A tale of two velocities: Threading versus slicing

Abstract

One of the important quantities in cosmology and astrophysics is the 3-velocity of an object. Specifically, when the gravitational fields are strong, one should require the employment of general relativity both in its definition and measurement. Looking into the literature for GR-based definitions of 3-velocity, one usually finds different ad hoc definitions applied according to the case under consideration. Here, we introduce and analyze systematically the two principal definitions of 3-velocity assigned to a test particle following the timelike trajectories in stationary spacetimes. These definitions are based on the [Formula: see text] (threading) and [Formula: see text] (slicing) spacetime decomposition formalisms and defined relative to two different sets of observers. After showing that Synge’s definition of spatial distance and 3-velocity is equivalent to those defined in the [Formula: see text] (threading) formalism, we exemplify the differences between these two definitions by calculating them for particles in circular orbits in axially symmetric stationary spacetimes. Illustrating its geometric nature, the relative linear velocity between the corresponding observers is obtained in terms of the spacetime metric components. Circular particle orbits in the Kerr spacetime, as the prototype and the most well known of stationary spacetimes, are examined with respect to these definitions to highlight their observer-dependent nature. We also examine the Kerr-NUT spacetime in which the NUT parameter, contributing to the off-diagonal terms in the metric, is mainly interpreted not as a rotation parameter but as a gravitomagnetic monopole charge. Finally, in a specific astrophysical setup which includes rotating black holes, it is shown how the local velocity of an orbiting star could be related to its spectral line shifts measured by distant observers.