# Thermodynamics and weak cosmic censorship conjecture in Reissner-Nordstr$\ddot{o}$m anti-de Sitter black holes with scalar field

## TL;DR

This paper investigates the thermodynamics and weak cosmic censorship in Reissner-Nordström anti-de Sitter black holes through scalar field scattering, confirming the laws hold and extremal black holes cannot be overcharged.

## Contribution

It demonstrates the validity of thermodynamics and cosmic censorship in these black holes during scalar field interactions, including extremal and near-extremal cases.

## Key findings

- First law of thermodynamics is recovered during scattering.
- Black holes cannot be overcharged, preserving cosmic censorship.
- Extremal black holes remain extremal after scattering.

## Abstract

The thermodynamics and weak cosmic censorship conjecture in Reissner-Nordstr$\ddot{o}$m anti-de Sitter black holes are investigated by the scattering of the scalar field. The first law of thermodynamics in the non-extremal Reissner-Nordstr$\ddot{o}$m anti-de Sitter black hole is recovered by the scattering. The increase of the horizon radius indicates that the singularity is not naked in this black hole. For the near-extremal and extremal black holes, the validity is tested by the minimum values of the function $f$ at their final states. It is found that both of the near-extremal and extremal black holes can not be overcharged. When $\omega=q\phi$, the final state of the extremal black hole is still an extremal black hole. When $\omega\neq q\phi$, it becomes a near-extremal black hole with new mass and charge.