The Null Geodesics of the Kerr Exterior (1910.12881v3)

Abstract: The null geodesic equation in the Kerr spacetime can be expressed as a set of integral equations involving certain potentials. We classify the roots of these potentials and express the integrals in manifestly real Legendre elliptic form. We then solve the equations using Jacobi elliptic functions, providing the complete set of null geodesics of the Kerr exterior as explicit parameterized curves.

• The paper provides a comprehensive classification of null geodesics in Kerr spacetime using explicit parametrized curves and elliptic functions.
• It reformulates the geodesic equations into Legendre elliptic form, ensuring real-valued results across the entire domain.
• The solutions offer practical tools for astrophysical applications, aiding in the accurate analysis of light trajectories around rotating black holes.

Insights into the Null Geodesics of the Kerr Exterior

The exploration of null geodesics within the Kerr spacetime is a vital component in understanding the trajectory of light in the vicinity of rotating black holes. The paper "Null Geodesics of the Kerr Exterior" by Samuel E. Gralla and Alexandru Lupsasca advances this understanding by providing a thorough classification and solution for these geodesics through explicit parametrized curves. This investigation builds on foundational work by Carter (1968) and subsequent studies on the null geodesic solution space.

Mathematical Framework

The core contribution of this paper is the classification of the roots of the null geodesic equations in Kerr spacetime expressed as a set of integrals involving specific potentials. These integrals are reformulated into Legendre elliptic form, and the equations are solved with Jacobi elliptic functions. Such a transformation ensures the mathematical expressions remain real and untainted by internal imaginary components across their domain. The approach taken does not limit the scope to partial scenarios but rather provides a comprehensive range of geodesics for the Kerr exterior.

Geodesic Classification

The research identifies distinct regions within the parametric space of the conserved quantities $(\lambda, \eta)$, which correspond to different geodesic behaviors. Each behavior is characterized by the nature and order of the radial roots. Real roots determine permissible ranges of motion, wherein transitions between allowed and forbidden regions are dictated by the physical constraints inherent in the Kerr geometry, notably the locations of event horizons.

1. Ordinary Motion: This type includes geodesics with trajectories oscillating across the equatorial plane, constrained by polar motion defined through positive $\eta$.
2. Vortical Motion: These geodesics, involving negative $\eta$, do not cross the equator and demonstrate a precession contained within each hemisphere.

Understanding these patterns is critical to deciphering path integrals valuable for determining specific parameters crucial for light propagation analysis around rotating black holes.

Solution Completeness and Practical Utility

The paper offers notable advancements over previous works by delivering both integration and inversion solutions devoid of auxiliary computations or discontinuous stitching at turning points. The explicit elucidation of special functions limited to elliptic integrals and Jacobi elliptic functions further attests to the method’s completeness and practical applicability.

Implications and Future Directions

The solutions found in this paper have broader implications for both theoretical and practical applications. They provide a toolset for physicists and astronomers seeking definite answers regarding the behavior of light around Kerr black holes. Aerospace and astrophysical observation missions, such as those captured by the Event Horizon Telescope, can utilize these solutions in interpreting observational data related to photon trajectories around black holes.

Furthermore, this work sets a precedent for future investigations into the Kerr metric's details, particularly as technologies improve and observations extend into regimes of previously inaccessible phenomena. Potential extensions of the research could involve accounting for perturbations or other astrophysical realisms affecting null geodesic paths.

In summary, this paper significantly enriches our toolkit for understanding the Kerr spacetime from both a mathematical and observational perspective, potentially steering future developments in both theoretical physics and astrophysical observations of black holes.