Non-Metric Gravity II: Spherically Symmetric Solution, Missing Mass and Redshifts of Quasars

We continue the study of the non-metric theory of gravity introduced in hep-th/0611182 and gr-qc/0703002 and obtain its general spherically symmetric vacuum solution. It respects the analog of the Birkhoff theorem, i.e., the vacuum spherically symmetric solution is necessarily static. As in general relativity, the spherically symmetric solution is seen to describe a black hole. The exterior geometry is essentially the same as in the Schwarzschild case, with power-law corrections to the Newtonian potential. The behavior inside the black-hole region is different from the Schwarzschild case in that the usual spacetime singularity gets replaced by a singular surface of a new type, where all basic fields of the theory remain finite but metric ceases to exist. The theory does not admit arbitrarily small black holes: for small objects, the curvature on the would-be horizon is so strong that non-metric modifications prevent the horizon from being formed. The theory allows for modifications of gravity of very interesting nature. We discuss three physical effects, namely, (i) correction to Newton's law in the neighborhood of the source, (ii) renormalization of effective gravitational and cosmological constants at large distances from the source, and (iii) additional redshift factor between spatial regions of different curvature. The first two effects can be responsible, respectively, for the observed anomaly in the acceleration of the Pioneer spacecraft and for the alleged missing mass in spiral galaxies and other astrophysical objects. The third effect can be used to propose a non-cosmological explanation of high redshifts of quasars and gamma-ray bursts.