Space-time and spatial geodesic orbits in Schwarzschild geometry. (English) Zbl 1393.83006
Summary: Geodesic orbit equations in the Schwarzschild geometry of general relativity reduce to ordinary conic sections of Newtonian mechanics and gravity for material particles in the non-relativistic limit. On the contrary, geodesic orbit equations for a proper spatial submanifold of Schwarzschild metric at any given coordinate-time correspond to an unphysical gravitational repulsion in the non-relativistic limit. This demonstrates at a basic level the centrality and critical role of relativistic time and its intimate pseudo-Riemannian connection with space. Correspondingly, a commonly popularised depiction of geodesic orbits of planets as resulting from the curvature of space produced by the Sun, represented as a rubber sheet dipped in the middle by the weighing of that massive body, is mistaken and misleading for the essence of relativity, even in the non-relativistic limit.
MSC:
| 83C10 | Equations of motion in general relativity and gravitational theory |
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
| 83C15 | Exact solutions to problems in general relativity and gravitational theory |
| 53C22 | Geodesics in global differential geometry |
Keywords:
general theory of relativity; curvature; space-time; curved space; gravitation; geodesic orbits; Schwarzschild geometryReferences:
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