Reconsidering Relativistic Causality

J. Butterfield

Trinity College
Cambridge CB2 1TQ

jb56@cam.ac.uk

2 August 2007

Forthcoming (after an Appendectomy) in
International Studies in the Philosophy of Science volume 21, issue 3, October 2007

Abstract

I discuss the idea of relativistic causality, i.e. the requirement that causal
processes or signals can propagate only within the light-cone. After briefly lo-
cating this requirement in the philosophy of causation, my main aim is to draw
philosophers’ attention to the fact that it is subtle, indeed problematic, in rela-
tivistic quantum physics: there are scenarios in which it seems to fail.

I consign to an Appendix two such scenarios, which are familiar to philoso-
phers of physics: the pilot-wave approach, and the Newton-Wigner represen-
tation. I instead stress two unfamiliar scenarios: the Drummond-Hathrell and
Scharnhorst effects. These effects also illustrate a general moral in the philosophy
of geometry: that the mathematical structures, especially the metric tensor, that
represent geometry get their geometric significance by dint of detailed physical
arguments.



Contents

1 Introduction 2

2 In the shadow of causal anti-fundamentalism 3

3 Formulating relativistic causality 7

3.1 Difficulties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Some precise formulations . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2.1 Minkowski spacetime . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2.2 Curved spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3 Causal loops? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Why is this tensor “read” by rods and clocks? Brown’s moral 16

4.1 The moral in general . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2 The moral in special relativity . . . . . . . . . . . . . . . . . . . . . . . 18

4.3 The moral in general relativity . . . . . . . . . . . . . . . . . . . . . . . 19

4.3.1 How the gravitational field gets its metric significance . . . . . . 19

4.3.2 The equivalence principles, weak and strong . . . . . . . . . . . 20

4.3.3 Motion along geodesics . . . . . . . . . . . . . . . . . . . . . . . 22

5 Two effects 24

5.1 The overall shape of the examples . . . . . . . . . . . . . . . . . . . . . 24

5.1.1 Limitations and alternatives . . . . . . . . . . . . . . . . . . . . 25

5.2 The Drummond-Hathrell effect . . . . . . . . . . . . . . . . . . . . . . 27

5.2.1 Faster than light? . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.2.2 Avoiding contradictions . . . . . . . . . . . . . . . . . . . . . . 31

5.3 The Scharnhorst Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.3.1 Faster than light? . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.3.2 Avoiding contradictions . . . . . . . . . . . . . . . . . . . . . . 34

6 Conclusion 35

7 Appendix: Two familiar examples 35

7.1 The pilot-wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1



7.1.1 The non-relativistic case . . . . . . . . . . . . . . . . . . . . . . 36

7.1.2 Action-at-a-distance? . . . . . . . . . . . . . . . . . . . . . . . . 37

7.1.3 The relativistic case . . . . . . . . . . . . . . . . . . . . . . . . . 39

7.2 The Newton-Wigner representation . . . . . . . . . . . . . . . . . . . . 43

8 References 44

1 Introduction

This paper concerns some cases where a relativistic quantum system apparently violates
relativistic causality, i.e. the requirement that causal processes or signals can travel at
most as fast as light. This is a large topic, both because there are several apparent
cases of such spacelike causal processes, and because there are open questions for both
physics and philosophy.

To save space, I will consign to an Appendix two cases which, though “heterodox”
traditions within physics, are already familiar to philosophers of physics. Namely: (i)
the pilot-wave approach, in whose relativistic versions the guidance equation and the
quantum potential yield non-local effects (analogous to their effects in non-relativistic
versions); and (ii) the Newton-Wigner representation, in which localized states prop-
agate superluminally (and which has other non-local aspects, such as the fact that
spectral projectors of the position operator associated with spacelike related spatial
regions do not in general commute).

I aim instead to draw philosophers’ attention to two unfamiliar cases, in which
a superluminal effect is predicted by an orthodox relativistic quantum theory, viz.
quantum electrodynamics (QED):

(i): the Drummond-Hathrell effect, in which a photon travels at a superluminal
speed in a curved (general relativistic) spacetime; and

(ii): the Scharnhorst effect, in which a photon travels at a superluminal speed
between two parallel plates in Minkowski spacetime.

Apart from these effects’ intrinsic interest, they are worth discussing for three
broader reasons.

(i): Their violation of relativistic causality is very different from that associated
with the familiar EPR-Bell correlations (since it occurs in a single quantum system,
not a pair of them).

(ii): They illustrate an important moral in the philosophy of geometry: a moral
which, incidentally, explains away the apparent contradiction in the above claim that
a photon travels at a superluminal speed.

(iii): Although these effects are uncontroversial in the physics community, our
present understanding of them is undoubtedly incomplete: there are open questions,
for both physics and philosophy, waiting to be addressed. In particular, there are open

2



philosophical, or at least conceptual, questions about: (i) relations between the vari-
ous formulations of relativistic causality, and thereby about which formulations these
effects violate; and (ii) how these effects avoid the causal loops, and hence (by the
“bilking argument”) the contradictions, that are traditionally meant to follow from
superluminal processes. So these open questions are an invitation to the reader for
future work!1

The plan is as follows. First, I place my topic within the general philosophy of
causation (Section 2; this general discussion is not needed later). In Section 3, I discuss
different formulations of relativistic causality, emphasising open questions about QED
in curved spacetime and about avoiding causal loops. Section 4 prepares us for the
two effects by discussing a general moral about the philosophy of (chrono)-geometry
which they illustrate: that the mathematical structures, especially the metric tensor,
that represent geometry get their geometric significance by dint of detailed physical
arguments. I learnt this moral from Brown (2005), and we need some details of his
argument for it, in order to understand the two effects. So in Section 4, I give those
details. Once we have them, we will be ready for Section 5’s punchline: the two
effects that violate relativistic causality.2 Finally, in the Appendix I discuss relativistic
causality, or rather is violation, in the pilot-wave approach and the Newton-Wigner
representation.

2 In the shadow of causal anti-fundamentalism

In the history both of physics, and of its relation to philosophy, causation is a peren-
nial player, though the role it plays of course changes—dramatically. Similarly in
contemporary physics, and philosophy of physics: the former seems to be up to its
neck in causal talk, and accordingly much of the latter focusses on causation—indeed,
an embarras de richesse.

