Gauge dependence and self-force from Galilean to Einsteinian free fall, compact stars falling into black holes, Hawking radiation and the Pisa tower at the general relativity centennial

Abstract

Obviously, in Galilean physics, the universality of free fall implies an inertial frame, which in turns implies that the mass [Formula: see text] of the falling body is omitted (because it is a test mass; put otherwise, the center of mass of the system coincides with the center of the main, and fixed, mass [Formula: see text]; or else, we consider only a homogeneous gravitational field). Conversely, an additional (in the opposite or same direction) acceleration proportional to [Formula: see text] would rise either for an observer at the center of mass of the system, or for an observer at a fixed distance from the center of mass of [Formula: see text]. These elementary, but overlooked, considerations fully respect the equivalence principle (EP) and the (local) identity of an inertial or a gravitational pull for an observer in the Einstein cabin. They value as fore-runners of the self-force and gauge dependency in general relativity. Because of its importance in teaching and in the history of physics, coupled to the introductory role to Einstein’s EP, the approximate nature of Galilei’s law of free fall is explored herein. When stepping into general relativity, we report how the geodesic free fall into a black hole was the subject of an intense debate again centered on coordinate choice. Later, we describe how the infalling mass and the emitted gravitational radiation affect the free fall motion of a body. The general relativistic self-force might be dealt with to perfectly fit into a geodesic conception of motion. Then, embracing quantum mechanics, real black holes are not classical static objects any longer. Free fall has to handle the Hawking radiation, and leads us to new perspectives on the varying mass of the evaporating black hole and on the varying energy of the falling mass. Along the paper, we also estimate our findings for ordinary masses being dropped from a Galilean or Einsteinian Pisa-like tower with respect to the current state of the art drawn from precise measurements in ground and space laboratories, and to the constraints posed by quantum measurements. Appendix A describes how education physics and high impact factor journals discuss the free fall. Finally, case studies conducted on undergraduate students and teachers are reviewed.