On the orbital motion description with a proper reference frame in a complete Schwarzschild field

Abstract

It is a fact that, none of the diverse coordinate systems unified in the Post-Newtonian formalism used to describe the exterior Schwarzschild field can be regarded as being materialized by a reference frame. Only the polar Gaussian coordinates (ρ, ϑ, ϕ, t), or their naturally associated Fermi coordinates, can be shown to have this property (Synge, 1960).

On the orbital motion description with a proper reference frame in a complete Schwarzschild field

“…(2) (Brumberg, 1991).The second term comes from the unique non-linear term taken in the Christoffel symbols and is shown to play an important role as a source of relativistic perturbations. For example, in standard coordinates, this term appears as the first one in the right hand side of the equations…”

“…It is a fact that, none of the diverse coordinate systems unified in the Post-Newtonian formalism used to describe the exterior Schwarzschild field can be regarded as being materialized by a reference frame. Only the polar Gaussian coordinates (ρ,ϋ,φ,ί), or their naturally associated Fermi coordinates, can be shown to have this property (Synge, 1960).…”

“…In effect, by using the general expression for the world function (Synge, 1960) in these coordinates, it is easy to see that for an observer at rest with respect to the Sun, the relativistic distance | p\ -P2 |, between two particles aligned with him, becomes their Euclidean distance; this is the first result. Now, starting from the lagrangian L, written again in these coordinates, corresponding to a test particle moving in the exterior field, and using the first integrals in the usual way, the equation of the trajectory (ΰ = π/2, ΰ' = 0) results to be = f(u) = a\ß 2 -l) + 2ma 2 ß 2 u + 4ma\ß 2 -l)ulogu +…”