But I must also admit to embarrassment, in the usual English sense! This arises
from my being inclined to endorse causal anti-fundamentalism in the sense espoused by
Norton (2003, 2006). Norton denies ‘that the world has a universal, causal character

1I confess at the outset that I set aside a recent line of argument which “goes in the opposite
direction” to what follows: i.e. which uses relativistic causality at high energies (mostly in the form
of an analytic S-matrix) to select low-energy effective theories—thereby concluding from effects like
those I shall discuss that QED is not embeddable in any causal high energy theory. Thanks to Hugh
Osborn for this, and for referring me to Adams, Arkani-Hamed et al. (2006). That paper also discusses
other superluminal effects, including one from a brane model—and how it might be detected by tiny
deviations in the moon’s orbit! Shore (2007) includes a response to this line of argument, drawing
on his work on the Drummond-Hathrell effect: details of which are in Section 5.2 below. So far as I
know, this line of argument awaits philosophers’ attention.

2Though these effects are unfamiliar, Brown is not alone in pointing to their philosophical impor-
tance. Weinstein mentions the Drummond-Hathrell effect as one of several examples, in his recent
judicious defence of superluminal signalling (2006, pp. 390, 393); and his (1996) urges a moral similar
to Brown’s. What follows is much indebted to them both.

3



such as would be expressed by a principle of causality that must be implemented in the
individual sciences’ (2006, Section 2). He supports this by surveying the efforts over
the centuries to articulate such a principle, concluding that there is ‘such a history of
persistent failure that only the rashest could possibly expect a viable, factual principle
still to emerge’ (2003, Section 2). Indeed, Norton’s summary of his survey provides a
helpful thumbnail sketch of some roles that causation has played in physics, and these
roles’ impact on natural philosophy, over the last four hundred years. I can best set the
scene for my discussion—and explain my embarrassment—by an extended quotation:

Highlights of this survey include ... Newton’s (1692/93, third letter) insis-
tence that unmediated action at a distance is “so great an absurdity, that
I believe no man, who has in philosophical matters a competent faculty of
thinking, can ever fall into it.” Yet the continued success of Newton’s own
theory of gravitation, with its lack of any evident mediation or transmission
time for gravitational action, eventually brought the grudging acceptance
that this absurdity was not just possible but actual. In the nineteenth cen-
tury, what was required of a process to be causal was stripped of all prop-
erties but one, the antecedent cause must determine its effect: “For every
event there exists some combination of objects or events, some given con-
currence of circumstances, positive and negative, the occurrence of which is
always followed by that phenomenon” (Mill, 1872, Bk. III, Ch.V, 2). The
advent of quantum mechanics in the early twentieth century established
that the world was not factually causal in that sense and that, in generic
circumstances, the present can at best determine probabilities for differ-
ent futures. So we retracted to a probabilistic notion of causation. Yet
the principles that we thought governed this probabilistic notion were soon
proved empirically to be false. For Reichenbach suggested that we could
still identify the common cause of two events, in this probabilistic setting,
by its ability to screen off correlations between the events. That too was
contradicted by the EPR pairs of quantum theory. (Norton, 2006, Section
2)

Thus Norton rejects in particular the idea that physics provides a reduction of causa-
tion, for example identifying it with an appropriate transmission of a conserved quan-
tity, especially energy or momentum, as in the process theory of causation of Salmon
(1984) or Dowe (2000). But he does admit (2003, pp. 5-6) that this is currently the
most promising candidate for a principle of causality.

I endorse much of this broad picture; (my disagreements with it will emerge below).
But it implies that the principal topic in the analysis of causation, on which the evo-
lution of physics can shed light, is determinism. And this is “embarrassing”, for two
reasons:

(i): In so far as determinism involves a notion of causation, it is a logically weak
(“thin”) notion. For example: the determining cause is the entire previous state of
the system, not some logically weaker, because localized, fact or event; (so the de-

4



termination claim is weaker). Also, determinism does not need, or even support, a
necessitarian view of laws (not least because determinism is a feature of a theory, and
so only needs a theory-relative notion of law); so that the notion of causation involved
in determinism will be non-necessitarian. So various philosophers of causation, keen on
facts or events as causal relata or on causal necessity, will find this notion of causation
unsatisfactory. Besides, as Norton notes: according to orthodox quantum theory, even
this weak notion fails.

(ii): Furthermore, some proposed weakenings of determinism, such as Reichenbach’s
Principle of the Common Cause, apparently fail in quantum theory.

But fortunately, there remains plenty of useful work to do. In the context of Nor-
ton’s anti-fundamentalism, we can usefully distinguish four kinds of work: this paper
will be an example of the fourth.

(1): Other sciences:— First, even if physics has thus lowered its sights as regards
causation, the other sciences might nevertheless use a concept of causation that is both
reasonably strong (“thick”) and in common among them, or many of them. (I think
Norton could and would agree with this.) Such a situation would again be embarrassing
for a philosopher of physics, in so far as physics, and its philosophy, apparently has
little to contribute to pinning down that concept, and fixing its scope and limits. But
in any case, doing so is of course the aim of many modern philosophers of causation:
more power to their elbow.

But there is also plenty to do within the philosophy of physics. Here I see three
kinds of work.

(2): Causation in classical physics:— First of all, one can deny that causation
within classical physics, or even some small fragment of it such as Newtonian gravi-
tational theory, boils down to determinism (as the quotation from Norton suggests).
This denial is plausible, even if one lacks a fully-fledged theory of causation such as
the process theory mentioned above. One can simply consider how Newton’s law of
universal gravitation, F = G m1m2

R2
, says that the gravitational force F between two

masses m1 and m2 depends on the distance R between them. Though the law is
strictly speaking a mathematical statement of instantaneous functional dependence,
not a causal statement, it has usually been taken to state action-at-a-distance—and
this has since Newton’s time been interpreted by many, I daresay most, physicists and
philosophers to be a matter of causation.

My own view is that this usual construal is right: but I agree that to defend it
would ultimately require a theory of causation—a large project which I will duck out
of! For this paper, I shall simply assume that instantaneous functional dependence can
amount to instantaneous causation. That is, turning to the relativistic context which
is our concern: I will allow that functional dependence between values associated with
spacelike-related points or regions can amount to a violation of relativistic causality—in
senses made more precise in Section 3.

