Rotating Black Holes with Primary Scalar Hair: Shadow Signatures in Beyond Horndeski Gravity

TL;DR

We address whether rotating black holes in beyond-Horndeski gravity with primary scalar hair can mimic Kerr shadows within EHT bounds. We construct rotating solutions via a revised Newman-Janis approach, analyze null geodesics with Hamilton-Jacobi separation, and derive photon-region and shadow observables (notably the angular diameter $θ_d$, shadow circularity deviation $ΔC$, and axis ratio $D_x$). We find that the independent hair parameter $Q$ can enlarge or shrink the shadow and alter its oblateness depending on the sign of $Q$, while current EHT bounds are compatible with a broad $(a,Q)$ range but constrain $Q>0$ more strongly. The results quantify degeneracies between spin and hair and highlight the potential of next-generation horizon-scale imaging to break them, placing scalar-tensor gravity tests within observational reach.

Abstract

The Event Horizon Telescope (EHT) image of M87* provides a direct test of strong-field gravity, measuring an angular shadow diameter $θ_d = 42 \pm 3~μ\mathrm{as}$ and a circularity deviation $ΔC \leq 0.1$. Such observations allow quantitative tests of the Kerr paradigm and of possible deviations from the no-hair theorem. In scalar-tensor extensions of gravity, black holes may possess primary scalar hair, introducing an additional independent parameter beyond mass and spin. In this work, we construct rotating black hole solutions with primary scalar hair in beyond Horndeski gravity and analyze their photon regions and shadow formation. We show that the scalar hair parameter $Q$ induces characteristic modifications of the shadow, and in particular negative $Q$ enlarges the shadow and reduces its oblateness, while positive $Q$ shrinks and enhances its distortion. Modeling M87* within this framework and imposing the EHT bounds on $θ_d$ and $ΔC$, we determine the viable $(a,Q)$ parameter space. We find that current observations do not exclude rotating black holes with primary scalar hair, although the allowed region is significantly restricted for $Q>0$. Finally, the scalar-hair-induced deviations are of order $\mathcal{O}(μ\mathrm{as})$, placing them near the sensitivity threshold of present instruments and within reach of next-generation horizon-scale imaging.

Figures (13)

• Figure 1: The radial behavior of the metric function $f(r)$. The upper panels correspond to $M=3\lambda$ and the lower panels to $M=5\lambda$, while the left panels correspond to $Q<0$ and the right panels to $Q>0$.
• Figure 2: The behavior of $\Delta$ versus $r$, for various values of $Q$ and $a=0.99$. The upper panels correspond to $M=3\lambda$ and the lower panels to $M=5\lambda$, while the left panels correspond to $Q<0$ and the right panels to $Q>0$.
• Figure 3: Angular velocity $\omega$ of photons as a function of the dimensionless radial coordinate $r/M$, for different values of the scalar hair parameter $Q$, in a rotating black hole spacetime with primary scalar hair.
• Figure 4: The behavior of SLS versus $r$ for various values of $Q$ with $a=0.99$.
• Figure 5: Cross-sectional profiles of the event horizon (blue), the SLS (red), and the enclosed ergoregion for various values of the parameters $\lambda$ and $Q$ with $a=0.99$.