Massive particle surfaces and black hole shadows from intrinsic curvature

Abstract

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.

Figures (7)

• Figure 1: We have plotted the Gaussian curvature \ref{['gaussiank:1']} for null trajectories of the Kerr spacetime \ref{['Kerr:1']}. In the left panel we present the Gaussian curvature for different values of the impact parameter $\sigma=0.1,1.1,2.1,3.1$ with fixed energy $E=1$. We have taken $a=1$ and $M=10$. The vertical gray dashed lines represent the black hole horizons. Similarly, in the right panel we present the same Gaussian curvature as in the left panel but setting $a=0.1$ and $M=1$.
• Figure 2: We have plotted the geodesic curvature \ref{['geo:1']} for null trajectories of the Kerr spacetime \ref{['Kerr:1']}. In the left panel we have set $a=0.1$ and $M=1$ for different values of the impact parameter $\sigma=0.1,1.1,2.1,3.1$ setting $E=1$. In the right panel we have a similar plot with $a=1$ and $M=10$. The vertical gray dashed lines represent the location of the horizons of the black hole.
• Figure 3: The graphic represents the Gaussian curvature corresponding to the null trajectories of the Kerr-Ads/dS spacetime metric \ref{['kerrds:1']}. In the left panel we present the Gaussian curvature of the Kerr-Ads ($e=-1$), for different values of the impact parameter $\sigma=0.05,1.05,2.05,3.05$ with fixed energy $E=1$. We have set $a=0.1$ and $M=1$. In the right panel we present the same Gaussian curvature corresponding to the null geodesics with $a=1$ and $M=10$. The vertical gray dashed lines represent the horizons of the black hole.
• Figure 4: We plot the geodesic curvature for the null trajectories of the Kerr-Ads spacetime metric \ref{['kerrds:1']}. We have set for both panels $a=0.1$ and $M=1$, and the impact factor $\sigma=0,0.5,1,1.1,2.1,4.1$. The dependence on the value of $\sigma$ is evident. For all curves we take $E=1$.
• Figure 5: We plot the Gaussian curvature for the null trajectories of the Kerr-de Sitter spacetime \ref{['kerrds:1']}. In both panels we have set $a=0.1$ and $M=1$. the values of the impact parameter $\sigma=0,0.5,1,1.1,3.1,5.1.$ The vertical gray dashed lines are located to represent the horizons of the metric. The value of the energy $E=1$ is fixed.