I say ‘can amount to’ since we must of course allow that such functional dependence
in some cases merely reflects the joint effects of a common cause, e.g. the values of an

5



electromagnetic field at an instant on a sphere around a radiating source. But again,
I shall duck out of trying to give a general criterion for when spacelike functional
dependence amounts to spacelike causation, or violation of relativistic causality, rather
than merely reflecting the joint effects of a common cause. For it will be clear that my
two examples involve the former, not merely the latter.3

(3): Denying the embarrassment:— Also, one can reasonably deny Norton’s as-
sertions, in (i) and (ii) above, that quantum theory gives up determinism and the
Principle of the Common Cause. As to (i), the pilot-wave approach to quantum theory
is deterministic, and unrefuted. And as to (ii), there are versions of the Principle of
the Common Cause which are not only unrefuted by quantum theory’s EPR pairs (i.e.
violations of the Bell inequalities), but provably satisfied by some rigorous formulations
of quantum theory (Redei 2002, Redei and Summers 2002, Butterfield 2007). So for
both (i) and (ii), there are issues that are yet to be settled.

(4): Accepting the embarrassment:— Finally, there is plenty to do, even if we accept
(i) and (ii). For first, the topic of determinism is very rich, for philosophical as well
as technical issues: witness many of the writings of John Earman (e.g. Earman 1986,
2004, 2006) . (Also, Section 3 of Norton (2003) describes how determinism can fail even
in a simple Newtonian example of a ball rolling on a dome; cf. also Norton (2006a).)

Second, there is plenty to say about relativistic causality, i.e. the requirement that
causal processes or signals can propagate only within the light-cone. Most physicists
and philosophers take modern (i.e. post-Einsteinian) physics to endorse this require-
ment wholeheartedly; and I agree that most modern physics does so. But I submit
that the requirement is subtler, indeed more problematic, than usually recognized.
Not only does it have various formulations; also, some examples violate some of the
formulations.

Finally, I should clarify the relation of these examples to Norton’s causal anti-
fundamentalism. The main point is that my endeavour—scrutinizing relativistic causal-
ity, and reporting violations of it—is of course compatible with causal anti-fundamentalism.
Although Norton sees relativistic causality as small beer in comparison with the am-
bitions of causal fundamentalism, he also thereby sees no conflict between modern
physics’ endorsement of relativistic causality and his own causal anti-fundamentalism.
Indeed, he links the two by claiming that within modern physics, the venerable “prin-
ciple of causality” (roughly: that every event has a cause) boils down to exactly the
requirement of relativistic causality (2006, Section 3). And Norton aside, there is ob-
viously no conflict between scrutiny of, or even violations of, relativistic causality, and

3The remarks in (2) apply equally to the Appendix’s two examples, the pilot-wave theory and
the Newton-Wigner representation. Thus for the pilot-wave theory: in the non-relativistic version,
the guidance equation is strictly speaking a mathematical statement of how the velocity of a particle
depends on the position of distant particles. But it is usually, and I think rightly, taken to indicate
action-at-a-distance—and not joint effects of a common cause. A similar remark applies to the quan-
tum potential; and there are similar equations, again indicative of action-at-a-distance, in relativistic
versions. For more discussion, cf. the Appendix.

6



causal anti-fundamentalism.4

On the other hand, what follows is opposed to the drift of Norton’s discussions, in
two ways. First: my endeavour does not positively support causal anti-fundamentalism.
Optimists who hope that physics might provide a “thick” notion of causation can look
to extract such a notion from how a precise formulation of relativistic causality treats
the notion of process or signal. (Below, we shall see some possible places to look.)
Second and more important: Norton’s discussions give the impression that relativistic
causality is both straightforward to understand, and endorsed by all post-Einsteinian
physics. And of course my main message is that this impression is false.

3 Formulating relativistic causality

There are three general reasons why formulating relativistic causality is difficult. And
even with precise formulations in hand, it is difficult to relate the theory of Section
5’s effects, viz. QED, to such formulations. I will first briefly list these difficulties
(Section 3.1). Then I will discuss some precise formulations (Section 3.2), and raise
the question of how to avoid causal loops (Section 3.3). But I must admit at the outset
that throughout we will see questions which I will not address.

3.1 Difficulties

The first reason for difficulty is philosophical. It is not just that there is always a
gap, and so room for contention, between a formalism and its physical interpretation.
There is also the more specific trouble (discussed in Section 2, especially (2)) that
causality, or more commonly in philosophy ‘causation’, is a much-contested concept. In
particular: when does functional dependence between values associated with spacelike-
related points or regions amount to spacelike causation, i.e. a violation of relativistic
causality, rather than, for example, joint effects of a common cause?

Such questions naturally prompt one to turn to wave physics: to fix on the idea of
the propagation of a wave (or more generally: a disturbance of a field), and to take
relativistic causality to prohibit the propagation having a superluminal velocity. But
again, there are good questions: for example, about the relation to the idea of a signal
(e.g. a signal might be required to be in some sense controllable, while a disturbance
in general need not be). Weinstein (2006) is a fine recent review of such questions. But
most relevant to us (cf. Section 4.3.3.2 et seq.) will be the fact that there are various
definitions of the velocity of a wave.

The third reason for difficulty is specific to quantum theory. For it introduces

4Given just his suspicion of causality, Norton could even rejoice at my examples violating relativistic
causality: he could see them as providing another nail in the coffin of (the modern representative of)
the principle of causality. But as noted in the next paragraph, Norton joins most other philosophers
of physics in being gung-ho about relativistic causality.

7



notions and considerations independent of classical physics (in particular: independent
of the physics of waves), in terms of which one can consider formulating relativistic
causality. The obvious example is the commutativity of spacelike related operators.

These difficulties no doubt mean that we should expect there to be several different
precise formulations of relativistic causality; and that we should not expect their mutual
logical relations to be clear, or easily ascertained. Indeed, we shall see this in Section
3.2: which will give three formulations drawn from algebraic quantum field theory.

