Upper bound on the radius of the innermost stable circular orbit of black holes

Abstract: In this work, we investigate a universal upper bound on the radius of the innermost stable circular orbit (ISCOs) for massive particles in static, spherically symmetric, and asymptotically flat black hole spacetimes. By analyzing the spacetime metrics with external matter fields, we derive the characteristic equation for ISCO via the effective potential method. By imposing appropriate energy conditions for the matter fields, we rigorously demonstrate that the ISCO radius is bounded by $r_{\mathrm{ISCO}}\le 6M$, where $M$ is the total ADM mass of the black hole. The Schwarzschild black hole saturates this bound ($r_{\mathrm{ISCO}}=6M$), and the Reissner-Nordstr\"om black hole, supergravity black holes, fluid sphere models, which satisfy the imposed energy conditions, also obey $r_{\mathrm{ISCO}}\le 6M$. The universality of this upper limit provides a theoretical benchmark for observational astrophysics: deviations from $6M$ in accretion disk observations or gravitational wave signals could indicate the presence of exotic matter fields. This work highlights the interplay between black hole geometry and external matter fields, paving the way for future studies in compact object dynamics.