## Abstract

Consider two material bodies, moving freely. One emits a light signal, the other reflects it back, the first reflects it again, and so on. When de Sitter's world is represented by the part of projective space outside a non-ruled quadric, the world lines of the material bodies are two secants and the world lines of the light signals form a zigzag of tangents between them. Let t$_{2n}$ be the proper time for the nth event at the first body. If the two world lines are `ultraparallel', so that the bodies approach each other to a minimal distance $\phi$ (at time zero) and then recede, we have t$_{2n}$ = log tan(n$\phi$+$\frac{1}{4}\pi$) for n < $\pi$/4$\phi$. If the world lines intersect at time zero and form a hyperbolic angle $\phi$ (depending on the relative velocity), we have $t_{2n} = log coth (\frac{1}{2} log coth \frac{1}{2}t_0-n\phi)$ or $-log coth (n\phi-\frac{1}{2} log tan h |\frac{1}{2}t_0|)$ according as t$_0$ is positive or negative. If the world lines are parallel (in the sense of hyperbolic geometry), so that the bodies approach each other asymptotically, we have t$_{2n}$ = log(2n+$\epsilon$) where $\epsilon$ = exp t$_0$. Finally, if the world lines are skew, t$_{2n}$ is given implicitly by a recursion formula which does not seem to have an elementary solution. The first two results were obtained by Wigner another way; the third and fourth are apparently new.

## Royal Society Login

Sign in for Fellows of the Royal Society

Fellows: please access the online journals via the Fellows’ Room

### Log in using your username and password

### Log in through your institution

Pay Per Article - You may access this article or this issue (from the computer you are currently using) for 30 days.

Regain Access - You can regain access to a recent Pay per Article or Pay per Issue purchase if your access period has not yet expired.