Opposite Thermodynamic Arrows of Time

The possibility of simultaneous opposite running thermodynamic arrows of time has been raised on several occasions, for didactic purposes [1], for general interest [2], and to confound by “obvious” counterexample [2]. A difficulty in these considerations is the absence of a welldefined framework. For example, one might argue against opposing arrows as follows. Let the systems be A and B. An observer in A will see a succession of small miracles in B as eggs uncrack, etc. It would seem that the tiniest interference by A, the smallest cry of amazement—transmitted to B—would destroy the monumental coordination needed for B’s reversed arrow. That this argument is flawed is apparent when one realizes that it is phrased from A’s perspective, and takes as natural that the images from B do not destroy the coordination that B would attribute to A. But whether the flaw is correctable or whether the conclusion is that both arrows would be destroyed is less clear. In [3–5] a framework for these questions was proposed. Here I use that framework to show that small interaction does not destroy the arrows. The question of whether the systems can communicate will be touched on. Signals are of interest because of causal paradoxes. One aspect of communication is electromagnetic radiation, and I will extend the Wheeler-Feynman absorber theory [6] to show that each system has its own retarded interactions, which appear advanced to the other. Our usual thermodynamic arrow can be phrased as the fact that when macroscopic (coarse grained) information is given it can be used, by averaging over the evolution of all microstates consistent with the macrostate, to estimate the future, but not (in that way) the past. As argued in [3–5], an unbiased treatment of thermodynamic arrow questions can be had by giving macroscopic information at two times (typically, cosmologically remote). It was found that despite the nonstandard conditioning, arrows emerge, consistent with a thesis correlating the thermodynamic arrow with the expansion of the Universe [7] (or at least with low entropy states at the remote eras [5]). In [5], p. 179, it was suggested that the 2-time formulation could be used to study opposing arrows, but the inquiry was dismissed as “science fiction.” However, in a time-symmetric universe this possibility should be considered (in fact this was a complaint in [2], so defense of the