Die Asymmetrie der kosmischen Zeit und RIEMANNS Gravitationstheorie1)

RIEMANN himself has considered his formulation of the differential geometry of curved spaces as a first step to a unified geometrical theory of “one ether of gravity, electricity and magnetism”. RIEMANN has pointed out that a fundamental point in such a theory of gravitation has to be the asymmetry of its sources: only positive masses exist. – According to RIEMANN this asymmetry of sources to be coupled with an asymmetry of gravitation field equation against the time-reversion t → - t. Therefore, the gravitation field equation is of the type of a continuity-equation of a velocity field v_i?g^ikθ _k Φ. RIEMANN 's ether is incompressible in empty space-domains: θ _k (g^1/2v^k) = o. But, in domains with a massdensity σ > o it is θg^1/2/θt = ?2 kcσ = ? 2 kcg^1/2σ₀ (with a universal constant kc). The matter-density defines depressions of the ether. In a general-relativistic approach RIEMANN 's ansatz means that in empty space-time domains the world-geometry is the purely metrical “RIEMANN ian” geometry. However, in domains with a non-vanishing matter-tensor T_μ^v ≠ o the geometry becomes “non-RIEMANN ian” affine connecting and is of the type of WEYL 's geometry or of the “EINSTEIN -CARTAN theories of gravitation”. Especially, RIEMANN 's field equation for the empty space θ _k ((g^1/2g^ikθ _k Φ) = o. is the EINSTEIN equation (-|g_μv|)^1/2 R₀⁰ = o with g₀₀ = - Φ²c^-4.