But for us, there is a further problem: even with Section 3.2’s precise formulations
in hand, there are difficulties about relating the theory used in Section 5, QED, to them.
Indeed, there are two sorts of difficulty here. First, there are important open questions
about formulating QED on Minkowski spacetime in a rigorous framework such as
algebraic quantum field theory (‘AQFT’): which is what we need for the Scharnhorst
effect. Second, there are further difficulties about formulating (i) quantum field theory,
in particular QED, and (ii) relativistic causality itself, in a curved spacetime: which
is what we need for the Drummond-Hathrell effect. Here I shall say a bit more about
the first sort of difficulty, postponing the second to Section 3.2.

So far as I know, a formulation of QED in the language of AQFT has not yet
been achieved.5 Of course, there has been impressive work. One major example is
Steinmann (2000): this book rigorously derives the perturbative formulation of QED,
and its scattering formalism, from axioms cast in the language of AQFT (2000, Parts
II and III). But as Steinmann stresses, there are plenty of open questions. And it
is not just a matter of important topics which are so far unexplored, or threaten to
be problematic, within his rigorous framework; (among the examples he mentions are
gauge invariance and path integrals). There is the deeper problem that no one is sure
that a rigorous QED exists. Here it is worth quoting a passage from his Introduction’s
summary of the book:

... Next, a description of what we understand under the name of QED is
arrived at in a partly heuristic way. The resulting definition is not as precise
as one might wish. This reflects the present state of knowledge. Not only is
it not known whether a rigorous theory deserving to be called QED exists,
it is not even exactly known what “deserving to be called QED” means. For
all we know, there may exist no rigorous QED, or a uniquely defined one, or
several distinct versions having equal claim to authenticity... Since it is not
known how to to fashion a coherent, exact, theory ... we set ourselves the
humbler task of carrying out this program in perturbation theory, which is
the most intensively studied and best understood approximation scheme in
QED. ... (Steinmann, 2000, p. 3)

But so much by way of listing difficulties. I turn to

5In this paragraph, I am indebted to Klaas Landsman and Fred Muller for correspondence and
references.

8



3.2 Some precise formulations

I shall begin with relativistic causality in classical physics, and then consider quantum
theories in Minkowski spacetime (Section 3.2.1). Then I consider curved spacetimes
(Section 3.2.2).

3.2.1 Minkowski spacetime

Most presentations of classical relativity theory assert that no “signal” or “information”
can propagate outside the light cone: there are to be no tachyons. There are various
more precise formulations of this prohibition, and justifications for it, in the physics
and philosophy literature; and comparisons of it with other “locality” principles: for
philosophical surveys cf. Earman (1987) and Weinstein (2006, Sections 1-4). But for
our purposes, we need only recall one way of making it precise: viz. by invoking
the hyperbolic character of the partial differential equations of a classical field theory.
The pre-eminent example is Maxwell’s equations for electromagnetism on Minkowski
spacetime.

For such equations, the state on a spacelike patch Σ determines the state on the
future domain of dependence D+(Σ) consisting of the spacetime points p such that
every past-inextendible causal (i.e. timelike or null) curve through p intersects Σ. This
determination indicates that the state of any field at p ∈ D+(Σ) cannot be influenced
by events so far away that an influence from them to p would have to be superluminal.
We do not need a precise statement of this determination claim, let alone its proof.
Here it suffices to say that:

(i) for an introduction to the initial value problem for classical fields obeying hy-
perbolic equations, cf. e.g. Wald (1984, Section 10.1); (Wald also defines ‘spacelike
patch’: p. 200);

(ii) the proof of this determination claim uses the theory of characteristics for the
equations concerned: which we will touch on at the end of Section 4.3.3.2;

(iii) note for future reference that one similarly defines (a) the past domain of de-
pendence D−(Σ); and (b) the domain of dependence as union of the two D(Σ) :=
D+(Σ) ∪ D−(Σ); and (c) the future etc. domains of dependence of an open region O
of Minkowski spacetime (rather than a spacelike patch).

Turning to relativistic quantum theories, the situation is more complicated. For the
interpretative problems of any quantum theory (relativistic or not, curved spacetime
or not) about non-locality and the measurement problem are widely regarded as se-
vere, and as threatening relativistic causality. In short: non-locality looks like “spooky
action-at-a-distance”; and if measurement involves a “collapse of the wave-packet”,
perhaps the collapse is superluminal. Besides, relativistic quantum theories raise fur-
ther issues, reflecting the embarrassment, well-known in foundations of physics, that
we do not have a developed theory of measurement for these theories. For example,
a recent line of work argues that in such a theory only a restricted class of quantities
can be ideally measured, on pain of superluminal signaling (Sorkin (1993), Beckman

9



et al. (2001); cf. Weinstein (2006, Section 6)). But suppose we set aside these in-
terpretative problems, say by endorsing a minimal instrumentalist interpretation, to
the effect that a relativistic quantum theory prescribes probabilities for appropriate
measurement results.

Given this supposition, most presentations of relativistic quantum theories in Minkowski
spacetime agree that the theories incorporate relativistic causality. But still, the sit-
uation is subtle. The idea has several precise formulations; and in various rigorous
formalisms, e.g. AQFT, these formulations are precise enough that one can prove e.g.
that one is logically independent of another.

As I see matters (and anyway: for the purposes of this paper), the three main
formulations are as follows: (the first is the analogue of the classical formulation above).

(i): Primitive causality; hyperbolicity: In a heuristic quantum field theory, using
the Heisenberg picture, operators indexed by spacetime points are subject to Heisen-
berg equations of motion, while the state is fixed once for all. But these equations are
hyperbolic, on analogy with classical field theories using hyperbolic dynamical equa-
tions; this means one can show, at least unrigorously, that for any state, all expectation
values are determined subluminally, in that the state’s restriction to the field opera-
tors in a region O determines all its expectation values for operators in the future
domain of dependence D+(Σ). In AQFT, this idea is made precise as the Diamond
Axiom. AQFT associates to each open bounded region O of Minkowski spacetime,
an algebra of operators A(O). We are to think of the Hermitian elements of A(O)
as the observables for that part of the field system lying in O, and so as measurable
by a procedure within O. We take states as linear expectation functionals ω on the
algebras: ω : A ∈ A(O) 7→ ω(A) ∈ |C. (We can recover Hilbert space representations
from this abstract setting, primarily by the GNS construction.) Then the Diamond
Axiom says that A(D(O)) = A(O). So the idea is: if O1 ⊂ D+(O), O1 ∩ O = ∅, i.e.
O1 lies in the top half of the “diamond” D

+(O), and A ∈ A(O1), so that we could
measure A by a procedure confined to O1, then we could also instead measure A by
a procedure confined to O. For thanks to the hyperbolic time-evolution, “the facts in
O1” are already determined by “the facts in O”.

