Images of the Thin Accretion Disk Around Kerr Black Holes coupled to time periodic scalar fields

Abstract

We investigate the orbital structure and observable appearance of rotating Kerr black holes endowed with synchronized scalar hair described by two time-periodic scalar fields with a flat target-space geometry. The presence of scalar hair enriches the geodesic structure of the spacetime relative to the Kerr case and significantly modifies the emission properties of geometrically thin Novikov-Thorne accretion disks. Combining an analysis of timelike circular orbits with backward ray tracing, we show that the normalized scalar charge governs the morphology and luminosity of both prograde and counter-rotating disks. In the strongly scalarized regime, additional light rings and modified circular-orbit regions produce multiple inner emitting zones and strongly enhanced redshift patterns that depart markedly from the Kerr prediction. The most pronounced deviations occur in the counter-rotating sector, where scalar hair generates inner retrograde radiative rings with substantially enhanced luminosity and distinctive frequency-shift signatures. Even when the spacetime approaches the Kerr geometry at weaker scalarization, the retrograde disk remains highly sensitive to the presence of scalar hair. Our results demonstrate that geometrically thin accretion disks can provide robust observational diagnostics of synchronized scalar hair and may offer a promising avenue for testing tensor-multi-scalar gravity with future horizon-scale black-hole imaging observations.

Figures (23)

• Figure 1: In the $M-\omega_s$ plane, we present solution curves for fixed $r_H$ from collodel2020rotating with $\kappa = 0$. Kerr black holes exists below the thick black line which represents extremal solutions, while hairy black holes exist in the yellow region. Red points indicate solutions with varying values of the normalized charge $q = \frac{mQ}{J}$. Configurations used for calculation of accretion disk images are labeled as $\textbf{X}^{\,0}_{\,v}$, where $0$ corresponds to $\kappa$ and $v$ represents the value of $r_\mathrm{H}$. Further details of these solutions can be found in Table \ref{['tab_7']} of the Appendix.
• Figure 2: The Keplerian angular velocities $\Omega_{\pm}$ for massive particles are shown for configuration $\mathbf{I}^{\,0}_{\,0.01}$ with parameters $\kappa = 0$, $\omega_s/\mu = 0.679241$, $M\mu = 0.881991$, and $q = 0.999875$. Red curves correspond to co-rotating orbits with $\Omega_{+} > 0$. Blue curves represent counter-rotating orbits when $\Omega_{-} < 0$ and co-rotating orbits when $\Omega_{-} > 0$. Long-dashed segments indicate regions where circular massive-particle orbits are dynamically unstable. The short-dashed curve delineates the domain in which timelike circular orbits are forbidden. Circular orbits terminate at the boundary where $(\partial_{r} g_{t\phi})^{2} < \partial_{r} g_{tt}\, \partial_{r} g_{\phi\phi}$. Vertical dashed lines together with the orange markers identify a sequence of light rings: a co-rotating unstable light ring (red), a counter-rotating unstable light ring (first blue), a co-rotating stable light ring (second blue), and a counter-rotating unstable light ring (third blue).
• Figure 3: The Keplerian angular velocities $\Omega_{\pm}$ for massive particles are shown for configuration $\mathbf{II}^{\,0}_{\,0.01}$ with parameters $\kappa = 0$, $\omega_s/\mu = 0.835272$, $M\mu = 0.648229$, and $q = 0.997293$. Red curves correspond to co-rotating orbits with $\Omega_{+} > 0$. Blue curves represent counter-rotating orbits when $\Omega_{-} < 0$ and co-rotating orbits when $\Omega_{-} > 0$. Long-dashed segments indicate regions where circular orbits of massive particles are dynamically unstable. The short-dashed curve delineates the domain in which timelike circular orbits are forbidden. Vertical dashed lines together with the orange markers identify a sequence of light rings: a co-rotating unstable light ring (red), a counter-rotating unstable light ring (first blue), a co-rotating stable light ring (second blue), and a counter-rotating unstable light ring (third blue).
• Figure 4: The Keplerian angular velocities $\Omega_{\pm}$ for massive particles are shown for configuration $\mathbf{III}^{\,0}_{\,0.05}$ with parameters $\kappa = 0$, $\omega_s/\mu = 0.690049$, $M\mu = 0.966595$, and $q = 0.994490$. Red and blue curves correspond to co-rotating and counter-rotating orbits, respectively. Short-dashed segments indicate regions where circular circular timelike orbits are dynamically unstable, while the long-dashed curve delineates the boundary beyond which timelike circular motion is forbidden. Vertical dashed lines together with the orange markers identify a sequence of light rings, including both stable and unstable ones, in accordance with the convention used in Fig. \ref{['fig:Omega_I']}.
• Figure 5: The plots present the Keplerian angular velocities $\Omega_{\pm}$ associated with massive particles in configuration IV$^{\,0}_{\,0.1}$ with $\kappa = 0$, $\omega_s/\mu = 0.738499$, $M\mu = 1.00043$, and $q = 0.9644770.96447$. The red and blue curves correspond to co-rotating and counter-rotating orbits, respectively. Long-dashed lines indicate regions of dynamical instability, whereas the short-dashed line delineates the boundary beyond which timelike circular orbits cease to exist. Dashed vertical lines indicate the positions of light rings, both stable and unstable, as illustrated in Fig. \ref{['fig:Omega_I']}.