Matter around Schwarzschild black holes in scalar-tensor theories: Absorption and Scattering

TL;DR

The paper addresses how a matter-coupled, position-dependent scalar mass $\mu^2(r)$ in scalar–tensor gravity alters scalar-wave absorption and scattering by a Schwarzschild black hole. It derives the Einstein-frame field equations, introduces two mass-profile models $\mu^2_{\text{I}}(r)$ and $\mu^2_{\text{II}}(r)$, and solves the Klein–Gordon equation via a partial-wave decomposition to compute the absorption cross section $\sigma_{\text{abs}}$ and differential scattering cross section $d\sigma/d\Omega$, highlighting resonance phenomena when a potential well is present. In Model I, absorption grows with $\lambda$ and $\mu_c$ and exhibits a zero-absorption band as $\omega \to \mu_c$, while Model II yields quasibound resonances and associated dips in scattering due to the well; at high frequencies, the cross sections converge to GR expectations, reducing sensitivity to the mass-profile parameters. These results point to potential observational discriminants between GR and scalar–tensor gravity in BH environments where matter around the BH induces an effective scalar mass.

Abstract

We investigate the absorption and scattering by a Schwarzschild black hole in scalar--tensor theories of gravity, where the coupling between matter and the scalar field induces different models for the effective mass of the scalar field. In model~I, a Bondi-type mass model described by the asymptotic mass $μ_c$, horizon mass $μ_H$, and profile slope $λ$, it is found that the absorption cross section increases with steeper $λ$, larger $μ_c$ (especially at higher frequencies), or smaller $μ_H$. The differential scattering cross section in this model shows the strongest dependence on the horizon mass $μ_H$. When $μ_H$ exceeds a critical value for a fixed incoming wave frequency $ω$, no partial wave transmits into the black hole, flattening the differential scattering cross section as a function of angle before it increases again with further increase of $μ_H$. Model~II, which considers a truncated accretion region outside some radius $r_0$, contains a potential well in its effective scattering potential. Its absorption cross section decreases in the low-frequency region as the accretion radius $r_0$ decreases, and more importantly, it shows resonance peaks at the quasibound wave frequencies due to resonances induced by the potential well. The differential scattering cross sections show dips around intermediate scattering angles when the parameters (mainly $μ_H$ and $ω$) are such that the resonantly scattered and non-resonant waves interfere destructively around these angles. In both models, absorption exhibits a zero-absorption band as $ω$ approaches $μ_c$ from above, and in both absorption and scattering, the effects of the parameters are found to diminish in the high-frequency limit.

Figures (7)

• Figure 1: The effective potential in model I as a function of $r$ for parameters $\mu_{c}$ (top left), $\lambda$ (top right), $\mu_{H}$ (bottom left), and $l$ (bottom right). The top inset shows $V_{\mathrm{eff}}$ for $l=0$ with $\mu_{H}=0$ and $\mu_{H}=1$. The bottom inset shows $V_{\mathrm{eff}}$ for $l=1$.
• Figure 2: The effective potential in model II as a function of $r$ for parameters $\mu_{c}$ (top left), $\lambda$ (top right), $\mu_{H}$ (middle), $r_{0}$ (bottom left), and $l$ (bottom right).
• Figure 3: $\sigma_{\mathrm{abs}}$ in model I as a function of $\omega$ for parameters $\lambda$ (top), $\mu_{c}$ (middle), and $\mu_{H}$ (bottom).
• Figure 4: $\sigma_{\mathrm{abs}}$ in model II as a function of $\omega$ for $\mu_{c}$ (top left), $\lambda$ (top right), $\mu_{H}$ (bottom left), and $r_{0}$ (bottom right). Red points and gray dashed lines mark the resonance peaks and frequencies, respectively. Note that the red point near $\omega=3.5$ belongs to the purple dotted curve.
• Figure 5: $\sigma_{l,\mathrm{abs}}$ (top), transmission coefficients (center), and effective potential (bottom) in model II for the parameters $\mu_{c}=0.1$ (left) and $\mu_{c}=0.42$ (right). The red points mark the resonance positions at specific values of $l$ and $\omega$, while the vertical gray dashed lines indicate the real parts of the quasibound frequencies $\omega_{R}$ listed in Table \ref{['tab:qsb']}.