New metrics of a spherically symmetric gravitational field passing classical tests of General Relativity

Abstract

A general form of a metric preserving all symmetries of a spherically symmetric gravitational field and angular momentum in spherical coordinates is obtained. Such metric may have $g_{01}(r)\neq 0$. The Newtonian limit uniquely defines $g_{00}(r)$. Geodesic motion under such metric exactly reproduces the precession of a planetary orbit, periastron advance of a binary, deflection of light and {Shapiro time delay} if the determinant of the time-radial parts of the metric is $-1$. In this model, the total time for a radial round trip of light is as in the Schwarzschild model, but it allows for light rays to have different speeds propagating toward or from the massive object. The value of $g_{01}(r) $ could be obtained by measuring these speeds. All of these metrics do satisfy Einstein's field equations