# Extended gravitational decoupling in $2+1$ dimensional space--times

## TL;DR

This paper extends the gravitational decoupling method to 2+1 dimensional spacetimes with a cosmological constant, enabling the generation of new black hole solutions and superpositions of sources.

## Contribution

It generalizes the minimal geometric deformation method to lower dimensions, showing conditions for decoupling are similar to 3+1 dimensions and applying it to construct novel black hole solutions.

## Key findings

- Decoupling conditions are consistent with 3+1 dimensions.
- Generated new 2+1 black hole solutions with specific equations of state.
- Constructed regular black holes using polytropic matter.

## Abstract

In this work we extend the so--called Minimal Geometric Deformation method in $2+1$ dimensional space--times with cosmological constant in order to deal with the gravitational decoupling of two circularly symmetric sources. We find that, even though the system here studied is lower dimensional and it includes the cosmological constant, the conditions for gravitational decoupling of two circularly symmetric sources coincides with those found in the $3+1$ dimensional case. We obtain that, under certain circumstances, the extended gravitational decoupling leads to the decoupling of the sources involved in the sense that both the isotropic and the anisotropic sector satisfy Einstein's field equations and the final solution corresponds to a non-linear superposition of two metric components. As particular examples, we implement the method to generate an exterior charged BTZ solution starting from the BTZ vacuum as the isotropic sector and new $2+1$ black hole solutions imposing a barotropic equation of state for the anisotropic sector. We also show that the imposition of a polytropic equation of state of the decoupler matter allows to construct a regular black hole solution in three--dimensional gravity.