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Tidal coupling between members of a compact binary system can have aninteresting and important influence on that binary's dynamical inspiral. Tidalcoupling also distorts the binary's members, changing them (at lowest order)from spheres to ellipsoids. At least in the limit of fluid bodies and Newtoniangravity, there are simple connections between the geometry of the distortedellipsoid and the impact of tides on the orbit's evolution. In this paper, wedevelop tools for investigating tidal distortions of rapidly rotating blackholes using techniques that are good for strong-field, fast-motion binaryorbits. We use black hole perturbation theory, so our results assume extrememass ratios. We develop tools to compute the distortion to a black hole'scurvature for any spin parameter, and for tidal fields arising from any boundorbit, in the frequency domain. We also develop tools to visualize thehorizon's distortion for black hole spin $a/M \le \sqrt{3}/2$ (leaving the morecomplicated $a/M > \sqrt{3}/2$ case to a future analysis). We then study how aKerr black hole's event horizon is distorted by a small body in a circular,equatorial orbit. We find that the connection between the geometry of tidaldistortion and the orbit's evolution is not as simple as in the Newtonianlimit.

Strong-field tidal distortions of rotating black holes: Formalism and results for circular, equatorial orbits