Geodesics of the five-dimensional Kerr-(A)dS black hole in closed form
Abstract
The Hamilton-Jacobi equation for geodesic motion in the five-dimensional Kerr-(A)dS black hole separates in Boyer-Lindquist coordinates, but the resulting first-order system has so far been integrated case by case, according to the real roots of the radial and the polar polynomial. We give the general solution in closed form. In dimensionless variables both polynomials are quartic, and a theorem of Biermann and Weierstrass then yields the radial and the polar coordinate as one pair of manifestly real formulas, parametrized by an arbitrary point of the orbit rather than by a turning point, and valid for timelike and null geodesics alike whatever the root structure. The time coordinate and the two azimuthal angles follow by quadratures that reduce to elliptic integrals of the third kind alone, and the affine parameter by one more of the same type. Along the way we correct the inverse metric and the radial potential as published.
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BibTeX
@misc{sarıoglu2026geodesics,
title = {{Geodesics of the five-dimensional Kerr-(A)dS black hole in closed form}},
author = {Özgür Sarıoğlu and Enes Yılmaz},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26242},
eprint = {2610.26242},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26242},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Geodesics of the five-dimensional Kerr-(A)dS black hole in closed form
AU - Özgür Sarıoğlu
AU - Enes Yılmaz
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26242
UR - https://arxiv.org/abs/2610.26242
AB - The Hamilton-Jacobi equation for geodesic motion in the five-dimensional Kerr-(A)dS black hole separates in Boyer-Lindquist coordinates, but the resulting first-order system has so far been integrated case by case, according to the real roots of the radial and the polar polynomial. We give the general solution in closed form. In dimensionless variables both polynomials are quartic, and a theorem of Biermann and Weierstrass then yields the radial and the polar coordinate as one pair of manifestly real formulas, parametrized by an arbitrary point of the orbit rather than by a turning point, and valid for timelike and null geodesics alike whatever the root structure. The time coordinate and the two azimuthal angles follow by quadratures that reduce to elliptic integrals of the third kind alone, and the affine parameter by one more of the same type. Along the way we correct the inverse metric and the radial potential as published.
ER -