n-DBI gravity
  n-DBI gravity is a gravitational theory introduced in arXiv:1109.1468 [hep-th], motivated by Dirac-Born-Infeld type conformal scalar theory and designed to yield non-eternal inflation spontaneously. It contains a foliation structure provided by an everywhere time-like vector field n, which couples to the gravitational sector of the theory, but decouples in the small curvature limit. We show that any solution of Einstein gravity with a particular curvature property is a solution of n-DBI gravity. Amongst them is a class of geometries isometric to Reissner-Nordstrom-(Anti) de Sitter black hole, which is obtained within the spherically symmetric solutions of n-DBI gravity minimally coupled to the Maxwell field. These solutions have, however, two distinct features from their Einstein gravity counterparts: 1) the cosmological constant appears as an integration constant and can be positive, negative or vanishing, making it a variable quantity of the theory; 2) there is a non-uniqueness of solutions with the same total mass, charge and effective cosmological constant. Such inequivalent solutions cannot be mapped to each other by a foliation preserving diffeomorphism. Physically they are distinguished by the expansion and shear of the congruence tangent to n, which define scalar invariants on each leave of the foliation. 
Carlos Herdeiro, Shinji Hirano and Yuki Sato
gr-qc hep-th