Completely integrable sector in 5D Einstein-Maxwell gravity and derivation of the dipole black ring solutions

Abstract

We consider 5D Einstein-Maxwell (EM) gravity in spacetimes with three commuting Killing vectors: one timelike and two spacelike Killing vectors one of them being hypersurface-orthogonal. Assuming a special ansatz for the Maxwell field we show that the 2-dimensional reduced EM equations are completely integrable by deriving a Lax-pair presentation. We also develop a solution generating method for explicit construction of exact EM solutions with considered symmetries. We also derive explicitly a new rotating six parametric 5D EM solution which includes the dipole black ring solution as a particular case.

More on five dimensional EVH black rings

“…where 0 < c ≤ 1 and the following coordinate transformations 12) and taking the µ → 1 limit, the solution (3.1) goes over to the MP black hole. Here the mass and rotation parameters are given by…”

“…A generalization of the double rotating black ring [3] containing dipole charges was introduced in [14] (see also [11][12][13]). This is a solution of Einstein-Maxwell-dilaton theory…”

“…Defining the following parameters 12) and performing the following coordinate transformations [14] 13) one can explicitly find the 5D MP black hole rotating in both φ and ψ directions 14) where ∆ = 1 r 2 r 2 + a 2 1 r 2 + a 2 2 − 2m, and ρ 2 = r 2 + a 2 1 cos 2 θ + a 2 2 sin 2 θ .…”

“…There are in particular black rings with conserved dipole charges, while having vanishing electric or magnetic charge [11][12][13][14]. The dipole charges can be defined as Noether-Wald conserved charge [15,16] and appear in the first law of black hole thermodynamics for the black rings [17].…”

Black ring with a positive cosmological constant

“…Microscopic analysis of entropy of black ring has been considered in [9]. Relation with integrable system was noted in [10]. See [11] for further developments and a comprehensive review.…”