  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.
