Existence of static spherically-symmetric objects in action-dependent Lagrangian theories

We study static symmetric solutions in the context of a gravitational theory based on a action-dependent Lagrangian. Such theory has been designed as a setup to implement dissipative effects into the traditional principle of least action. Dissipation appears therefore from the first principles and has a purely geometric origin. An interesting feature of this theory is the existence of a coupling four-vector ${\lambda }_{\mu }$, which in an expanding background is related to cosmological bulk viscosity. General relativity is recovered with a vanishing ${\lambda }_{\mu }$. We analyze the existence of equilibrium solutions of static configurations aiming to describe astrophysical objects. We find out that the existence of static spherically symmetric configurations occurs only in the particular scenario with vanishing ${\lambda }_t$, ${\lambda }_r$ and ${\lambda }_{\varphi }$ components i.e, ${\lambda }_{\mu }=\{0,0,{\lambda }_{\theta },0\}$. Thus, the component ${\lambda }_{\theta }$ is the unique available parameter of the theory in the astrophysical context. This result severely constrains the existence of this sort of gravitational theories. We proceed then verifying the impact of ${\lambda }_{\theta }$ on the stability and the mass-radius configurations for a reasonable equation of state for the cold dense matter inside compact stars. We further investigate the relativistic spherical collapse in order to track the structure of geometrical singularities appearing in the theory.

Figure 1

Mass-radius diagram. The black line represents the GR configurations. For the red (blue) line the parameter value ${\lambda }_0=-0.3(+0.3)$ has been adopted.

Figure 2

Neutron star mass versus its central density. Stable configurations are represented by the solid lines. Unstable configurations are shown in the dotted lines.