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On the time reversal invariance of classical electromagnetic theory
Studies in History and Philosophy of
Modern Physics 35 (2004) 295315
On the time reversal invariance of classical
electromagnetic theory
David B. Malament
Department of Logic and Philosophy of Science, University of California, Irvine, CA 92697-5100, USA
Received 4 July 2003; accepted 10 September 2003
This article is dedicated to the memory of Rob Clifton.
Abstract
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 elds. On
that account, electric elds are left intact by the operator, but magnetic elds are inverted.
Albert sees no reason for the asymmetric treatment, and insists that neither eld should be
inverted. I argue, to the contrary, that the inversion of magnetic elds 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 theoryone that makes use of
the invariant four-dimensional formulation of the theorythat makes no reference to
magnetic elds 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.
r
2004 Elsevier Ltd. All rights reserved.
Keywords: Time reversal invariance; Electromagnetic theory; Relativity theory
1. Introduction
In the rst chapter of Time and Chance, David
Albert (2000)
argues that
classical electromagnetic theory (in contrast, for example, to Newtonian mechanics)
ARTICLE IN PRESS
E-mail address:
dmalamen@uci.edu (D.B. Malament).
1355-2198/$ - see front matter r 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.shpsb.2003.09.006 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, acts on magnetic
elds. 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 do not believe it is here. Physics
books tell us that the time reversal operation leaves the electric eld E intact, but
inverts the magnetic eld B: Albert sees no reason for the asymmetric treatment, and
insists that neither eld 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 will argue, to the contrary (Section 6), that
the inversion of B makes good sense and is, in fact, forced by elementary
geometric considerations. The argumentreally just a version of one that can be
found in any book on the subjecttraces the asymmetric treatment of E and B to
the fact that the latter, unlike the former, is not a vector eld in the usual sense.
(In traditional language, E is a polar vector eld, while B is an axial vector
eld.)
Before giving this response to Alberts claims, I will make a somewhat different
point. It seems to me that the inversion of magnetic elds by the time reversal
operator is really something of a distraction. One can formulate and argue for the
claim that classical electromagnetic theory is time reversal invariant without making
reference to magnetic elds at all. I will do so in Sections 3 and 4, using the invariant
four-dimensional formulation of the theory. It is my hope that the proposed way of
thinking about time reversal invariance will be of interest in its own right, Albert
aside.
The key idea is this. The tensor elds ðF
ab
; J
a
Þ that represent the electromagnetic
eld and its charge-current source eld are only determined relative to a choice of
temporal orientation. I will construe time reversal as an operation taking pairs
ðF
ab
; J
a
Þ as determined relative to one orientation to pairs ð
T
F
ab
;
T
J
a
Þ as determined
relative to the other. This approach 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. (In contrast, the standard approach presupposes that the background
spacetime structure exhibits special time reection symmetries.
1
) At the same time, it
is fully equivalent to the standard approach when the symmetries are present, as in
Minkowski spacetime.
2
ARTICLE IN PRESS
1
The standard approach leaves the background temporal orientation xed, but inverts dynamical
histories under the action of the symmetries.
2
My discussion of Alberts views is closely related to those of
John Earman (2002)
and
Frank
Arntzenius (2004)
. They too dispute his claims about the (non) temporal invariance of classical
electromagnetic theory, but offer somewhat different arguments in response.
D.B. Malament / Studies in History and Philosophy of Modern Physics 35 (2004) 295315
296 2. Alberts argument
I will start by presenting the standard account of time reversal invariance and then
reconstructing Alberts argument.
The standard account goes something like this.
3
A physical theory is said to be
time reversal invariant if, for any sequence of instantaneous states S
I
; y; S
F
allowed
by the theory, the time reversed sequence
R
ðS
F
Þ; y;
R
ðS
I
Þ is allowed as well. (Here R
is the time reversal operator, and the temporal order of states is understood to run
fromleft to right.) If a time coordinate t is given, we can formulate the dening
condition, somewhat more precisely, this way: for any history t
/SðtÞ allowed by the
theory,
the
time
reversed
history
t
/ð
T
SÞðtÞ
is
allowed
as
well,
where
ð
T
SÞðtÞ ¼
R
ðSð tÞÞ:
4
For these characterizations to make full sense in any particular
case, of course, we have to know what count as instantaneous states, and how the
time reversal operator R acts on them.
Consider, for example, the case of a point particle in Newtonian mechanics. Here
(on the standard account) the instantaneous states are pairs ðx; vÞ; where x is the
particles position and v its velocity; and the time reversal operator R takes the state
ðx; vÞ to the state ðx; vÞ: It follows that the induced operator T takes the history
t
/SðtÞ ¼ ðxðtÞ; vðtÞÞ to the time reversed history
t
/ð
T
SÞðtÞ ¼
R
ðxð tÞ; vð tÞÞ ¼ ðxð tÞ;
vð tÞÞ:
The latter reverses the order in which the particle occupies particular positions, and
inverts its velocity at every one. The latter inversion (turning v to
v) makes sense. If
we watch a movie of a particle moving from left to right, and then run the movie
backwards, we see it moving from right to left. Since the velocity of a particle is the
time derivative of its position, R must invert v:
5
Next consider (the standard account of) classical electromagnetic theory. Here the
instantaneous states are quadruples ðE; B; r; jÞ; where E is the electric eld, B is the
magnetic eld, r is the charge density eld, and j is the current density eld. The
latter two characterize the instantaneous state of the charge distribution that serves
as a source for E and B: The time reversal operator R (at least according to the
standard account) has the following action on these objects:
ðE; B; r; jÞ
/
R
ðE;
B; r;
jÞ:
ð1Þ
ARTICLE IN PRESS
3
I will, for the moment, take for granted that we have a well-dened notion of space at a given time and
ignore complications involving relativity theory.
4
The notation may be confusing here. I am taking R to be an operator acting on individual
instantaneous states, and taking T to be an operator acting on histories that is induced by R: The
time reversed history t
/ð
T
SÞðtÞ runs the states of t
/SðtÞ in reverse temporal order and applies R to
each one.
5
Since the position of the particle in the time reversed trajectory is ð
T
xÞðtÞ ¼ xð tÞ at time t, its velocity
at t is
ð
T
vÞðtÞ ¼ d
dt ð
T
xÞðtÞ ¼ d
dt xð tÞ ¼
vð tÞ:
D.B. Malament / Studies in History and Philosophy of Modern Physics 35 (2004) 295315
297 Hence the induced operator T takes the composite history
t
/ðEðt; xÞ; Bðt; xÞ; rðt; xÞ; jðt; xÞÞ
ð2Þ
to the time reversed history
t
/ðð
T
EÞðt; xÞ; ð
T
BÞðt; xÞ; ð
T
rÞðt; xÞ; ð
T
jÞðt; xÞÞ;
ð3Þ
where
ð
T
EÞðt; xÞ ¼ Eð t; xÞ;
ð4Þ
ð
T
BÞ