Gravitational-wave imprints of Kerr--Bertotti--Robinson black holes: frequency blue-shift and waveform dephasing

Abstract: We investigate the orbital dynamics and gravitational-wave signatures of extreme mass-ratio inspirals (EMRIs) in the spacetime of a Kerr black hole immersed in an asymptotically uniform magnetic field, described by the exact Kerr--Bertotti--Robinson (Kerr--BR) solution~\cite{Podolsky:2025tle}. In contrast to the widely used Kerr--Melvin metric, the Kerr--BR spacetime is of algebraic type~D, admits a clear asymptotic structure, and allows for a systematic analytic treatment of geodesics. By analyzing the innermost stable circular orbit (ISCO), we find that the external magnetic field consistently pushes the ISCO to larger radii (r_{\rm ISCO}) for all spin configurations considered. Counterintuitively, despite this outward radial shift, the ISCO orbital frequency (Ω_{\rm ISCO}) increases monotonically with the magnetic-field strength, leading to a robust ``blue-shift'' of the gravitational-wave cutoff frequency. We further show that retrograde orbits are significantly more sensitive to magnetic fields than prograde orbits, and identify a frequency crossover phenomenon in which magnetic corrections can invert the usual spin--frequency hierarchy at the ISCO. Finally, employing a semi-analytic adiabatic evolution scheme driven by exact geodesic relations and a leading-order quadrupole flux, we generate inspiral waveforms and quantify the substantial dephasing induced by the magnetic field. Our results indicate that large-scale magnetic environments can leave observable imprints in EMRI signals for future space-based detectors such as LISA, TianQin, and Taiji, and that neglecting such effects in waveform models may introduce non-negligible biases in parameter estimation, particularly for the black-hole spin.