Marzke-Wheeler coordinates for accelerated observers in special relativity

Abstract

In special relativity, the definition of coordinate systems adapted to generic accelerated observers is a long-standing problem, which has found unequivocal solutions only for the simplest motions. We show that the Marzke-Wheeler construction, an extension of the Einstein synchronization convention, produces accelerated systems of coordinates with desirable properties: (a) they reduce to Lorentz coordinates in a neighborhood of the observers' world-lines; (b) they index continuously and completely the causal envelope of the world-line (that is, the intersection of its causal past and its causal future: for well-behaved world-lines, the entire space-time). In particular, Marzke-Wheeler coordinates provide a smooth and consistent foliation of the causal envelope of any accelerated observer into space-like surfaces. We compare the Marzke-Wheeler procedure with other definitions of accelerated coordinates; we examine it in the special case of stationary motions, and we provide explicit coordinate transformations for uniformly accelerated and uniformly rotating observers. Finally, we employ the notion of Marzke-Wheeler simultaneity to clarify the relativistic paradox of the twins, by pinpointing the local origin of differential aging.