Conformal Structures and Reference Frames in the Post-Newtonian Approximation to General Relativity

Abstract

According to general relativity, the gravitational field of a body is described by the metric tensor field g on the spacetime manifold. With respect to some given local chart (x ^μ) ≡ (x ⁰, x ¹, x ², x ³) of spacetime, the components (“potentials”) g _μv of g contain ten independent elements in general. That is, in order to describe the relativistic gravitational field of the Earth, the knowledge of the multipole structure of each of these “potentials” is required. The gravitational field of the Earth, however, is weak so that the post-Newtonian approximation to general relativity can be used in this context. To be more specific, let U be the Newtonian potential in the vicinity of the Earth and c the velocity of light. Then one obtains U/c ² ≈ 10^-10; which expresses essentially the weakness of the gravitational field.