[Paper Review] Characterization of Null Geodesics on Kerr Spacetimes

Claudio F. Paganini, Błażej Ruba+1 more|arXiv (Cornell University)|Nov 17, 2016

Astrophysical Phenomena and Observations14 references3 citations

TL;DR

This paper provides a unified phase-space visualization of null geodesics in subextremal Kerr spacetimes, demonstrating that trapping and the ergoregion are topologically distinct. It shows that, from any timelike observer's perspective, the trapped null geodesics form a smooth circle on the celestial sphere, offering a geometric foundation for understanding black hole shadows and wave decay in Kerr spacetimes.

ABSTRACT

We consider null geodesics in the exterior region of a subextremal Kerr spacetime. We show that most well-known fundamental properties of null geodesics can be represented on one plot. In particular, one can see immediately that the ergoregion and trapping are separated in phase space. Furthermore, we show that from the point of view of any timelike observer outside of the black hole, trapping can be understood as a smooth set of spacelike directions on the celestial sphere of the observer. Finally, we discuss some applications of these insights.

Motivation & Objective

• To provide a unified, accessible visualization of fundamental properties of null geodesics in subextremal Kerr spacetimes.
• To clarify the topological separation between the ergoregion and the trapped set in phase space.
• To demonstrate that the trapped set appears as a smooth circle on the celestial sphere for any external timelike observer.
• To support the analysis of black hole shadows and wave decay in Kerr spacetimes through geometric intuition.
• To provide numerical and analytical evidence that the shadow's shape breaks radial degeneracy away from symmetry axes, enabling full parameter extraction in principle.

Proposed method

• Uses a single phase-space plot to represent the pseudo-potentials $ V_{/pm} $ and the trapping curve $ \mathcal{E}_{\text{trap}}(r) $, visualizing forbidden regions and turning points.
• Applies the eikonal approximation to relate the scalar wave equation to null geodesics in the high-frequency limit, linking $ \omega \sim E $, $ m \sim L_z $, and $ \lambda_{lm} \sim L^2 $.
• Employs the celestial sphere formalism to map trapped null geodesics as topological circles, independent of observer location in the exterior region.
• Utilizes a Mathematica notebook to explore parameter dependence of $ a/M $, $ \mathcal{Q} $, and observer position $ \{r(p), \theta(p)\} $, enabling interactive intuition.
• Applies turning point analysis of dynamical systems, following Wilkins' method, to infer global geodesic behavior from radial and $ \theta $-equations.
• Combines results from co- and counter-rotating geodesics to show that trapping lies outside the ergoregion in both cases, confirming phase-space separation.

Experimental results

Research questions

• RQ1 How can all fundamental properties of null geodesics in subextremal Kerr spacetimes be visualized on a single phase-space plot?
• RQ2 What is the topological structure of the trapped set on the celestial sphere as seen by any external timelike observer?
• RQ3 How does the separation between trapping and the ergoregion manifest in phase space, and can it be demonstrated independently of rotation direction?
• RQ4 To what extent does the shadow of a Kerr black hole break radial degeneracy in its shape, and can all black hole parameters be extracted from a precise shadow measurement?
• RQ5 How do the pseudo-potentials $ V_{\pm} $ and the trapping curve $ \mathcal{E}_{\text{trap}}(r) $ encode the global behavior of null geodesics?

Key findings

• The trapped set in Kerr spacetime appears as a smooth, topologically circular set of spacelike directions on the celestial sphere for any external timelike observer.
• Trapping and the ergoregion are strictly separated in phase space, with no overlap, as confirmed by the disjoint regions of negative-energy and trapped geodesics.
• For all subextremal Kerr spacetimes with $ a < M $, the trapped geodesics in the counter-rotating case are confined to $ r \in (r_3, r_2] $, where $ r_3 > 2M \geq r_{\text{ergo}} $, ensuring ergoregion-trapping separation.
• The celestial sphere representation reveals that the shadow's shape breaks radial degeneracy away from the symmetry axis, implying full black hole parameter extraction is possible in principle.
• The pseudo-potential plot with $ \mathcal{E}_{\text{trap}}(r) $ allows intuitive identification of where mode currents change sign in wave decay analysis, linking to the results in [15].
• Numerical evidence suggests that extracting all black hole parameters from a shadow measurement would require precision likely unattainable for decades, despite theoretical feasibility.