Soft-photon theorem near static spherical horizons in generic dimensions

We study Maxwell theory near a nonextremal bifurcate Killing horizon in a general static, spherically symmetric spacetime of dimension $D>3$. We impose near-horizon boundary conditions in a regular Eddington–Finkelstein chart that retain both radiative and Coulombic data, and construct the associated horizon phase space and large-$U(1)$ charge. At null infinity, the radiative falloff depends on the spacetime dimension and involves half-integer powers in odd $D$, whereas a nonextremal horizon at finite radius supports radiative data with an ordinary integer-power expansion. Expanding the horizon fields in Killing frequency and spherical harmonics, we derive the associated soft-photon relation from the large-$U(1)$ Ward identity. For comparison, we express the flat-space leading soft-photon theorem using the same spherical harmonics and commutator normalization as the horizon modes. In $D=4$, the two relations have the same soft-frequency dependence and the same angular dependence of the hard-charge terms, whereas for $D>4$ the flat-space relation contains additional dimension-dependent frequency and angular factors.