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In the first part of this article I present a system of retarded coordinatesbased at an arbitrary world line of an arbitrary curved spacetime. Theretarded-time coordinate labels forward light cones that are centered on theworld line, the radial coordinate is an affine parameter on the null generatorsof these light cones, and the angular coordinates are constant on each of thesegenerators. The spacetime metric in the retarded coordinates is displayed as anexpansion in powers of the radial coordinate and expressed in terms of theworld line's acceleration vector and the spacetime's Riemann tensor evaluatedat the world line. The formalism is illustrated in two examples, the firstinvolving a comoving world line of a spatially-flat cosmology, the otherfeaturing an observer in circular motion in the Schwarzschild spacetime. Themain application of the formalism is presented in the second part of thearticle, in which I consider the motion of a small black hole in an emptyexternal universe. I use the retarded coordinates to construct the metric ofthe small black hole perturbed by the tidal field of the external universe, andthe metric of the external universe perturbed by the presence of the blackhole. Matching these metrics produces the MiSaTaQuWa equations of motion forthe small black hole.Comment: 20 pages, revtex4, 2 figure

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