Masking a singularity with k-essence fields in an emergent gravity metric

It is known that dynamical solutions of the $k$-essence equation of motion change the metric for the perturbations around these solutions and the perturbations propagate in an emergent spacetime with metric $\tilde G^{\mu\nu}$ different from the gravitational metric $g^{\mu\nu}$. We show that for observers travelling with the perturbations, there exist field configurations for the lagrangian $L=[{1\over 2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi]^{1\over 2}$ for which a singularity in the gravitational metric $g^{\mu\nu}$ can be masked or hidden for such observers. This is shown for the Schwarzschild and the Reissner-Nordstrom metrics.