⚛️ general relativity

Frequency shift in gravitational lensing by Kerr metric

This study calculates the relative frequency shift caused by the transverse motion of a spinning lens described by the Kerr metric, offering a novel method to measure a lens's transverse velocity or spin parameter and thereby resolve degeneracies in gravitational microlensing observations of astrophysical black holes.

Original authors: Samaneh Sarbaz, Sohrab Rahvar

Published 2026-09-15

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Samaneh Sarbaz, Sohrab Rahvar

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Gravity is not merely a force that pulls objects together; according to Einstein's theory of general relativity, it is a curvature of space and time itself. When a massive object, like a star or a black hole, sits in the universe, it warps the fabric around it. If light from a distant source passes near this object, the light follows the curve of space, bending its path. This phenomenon, known as gravitational lensing, acts like a cosmic magnifying glass, often creating multiple images of the same background object. For decades, scientists have used these bent images to map the distribution of dark matter and discover planets around other stars. However, most of these studies have treated the massive objects causing the bend as simple, non-rotating spheres. In reality, almost all massive objects in the universe, from stars to the supermassive black holes at the centers of galaxies, are spinning. This rotation drags the very fabric of space around with it, a subtle effect that changes how light travels and how time passes near the object.

A team of researchers has now taken a closer look at how this spin influences the light we receive. While previous work had examined how the motion of a lens affects the frequency of light, this new study specifically investigates the role of the lens's rotation, or spin, in the context of the Kerr metric. The Kerr metric is the mathematical description of the space around a spinning, uncharged black hole. The researchers wanted to know if the spin of a black hole leaves a unique fingerprint on the frequency of light passing by it, distinct from the effects caused by the object's mass alone. They focused on the "frequency shift," which is the change in the color or pitch of light as it moves through a gravitational field. Just as a siren sounds different as an ambulance drives past you, light from a distant star changes its frequency as it navigates the warped space around a moving, spinning lens.

The researchers developed a detailed theoretical framework to calculate exactly how much the frequency of light shifts when it passes a spinning black hole. They modeled a scenario where a distant source, such as a quasar, sends light toward an observer on Earth, but the path is blocked by a rotating black hole. Because the black hole is spinning, it creates a complex environment where space itself is twisted. The team calculated the time it takes for light to travel from the source to the observer along two different paths, which create two separate images of the source. They found that the time delay between these two paths, and the resulting frequency shift, depends not just on the mass of the black hole, but significantly on how fast it is spinning. By using a specific mathematical approach that avoids the singularities found in other models, they were able to derive precise formulas for this shift.

Their analysis reveals that the spin of the black hole creates a measurable difference in the frequency shift of the light, particularly for images that form very close to the black hole. In their simulations, they considered a supermassive black hole with a mass of 10 million times that of our Sun, located 1 megaparsec away, and a background quasar 1 gigaparsec away. They found that for light passing very close to the event horizon, the frequency shift is dominated by the spin of the black hole rather than its mass. This is a crucial finding because it suggests a new way to measure the spin of black holes. Usually, the mass and spin of a black hole are difficult to separate in observations because they often produce similar effects. However, this study shows that by looking at the frequency shift of light that passes extremely close to the black hole, astronomers can isolate the spin parameter.

The researchers also explored a method involving two different background sources. They simulated a situation where two distinct sources are lensed by the same spinning black hole, creating two images on opposite sides of the lens. By comparing the frequency shifts of these two images, they found that the difference between them depends entirely on the spin of the black hole and is completely independent of its mass. This offers a powerful tool for breaking the "degeneracy" that often plagues gravitational lensing observations, where it is hard to tell if a change in the data is due to the object's mass or its spin. In their models, they showed that even for a stellar-mass black hole, the frequency shift at a close distance is governed by the spin, whereas at larger distances, the mass becomes the dominant factor.

To ensure their results were robust, the team compared their findings with other established methods of calculating light deflection in general relativity. They found that their results matched perfectly with other theories at the most basic level of approximation, confirming that their approach correctly captures the fundamental physics. Any small differences that appeared at higher levels of precision were traced back to the different mathematical coordinates used to describe the space around the black hole, rather than any physical disagreement. This validation gives confidence that their calculations are accurate and that the effects they describe are real consequences of the geometry of spacetime around a spinning object.

The implications of this work are significant for the future of astrophysics. As telescopes become more powerful and capable of measuring the frequency of light with extreme precision, this method could allow astronomers to directly measure the spin of black holes, including the supermassive ones at the centers of galaxies. By observing the subtle shifts in the light from background sources, scientists could determine how fast these cosmic giants are spinning without needing to rely on indirect methods. This study provides a clear roadmap for how to use the frequency of light as a probe to understand the intrinsic properties of the most extreme objects in the universe, turning the bending of light into a precise instrument for measuring the spin of the cosmos.

