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

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 laws for the configurations under consideration using the Noether current approach. The novelty is the appearance of new terms in the Smarr-like relations and the first laws containing the magnetic flux. The solutions we consider are also 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).