On the horizons in a viable vector-tensor theory of gravitation

Abstract

A certain vector-tensor (VT) theory of gravitation was tested in previous papers. In the background universe, the vector field of the theory has a certain energy density, which is appropriate to play the role of vacuum energy (cosmological constant). Moreover, this background and its perturbations may explain the temperature angular power spectrum of the cosmic microwave background (CMB) obtained with WMAP (Wilkinson Map Anisotropy Probe), and other observations, as e.g., the Ia supernova luminosities. The parametrized post-Newtonian limit of the VT theory has been proved to be identical to that of general relativity (GR), and there are no quantum ghosts and classical instabilities. Here, the stationary spherically symmetric solution, in the absence of any matter content, is derived and studied. The metric of this solution is formally identical to that of the Reissner-Nordstr\"om-de Sitter solution of GR, but the role of the electrical charge is played by a certain quantity $\Gamma $ depending on both the vector field and the parameters of the VT theory. The black hole and cosmological horizons are discussed. The radius of the VT black hole horizon deviates with respect to that of the Kottler-Schwarzschild-de Sitter radius. Realistic relative deviations depend on $\Gamma $ and reach maximum values close to 30 per cent. For large enough $\Gamma $ values, there is no any black hole horizon, but only a cosmological horizon. The radius of this last horizon is almost independent of the mass source, the vector field components, and the VT parameters. It essentially depends on the cosmological constant value, which has been fixed by using cosmological observational data (CMB anisotropy, galaxy correlations and so on).