# On the existence of static spherically-symmetric objects in action-dependent Lagrangian theories

## TL;DR

This paper investigates static spherically symmetric solutions in an action-dependent Lagrangian gravitational theory, revealing constraints on the coupling vector and exploring implications for astrophysical objects and gravitational collapse.

## Contribution

It demonstrates that static solutions require a specific form of the coupling vector, constraining the theory's applicability to astrophysical objects and analyzing effects on star stability and collapse.

## Key findings

- Static solutions exist only with specific coupling vector components.
- The coupling parameter influences star stability and mass-radius relations.
- Constraints on the theory limit its astrophysical relevance.

## Abstract

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 analyse 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 $\lambda_{\phi}$ 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.