Technical Summary: Frequency Shift in Gravitational Lensing by the Kerr Metric

Problem Statement
Gravitational lensing is a primary tool in modern astrophysics, yet standard analyses often rely on the Schwarzschild metric, which describes non-rotating spherical masses. However, many astrophysical lenses, particularly black holes formed via stellar evolution or residing at galactic centers, possess intrinsic angular momentum and are better described by the Kerr metric. While the effect of transverse lens velocity on frequency shift has been investigated for Schwarzschild lenses, the specific contribution of the lens's spin parameter ( $a$ ) to this frequency shift in the weak-field regime remains unestablished. This gap limits the ability to measure transverse velocities or break degeneracies in spin parameters during gravitational microlensing observations.

Methodology
The authors develop a theoretical framework to calculate the relative frequency shift ( $\Delta\nu/\nu$ ) induced by gravitational lensing from a spinning lens described by the Kerr metric.

Metric Formulation: The study utilizes the Kerr-Schild coordinate system, which is horizon-penetrating and regular, to describe the spacetime geometry. The metric is expanded in the weak-field limit ( $r \gg GM/c^2$ ) to isolate contributions from mass ( $M$ ) and spin ( $a$ ).
Time Delay Derivation: The total time delay for a photon is decomposed into geometric and gravitational (Shapiro) components. The authors derive the arrival time function $\tau(\vec{\theta})$ and its time derivative $\dot{\tau}$ , accounting for the transverse motion of the source relative to the lens.
Frequency Shift Calculation: By taking the total time derivative of the integrated photon travel time along the trajectory (from $-\infty$ to $+\infty$ ), the authors derive the fractional frequency shift. This involves differentiating the integrand of the time integral with respect to spatial coordinates and applying chain rules to account for the implicit dependence of the radial coordinate on the lens plane coordinates.
Numerical Analysis: The derived analytical expressions are evaluated numerically for specific astrophysical scenarios, including supermassive black holes ( $M = 10^7 M_\odot$ ) and stellar-mass black holes. The analysis considers three source motion scenarios: motion along the y-axis, z-axis, and arbitrary directions.
Validation: The results are cross-compared with the second-order post-Minkowskian (2PM) approximation derived by Jiang et al. [21] to verify analytical integrity.

Key Contributions and Results

Spin-Dependent Frequency Shift: The study demonstrates that the spin parameter of the lens introduces a distinct, measurable modification to the frequency shift of lensed images, beyond the effects predicted by the Schwarzschild metric.
Dual-Diagnostic Capability: Numerical results reveal a decoupling of mass and spin effects based on the image's proximity to the lens:
- Near-Lens Images: For images located in the strong-field regime (e.g., at $10R_s$ ), the frequency shift is predominantly sensitive to the spin parameter ( $a$ ) and is largely independent of the lens mass ( $M$ ).
- Far-Lens Images: For images formed at larger distances (e.g., $0.5R_E$ ), the frequency shift is dictated by the lens mass.
Proposed Measurement Techniques: The authors propose two methods to constrain the black hole spin:
1. Analyzing the frequency shift of a negative-parity image near the lens and cross-referencing it with a far-field image to separate mass and spin contributions.
2. Comparing the differential frequency shift between two symmetric negative-parity images from distinct sources. The study finds that this differential shift is explicitly dependent on the spin parameter while being entirely independent of the lens mass, offering a robust method to break parameter degeneracy.
Coordinate System Comparison: The comparison with the 2PM approximation confirms that both frameworks agree perfectly at the first-order linear approximation ( $O(m)$ ). Discrepancies observed at the second order ( $O(m^2)$ ) are attributed to coordinate artifacts arising from the difference between the harmonic gauge used in prior work and the Kerr-Schild coordinates used in this study, rather than physical contradictions.

Significance
The paper establishes that the frequency shift in gravitational lensing serves as a viable probe for the spin parameter of rotating black holes, a quantity often difficult to measure due to degeneracies in microlensing observations. By demonstrating that the Kerr metric induces specific, spin-dependent signatures in the frequency shift—particularly for images in the strong-field regime—the study provides a theoretical basis for using high-resolution spectroscopy of lensed sources to directly measure the angular momentum of massive cosmic bodies. This approach offers a new window for testing the observational effects of the Kerr metric in astrophysical environments, such as the centers of galaxies.

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