(ii): Spacelike commutativity (also called micro-causality): Operators associ-
ated with spacelike-related regions commute. In heuristic quantum field theory, treat-
ing fermions requires one to also allow anti-commutation; but in AQFT, one dis-
tinguishes field algebras and observable algebras, and for the latter imposes only
spacelike commutativity. Thus one requires: if O1, O2 are spacelike, then for all
A1 ∈ A(O1), A2 ∈ A(O2) : [A1, A2] = 0. The physical idea is of course that such
spacelike observables should be co-measurable, and so should commute. (Think of how
in elementary quantum theory, one proves the no-signalling theorem, viz. that a non-
selective Lüders rule measurement of A cannot affect the measurement probabilities of
B, provided [A, B] = 0.)

(iii): Spectrum: The field system’s energy-momentum operator has a spectrum
(roughly: set of eigenvalues) confined to the future light-cone. Of the three condi-

10



tions (i)-(iii), this is perhaps the most direct expression of the prohibition of spacelike
processes.

So we have three different senses of relativistic causality. And indeed, in most
relativistic quantum theories, and models constructed within them, these three senses
(and even others) hold good. Besides, there are various arguments connecting these
sorts of formulation. For example, one can argue for (ii), spacelike commutativity, from
an analogue of (i) which one might call a “signal principle”, viz. that “turning on”
any unitary evolution U of the field system within region O1 should leave unaffected
the measurement probabilities of any observable A associated with a spacelike region
O2; along the following lines. If O1, O2 are spacelike, then this signal principle requires
that for all unitary U ∈ A(O1), all A ∈ A(O2), and all states ω, ω(U AU∗) = ω(A). So
U AU∗ = A, i.e. [U, A] = 0. Since the unitary operators span the algebras concerned,
this means that the algebras commute, i.e. for all A1 ∈ A(O1), A2 ∈ A(O2), we have
[A1, A2] = 0.

6

On the other hand, there are no corresponding rigorous implications between a pair
drawn from (i) to (iii). That is: truly precise statements of (i) to (iii), in the language
of AQFT, are logically independent of each other. Indeed, they are independent even
in the presence of AQFT’s other axioms. And more is true: Horuzhy (1990, pp. 19-
21) reports that all six of his basic axioms of AQFT (three of which are his versions
of (i)-(iii)) are independent: for each subset comprising five axioms, there is a model
satisfying all five—but not the remaining sixth axiom. And although some of these
models are very unphysical, others are not: again we see that relativistic causality—
formulate it as you like!—is a subtle matter in relativistic quantum theories.

3.2.2 Curved spacetime

I turn to curved spacetimes. The first thing to say is that in the light of our not having
a rigorous formulation of QED on Minkowski spacetime (cf. the end of Section 3.1),
a fortiori we lack such a formulation on curved spacetimes. But on the other hand,
a great deal is now understood about formulating heuristic quantum field theory on
such spacetimes. I shall sketch the general situation, emphasising the case of a non-
interacting field (nowadays well understood), and then reporting some recent progress
on the (much harder) interacting case. In both cases, I shall emphasise ideas we need
for our main question, to which I will turn at the end of the Section: how to adapt our
three precise conditions, (i) to (iii), to curved spacetimes.

Broadly speaking, by the mid 1990s quantum field theory on curved spacetime
could be formulated in as satisfactory a manner as heuristic quantum field theory on
Minkowski spacetime, subject to three conditions. (This summary is based on Wald
(1994).) These conditions are:

(a): the curved spacetime is fixed, i.e. there is no back-reaction of the field on the

6I learnt this argument, apparently common in the folklore of AQFT, from K. Fredenhagen and S.
Summers, in conversation: to whom my thanks.

11



spacetime geometry; (though the curvature can be non-constant);
(b): the field is linear (i.e. not self-interacting);
(c): the spacetime is such that the corresponding classical field theory has a well-

posed initial value problem.

For our purposes, condition (c) prompts two comments. First, ‘well-posed initial
value problem’ refers to the determination of solutions by initial data, as in the classical
discussion at the start of Section 3.2.1. And the most common way to satisfy condition
(c) is to restrict attention to globally hyperbolic spacetimes. These are spacetimes
with a Cauchy surface, i.e. a spacelike slice Σ whose domain of dependence D(Σ) is
the whole spacetime. In fact, global hyperbolicity is a strong condition of causal “good
behaviour”; it implies that spacetime is foliated by Cauchy surfaces, and implies several
other causality conditions, including stable causality—which we will return to in Sec-
tion 5. The reason for this restriction is that the main theorems securing a well-posed
initial value problem for hyperbolic classical equations assume global hyperbolicity
(Wald 1984, Theorems 10.1.2-3, p. 250.)7

Second, we need to explain ‘the corresponding classical theory’. This phrase indi-
cates a standard construction of a heuristic quantum field theory (with e.g. a Fock
space of states built up from a vacuum) from the solution space of a classical theory.
(The construction for Minkowski spacetime is in Wald (1994, Sections 3.1-3.2); the
adaptation to curved spacetimes is in his Section 4.2.) When this construction is given
for Minkowski spacetime, but in a form that is suitable for generalization to curved
spacetimes, one sees that:

(i): it involves the choice of a Hilbert space (essentially, of a set of complex solu-
tions of the classical theory), with different Hilbert spaces giving unitarily inequivalent
theories; (the Stone-von Neumann theorem asserting unitary equivalence of represen-
tations of commutation relations depends on the system being finite-dimensional i.e.
not a field); and

(ii): this freedom of choice is characterized by a choice of a bilinear map (inducing
an inner product, and so a complex structure); but that

(iii): the Poincaré symmetry of Minkowski spacetime gives a preferred bilinear map
and so Hilbert space: equivalently, a preferred vacuum with the Hilbert space being
the Fock space built up from that vacuum; (Wald 1994, pp. 27-29, 39-42).

