A massive photon — described by the Proca equation — on a charged black hole background (Reissner–Nordström) decays in time. The late-time behavior is controlled by spectral thresholds: points where the resolvent operator changes character.
The paper on threshold asymptotics and decay for massive Maxwell (arXiv: 2603.23905) derives the complete late-time picture. After spherical harmonic decomposition, the odd sector reduces to a scalar equation while the even sector remains a coupled 2×2 system. An exact asymptotic polarization splitting at spatial infinity generates three channels with effective angular momenta l−1, l, and l+1.
The universal late-time tail follows a t^{−5/6} power law from the massive branch cut — slower than the massless case because the mass creates a spectral singularity at the threshold. Quasibound resonance branches from timelike trapping (the massive field orbits the black hole before escaping) produce oscillatory contributions that compete with the polynomial tail.
The full Proca field exhibits logarithmic decay in compact regions (near the black hole), while the radiative component retains explicit polynomial asymptotics at large distances.
The through-claim: mass creates a universal tail through the branch-cut singularity. Massless fields on black holes decay as inverse polynomials (Price's law). Massive fields decay differently — the mass threshold introduces a branch cut in the spectral plane, and the t^{−5/6} tail is the universal signature of this singularity. The exponent 5/6 comes from the threshold geometry, not from the specific black hole.
2603.23905. Mathematical physics / black holes / Proca equation / spectral theory / late-time tails.