Visual Appearance of Extended objects in Special Relativity

TL;DR

This paper investigates the visual appearance of extended objects in special relativity, analyzing apparent shapes, speeds, and effects like Doppler shift, providing mathematical proofs and visualizations to enhance intuitive understanding.

Contribution

It offers a comprehensive mathematical and visual analysis of how extended objects appear in special relativity, including shape preservation, apparent superluminal speeds, and relativistic effects, extending previous simpler models.

Findings

A sphere retains its circular silhouette under relativistic transformation.

Apparent speed of points can exceed light speed approaching the observer.

Derived generalized equations for Doppler effect and aberration.

Abstract

The strange visual appearance of objects is one of the puzzling predictions of Einstein's relativity. This is mainly due to the distinction between measuring and seeing, where the former is described by the Lorentz Transformation and the latter considers the time light rays (emitted by each point on the object) take to reach the observer. We compute the apparent position of a point given its velocity, initial position, and observation time. The apparent speed of a point is calculated, and we obtain that it exceeds the speed of light when approaching the observer, similar to superluminal motion. For parameterizable surfaces, we analyze properties (such as curvature and torsion) of apparent shapes. The observation that a sphere retains its circular silhouette when transformed to its apparent shape, independent of the initial conditions, is proved mathematically. Plots describing the apparent speed and length of objects are made, and the metric tensor for a distorted sphere is calculated. A generalized equation for the Doppler effect and relativistic aberration is derived to analyze regions of redshift and blueshift. Using the Born-rigidity conditions, we compute the hyperbolic trajectories of each point on an extended object given an initial velocity, position, and proper acceleration for any reference point. The claim that a rigid body, accelerating in Special Relativity, cannot exceed a given length in certain circumstances is justified. We obtain many non-trivial results, which are proved algebraically and using light cones, that are tested by taking the limit of acceleration approaching 0 to retrieve results in the constant velocity scenario. In conclusion, these visualizations may be used by teachers to explain SR intuitively. Finally, we provide an overview of extending the same problem to curved spacetime and explain the potential applications of this project.