Dynamics of photons and shadows for black holes haired with parity-odd fields

TL;DR

The paper tackles whether parity-odd scalar hair on rotating black holes leaves observable imprints in shadows. It builds HBH solutions in Einstein-Klein-Gordon theory using the action $S=\int d^4x\sqrt{-g}\left(\frac{R}{16\pi}-\mathcal{L}_m\right)$ with the Klein-Gordon equation $(\Box-\mu^2)\Psi=0$, enforcing a synchronization condition $w=m\Omega_h$ and exploring parity choices; photon trajectories are computed via ray-tracing of null geodesics with $\mathcal{H}=\frac{1}{2}g^{\mu\nu}p_\mu p_\nu=0$ for models matched in $M_{\text{ADM}}$ and $J_{\text{ADM}}$ to Kerr. The study reveals that HBH shadows can deviate from Kerr by up to roughly $15\%$, with some configurations producing multiple disconnected shadows or human-face-like patterns, and shows that one tiny-hair case lies within EHT uncertainties for M87*, probing scalar masses of $1.02\times10^{-20}$ eV. These results provide observational templates and benchmarks for constraining parity-odd hair and scalar-field dark matter through black hole shadow imaging, guiding future high-resolution shadow measurements.

Abstract

Strong self-gravitational fields enable the realization of macroscopic odd-parity quantum objects. Using ray-tracing methods, we systematically analyze the dynamics of photons and the shadow features of rotating black holes with parity-odd scalar hair and contrast them with those of Kerr black holes. Our results demonstrate measurable distinctions between scalar-haired black hole shadows and their Kerr counterparts. Notably, even for tiny scalar charge and negligible scalar hair mass contributions, these differences remain quantitatively resolvable. In particular, one of the hairy black hole reported here lies within the Event Horizon Telescope observational uncertainties, probing the scalar masses of $1.02\times10^{-20}$eV with M87*. These findings may provide related theoretical benchmarks for future observational campaigns targeting scalar-field dark matter candidates through black hole shadow imaging.

Figures (10)

• Figure 1: ADM mass $M$ vs frequency $w$ diagram for HBH solutions for $m=1$. Solid black curve: extremal Kerr BHs; solid red curve: parity-odd boson states. Five marked points correspond to solutions studied in detail.
• Figure 2: An example of a light ray that crosses the pole at $\theta=\pi$. As the light ray gets closer to the pole, the step size $\Delta\lambda$ is automatically reduced to ensure numerical accuracy. Throughout the crossing, the Hamiltonian constraint is kept below $10^{-8}$.
• Figure 3: The full celestial sphere with event horizon marked by orange sphere at center.
• Figure 4: Images of HBH 3 (left) and a Kerr BH (right) with identical ADM mass and angular momentum for $\theta_{\text{obs}}=90^\circ$. Background image adopted from hubble_image.
• Figure 5: Top: images of HBH 1 (left) and Kerr BH (right). Bottom: images of HBH 2 (left) and its Kerr counterpart (right). Both Kerr BHs have identical ADM mass and angular momentum as their corresponding HBHs.