In a curved spacetime, property (iii) fails, and so one must either: (a): seek some
other criterion for choosing the bilinear map (or at least a set of them corresponding to
a unitary equivalence class of representations); or (b): treat the different choices (and
so unitarily inequivalent representations) on a par.

In some cases, tactic (a) is sensible. For example, for stationary spacetimes, there is
a natural unique choice of map; and for a spacetime with a compact Cauchy surface, a
condition of physical reasonableness for the state (the Hadamard condition, discussed
below) constrains the bilinear maps so as to fix a unitary equivalence class. But in

7For an approach to quantum field theory on non-globally hyperbolic spacetimes, cf. Kay (1992,
especially Section 6).

12



general, there is no such choice and one must opt for (b) (Wald 1994, pp. 58-60).

This suggests that we should adopt the framework of algebraic quantum theory,
which (as mentioned in (i) of Section 3.2.1) takes abstract algebras of observables as
the primary notion, with states being linear functionals on them. And indeed, adapting
the standard construction mentioned above to the algebraic approach, we find that the
(Weyl) algebra of observables that naturally arises is independent of the choice of
bilinear map (Wald 1994, p. 74f.). Furthermore, this approach is at first sight very
promising for our own topic of precise formulations of relativistic causality, since this
approach again associates abstract algebras of observables with open bounded regions
of any globally hyperbolic spacetime (Wald, 1994 p. 84). Thus we naturally hope
to carry over directly to such spacetimes the three Minkowski formulations of Section
3.2.1.

Indeed, there is no problem about the first two conditions, primitive causality and
spacelike commutativity. The global hyperbolicity assumption prevents any “funny
business” in the causal structure, such as closed causal curves, so that these conditions
can be carried over word for word: ‘domain of dependence’, ‘spacelike’ etc. now just
refer to the curved spacetime’s structure.

Besides, though I have so far confined my summary to the easier, and better under-
stood, case of non-interacting fields, the same considerations apply to the interacting
case. As we shall see shortly, recent formulations of interacting quantum field theory
on curved spacetimes use the algebraic approach, and again there is nothing to prevent
carrying over these two conditions intact.

But there is a problem about the third condition, the spectrum condition: though
it is a problem that has recently been largely solved. In effect, the problem was that
no one knew how to define the spectrum condition’s topic, i.e. the energy-momentum
operator, in a curved spacetime: all one knew was how to define a class of physically
reasonable states that gave a well-defined expectation value. But in recent years, the
problem has been solved by exploiting a mathematical theory, microlocal analysis.
Though the details are not needed for later discussion, I should summarize them since
the problem is much more general, and thus its solution much more impressive, than
my mentioning just one observable can suggest. Indeed, as I understand matters,
the solution secures a perturbative formulation of interacting heuristic quantum field
theory on a globally hyperbolic spacetime that is about as precise as those we have for
Minkowski spacetime. So, even apart from Section 5’s two effects, this achievement is
worth reporting.8

The problem of definability arises from the fact that the energy and momentum of
the field are encoded in the stress-energy tensor T̂ , which involves the square of the
quantum field φ̂. But φ̂ is a distribution, and the product of distributions at a single

8I am very grateful to Bob Wald for teaching me what follows. One caveat: to date, the framework
is secured for scalar fields only. Wald tells me he is very sure it can be extended to Dirac fields, and
pretty sure for QED: a happy prospect, also for this paper’s topic, since it is what we need for fully
understanding Section 5’s effects.

13



spacetime point is mathematically undefined; so that some prescription is needed in
order that T̂ make sense. Until about 2000, it was not known how to do this “di-
rectly”, i.e. by enlarging the algebra of observables to include some suitably smeared
version of T̂ ; so one aimed only to characterize a class of physically reasonable states
ω for which the expectation value < T̂ >ω was well-defined. This was work enough
since, in particular, the standard prescription for Minkowski spacetime (normal order-
ing, which corresponds to subtracting off the infinite sum of the zero point energies of
the oscillators comprising the field) depends on a preferred vacuum—which is generally
unavailable in a globally hyperbolic spacetime. In fact, there is a compelling characteri-
zation of such states. Since it builds on Hadamard’s work on distributional solutions to
hyperbolic equations, they are called ‘Hadamard states’. A bit more precisely: one re-
quires the short-distance singularity structure of the two-point function < φ̂(x)φ̂(x′) >
to be “as close as possible” to the corresponding structure of the two-point function
of the Minkowski vacuum (Wald 1994, p. 94). This definition turns out to be very
successful, as shown by both existence and uniqueness results. That is: (i) any globally
hyperbolic spacetime has a large class of Hadamard states, and (ii) Hadamard states
satisfy some natural axioms, and do so uniquely up to a local curvature term; (Wald,
1994: pp. 89-95 for (ii), and pp. 95-97 for (i)).

But in recent years, various authors have exploited microlocal analysis so as to
achieve the original goal (“direct” in the preceding paragraph). Indeed, they have
defined, not just the energy-momentum, and stress-energy operators; but also the other
products of field operators and their derivatives, and polynomials of such products, and
time-ordered products, that are crucial in order to formulate the perturbation theory
of an interacting quantum field theory. I will just gesture at what is involved, with
an eye on our interest in the spectrum condition. For more detail, cf. Hollands and
Wald (2001, 2002), who build on previous work, especially by Brunetti, Fredenhagen
and collaborators.

The fundamental idea is to use microlocal analysis’ definition of the set of directions
(at each point of the spacetime) in which a distribution is singular: it is called the ‘wave
front set’ of the distribution. Using this and related ideas, one can define the above
operators so as to satisfy appropriate properties, in particular locality and covariance.
To give the flavour, we need only note the following characterization of Hadamard
states in any globally hyperbolic spacetime (and thereby of the spectrum condition in
Minkowski spacetime): the two point function of a Hadamard state has a wave front
set consisting of pairs comprising: (i) at any given spacetime point x, a future-pointing
null vector k; and (ii) at any other point x′ on the null geodesic through x generated
by parallel-transporting k, the corresponding tangent vector, i.e. the parallel-transport
of k from x to x′. (This is, modulo some technicalities, Radzikowski 1996, Theorem
5.1.) Hollands and Wald build on this sort of characterization so as to require of
their local and covariant operators, a generalized microlocal spectrum condition, which
is a microlocal analogue of the translation invariance of Minkowski spacetime (2001,
Definitions 3.3, 4.1). Thus the spectrum condition, (iii) of Section 3.2.1, is carried over
to curved spacetimes.

