n-DBI gravity - Herdeiro, Carlos (2011)
n-DBI gravity is a gravitational theory introduced in [C. Herdeiro and S. Hirano, arXiv: 1109.1468.], motivated by Dirac-Born-Infeld type conformal scalar theory and designed to yield noneternal inflation spontaneously. It contains a foliation structure provided by an everywhere timelike 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. Among them is a class of geometries isometric to a 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; and (2) there is a nonuniqueness 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 leaf of the foliation.