We carry on a comprehensive study on static fluid distributions endowed with hyperbolical symmetry. Their physical properties are analyzed in detail. The energy density appears to be necessarily negative, which suggests that any possible application of this kind of fluids requires extreme physical conditions where quantum effects are expected to play an important role. Also, it is found that the fluid distribution cannot fill the region close to the center of symmetry. Such a region may be represented by a vacuum cavity around the center. A suitable definition of mass function, as well as the Tolman mass are explicitly calculated. While the former is positive defined, the latter is negative in most cases, revealing the repulsive nature of gravitational interaction. A general approach to obtain exact solutions is presented and some exact analytical solutions are exhibited.

• Received 9 November 2020
• Accepted 6 January 2021

DOI:https://doi.org/10.1103/PhysRevD.103.024037