# No Violation of the Second Law in Extended Black Hole Thermodynamics

## TL;DR

This paper clarifies that the second law of extended black hole thermodynamics remains valid when considering enthalpy instead of internal energy, countering claims of entropy reduction in near-extremal black holes.

## Contribution

It demonstrates that the apparent violation of the second law is due to an incorrect assumption about energy change, emphasizing the importance of using enthalpy in black hole thermodynamics.

## Key findings

- The second law holds when considering enthalpy rather than internal energy.
- Dropping particles into black holes does not violate the second law under proper assumptions.
- Previous claims of entropy decrease are based on an oversimplified energy change assumption.

## Abstract

Recently a number of papers have claimed that the horizon area - and thus the entropy - of near extremal black holes in anti-de Sitter spacetimes can be reduced by dropping particles into them. In this note we point out that this is a consequence of an underlying assumption that the energy of an infalling particle changes only the internal energy of the black hole, whereas a more physical assumption would be that it changes the enthalpy (mass). In fact, under the latter choice, the second law of extended black hole thermodynamics is no longer violated.