Research on Geometry of Gravitation by Distinguishing Between Spatial and Temporal Part of Spherically Symmetric Metric - Trenčevski, Kostadin; Celakoska, Emilija (Communications in Mathematics and Applications, 2025)
We prove a theorem that in case of a spherically symmetric metric the curvature tensor of the spatial metric (by ignoring the temporal component) must be 0, because otherwise it leads to a paradox. We consider gravitation, assuming that a point-mass body with mass \(M\) radially translates each point by a 3-vector with magnitude \(GM/c^2\). The temporal metric coefficient \(g_{44}\) is uniquely determined by \(\boldsymbol{g}\), so the Riemann curvature tensor is non-zero. The experimental tests apply and the basic results remain the same as in GR. The obtained equations for \(n\)-body problem differ from the Einstein-Infeld-Hoffmann equations by Lorentz invariant addends. While in GR the curvature scalar \(\mathcal{R}\) is proportional to the density of matter, here \(\mathcal{R}\) is proportional to the density of energy and also to the density of matter.
Keywords: 4D metric; Field equations; Experimental tests of General Relativity