⚛️ general relativity

Frequency shift and viewing direction variations in gravitational lensing

This paper presents a rigorous theoretical framework for gravitational lensing in cosmological spacetimes that derives exact formulas for frequency shifts and viewing direction variations induced by the relative transverse velocities of the lens, source, and observer, including in highly relativistic regimes.

Original authors: Mikołaj Korzyński, Mateusz Kulejewski

Published 2026-09-09

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CC BY 4.0

Original authors: Mikołaj Korzyński, Mateusz Kulejewski

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 acts as a cosmic lens, bending the path of light as it travels across the universe. When a massive object like a galaxy or a black hole sits between a distant light source and an observer on Earth, it distorts the image of that source, often splitting it into multiple copies. For decades, astronomers have relied on this phenomenon to map the invisible mass of the universe. In the standard view of this process, the energy of the light remains constant as it bends; the frequency of the wave, which determines its color or pitch, is thought to be preserved during the deflection. This assumption holds true if everything involved—the light source, the lensing object, and the observer—is stationary relative to one another. However, the universe is rarely still. Stars orbit, galaxies collide, and black holes move at tremendous speeds. When these components are in motion, the interaction between the light and the gravitational field becomes more complex, potentially altering the frequency of the light in ways that standard models have not fully captured.

A team of physicists has now developed a rigorous theoretical framework to describe exactly how this frequency shift occurs when the lens, the source, or the observer are moving. Their work moves beyond the simplified approximations used in previous studies, which often assumed that all objects were moving slowly compared to the speed of light. Instead, they derived a general formula that remains valid even when the lensing object is traveling at relativistic speeds, approaching the velocity of light. They also expanded the theory to include the effects of the expanding universe, placing the lensing system within a realistic cosmological model rather than a static, empty space. The researchers found that the motion of the lens introduces a shift in the frequency of the observed radiation, a phenomenon that applies equally to light and gravitational waves. This shift is not uniform; it depends on the specific direction of the motion and the relative positions of the source, the lens, and the observer.

One of the most significant findings of this study is the discovery of a subtle change in the viewing direction of the source. In standard lensing theory, the position of a source is mapped to the position of its images on the sky, but the direction from which the light appears to arrive is usually assumed to be fixed. The authors demonstrated that when the lens is moving, the apparent direction from which the light reaches the observer actually changes. This effect is particularly important for highly directional sources, such as binary systems emitting gravitational waves. For these objects, a slight change in viewing angle corresponds to a measurable change in the phase and polarization of the wave, offering a new way to extract information about the system. The researchers also quantified how the time delay between different images of the same source can cause a misalignment. Because the light takes different paths to reach the observer, the images are formed at slightly different moments in time. If the source is moving, it will be in a different physical location when each image is formed, causing the images to appear slightly out of line with the lens itself, breaking the perfect symmetry usually expected in such systems.

The magnitude of these frequency shifts is generally very small, often too tiny to detect with current instruments for typical astrophysical events. However, the authors show that the effect becomes significant in specific scenarios, such as when a massive object moves at high speed or when the lens is very close to the source or the observer. They calculated that for a lens moving at relativistic speeds, the frequency shift can vary depending on the angle of motion, vanishing completely if the lens moves directly toward the observer but reaching a maximum when it moves sideways. This sensitivity to direction and speed means that in the future, with more precise detectors, astronomers could potentially measure the transverse velocities of lenses and sources by analyzing the frequency differences between multiple images. This would provide a new method for determining the motion of cosmic objects, complementing existing techniques like measuring the apparent position of a source or observing the parallax effect.

The study also highlights that these effects are not limited to electromagnetic radiation but apply equally to gravitational waves. As detectors become more sensitive, they may begin to observe the interference patterns created when multiple lensed gravitational wave signals arrive at slightly different frequencies. The researchers suggest that for compact binary systems, the combination of frequency shifts and viewing direction changes could lead to observable distortions in the wave signal, such as a stretching or squeezing of the waveform over time. While the paper does not claim that these effects have been observed yet, it provides the necessary theoretical tools to identify them. By offering a complete description of the frequency shift and viewing direction variation for any velocity and any distance, the work lays the groundwork for a new era of precision lensing, where the motion of the universe itself becomes a measurable part of the gravitational lensing equation.

Technical Summary: Frequency Shift and Viewing Direction Variations in Gravitational Lensing

Problem Statement
In standard gravitational lensing theory, the deflection of light by a massive body is typically treated in a static framework where photon energy is conserved during the interaction, resulting in no frequency shift. However, when the source, lens, or observer possess relative transverse velocities, the lensing process induces a frequency shift (spectroscopic effect) in the observed radiation. While this effect has been studied in specific contexts—such as moving lenses in flat spacetime or within the first-order-in-velocity (FOV) approximation—existing literature lacks a rigorous, fully relativistic treatment that accommodates arbitrary velocities and cosmological distances. Furthermore, standard formalisms often neglect the variation in the source's viewing direction, an effect that is negligible for isotropic electromagnetic emitters but potentially critical for highly anisotropic sources like compact binary systems emitting gravitational waves (GWs).

