A dark-matter-sensitive orientation clock near rotating black holes

Abstract

A coherent triaxial structure has an intrinsic orientation clock because rotation about its intermediate principal axis is unstable. I ask how this clock changes when the structure moves on a circular equatorial orbit near a rotating black hole in an effective dark matter (DM) environment. The calculation uses a comoving orthonormal tetrad, the electric part of the Riemann tensor, Euler--quaternion dynamics, and the conversion from proper time to stationary coordinate time. The clock shift separates exactly into a local proper-time response and a geometric time-conversion response. I compute a direct grid of 600 DM models for Einasto, regularized cored Navarro--Frenk--White, and Hernquist profiles. Across the scanned domain, the total fractional frequency shift lies between about $-3\times10^{-7}$ and $-3\times10^{-4}$. For the fiducial body model, most of the shift comes from the time conversion. Synthetic timing tests show that a shift near $10^{-4}$ is recoverable only when the clock is highly coherent and its Kerr baseline is known independently. The result is a theoretical timing diagnostic with a synthetic observational forecast; I do not fit real data.