David Albert claims that classical electromagnetic theory is not time reversal invariant.He acknowledges that all physics books say that it is, but claims they are "simply wrong" because they rely on an incorrect account of how the time reversal operator acts on magnetic fields. On that account, electric fields are left intact by the operator, but magnetic fields are inverted. Albert sees no reason for the asymmetric treatment, and insists that neither field should be inverted. I argue, to the contrary, that the inversion of magnetic fields makes good sense and is, in fact, forced by elementary geometric considerations. I also suggest a way of thinking about the time reversal invariance of classical electromagnetic theory -one that makes use of the invariant four-dimensional formulation of the theory -that makes no reference to magnetic fields at all. It is my hope that it will be of interest in its own right, Albert aside. It has the advantage that it allows for arbitrary curvature in the background spacetime structure, and is therefore suitable for the framework of general relativity. The only assumption one needs is temporal orientability.Keywords: time reversal invariance; electromagnetic theory; relativity theory.
IntroductionIn the first chapter of Time and Chance, David Albert (2000) argues that classical electromagnetic theory (in contrast, for example, to Newtonian mechanics) is not time reversal invariant.He acknowledges that all physics books say that it is, but claims they are "simply wrong" because they rely on an incorrect account of how the time reversal operator, properly understood, * E-mail address: dmalamen@uci.edu. 1 acts on magnetic fields. Once that account is corrected, he believes, it is perfectly obvious that the theory is not time reversal invariant. No deep mathematics or physics is called for, only a clear understanding of the nature of the time reversal operation.Received opinion, no doubt, is often wrong. But I don't believe it is here. Physics books tell us that the time reversal operation leaves the electric field E intact, but inverts the magnetic field B. Albert sees no reason for the asymmetric treatment, and insists that neither field should be inverted. He even suggests (p. 18) that the inversion of B is nothing but an ad hoc maneuver to save the time reversal invariance of classical electromagnetic theory. I'll argue, to the contrary (section 6), that the inversion of B makes good sense and is, in fact, forced by elementary geometric considerations. The argument -really just a version of one that can be found in any book on the subject -traces the asymmetric treatment of E and B to the fact that the latter, unlike the former, is not a vector field in the usual sense. (In traditional language, E is a "polar" vector field, while B is an "axial" vector field.)Before giving this response to Albert's claims, I'll make a somewhat different point. It seems to me that the inversion of magnetic fields by the time reversal operator is really something of a distraction. One ca...