# The role of the cosmological constant as a pressure in the (2+1)-dimensional black string

## TL;DR

This paper investigates whether the cosmological constant can be interpreted as a pressure in (2+1)-dimensional black strings, demonstrating consistency with thermodynamic laws under certain assumptions despite the dual solution's asymptotic flatness.

## Contribution

It shows that assuming a proportional relationship between pressure and the cosmological constant yields consistent thermodynamics for black strings, extending the pressure interpretation beyond AdS black holes.

## Key findings

- Thermodynamic first law and Smarr relation are consistent under the pressure assumption.
- Thermodynamic quantities remain well-behaved under dual transformation.
- The pressure form is analogous to the AdS black hole case despite asymptotic flatness.

## Abstract

It has been claimed that the cosmological constant in AdS black holes such as the BTZ black hole plays the role of the thermodynamic variable of a pressure in the thermodynamic first law and the Smarr relation from the scaling law of the Christodoulou-Ruffini formula. However, the dual solution of the BTZ black hole is the black string which is asymptotically flat despite the presence of the cosmological constant, and so the explicit form of the pressure with the role of the cosmological constant is unclear in the black string since the pressure is subject to the choice of the energy-momentum tensor. Thus, we show that if the pressure of the black string is still assumed to be proportional to the cosmological constant similar to the case of the BTZ black hole, then the thermodynamic first law is consistent with the Smarr relation from the Christodoulou-Ruffini formula, and the thermodynamic quantities for the pressure are well-behaved under the dual transformation.