Starting with geometrical premises, we infer the existence of fundamental cosmological scalar fields. We then consider physically relevant situations in which spacetime metric is induced by one or, in general, by two scalar fields, in accord with the Papapetrou algorithm. The first of these fields, identified with dark energy, has exceedingly small but finite (subquantum) Hubble mass scale (≈ 10 −33 eV), and might be represented as a neutral superposition of quasi-static electric fields. The second field is identified with dark matter as an effectively scalar conglomerate composed of primordial neutrinos and antineutrinos in a special tachyonic state.

The Janis-Newman-Winicour and Papapetrou metrics represent counterparts to the Schwarzschild black hole with scalar and antiscalar background fields, correspondingly (where "anti" is to be understood as in "anti-de Sitter"). There is also a scalar counterpart (the Krori-Bhattacharjee metric) to the Kerr black hole. Here we study analytical connections between these solutions and obtain the exact rotational generalization of the antiscalar Papapetrou spacetime as a viable alternative to the Kerr black hole. The antiscalar metrics appear to be the simplest ones as they do not reveal event horizons and ergospheres, and they do not involve an extra parameter for scalar charge. Static antiscalar field is thermodynamically stable and self-consistent, but this is not the case for the scalar Janis-Newman-Winicour solution; besides, antiscalar thermodynamics is reducible to black-hole thermodynamics. Lensing, geodetic and Lense-Thirring effects are found to be practically indistinguishable between antiscalar and vacuum solutions in weak fields. Only strongfield observations might provide a test for the existence of antiscalar background. In particular, the antiscalar solution predicts 5% larger shadows of supermassive compact objects, as compared to the vacuum solution. Another measurable aspect is the 6.92% difference in the frequency of the innermost stable circular orbit, characterizing the upper cut-off in the gravitational wave spectrum.

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It is shown that the timelike, spacelike and null versions of the Ehlers identity, as well as ensuing Raychaudhuri equations, might be all derived within a single geometrical approach based on the definition of the Riemann curvature tensor specified with respect to the corresponding congruence. Still, spacelike and null cases have a number of non-trivial peculiarities deserving special attention.