The setup is the classic one of textbook optics. In a laboratory frame, denoted (L), a laser beam of wavelength λ passes through a slit of width d, where the slit dimensions are on the order of the wavelength so that diffraction is strong. The diffracted intensity as a function of observation angle θ follows the standard Fraunhofer result, proportional to the square of a sinc function, with the argument set by the transverse wave-vector mismatch. Minima occur when the sine of the observation angle equals integer multiples of λ divided by d. The angular width of the central peak can be estimated from Heisenberg’s uncertainty principle, and Bourdier shows this estimate agrees with the standard Fourier optics result. In the laboratory frame, expressed in the appropriate angular variable, the two first minima on either side of the central maximum sit symmetrically about the peak.

The novelty of the work lies in introducing a second inertial frame, (L′), moving at constant velocity V along the z-axis relative to the laboratory. Bourdier applies the Lorentz transformation to the four-velocity of diffracted photons and to the wave four-vector, deriving the relativistic aberration relations that connect propagation angles in the two frames. These relations are nonlinear: equal angular intervals in the laboratory frame do not map to equal angular intervals in the moving frame. It is precisely this nonlinearity that breaks the symmetry of the diffraction pattern. The transformation is expressed in compact tensorial form using the Lorentz matrix, and the author emphasizes that the same spacetime transformation applies whether the light propagates in vacuum or in a material medium, although the refractive index must then be treated carefully as a property of the medium’s rest frame rather than as a simple Lorentz scalar.

For the case of normal incidence, the analysis uncovers a striking special situation. There exists a particular boost velocity, determined by the condition that the sine of the first diffraction minimum angle equals V divided by c, in which one of the low-intensity directions adjacent to the central peak becomes exactly perpendicular to the boost axis in the moving frame. Combining this with the diffraction condition yields the elegant result that V/c equals λ/d. Since significant diffraction requires the slit width to be no more than about ten wavelengths, the relevant velocities are at least a tenth of the speed of light, placing the effect firmly in the relativistic regime. Under this condition, the transformed wave vector of photons diffracted toward that particular minimum becomes parallel to the transverse axis in the moving frame, a result confirmed independently by transforming the energy-momentum four-vector of the photon.

The direction of peak intensity transforms as well. In the laboratory frame, the central maximum corresponds to forward propagation along the original beam direction. After the Lorentz boost, this direction is tilted, with the sine of the new peak angle equal to minus V/c. Bourdier verifies this using the transformation of the electromagnetic fields themselves: the Poynting vector, which describes the flow of energy in the wave, acquires a transverse component in the moving frame. The energy flow is therefore no longer aligned with the original propagation direction but propagates obliquely. This is not a violation of relativity but a direct manifestation of relativistic aberration, the same effect responsible for the well-known Penrose-Terrell appearance of relativistically moving objects.

A crucial point established in the paper’s appendices is that the diffraction mechanism itself is untouched. The phase factor governing interference between contributions from different points of the aperture is the Lorentz-invariant scalar product of the wave four-vector and the spacetime position. Because this phase is invariant, the diffraction integral in the boosted frame has exactly the same sinc-type functional form as in the laboratory frame. What changes is only the mapping between the observation angle and the transverse wave-vector component, which is distorted by aberration. The maxima and minima of the pattern remain well defined under the transformation, but their angular spacing is redistributed. Most of the diffracted power in the moving frame lies between two directions that are no longer equidistant from the peak, and a relativistic correction term breaks the symmetry that existed in the laboratory frame.

The paper also treats oblique incidence, where the incoming wave strikes the slit at an angle. In the laboratory frame, the central maximum then lies along the incidence direction, and the two adjacent minima are again symmetric in sine space. Choosing the boost velocity so that the central maximum becomes normal to the slit in the moving frame, Bourdier derives the transformed positions of the two minima and shows they are no longer symmetric about the peak. Their sine values differ by a relativistic correction, and the asymmetry again traces back to the nonlinearity of the aberration formula. In both configurations, normal and oblique incidence, the conclusion is the same: the symmetry is not destroyed but reparametrized, and the apparent asymmetry is a kinematic effect of observation from a moving frame.

The magnitude of the effect is quantified explicitly. Applying the aberration transformation to the two symmetric minima at plus and minus the first minimum angle, and expanding to first order in V/c for small diffraction angles, the author finds that the angular separation between the transformed minima is approximately minus two times V/c. The asymmetry therefore scales linearly with the ratio of the boost velocity to the speed of light. The transformation of the intensity distribution itself is handled through photon number conservation, which requires that the intensity times the angular interval be preserved. This yields a Jacobian factor relating the intensity in the moving frame to that in the laboratory frame, and the invariance properties of the radiation distribution function ensure that maxima and minima are preserved under the transformation.

Although the study is analytical, Bourdier outlines how the predictions could be tested. A numerical reconstruction of the diffraction pattern in both frames, applying the derived angular transformation and Jacobian to the Fraunhofer profile, would directly visualize the distortion and confirm the predicted asymmetry of the minima. Experimentally, observing the effect with a genuinely relativistically moving slit would be extraordinarily difficult, but the author suggests analogue approaches. Optical systems involving moving or effectively moving interfaces, such as plasma environments or time-dependent photonic media with a drift velocity, could produce angular redistributions formally analogous to those derived here. Alternatively, a dynamically controlled optical setup could deliberately reparametrize the outgoing angular distribution according to the aberration law, emulating the observable consequence of a Lorentz boost without physically accelerating an aperture.

The broader significance of the work is conceptual as much as practical. Wherever diffraction occurs in the presence of substantial relative motion between a radiating structure and the observer, whether an optical aperture, a radiating interface, or a drifting plasma structure, the observed profile may lose its symmetry even though the underlying diffraction law is unchanged in the structure’s own rest frame. Relativistic distortions of angular radiation patterns could in principle serve as diagnostic signatures of motion in optical and plasma systems. The study offers a clean, simple example of how a familiar wave-optics phenomenon is reshaped by the kinematics of special relativity, reminding physicists that even the most textbook patterns carry the fingerprints of spacetime structure when viewed from a moving frame.

Subject of Research: Relativistic distortion of slit diffraction patterns under Lorentz transformation

Article Title: Relativistic asymmetry of diffraction patterns induced by Lorentz transformation

Article References: Bourdier, A. (2026). Relativistic asymmetry of diffraction patterns induced by Lorentz transformation. Results in Physics, 88, Article 108750. https://doi.org/10.1016/j.rinp.2026.108750

Image Credits: AI Generated

DOI: 10.1016/j.rinp.2026.108750

Keywords: diffraction, special relativity, Lorentz transformation, relativistic aberration, Fraunhofer pattern, wave optics, sinc function, Poynting vector, photon four-vector, laser propagation, plasma optics, asymmetry