Probing near-zone magnetic fields with extreme mass-ratio inspirals

We investigate whether weak near-zone magnetic fields can leave observable imprints on extreme-mass-ratio inspiral (EMRI) waveforms. The central massive black hole is modeled by the magnetized Schwarzschild, or Ernst, solution, and the secondary compact object is treated as a neutral point particle on equatorial circular geodesics. We compute the magnetic corrections to the circular-orbit quantities and the innermost stable circular orbit, and then evolve the inspiral using a hybrid, source-corrected Regge--Wheeler--Zerilli approximation, in which the Schwarzschild wave-propagation potentials are kept fixed while the source is evaluated on the magnetized orbit. For a fiducial system with $M=10^6M_\odot$ and $\mu=10M_\odot$, a field strength $B\simeq 4\times10^{-5}M^{-1}$, corresponding to $B_{\rm phys}\sim10^9\,{\rm G}$, produces a one-year dephasing of about $1.3$ rad and reaches the adopted LISA-noise-weighted mismatch threshold. Our results suggest that EMRIs can in principle probe extremely strong near-zone magnetic fields, whereas ordinary magnetic environments around massive black holes are likely too weak to produce detectable effects within the present approximation.