Bell's Spaceships: The Views from Bow and Stern

TL;DR

This paper analyzes Bell's spaceship paradox by examining the relative motion and visual perspectives of two uniformly accelerating spaceships, revealing asymmetric observations and horizon effects in special relativity.

Contribution

It provides a detailed analysis of the visual and observational differences from each spaceship's perspective during acceleration, highlighting asymmetries and horizon phenomena.

Findings

Leading spaceship loses sight of the trailing one due to redshift and horizon effects.

Trailing spaceship sees the leading spaceship with initial blueshift, then decreasing blueshift.

Asymptotic viewing properties can match initial conditions under certain parametrizations.

Abstract

Unravelling apparent paradoxes has proven to be a powerful tool for understanding the complexities of special relativity. In this paper, we focus upon one such paradox, namely Bell's spaceship paradox, examining the relative motion of two uniformly accelerating spaceships. We consider the view from either spaceship, with the exchange of photons between the two. This recovers the well known result that the leading spaceship loses sight of the trailing spaceship as it is redshifted and disappears behind what is known as the `Rindler horizon'. An immediate impact of this is that if either spaceship tries to measure the separation through `radar ranging', bouncing photons off one another, they would both eventually fail to receive any of the photon `pings' that they emit. We find that the view from this trailing spaceship is, however, starkly different, initially, seeing the leading spaceship with an increasing blueshift, followed by a decreasing blueshift. We conclude that, while the leading spaceship loses sight of the trailing spaceship, for the trailing spaceship the view of the separation between the two spaceships, and the apparent angular size of the leading spaceship, approach asymptotic values. Intriguingly, for particular parametrization of the journey of the two spaceships, these asymptotic values are identical to those properties seen before the spaceships began accelerating, and the view from the trailing spaceship becomes identical to when the two spaceships were initially at rest.