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

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).Comment: 13 page

Sequences of dipole black rings and Kaluza-Klein bubbles

We construct new exact solutions to 5D Einstein-Maxwell equations describing sequences of Kaluza-Klein bubbles and dipole black rings. The solutions are generated by 2-soliton transformations from vacuum black ring - bubble sequences. The properties of the solutions are investigated. We also derive the Smarr-like relations and the mass and tension first laws in the general case for such configurations of Kaluza-Klein bubbles and dipole black rings. The novel moment is the appearance of the magnetic flux in the Smarr-like relations and the first laws.Comment: 26 pages, 1 figur