Geodesics in Black-Hole Spacetimes: A Qualitative Classification Using Vieta's Formulae
Abstract
This monograph classifies the geodesics of the Schwarzschild, Kerr and Schwarzschild-de Sitter black holes without solving for their turning points. In all three spacetimes, the radial and polar motions are governed by polynomials whose coefficients depend only on the constants of motion, and the qualitative behavior of a geodesic is fixed by the pattern of their roots. We read this pattern directly off the coefficients, using Vieta's formulae together with the discriminant, Descartes' rule of signs, and the values of the polynomials at the horizons and at infinity. In Schwarzschild spacetime, the reciprocals of the radial roots of every non-radial geodesic add up to 1/(2M), which gives the third root of a bound orbit in closed form and the separatrix p=6+2e. In Kerr spacetime, the signs of the coefficients show that no radially bound orbit has ε2≥1 and that vortical geodesics, timelike or null, have no radial turning point; a positive discriminant is necessary but not sufficient for bound motion, and Descartes' rule applied at the event horizon gives the exact criterion. Spherical orbits and their stability, the equatorial ISCO and light rings, and the photon shell follow from the same relations. In Schwarzschild-de Sitter spacetime, the timelike radial polynomial is a quintic with exactly one negative root, a bound branch requires ε2+ΛL2/3<1, and both marginally stable circular orbits lie beyond 6M. Most of these results are known; the aim is to show that a single, elementary algebraic route leads to all of them. The text is meant for graduate students who have taken a first course in general relativity, and each chapter ends with worked examples.
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BibTeX
@misc{piattella2026geodesics,
title = {{Geodesics in Black-Hole Spacetimes: A Qualitative Classification Using Vieta's Formulae}},
author = {Oliver F. Piattella},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22399},
eprint = {2610.22399},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22399},
}
RIS
TY - UNPB
TI - Geodesics in Black-Hole Spacetimes: A Qualitative Classification Using Vieta's Formulae
AU - Oliver F. Piattella
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.22399
UR - https://arxiv.org/abs/2610.22399
AB - This monograph classifies the geodesics of the Schwarzschild, Kerr and Schwarzschild-de Sitter black holes without solving for their turning points. In all three spacetimes, the radial and polar motions are governed by polynomials whose coefficients depend only on the constants of motion, and the qualitative behavior of a geodesic is fixed by the pattern of their roots. We read this pattern directly off the coefficients, using Vieta's formulae together with the discriminant, Descartes' rule of signs, and the values of the polynomials at the horizons and at infinity. In Schwarzschild spacetime, the reciprocals of the radial roots of every non-radial geodesic add up to 1/(2M), which gives the third root of a bound orbit in closed form and the separatrix p=6+2e. In Kerr spacetime, the signs of the coefficients show that no radially bound orbit has ε^2≥1 and that vortical geodesics, timelike or null, have no radial turning point; a positive discriminant is necessary but not sufficient for bound motion, and Descartes' rule applied at the event horizon gives the exact criterion. Spherical orbits and their stability, the equatorial ISCO and light rings, and the photon shell follow from the same relations. In Schwarzschild-de Sitter spacetime, the timelike radial polynomial is a quintic with exactly one negative root, a bound branch requires ε^2+ΛL^2/3<1, and both marginally stable circular orbits lie beyond 6M. Most of these results are known; the aim is to show that a single, elementary algebraic route leads to all of them. The text is meant for graduate students who have taken a first course in general relativity, and each chapter ends with worked examples.
ER -