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.