Quasistationary state gravitationally bound to Lense-Thirring Bumblebee-ModMax black hole

Abstract

In this paper, we study the propagation of a relativistic scalar field in a rotating Einstein-Bumblebee spacetime in (3+1) dimensions, focusing on the slowly rotating Bumblebee-ModMax black hole. We first derive the covariant Klein–Gordon equation explicitly by computing its components in the given background. By adopting a separation-of-variables ansatz, the equation is decoupled into independent polar and radial parts. The angular equation admits exact solutions in terms of spherical harmonics, while the radial equation is solved exactly using confluent Heun functions. Quasibound states of a massive scalar field are obtained by imposing the polynomial condition on the confluent Heun function, which leads to a discrete spectrum of complex energy eigenvalues. The real part of the energy represents the relativistic energy of the scalar particle, whereas the imaginary part encodes the stability properties of the bound configuration. We further analyze Hawking radiation emitted from the outer horizon of the slowly rotating Bumblebee-ModMax black hole using the Damour–Ruffini method, based on the exact scalar wave solutions derived in this work. We find that the radiation spectrum retains the standard Bose–Einstein functional form, while the effective exponent ζ is modified by rotation, Lorentz-violating effects, and nonlinear electrodynamics. In the limiting case l<inf>B</inf>→0 and Q→0, the spectrum reduces to the standard Schwarzschild result with a well-defined Hawking temperature.