light Galileon

We have been studied the model of light Galileon with translational shift symmetry $\varphi \to \varphi +c$. The matter Lagrangian is presented in the form $$\mathcal{L}${\phi}= -\eta (\partial \phi)^2+\beta G^{\mu\nu}\partial{\mu}\phi\partial_{\nu}\phi$. We have been addressed two issues: the first is that, we have been proven that, this type of Galileons belong to the modified matter-curvature models of gravity in type of $f(R,R^{\mu \nu }{T}_{\mu \nu }^m)$. Secondly, we have been investigated exact solution for spherically symmetric geometries in this model. We have been found an exact solution with singularity at $r=0$ in null coordinates. We have been proven that the solution has also a non-divergence current vector norm. This solution can be considered as an special solution which has been investigated in literature before, in which the Galileon’s field is non-static (time dependence). Our scalar-shift symmetrized Galileon has the simple form of $\varphi =t$, which it is remembered by us dilaton field.