Massive particle surfaces and black hole shadows from intrinsic curvature

In a recent article PRD 111, 064001 (2025) a new geometric a approach for studying massive particle surfaces was proposed. Using the Gaussian and geodesic curvatures of a two dimensional Riemannian metric a criteria for the existence of massive particle surfaces was provided. In this work we generalize these results by including stationary spacetime metrics. We surmount the difficulty of having a Jacobi metric of the Randers-Finsler type by using a $2$-dimensional Riemannian metric that is obtained by projecting the spacetime metric over the directions of its Killing vectors. We provide a condition for the existence of massive particle surfaces and a simple characterization for null and timelike trajectories only by using intrinsic curvatures of that $2$-dimensional Riemannian surface. We study the massive particle surfaces of spacetimes that are not an asymptotically flat. We show that the Riemannian formalism can be used to study the shadows of the associated black holes. We show the existence of massive particle surfaces for the Kerr metric, the Kerr-(A)dS metric and for a solution of the Einsten-Maxwell-dilaton theory.

💡 Research Summary

In this paper the authors extend the concept of massive particle surfaces (MPS) – timelike analogues of photon spheres – from static spacetimes to the much broader class of stationary (i.e., rotating or otherwise non‑static) geometries. The key technical obstacle in the stationary case is that the usual Jacobi metric, obtained by fixing the particle’s energy, becomes a Randers‑Finsler metric rather than a purely Riemannian one, which prevents a straightforward application of intrinsic curvature tools. To bypass this difficulty the authors adopt a dimensional‑reduction scheme introduced in earlier work: they project the full four‑dimensional Lorentzian metric onto the two‑dimensional subspace spanned by the Killing vectors (typically ∂ₜ for stationarity and ∂φ for axial symmetry). By fixing the conserved energy E and angular momentum L, they obtain a two‑dimensional Riemannian metric h{ij} that encodes the dynamics of both null and timelike geodesics.

With h_{ij} in hand, the authors compute two intrinsic curvature quantities: the Gaussian curvature K and the geodesic curvature κ_g of arbitrary curves on the reduced surface. The condition κ_g = 0 is equivalent to the vanishing of the geodesic curvature of a curve, i.e., the curve is a geodesic of h_{ij}. This condition reproduces the “master equation” previously derived for MPS, providing a simple geometric criterion for the existence of circular timelike or null orbits. Moreover, the sign of K determines the stability of such orbits: K > 0 signals a locally convex geometry and corresponds to stable circular orbits (e.g., the innermost stable circular orbit, ISCO), whereas K < 0 indicates a saddle‑like geometry associated with unstable light rings.

The formalism is applied to three concrete spacetimes:

1. Kerr spacetime – The reduced metric h_{ij} depends on the mass M and spin parameter a. By evaluating K and κ_g the authors recover the well‑known photon‑ring radius r_ph and the ISCO radius r_ISCO, and they demonstrate how the curvature signs change across the transition from stable to unstable orbits.

2. Kerr–(A)dS spacetime – Inclusion of a cosmological constant Λ adds extra terms to the curvature expressions. The analysis shows that a positive Λ (de Sitter) enlarges the shadow and pushes the light ring outward, while a negative Λ (Anti‑de Sitter) has the opposite effect. The curvature‑based method cleanly captures these Λ‑dependent shifts without solving the full geodesic equations.

3. Einstein‑Maxwell‑Dilaton (EMD) solution – This non‑asymptotically flat, non‑trivial dilaton background does not admit a conventional optical metric. Nevertheless, the projection onto the Killing directions yields a well‑defined h_{ij}, allowing the same curvature analysis. The authors find that both massive particle surfaces and light rings exist in this spacetime, illustrating the broad applicability of their technique.

Beyond orbit analysis, the paper connects the Gaussian curvature of h_{ij} to the observable black‑hole shadow. Since the shadow boundary can be interpreted as the set of critical null geodesics, its size and shape are directly encoded in the curvature of the reduced surface. Positive curvature leads to a smaller, more compact shadow, while negative curvature produces a larger, more distorted silhouette. This provides a geometric bridge between intrinsic curvature calculations and astrophysical observations such as those performed by the Event Horizon Telescope.

In summary, the authors present a powerful, purely geometric framework for studying massive particle surfaces, light rings, ISCOs, and black‑hole shadows in stationary spacetimes. By reducing the problem to a two‑dimensional Riemannian surface, they avoid the complications of Randers‑Finsler geometry and the direct integration of geodesic equations. Their method works for asymptotically flat, (A)dS, and even non‑asymptotically flat dilaton spacetimes, offering a unified and computationally efficient tool for probing strong‑gravity phenomena.