14



To sum up Section 3.2:— In Section 3.2.1, we saw three precise formulations of
relativistic causality for quantum theories on Minkowski spacetime, which prescind
from interpretative controversies about such notions as causation, or signal, or quan-
tum measurement, and which are logically independent. In Section 3.2.2, these three
formulations were adapted to globally hyperbolic spacetimes. But for both kinds of
spacetime, we do not yet have a rigorous formulation of perturbative interacting QED,
in which to study the fate of these conditions (and so better understand Section 5’s
effects). But there are good prospects for soon getting such formulations.

3.3 Causal loops?

Given an apparent violation of relativistic causality, there are various questions to ask.
First: which of the various precise formulations of relativistic causality is false? (Of
course, if there are logical connections between them, some may fall as a consequence
of others, by modus tollens.) Section 3.2 showed that even when one sets aside inter-
pretative problems (as in Section 3.1), this is a hard technical question for either of
Section 5’s effects: hardly a question for philosophers, and so a question I must leave
to the reader!

But other questions are more philosophical. An obvious one is: how does the
superluminal propagation avoid leading to outright contradictions via causal loops?
Although Section 5 will not pursue this question, the general strategy which the inves-
tigators of the effects have used to address it, is noteworthy—and I can already explain
it. Thus the investigators of each example are of course aware of the threat that:

(i): superluminal propagation would imply that causal processes could go in a loop
(technically: closed timelike or null curves); and that:

(ii): such loops would yield, by a “bilking argument”, contradictions.
Given this awareness, it is no surprise that they argue that the threat can be avoided:
if it could not be, the example would be hopeless. But their arguments are worth
noting.

For they do not adopt the standard philosophical tactic for avoiding the threat.
That tactic addresses only (ii). Namely, by saying that a “causal loop” merely implies
a severe consistency condition on the state of the system at an initial time, namely that
it must evolve around the loop back to itself. Thus, taking the standard philosophical
example: a contradiction seems to threaten if one can travel back in time and kill
one’s grandparents before one’s parents were conceived. The standard philosophical
reply is that indeed one could not kill them; but that means, not that time-travel is
impossible, but that it is severely constrained: any initial state must evolve back to
itself; (e.g. Putnam 1962, pp. 243-247; and Lewis 1976, pp. 75-79: Earman (1995,
pp. 170-188), Berkovitz (2002) and Arntzenius and Maudlin (2005, Section 8) are
sophisticated discussions of this reply).

Instead, the defences of these examples address (i). That is: they argue that the
superluminal propagation which the example countenances could not be exploited to

15



make a spacelike zig-zag, “there and back”, into the causal past of an initial event:
a refreshing change from the philosophical literature’s repeatedly addressing only the
second aspect, (ii), by invoking the idea of a consistency constraint.

Another obviously philosophical question is: given superluminal propagation, how
should we think of the light-cone—what is its significance, now that it is not the locus
of light rays, or in a quantum context, of photon propagation? This leads to the
more general question, which I address in the next Section: how do the mathematical
representatives of metric structure earn their physical significance?

4 Why is this tensor “read” by rods and clocks?

Brown’s moral

This general question deserves a Section of its own, for two reasons. First, what I
take to be the right answer—which I learnt from Brown (2005)—is controversial. So
I shall spell out what I shall call ‘Brown’s moral’, where ‘moral’ connotes both its
being controversial, and my endorsing it. Second, we need some details of this moral
as preparation for Section 5’s effects; indeed, Brown takes his moral to be illustrated
by them (2005, Chapter 9.4.1, 9.5.2).9

Brown’s moral is a general doctrine in the philosophy of (chrono)-geometry, though
he develops it mostly for special relativity, and briefly for general relativity. I will first
state the moral in general terms (Section 4.1), then report his treatment of it in special
relativity (Section 4.2), and then turn to the case we need: general relativity (Section
4.3).

4.1 The moral in general

We can think of the moral as having two aspects, “negative” and “positive”. It will
be clearer to start with the negative aspect, since the positive aspect explains it. Neg-
atively, the rough idea is that we should not simply postulate that a quantity in a
physical theory has (chrono)-geometric significance. The point here is not just that
it would be wrong to infer from a quantity’s being called a metric that it mathemat-
ically represents (what the theory predicts about) the readings of rods, and-or clocks
and-or other instruments for measuring lengths and time-intervals. That is obvious

9Though the words are mine, this Section is due to Brown. He urges this moral, and related ones,
in various passages of his (2005); cf. also his associated papers, especially Brown and Pooley (2001,
2006). Since drafting this paper, I have read Weinstein’s brief but rich (1996). This paper: (i) floats
a moral like Brown’s, for general relativity and kindred theories, and lists some adjacent issues; (ii)
illustrates the moral with several examples of non-minimal coupling, not just two as in our Section 5;
and (iii) discusses coupling in terms of action principles. By the way: though the moral is controverted
by philosophers, my impression in discussion with physicists is that they endorse it—at least the way
I say it!

16



enough: after all, a quantity might be given an undeserved, even tendentious, name.
But also: we should not infer from the fact that in the theoretical context, the quan-
tity is mathematically appropriate for representing such behaviour, that it does so.
For example, on the Gauss-Riemann conception of length as given by line-integrals of
ds =

√
(gijdx

idxj), the symmetric tensor gij is appropriate. More specifically, for rela-
tivity’s spatiotemporal lengths, gij is to have Lorentzian signature. But such a tensor
might well not represent measured lengths or times. After all, a theory might contain
two such tensors, just one of which represents such matters. (We shall see such an
example in Section 5.1.1.3.)

The reason these inferences are invalid is of course that any physical (chrono)-
geometry should take rods, clocks and other instruments as composite bodies whose
behaviour is determined by the laws governing matter. In practice, these bodies are
usually very complex and so the determination of their behaviour by laws governing
their micro-constituents will be very complex. But this is not to suggest that a term
like ‘metric tensor’ cannot be justified. We can indeed write down a idealized (“toy”)
model of a rod or clock etc. in terms of our theory of matter (nowadays a relativistic
and quantum theory), and thereby deduce that a certain quantity in our theory—in a
relativistic theory, a (0,2) symmetric tensor gij with Lorentzian signature—represents
their readings; and thereby earns the name ‘metric’.

