Gravitational Lensing in the Kerr Spacetime: An Analytic Approach for Light and High-Frequency Gravitational Waves

The Kerr spacetime is one of the most widely known solutions to Einstein’s vacuum field equations and is commonly used to describe a black hole with mass $m$ and spin $a$. Astrophysical observations in the electromagnetic spectrum as well as detected gravitational wave signals indicate that it can be used to describe the spacetime around candidates for rotating black holes. While the geodesic structure of the Kerr spacetime is already well known for decades, using exact analytic solutions to the equations of motion for applications to astrophysical problems has only attracted attention relatively recently. Here, these applications mainly focus on predicting observations for the shadow, the photon rings, and characteristic structures in the accretion disk. Using the exact analytic solutions to investigate exact gravitational lensing of light and gravitational waves emitted by sources outside the accretion disk has only received limited attention so far. Therefore, the focus of this paper will be to address this question. For this purpose we assume that we have a standard observer in the domain of outer communication. We introduce a local orthonormal tetrad to relate the constants of motion of light rays and high-frequency gravitational waves detected by the observer to latitude-longitude coordinates on the observer’s celestial sphere. In this parameterisation we derive the radius coordinates of the photon orbits and their latitudinal projections onto the observer’s celestial sphere as functions of the celestial longitude. We use the latitude-longitude coordinates to classify the different types of motion, and solve the equations of motion analytically using elementary and Jacobi’s elliptic functions as well as Legendre’s elliptic integrals. We use the analytic solutions to write down an exact lens equation, and to calculate the redshift and the travel time.

💡 Research Summary

The paper presents a comprehensive analytic treatment of gravitational lensing of both light rays and high‑frequency gravitational waves (HFGWs) in the Kerr spacetime, which describes a rotating black hole of mass (m) and spin (a). The authors adopt a “standard observer” (also known as the Carter observer) located in the domain of outer communication, and construct a local orthonormal tetrad that links the conserved quantities of null geodesics—energy (E), axial angular momentum (L_z), and Carter constant (K)—to latitude–longitude coordinates on the observer’s celestial sphere. This mapping replaces the traditional constant‑based description with a directly observable angular parametrisation, facilitating ray‑tracing and visualisation.

Using this parametrisation, the authors derive explicit expressions for the radial coordinate of photon (or HFGW) orbits and their latitudinal projections as functions of celestial longitude. The condition for unstable photon orbits is cast as a sixth‑order polynomial in (r) that depends on (m), (a), and an effective inclination angle. Solving this polynomial yields the radii of the spherical photon orbits, which are then inserted into the geodesic equations.

The core technical contribution lies in solving the radial and polar equations analytically. Two families of methods are discussed: (i) Jacobi elliptic functions together with Legendre elliptic integrals, and (ii) the Weierstrass (\wp) function with its associated (\zeta) and (\sigma) functions. The authors argue that the Jacobi‑Legendre approach is more convenient for lensing applications because it yields manifestly real expressions without the need for intricate branch‑cut handling. Consequently, all solutions for the four possible motion types (direct, one‑turn, multi‑turn, and plunging trajectories) are expressed in terms of elementary functions, Jacobi (\operatorname{sn},\operatorname{cn},\operatorname{dn}), and Legendre integrals.

With the analytic solutions in hand, an exact lens equation is constructed. The equation relates the source position on a distant two‑sphere to the observed angles ((\theta,\phi)) on the celestial sphere, incorporating the full dependence on spin (a) and observer inclination. From the same formalism the redshift factor (z) and the travel time (\Delta t) are derived analytically. The redshift includes contributions from both gravitational potential and frame‑dragging, while the travel time exhibits characteristic delays for trajectories that wind around the black hole multiple times or cross the ergosphere.

The paper then applies the formalism to several astrophysical scenarios. For isolated sources such as stars, neutron stars, or compact binary systems orbiting a supermassive black hole, the authors compute the positions, magnifications, and time delays of the resulting images. They show how the spin of the black hole induces asymmetries in the image distribution and in the timing of HFGW signals, offering a potential probe of the Kerr parameter that complements electromagnetic shadow and photon‑ring observations. The authors also discuss the relevance for upcoming facilities: space‑VLBI missions (e.g., the proposed Black Hole Explorer) that aim to resolve photon rings, and third‑generation gravitational‑wave detectors (Einstein Telescope, Cosmic Explorer) that will be sensitive to HFGWs lensed by supermassive black holes.

In summary, the work delivers a fully analytic, observer‑centric description of null geodesics in Kerr spacetime, provides explicit lens, redshift, and time‑delay formulas, and demonstrates their applicability to both electromagnetic and high‑frequency gravitational‑wave observations. This bridges a gap between exact theoretical results and practical astrophysical modeling, paving the way for precision tests of general relativity and black‑hole spin measurements with next‑generation observational platforms.