Orbital dynamics in GR are calculated from the Schwarzschild metric, which is the spherically symmetric solution embedded in a static Minkowskian space-time. Might not a solution embedded in a cosmological metric be expected to expand with the Hubble flow?The overall theme of this paper concerns the question of whether the global cosmological expansion has any influence on the local dynamics and kinematics within the Solar System. Despite many efforts in the past, this problem is still debated controversially in the current literature and hardly anything is written about it in standard textbooks.
This is rather strange as the question seems to be of obvious interest. Hence there is room for speculations that such an influence might exist and be detectable with current experimental means. It has even be suggested that it be (partly) responsible for the apparently anomalous acceleration of the Pioneer spacecrafts, the so-called ‘Pioneer- Anomaly’ [74, 75, 78, 79, 80, 81], henceforth abbreviated by PA.
Existing investigations in this direction arrive at partially conflicting conclusions. To resolve this issues at a fundamental level one may think to start from an investigation of the embedding-problem for the Solar System.