An Orbital Curvature for ISCOs
Abstract
Building on the fact that circular orbits are the fixed points of the phase portraits arising from the geodesic equations on a black hole spacetime, we develop a general orbital geometry, which captures the stability structure of orbits. The construction uses the orbital energy, angular momentum and angular velocity as state space variables, with response functions encoding how the circular orbits react to changes in the black hole background and in their angular motion. The associated scalar curvature provides a geometric diagnostic of certain distinguished points of the circular orbit family, including, the innermost stable circular orbit (ISCO), where it becomes singular. Across the ISCO, the relevant orbital response functions change sign as the system passes from the stable to the unstable circular orbit branch, and typically vanish at marginal stability. We illustrate the framework for Schwarzschild black holes in asymptotically flat four dimensions, and five dimensional Schwarzschild AdS spacetimes. In the four-dimensional case, the ISCO is associated with a degeneration of the energy response sector, while in the five dimensional AdS case it is detected through the angular response. The AdS example also exhibits a separate curvature singularity at a point which coincides with the minimum temperature of the black hole background. These results show that the orbital response geometry provides a unified geometric framework for characterizing the stability of circular orbits and criticality of ISCOs.
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BibTeX
@misc{mukherjee2026orbital,
title = {{An Orbital Curvature for ISCOs}},
author = {Poulami Mukherjee and Chandrasekhar Bhamidipati and Sudipta Mukherji},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07875},
eprint = {2611.07875},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07875},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Orbital Curvature for ISCOs
AU - Poulami Mukherjee
AU - Chandrasekhar Bhamidipati
AU - Sudipta Mukherji
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07875
UR - https://arxiv.org/abs/2611.07875
AB - Building on the fact that circular orbits are the fixed points of the phase portraits arising from the geodesic equations on a black hole spacetime, we develop a general orbital geometry, which captures the stability structure of orbits. The construction uses the orbital energy, angular momentum and angular velocity as state space variables, with response functions encoding how the circular orbits react to changes in the black hole background and in their angular motion. The associated scalar curvature provides a geometric diagnostic of certain distinguished points of the circular orbit family, including, the innermost stable circular orbit (ISCO), where it becomes singular. Across the ISCO, the relevant orbital response functions change sign as the system passes from the stable to the unstable circular orbit branch, and typically vanish at marginal stability. We illustrate the framework for Schwarzschild black holes in asymptotically flat four dimensions, and five dimensional Schwarzschild AdS spacetimes. In the four-dimensional case, the ISCO is associated with a degeneration of the energy response sector, while in the five dimensional AdS case it is detected through the angular response. The AdS example also exhibits a separate curvature singularity at a point which coincides with the minimum temperature of the black hole background. These results show that the orbital response geometry provides a unified geometric framework for characterizing the stability of circular orbits and criticality of ISCOs.
ER -