The intrinsic derivative and centrifugal forces in general relativity: I. Theoretical foundations

TL;DR

This paper develops a theoretical framework in general relativity to analyze centrifugal and gravitational forces using covariant derivatives along world lines, clarifying the concept of inertial forces in curved spacetimes.

Contribution

It introduces a covariant derivative-based approach to define and analyze inertial forces, including centrifugal forces, in arbitrary spacetimes within general relativity.

Findings

Clarifies the mathematical structure of inertial forces in curved spacetime.

Provides a foundation for interpreting optical centrifugal forces in static axisymmetric spacetimes.

Prepares for applications to stationary axisymmetric spacetimes and black hole physics.

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.