Retarded coordinates based at a world line, and the motion of a small black hole in an external universe

TL;DR

This paper develops a system of retarded coordinates based on a world line in curved spacetime, and applies it to derive equations of motion for a small black hole influenced by external tidal fields.

Contribution

It introduces a new retarded coordinate system centered on an arbitrary world line and uses it to derive the MiSaTaQuWa equations for a small black hole in an external universe.

Findings

Retarded coordinates are expressed as an expansion involving acceleration and Riemann tensor.

The formalism is illustrated with cosmological and Schwarzschild examples.

Derived the equations of motion for a small black hole in an external universe.

Abstract

In the first part of this article I present a system of retarded coordinates based at an arbitrary world line of an arbitrary curved spacetime. The retarded-time coordinate labels forward light cones that are centered on the world line, the radial coordinate is an affine parameter on the null generators of these light cones, and the angular coordinates are constant on each of these generators. The spacetime metric in the retarded coordinates is displayed as an expansion in powers of the radial coordinate and expressed in terms of the world line's acceleration vector and the spacetime's Riemann tensor evaluated at the world line. The formalism is illustrated in two examples, the first involving a comoving world line of a spatially-flat cosmology, the other featuring an observer in circular motion in the Schwarzschild spacetime. The main application of the formalism is presented in the second part of the article, in which I consider the motion of a small black hole in an empty external universe. I use the retarded coordinates to construct the metric of the small black hole perturbed by the tidal field of the external universe, and the metric of the external universe perturbed by the presence of the black hole. Matching these metrics produces the MiSaTaQuWa equations of motion for the small black hole.