The Red Queen visits Minkowski Space

When Alice went Through the Looking Glass [1], she found herself in a situation where she had to run as fast as she could in order to stay still. In accordance with the dictum that truth is stranger than fiction, we will see that it is possible to find a situation in special relativity where running towards one’s target is actually counter-productive. Although the situation is easily analysed algebraically, the qualitative properties of the analysis are greatly illuminated by the use of spacetime diagrams. Although tachyons (particles which travel faster than light) are not at present observed experimentally, they arise naturally in superstring theory, where their consequences require investigation: one example of such an inquiry is found in [2]. Outside this context, tachyons have also been considered from advanced viewpoints, as in [3], in which it was found that the obvious problems associated with causality might be illusory; and from elementary viewpoints, as in [4] where simple geometrical properties of a tachyonic wavefront were considered. This article takes a brief look at how tachyons appear to move from the point of view of various inertial observers in special relativity. The results are reminiscent of Alice’s experience through the looking glass, where she had to run as fast as she could just to stay still. Here we will find that the situation can be worse even than that: it is possible for a target to recede faster, the faster you chase it. Although the results are easy to obtain algebraically, it is the use of space-time diagrams that renders the situation intelligible. Finally, the relative strengths of the algebraic and diagrammatic approaches are briefly discussed. The article is presented in a discursive manner, and should be accessible to students who have taken a course in special relativity. ∗[email protected]