# The First Law for Boosted Kaluza-Klein Black Holes

## TL;DR

This paper explores the thermodynamics of boosted Kaluza-Klein black holes, deriving the first law and Smarr formulas, and introduces an effective tension concept that differs from the ADM tension, enriching black hole thermodynamics in higher dimensions.

## Contribution

It presents the first derivation of the first law for boosted Kaluza-Klein black holes and introduces an effective tension parameter distinct from ADM tension.

## Key findings

- Effective tension is positive for boosted black strings.
- Derived two Smarr formulas for these black holes.
- Extended tension first law to stationary, non-rotating cases.

## Abstract

We study the thermodynamics of Kaluza-Klein black holes with momentum along the compact dimension, but vanishing angular momentum. These black holes are stationary, but non-rotating. We derive the first law for these spacetimes and find that the parameter conjugate to variations in the length of the compact direction is an effective tension, which generally differs from the ADM tension. For the boosted black string, this effective tension is always positive, while the ADM tension is negative for large boost parameter. We also derive two Smarr formulas, one that follows from time translation invariance, and a second one that holds only in the case of exact translation symmetry in the compact dimension. Finally, we show that the `tension first law' derived by Traschen and Fox in the static case has the form of a thermodynamic Gibbs-Duhem relation and give its extension in the stationary, non-rotating case.