Curved space, curved time, and curved space-time in Schwarzschild geodetic geometry

Abstract

We investigate geodesic orbits and manifolds for metrics associated with Schwarzschild geometry, considering space and time curvatures separately. For ‘a-temporal’ space, we solve a central geodesic orbit equation in terms of elliptic integrals. The intrinsic geometry of a two-sided equatorial plane corresponds to that of a full Flamm’s paraboloid. Two kinds of geodesics emerge. Both kinds may or may not encircle the hole region any number of times, crossing themselves correspondingly. Regular geodesics reach a periastron greater than the $r_S$ Schwarzschild radius, thus remaining confined to a half of Flamm’s paraboloid. Singular or s-geodesics tangentially reach the $r_S$ circle. These s-geodesics must then be regarded as funneling through the ‘belt’ of the full Flamm’s paraboloid. Infinitely many geodesics can possibly be drawn between any two points, but they must be of specific regular or singular types. A precise classification can be made in terms of impact parameters. Geodesic structure and completeness is conveyed by computer-generated figures depicting either Schwarzschild equatorial plane or Flamm’s paraboloid. For the ‘curved-time’ metric, devoid of any spatial curvature, geodesic orbits have the same apsides as in Schwarzschild space-time. We focus on null geodesics in particular. For the limit of light grazing the sun, asymptotic ‘spatial bending’ and ‘time bending’ become essentially equal, adding up to the total light deflection of 1.75 arc-seconds predicted by general relativity. However, for a much closer approach of the periastron to $r_S$, ‘time bending’ largely exceeds ‘spatial bending’ of light, while their sum remains substantially below that of Schwarzschild space-time.