Kerr metric, geodesic motion, and Flyby Anomaly in fourthorder Conformal Gravity
Abstract
In this paper we analyze the Kerr geometry in the context of Conformal Gravity, an alternative theory of gravitation, which is a direct extension of General Relativity (GR). Following previous studies in the literature, we introduce an explicit expression of the Kerr metric in Conformal Gravity, which naturally reduces to the standard GR Kerr geometry in the absence of Conformal Gravity effects. As in the standard case, we show that the HamiltonJacobi equation governing geodesic motion in a spacetime based on this geometry is indeed separable and that a fourth constant of motion—similar to Carter's constant—can also be introduced in Conformal Gravity. Consequently, we derive the fundamental equations of geodesic motion and show that the problem of solving these equations can be reduced to one of quadratures. In particular, we study the resulting timelike geodesics in Conformal Gravity Kerr geometry by numerically integrating the equations of motion for Earth flyby trajectories of spacecraft. We then compare our results with the existing data of the Flyby Anomaly in order to ascertain whether Conformal Gravity corrections are possibly the origin of this gravitational anomaly. Although Conformal Gravity slightly affects the trajectories of geodesic motion around a rotating spherical object, we show that these corrections are minimal and are not expected to be the origin of the Flyby Anomaly, unless conformal parameters are drastically different from current estimates. Therefore, our results confirm previous analyses, showing that modifications due to Conformal Gravity are not likely to be detected at the Solar System level, but might affect gravity at the galactic or cosmological scale.
 Publication:

General Relativity and Gravitation
 Pub Date:
 June 2014
 DOI:
 10.1007/s107140141741z
 arXiv:
 arXiv:1401.6503
 Bibcode:
 2014GReGr..46.1741V
 Keywords:

 Conformal gravity;
 Kerr metric;
 Geodesics;
 Flyby Anomaly;
 General Relativity and Quantum Cosmology;
 Astrophysics  Cosmology and Nongalactic Astrophysics
 EPrint:
 Revised version: improved notation, added references, comments, and one equation. 31 pages, including 2 figures. Accepted for publication in General Relativity and Gravitation