Analytic solutions for parallel transport along generic bound geodesics in Kerr spacetime

Abstract

We provide analytical closed form solutions for the parallel transport along a bound geodesic in Kerr spacetime. This can be considered the lowest order approximation for the motion a spinning black hole in an extreme mass-ratio inspiral.As an illustration of the usefulness of our new found expressions we scope out the locations of spin-spin resonances in quasi-circular EMRIs. All solutions are given as functions of Mino time, which facilitates the decoupling of the equations of motion. To help physical interpretation, we also provide an analytical expression for the proper time along a geodesic as a function of Mino time.

Adiabatic waveforms for extreme mass-ratio inspirals via multivoice decomposition in time and frequency

“…We begin by discussing bound Kerr geodesics. The most important aspects of this content are discussed in depth elsewhere [24][25][26][27][28]; we briefly review this material for this paper to be self contained, as well as to introduce notation and conventions that we use. Certain lengthy but important formulas are given in Appendix A.…”

“…We have found that x I ≡ cos I is a particularly good parameter to describe inclination: x I varies smoothly from 1 to −1 as orbits vary from prograde equatorial to retrograde equatorial, with L z having the same sign as x I . Schmidt [25] first showed how to compute (E, L z , Q) for generic Kerr orbits; a particularly clean representation is provided by van de Meent [28]. We summarize his formulas in Appendix A, tweaking them slightly to use our preferred parameter set (p, e, x I ).…”

“…In this appendix, we list formulas for the geodesic constants of the motion E, L z , and Q as functions of our preferred parameters p, e, and x I ≡ cos I. Formulas for these constants were first worked out by Schmidt [25], and are provided in particularly clean form for generic orbits by van de Meent [28]. We list them here using our preferred parameterization, and also include formulas for spherical orbits.…”

“…[25] and [28]: we use in these lists rather than to avoid notational collision with = √ M 2 − a 2 /2M r + , as well as with the very similar ε ≡ µ/M .…”

Classification of radial Kerr geodesic motion

“…In the last two years, several novel analytic results on Kerr geodesics were achieved [40,41,[51][52][53][54][55][56][57][58][59][60]. Explicitly real, fully explicit, "initial data-dependent" analytical solutions in terms of elliptic functions were given for (i) radial and polar motion for timelike bounded orbits [52]; (ii) generic (i.e. excluding zero measure sets) polar motion for null or timelike orbits [40,41]; (iii) generic radial motion for null orbits [53] and (iv) general (i.e.…”