Multipole Decomposition of the Gravitational Field of a Point Mass at the Kerr Horizon: Singular-Field Origin of the Divergent Absorbed Energy

We show that the gravitational energy absorbed by a Kerr black hole from an infalling point mass diverges at leading order in perturbation theory. Using a horizon-regular Newman–Penrose tetrad in ingoing Kerr coordinates, we derive the distributional Teukolsky source for $\psi_4$ and analyze the Detweiler–Whiting singular field in a local rest frame at horizon crossing. The singular field induces a horizon-localized quadrupolar excitation whose Teukolsky amplitude has a mode-independent large-$\ell$ limit. By combining the asymptotics of spin-weighted spheroidal harmonics with near-horizon matched expansions of the radial Teukolsky equation, we obtain \[ \mathcal{F}_\ell^H(a)=C_H(a)+O(1/\ell), \qquad C_H(a)>0, \] for all $a\in[0,M)$. Consequently, $E^H(a)=\sum_{\ell} \mathcal{F}_\ell^H(a)$ is divergent for every Kerr spin. The divergence reflects that the Detweiler–Whiting singular field does not lie in the $L^2$-domain of the Kerr Teukolsky operator. This places the phenomenon on a general operator-theoretic footing and shows that the Schwarzschild result is the $a\to 0$ limit of a broader, spin-independent mechanism.