Massive vector fields on the Schwarzschild spacetime: quasinormal modes and bound states

TL;DR

This work analyzes massive vector (Proca) perturbations on Schwarzschild spacetime, deriving a separation into a single odd-parity equation and a coupled even-parity system. It computes the quasinormal-mode and quasi-bound-state spectra using continued-fraction (Leaver) and forward-integration methods, and develops small-mass analytical matches to reveal hydrogenic leading behavior and spin-dependent decay rates. The results show a rich interplay between vector and scalar-like degrees of freedom, with parity-dependent splittings and bound-state lifetimes that depend on both orbital and spin quantum numbers, signaling spin-orbit–like coupling. The findings have implications for beyond-Standard-Model scenarios (e.g., hidden photons) and motivate extensions to Kerr geometries, where superradiant dynamics could reveal ultralight vector fields around astrophysical black holes.

Abstract

We study the propagation of a massive vector or Proca field on the Schwarzschild spacetime. The field equations are reduced to a one-dimensional wave equation for the odd-parity part of the field and two coupled equations for the even-parity part of the field. We use numerical techniques based on solving (scalar or matrix-valued) three-term recurrence relations to compute the spectra of both quasi-normal modes and quasi-bound states, which have no massless analogue, complemented in the latter case by a forward-integration method. We study the radial equations analytically in both the near-horizon and far-field regions and use a matching procedure to compute the associated spectra in the small-mass limit. Finally, we comment on extending our results to the Kerr geometry and its phenomenological relevance for hidden photons arising e.g. in string theory compactifications.

Figures (8)

• Figure 1: Quasinormal mode frequencies of the Proca field, for $l=0$ (monopole), $l=1$ (dipole) modes, $l = 2$ (quadrupole) modes, for a range of field masses $M \mu = 0, 0.04, \ldots, 0.2$. The fundamental ($n=0$) and first overtones ($n=1$) are shown. For a given $l$, $n$ there are two even-parity modes, and one odd-parity mode. In the massless limit, the 'scalar' even-parity mode has the same QN frequency as the scalar ($s=0$) field, whereas the 'vector' even-parity and odd-parity modes have the same QN frequency as the electromagnetic field.
• Figure 2: Polarization state of even-parity quasinormal modes. The plot shows the complex number $\mathcal{P}$, i.e. the ratio $u_{(3)} / u_{(2)}$ far from the black hole [defined in Eq. (\ref{['eq-pol']})], as a function of the mass coupling $M\mu = 0 \ldots 0.2$, for $l=1,2,3$ and $n = 0,1$. For scalar (vector) modes, $\mathcal{P} \rightarrow 0$ ($\mathcal{P} \rightarrow 1$) as $M\mu \rightarrow 0$.
• Figure 3: Bound state levels of the Proca field on the Schwarzschild spacetime. The upper plot shows the real part of the frequency $\text{Re}(\omega / \mu)$ as a function of the mass coupling $M \mu$, and the lower plot shows (the negative of) the imaginary part $\text{Im}(\omega / \mu)$ on a logarithmic scale. The modes are labeled by their angular momentum number $l$, overtone number $n$, spin projection $S$ and parity (odd or even).
• Figure 4: Numerical data for the exponent $\eta$ in the power-law relationship $\text{Im} (\omega / \mu) \propto -(M\mu)^{\eta}$ which determines the decay rate of the quasi-bound states in the small-coupling regime $M\mu \ll l$. Here $l = 0,1, \ldots$ is the angular momentum number, and $S \in \{-1, 0, +1\}$ is the spin projection of the state in the large-$r$ regime. The data shown were obtained by numerically evaluating the function $\eta = -\frac{\partial \ln[\text{Im}(\omega / \mu)]}{\partial \ln [ M \mu ]}$. Note that numerical evaluation becomes increasingly difficult in the small-$M\mu$ regime, where $|\text{Im}(\omega / \mu)|$ is tiny ($\lesssim 10^{-12}$). The data strongly suggests that modes $l=L, S=+1$ and $l=L+1, S=-1$ share the same exponent $\eta$. The data is consistent with Eq. (\ref{['eq-exponent']}), which implies that, in the limit $M \mu \rightarrow 0$, the exponent tends to $\eta = 7$ ($l=0,\ S=+1$ and $l=1,\ S=-1$), $\eta = 9$ ($l=1,\ S=0$) and $\eta = 11$ ($l=1,\ S=+1$ and $l=2,\ S=-1$).
• Figure 5: Polarization state of even-parity bound states. The plot shows the complex number $\mathcal{P}$, i.e. the ratio $u_{(3)} / u_{(2)}$ evaluated asymptotically [see Eq. (\ref{['eq-pol']})], as a function of the mass coupling $M\mu = 0 \ldots 0.8$, for a selection of the lowest modes ($l=0,1,2$) and overtones ($n = 0,1$). The points show the values of $\mathcal{P}$ at $M \mu = 0, 0.2, 0.4, 0.6$ and $0.8$. In the limit of vanishing mass, we find $P \rightarrow -l$ for $S = +1$ modes, and $\mathcal{P} \rightarrow l+1$ for $S = -1$ modes (where $S$ is the spin projection).