Scalar decoupling and critical-curve invariance in a reduced MOG black-hole theory

Abstract

Linear perturbations of a Schwarzschild-MOG black hole separate into scalar and gravito-vector sectors in the reduced strong-field theory at leading order in $\mu \bar{G}M\ll 1$. The field redefinition ${\hat{h}}_{\mu \nu }=h_{\mu \nu }-\psi {\bar{g}}_{\mu \nu }$ gives a decoupled Klein--Gordon equation for the scalar and the Reissner--Nordström equations for the gravito-vector sector. The scalar is stable under the stated spectral boundary conditions for $m_G^2\ge 0$. A scalar-only excitation produces a conformal metric perturbation with null Hamiltonian contribution $H_1^{(s)}=-\psi H_0$. For fixed observer worldlines and conformally corresponding orthonormal frames, every linear scalar-to-critical-curve transfer coefficient vanishes, including monopole breathing, provided the conformal factor is smooth, positive, and preserves the capture boundary. The exceptional charged axial dipole instead induces a constrained metric perturbation. A first-order radial integral reconstructs this perturbation in a gauge regular at the future horizon and on the sphere, separating the radiative amplitude from stationary angular momentum without division by frequency. For a unit-incident canonical vector amplitude with zero angular momentum, matched asymptotic solutions give an explicit charge-dependent local metric response that vanishes as ${\omega }^2$ at fixed radius and fixed subextremal background. A separate infinitesimal Kerr--Newman benchmark yields the stationary impact-parameter midpoint shift and recovers $\delta C_b=-2a_{\mathrm{rot}}$ in the neutral limit. These results exclude linear scalar breathing of the specified angular critical curve and distinguish the low-frequency vector-induced metric response from a stationary rotation effect.