Frenet--Serret invariants of timelike worldlines in Kerr spacetime: chart independence, horizon crossing and null-endpoint limits
Abstract
We examine the Frenet-Serret (FS) curvature kappa and first torsion tau_1 of timelike worldlines in Kerr spacetime, focusing mainly on non-Killing equatorial trajectories. Since these quantities are scalars associated with a worldline, a coordinate transformation cannot remove a divergence for the same curve. We demonstrate this by transforming a single worldline between five coordinate charts and verifying agreement of its invariants, including in an off-equatorial test. We also show why reusing the same coordinate components of the tangent vector in two charts generally describes different curves. For a non-unit tangent v with norm S = g(v,v), we derive the identity kappa squared = Q squared / S cubed - (q dot q) / S squared, where q = nabla_v v and Q = q dot v. When Q approaches a nonzero limit and q dot q remains finite, the curvature scales as |Q_0| S to the power -3/2 as the curve approaches a null endpoint. Under the additional condition that S' is nonzero, we obtain a closed expression for the limiting value of tau_1. We verify these results using exact symbolic calculations for two Kerr-Schild spirals, including one with spin parameter a greater than the mass M, for which no event horizon exists. For Killing observers, we derive closed-form expressions for kappa and tau_1 and show that both become singular at light surfaces. Finally, we distinguish FS torsion, which measures rotation relative to the acceleration direction, from gyroscope rotation rates defined using regular reference frames. For a geodesic plunge, the rotation rate relative to the principal-null frame remains finite across the horizon, while the rate relative to the Carter-plane axis agrees with Marck's precession rate. Our analysis focuses mainly on equatorial examples; a numerical comparison in double-null coordinates is left for future work.
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BibTeX
@misc{majumder2026frenetserret,
title = {{Frenet--Serret invariants of timelike worldlines in Kerr spacetime: chart independence, horizon crossing and null-endpoint limits}},
author = {Paulami Majumder},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.29759},
eprint = {2605.29759},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.29759},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Frenet--Serret invariants of timelike worldlines in Kerr spacetime: chart independence, horizon crossing and null-endpoint limits
AU - Paulami Majumder
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2605.29759
UR - https://arxiv.org/abs/2605.29759
AB - We examine the Frenet-Serret (FS) curvature kappa and first torsion tau_1 of timelike worldlines in Kerr spacetime, focusing mainly on non-Killing equatorial trajectories. Since these quantities are scalars associated with a worldline, a coordinate transformation cannot remove a divergence for the same curve. We demonstrate this by transforming a single worldline between five coordinate charts and verifying agreement of its invariants, including in an off-equatorial test. We also show why reusing the same coordinate components of the tangent vector in two charts generally describes different curves. For a non-unit tangent v with norm S = g(v,v), we derive the identity kappa squared = Q squared / S cubed - (q dot q) / S squared, where q = nabla_v v and Q = q dot v. When Q approaches a nonzero limit and q dot q remains finite, the curvature scales as |Q_0| S to the power -3/2 as the curve approaches a null endpoint. Under the additional condition that S' is nonzero, we obtain a closed expression for the limiting value of tau_1. We verify these results using exact symbolic calculations for two Kerr-Schild spirals, including one with spin parameter a greater than the mass M, for which no event horizon exists. For Killing observers, we derive closed-form expressions for kappa and tau_1 and show that both become singular at light surfaces. Finally, we distinguish FS torsion, which measures rotation relative to the acceleration direction, from gyroscope rotation rates defined using regular reference frames. For a geodesic plunge, the rotation rate relative to the principal-null frame remains finite across the horizon, while the rate relative to the Carter-plane axis agrees with Marck's precession rate. Our analysis focuses mainly on equatorial examples; a numerical comparison in double-null coordinates is left for future work.
ER -