Magnetized configurations with black holes and Kaluza-Klein bubbles: Smarr-like relations and first law

TL;DR

This paper derives Smarr-like relations and a first law for magnetized black hole and Kaluza-Klein bubble configurations in Einstein-Maxwell-dilaton gravity, highlighting the importance of magnetic fluxes for solution classification.

Contribution

It introduces a new class of exact solutions with magnetic fluxes in Kaluza-Klein spacetimes and establishes Smarr-like relations and a first law for these configurations.

Findings

Derived Smarr-like relations for mass and tension.

Established a first law incorporating magnetic fluxes.

Showed that interval structure alone is insufficient for solution classification.

Abstract

We present a general class of exact solutions in Einstein-Maxwell-dilaton gravity describing configurations of black holes and Kaluza-Klein bubbles magnetized along the compact dimension. Smarr-like relations for the mass and the tension are found. We also derive the mass and tension first law for the configurations under consideration using the Noether current approach. The solutions we consider are explicit examples showing that in Kaluza-Klein spacetimes the interval (rod) structure and the charges (which are zero by construction for the solutions here), are insufficient to classify the solutions and additional data is necessary, namely the magnetic flux(es).