The images of Brans-Dicke-Kerr type naked singularities

We have studied the images of the Brans-Dicke-Kerr spacetime with a dimensionless Brans-Dicke parameter $ω$, which belongs to axisymmetric rotating solutions in the Brans-Dicke theory. Our results show that the Brans-Dicke-Kerr spacetime with the parameter $ω>-3/2$ represents naked singularities with distinct structures. For the case with $a \leq M$, the shadow in the Brans-Dicke-Kerr spacetime persists, gradually becomes flatter and smaller as $ω$ decreases. Especially when $ω<1/2$, the shadow in the image exhibit a very special ``jellyfish" shape and possesses a self-similar fractal structure. For the case with $a > M$, a distinct gray region consisting of two separate patches appears in the image observed by equatorial observers. This indicating that the Brans-Dicke-Kerr spacetime can be distinguished from the Kerr and Kerr-de Sitter cases based on its image. These effects of the Brans-Dicke parameter could help us to reveal the intrinsic structure of the Brans-Dicke-Kerr spacetimes and provide a foundation for testing Brans-Dicke theory through future high-precision observations.

💡 Research Summary

The paper investigates the observational signatures of rotating Brans‑Dicke–Kerr (BD‑Kerr) spacetimes, focusing on how the dimensionless Brans‑Dicke coupling parameter ω influences the appearance of shadows and photon rings. Starting from the Brans‑Dicke action with a vanishing scalar potential, the authors derive the field equations and obtain a Kerr‑like solution in the Jordan frame. By applying a conformal transformation to the Einstein frame, they express the metric (Eq. 7) and scalar field in terms of ω, the mass M, and the spin parameter a. The solution reduces to the standard Kerr metric as ω → ∞, but for finite ω the metric components g_rr and g_θθ acquire explicit ω‑dependence, which directly modifies null geodesics.

Two types of curvature singularities are identified. The first occurs at Δ = r² + a² − 2Mr = 0, where the Kretschmann scalar κ diverges, forming a secondary curvature singularity that coincides with the usual event horizon when ω > ½. For −3/2 < ω ≤ ½ the horizon disappears, yet the secondary singularity remains, yielding a naked singularity. The second singularity is the ring singularity at Σ = r² + a²cos²θ = 0 (r = 0, θ = π/2). When a ≤ M both singularities coexist, while for a > M the equation Δ = 0 has no real roots, leaving only the ring singularity.

Photon motion is governed by the Hamiltonian H = ½ g^{μν}p_μp_ν = 0 together with conserved energy E and axial angular momentum L_z. The resulting equations of motion (13‑16) contain ω through the metric components, causing photon trajectories to deviate from those in pure Kerr spacetime. To visualize these effects, the authors employ backward ray‑tracing: they integrate the null geodesics from an observer located at r_obs = 8M in the equatorial plane (θ_obs = π/2) and map each photon’s locally measured four‑momentum to celestial coordinates (x, y) via Eq. 23.

The numerical results are organized into two regimes. For a ≤ M, decreasing ω progressively flattens the shadow, turning the familiar circular silhouette into an increasingly elliptical shape. When ω drops below ½, the shadow develops a striking “two‑headed jellyfish” morphology with self‑similar fractal edges; the left half of the shadow deforms more strongly than the right, producing a footprint‑like outline. This behavior reflects the enhanced bending of light caused by the ω‑dependent metric components and the presence of a secondary curvature singularity that replaces the event horizon.

For a > M, the spacetime lacks any horizon, resembling a Kerr naked singularity but with an additional “hair” parameter ω. The image no longer shows a closed dark region; instead a bright background is interrupted by a thin black line (the ring singularity) and, notably, by two gray patches that appear on the left side of the image for smaller ω. These gray regions correspond to photons that originate from a hypothetical negative‑radius region (r < 0) and traverse through r = 0, θ ≠ π/2 to reach the observer. Their size grows as ω decreases, indicating that the scalar coupling enhances the accessibility of the negative‑radius domain. The presence and shape of the gray patches also depend on the observer’s inclination and on the spin magnitude a.

The authors argue that these ω‑induced features—especially the jellyfish‑shaped shadow for a ≤ M and the gray patches for a > M—provide clear observational discriminants between BD‑Kerr spacetimes, standard Kerr, and Kerr–de Sitter solutions. They suggest that forthcoming high‑resolution interferometric observations (e.g., next‑generation Event Horizon Telescope) could detect such signatures, thereby placing constraints on the Brans‑Dicke coupling parameter and testing scalar‑tensor theories of gravity in the strong‑field regime. The paper concludes that the Brans‑Dicke parameter fundamentally alters the geometry of rotating compact objects and leaves imprints on photon trajectories that are, in principle, observable.