The Intrinsic Derivative and Centrifugal Forces in General Relativity: I.

Abstract

Everyday experience with centrifugal forces has always guided thinking on the close relationship between gravitational forces and accelerated systems of reference. Once spatial gravitational forces and accelerations are introduced into general relativity through a splitting of spacetime into space-plus-time associated with a family of test observers, one may further split the local rest space of those observers with respect to the direction of relative motion of a test particle world line in order to define longitudinal and transverse accelerations as well. The intrinsic covariant derivative (induced connection) along such a world line is the appropriate mathematical tool to analyze this problem, and by modifying this operator to correspond to the observer measurements, one understands more clearly the work of Abramowicz et al who define an "optical centrifugal force" in static axisymmetric spacetimes and attempt to generalize it and other inertial forces to arbitrary spacetimes. In a companion article the application of this framework to some familiar stationary axisymmetric spacetimes helps give a more intuitive picture of their rotational features including spin precession effects, and puts related work of de Felice and others on circular orbits in black hole spacetimes into a more general context.

Gravitomagnetism and relative observer clock effects

“…The first inequality is discussed in equations (4.24) and (4.25) of [3], while an explicit formula for the extremely accelerated observer angular velocity has been given by Semerák [6] …”

“…which always has two real solutions, one subluminal and the other superluminal [3]. One can rewrite this equation in the form 5) which is the familiar relativistic addition of velocity formula applied to the two relative velocity differences, equivalent to…”

“…(a) the observer-dependent single-clock clock effect: the difference between the periods of the oppositely rotating geodesic test particles as measured by the observer's clock [3]; (b) the observer-dependent two-clock clock effect: the difference between the periods of the oppositely rotating geodesic test particles as measured by their own clocks for one revolution with respect to an observer [4][5][6]; and (c) the observer-independent two-clock clock effect: the difference between the periods of the oppositely rotating geodesic test particles as measured by their own clocks between two crossing events [7][8][9][10].…”

Inertial forces and photon surfaces in arbitrary spacetimes

“…However, their suggested generalizations (see Abramowicz, Nurowski and Wex [5]) have been partially criticized by several authors (see for example de Felice [6], Sonego and Massar [7] and Bini, Carini and Jantzen [8]). Some of the critical comments refer to the fact that the inertial forces, as defined in [5], have unwanted features, others to the fact that they are in a certain sense ambiguous.…”

Position determination and strong field parallax effects for photon emitters in the Schwarzschild spacetime

“…Note that the intrinsic curvature of this circular orbit in the curved Schwarzschild spacetime can be described in terms of the associated Lie relative curvature and radius of curvature defined in [11,12] …”