Methodology
The authors develop a general theoretical framework for gravitational lensing embedded in a Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime. The derivation relies on two primary pillars:

Geometric Optics and Geodesic Deviation: The propagation of light between the source, lens, and observer is modeled using the first-order geodesic deviation equation (GDE). The authors utilize the bilocal geodesic operators (BGOs), specifically the Jacobi propagator, to relate perturbations in the geodesic at the observer to those at the source and lens.
Frame-Invariant Formalism: To ensure validity across arbitrary velocities (including highly relativistic regimes), the authors employ a frame-invariant approach. They define transverse 4-velocity differences ( $T_{l,U}(u)$ ) between the local rest frames of the observer, lens, and source and the comoving cosmological frame. This avoids the limitations of the FOV approximation used in prior works (e.g., Wucknitz and Sperhake [28]) and allows for the inclusion of cosmological expansion effects.

The derivation assumes the thin lens approximation and small deviation angles (first-order-in-deviation, FOD). Crucially, the metric in the vicinity of the lens is assumed to be static in the lens's rest frame, ensuring energy conservation within that frame, while the background spacetime is a general FLRW metric.

Key Contributions and Results

General Frequency Shift Formula:
The paper derives a general expression for the redshift variation ( $\Delta \zeta$ ) of a lensed image (Eq. IV.4). This formula expresses the frequency shift in terms of the lensed angular position ( $\theta_A$ ), the unlensed source position ( $\beta_A$ ), and the transverse 4-velocity differences of the observer, lens, and source relative to the cosmological flow.
- The result generalizes previous findings by lifting the FOV approximation, allowing for arbitrary velocities.
- It explicitly includes a term dependent on the unlensed angle $\beta_A$ , which accounts for the frequency gradient across extended sources or sources with large angular extensions. This term is absent in analyses assuming the source lies exactly on the line of sight.
- The total effect is decomposed into a "dipole" term (dependent on $\theta_A$ and all three velocities) and a "proper lensing" term (dependent on $\beta_A$ and the relative motion of the lens and source).
Variation of Viewing Direction:
A novel contribution is the rigorous derivation of the variation in the source's viewing direction ( $\iota_A$ ) caused by lensing (Eq. III.12).
- This variation is expressed as a function of the lensed angle $\theta_A$ and the unlensed angle $\beta_A$ , incorporating cosmological distance factors and scale factors.
- The authors note that while this effect is negligible for isotropic electromagnetic sources, it is significant for anisotropic GW sources, where a change in viewing direction directly alters the observed phase and polarization of the wave.
Relativistic Lens Analysis:
The paper analyzes the spectroscopic effect for a point mass lens moving at relativistic speeds.
- The characteristic redshift difference between images is found to depend on a kinematic factor $f(\alpha, v_L)$ , where $\alpha$ is the angle between the lens velocity and the line of sight.
- The effect vanishes if the lens moves directly toward the observer ( $\alpha=0$ ), regardless of velocity. For ultra-relativistic lenses moving transversely, the effect scales with the square root of the Lorentz factor ( $\gamma_L^{1/2}$ ), suggesting a modest growth even at high velocities.
Flat Spacetime and Misalignment:
In the limit of flat spacetime (relevant for Galactic microlensing), the authors derive simplified expressions for frequency shifts and viewing direction variations.
- They demonstrate that for moving sources behind a stationary axisymmetric lens, time-delay differences cause the images to be misaligned with the lens and the unlensed source position, breaking the axial symmetry typically assumed in static lensing.
- The maximal asymmetry angle is estimated to be of the order of microarcseconds for typical astrophysical parameters, currently below observational resolution for single images.

Significance and Claims
The authors claim that this work provides the most general formula for the frequency effect of gravitational lensing to date, valid for arbitrary motions and cosmological distances.

Theoretical Rigor: The derivation is explicitly Lorentz-invariant and valid beyond the FOV approximation, correcting inconsistencies in previous velocity-correction formulas.
Extension of Standard Theory: The inclusion of the viewing direction variation extends standard lensing theory, predicting additional phase differences between multiple lensed images. This is highlighted as particularly relevant for next-generation gravitational wave detectors observing compact binary systems.
Observational Prospects: The paper quantifies the magnitude of these effects, noting that while generally small (scaling with the square root of the Schwarzschild radius over distance and transverse velocity), they may become observable in specific regimes, such as "self-lensing" scenarios where a source orbits a massive black hole, or in the interference patterns of strongly lensed GW signals.

The paper concludes that the frequency shift and viewing direction variations constitute a distinct, measurable spectroscopic effect of lensing, offering a potential method to constrain transverse velocities of lenses and sources, complementing existing techniques like astrometric microlensing and interferometry.

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