[Paper Review] Einstein's mirror revisited

Aleksandar Gjurchinovski, Aleksandar Skeparovski|arXiv (Cornell University)|Jan 19, 2007

Relativity and Gravitational Theory4 citations

TL;DR

This paper presents a geometric derivation of the relativistic reflection law for light off a uniformly moving plane mirror using only special relativity postulates and elementary plane geometry, without relying on Lorentz transformations. It derives the reflection angle as a function of incident angle and mirror speed, showing that reflection deviates from the classical law when the mirror moves perpendicularly, with a novel forward reflection effect for high-speed mirrors.

ABSTRACT

We describe a simple geometrical derivation of the formula for reflection of light from a uniformly moving plane mirror directly from the postulates of special relativity.

Motivation & Objective

• To derive the relativistic reflection law for a moving mirror using only the postulates of special relativity and basic geometry.
• To demonstrate that motion parallel to the mirror's surface preserves the classical law of reflection (incident angle = reflected angle).
• To analyze how perpendicular motion alters the reflection law, particularly the emergence of forward reflection.
• To generalize the result to arbitrary mirror motion directions by decomposing velocity into normal and tangential components.

Proposed method

• Compare the geometry of photon reflection in the mirror's rest frame and in a frame where the mirror moves at velocity $v$.
• Use triangle similarity and the sine theorem in spacetime diagrams to relate angles and path lengths in the moving frame.
• Apply Einstein's second postulate (constant speed of light $c$ in all inertial frames) to equate photon path lengths to $ct_1$ and $ct_2$.
• Derive the time-of-flight ratio $t_1/t_2$ from geometric constraints and the mirror's displacement during photon travel.
• Substitute time ratios into trigonometric relations to express the reflected angle $\beta$ in terms of incident angle $\alpha$ and mirror speed $v$.
• Generalize the result for arbitrary mirror motion by replacing $v$ with its normal component $v\cos\varphi$.

Experimental results

Research questions

• RQ1 How does the reflection of light from a uniformly moving mirror deviate from the classical law of reflection?
• RQ2 What geometric and relativistic principles govern the reflection when the mirror moves perpendicularly to its surface?
• RQ3 Under what conditions does forward reflection (reflection angle > 90°) occur, and what determines its onset?
• RQ4 Can the relativistic reflection formula be derived without using Lorentz transformations or wave optics?

Key findings

• When the mirror moves parallel to its surface, the incident and reflected angles remain equal, preserving the classical law of reflection.
• For perpendicular motion, the reflection angle $\beta$ is given by $\cos\beta = \frac{-2(v/c) + (1 + v^2/c^2)\cos\alpha}{1 - 2(v/c)\cos\alpha + v^2/c^2}$, which deviates from the classical law.
• Forward reflection occurs when $\alpha > \alpha_c = \arccos\left(\frac{2v/c}{1 + v^2/c^2}\right)$, causing the photon to reflect in the same direction as the mirror’s motion.
• The maximum incident angle for reflection is $\alpha_{\text{max}} = \arccos(v/c)$; beyond this, the photon cannot catch up to the mirror.
• The derived formula reduces to the classical case ($\alpha = \beta$) when $v = 0$, confirming consistency with stationary mirrors.
• The general formula for arbitrary mirror motion direction is $\cos\beta = \frac{-2(v/c)\cos\varphi + [1 + (v^2/c^2)\cos^2\varphi]\cos\alpha}{1 - 2(v/c)\cos\alpha\cos\varphi + (v^2/c^2)\cos^2\varphi}$, with $\varphi$ as the angle between velocity and mirror normal.