We need three further general points about this moral.
(1): This is not to say that the quantity must itself be derived, perhaps as an

effective or phenomenological aspect of a complex instrument. Our theory can postulate
the quantity “on the ground floor” in its model of the instrument; indeed, most theories
do so. The point is just that the quantity only earns the name ‘metric’ when we “close
the circle” by exhibiting how instruments’ readings display it. (This point is developed
in Butterfield (2001, Section 2.1.2).)

(2): Nor is it to say that only by representing the readings of measuring instruments
for lengths and times (such as rods and clocks), can a quantity have chrono-geometric
significance. For in both Newtonian and relativistic theories, part of the metric ten-
sor’s significance is that massive test-particles and light-rays travel along appropriate
geodesics. (Here, I have put the point in relativistic jargon; and ‘appropriate’ covers
relativity’s timelike/null distinction.)

(3): Brown argues that this moral needs to be stressed, because in recent decades
the philosophical literature about relativity theory has tended to ignore it. He is
especially concerned with the moral in special relativity: it is a theme of his Chapters
2-8. For the sake of completeness, I will now report his discussion. But this report is
not needed for the rest of this paper’s argument: for that, one can proceed directly to
the moral in general relativity (Section 4.3).

17



4.2 The moral in special relativity

In special relativity, the tensor gij get its chrono-geometric significance principally
through the behaviour of rods and clocks, especially the length contraction and time
dilation effects.10 So as evidence for his moral being ignored, Brown describes the
current tendency to call them ‘kinematical effects’, where ‘kinematical’ is taken to
connote ‘prior to dynamics, and so not needing a dynamical explanation’. To put
the point in more philosophical terms: many philosophical commentators on special
relativity apparently conceive the Minkowski metric as encoding a property (or better:
structured family of properties) of spacetime that (a) is intrinsic to it, in the sense that
spacetime would have the property even in the absence of matter, and (b) suffices to
explain the effects.

Brown admits that this tendency has roots, both historical and conceptual. Histori-
cally, Einstein himself called the opening Sections of his 1905 paper ‘Kinematical Part’,
and he called special relativity a ‘principle’ theory as against a ‘constructive’ one, i.e.
as not concerned with any detailed mechanisms bringing about length contraction and
time dilation. And conceptually, the account of length contraction and time dilation
based on a spacetime diagram with hyperbolae of constant Minkowski interval from
a given point,11 is undeniably striking. Every student feels that the diagram greatly
clarifies algebraic derivations based on the Lorentz transformations; and that it also
makes unmysterious the reciprocity of the effects, i.e. the fact that each of two inertial
observers judges the other’s rods to be contracted and their clocks to be slowed.

But, says Brown, these roots do not justify the tendency. Einstein himself later
admitted that the kinematical part of the theory did not pre-empt the need for a
dynamical explanation of the effects, and accordingly downplayed the idea of special
relativity as a principle theory. And in order for the account based on a spacetime dia-
gram and hyperbolae to explain the effects, one needs to accept (or to have previously
explained) that the primed variables do indeed represent the readings of the moving
rods and clocks: for only via this fact can the diagram’s hyperbolae be connected to
those readings. And, according to Brown’s moral, it is of course just this fact that
needs a dynamical explanation.

So much by way of summarising Brown’s critique of a current philosophical ten-
dency. Some highlights of this critique are at: pp. 22-25, 89-92, 99-102, 129-131,
132-139, and 143-148. For example: the critique of the Minkowskian diagrammatic
‘explanation’, i.e. Brown’s demand for a dynamical explanation of the physical in-
terpretation of the primed variables, is at pp. 129-131. And on pp. 132-139, Brown
argues by analogy with the geometric structure attributed to other state spaces in
physics, viz. curved configuration manifolds in analytical mechanics, the curvature of
projective Hilbert space in quantum theory, and Carathéodory’s postulates for classical

10One might add: and through massive test-particles and light rays following gij ’s timelike and null
geodesics, respectively. But I shall set aside this aspect, for brevity: compare (2) in Section 4.1.

11Originally by Minkowski (1908, pp. 77-78, 81-82), and oft repeated since, e.g. Born (1962, pp.
247-249) Torretti (1983, p. 97).

18



thermodynamical state-space. In these and similar cases, we naturally interpret the
geometry not as causing or explaining the system’s behaviour, but as codifying it: so
why not also in spacetime theory?

More positively, Brown gives historical and technical details about the dynamical
explanations of the effects. He describes how over the decades several authors, includ-
ing Einstein: (i) have seen the need for such explanations; and (ii) have even spelt out
accurately just what such an explanation requires—viz. the Lorentz-covariance of the
laws responsible for the cohesion of matter, laws which after the 1920s were of course
recognized to be quantum-theoretic. Some highlights of this positive story are: for (i),
Einstein (pp. 113-114) and Pauli (p. 118); and for (ii), Swann (pp. 119-122) and Bell
(pp. 124-126). Brown describes how both Swann and Bell realize that:

(a): The dynamical explanation does not require one to know what the laws are,
but only that they are Lorentz-covariant. And:

(b): The dynamical explanation does not require a transformation to moving coor-
dinates. For example, a Lorentz boost is interpreted as active, mapping a given solution
describing a “stationary” rod in internal equilibrium, to another solution describing a
longitudinally contracted rod.

4.3 The moral in general relativity

4.3.1 How the gravitational field gets its metric significance

Broadly speaking, Brown’s discussion of general relativity (Chapter 9, and pp. 141-
143) confirms the moral he gathered from special relativity: that the metric tensor gij
gets its chrono-geometrical significance, not by fiat, but by detailed physical arguments.
For Section 5, we need the following details.

All agree that in general relativity, the metric tensor gij is (or better: represents a
field that is) dynamical: it acts and is acted on. They also agree that it is a special
field since it couples to every other one, and also cannot vanish anywhere in spacetime.
Many authors go on to say that the metric tensor represents geometry, or spacetime
structure, so that geometry or spacetime structure acts and is acted on. But Brown
resists this. He says that the metric tensor represents primarily the gravitational field,
‘which interacts with every other [field] and thus determines the relative motion of the
individual components we want to use as rod or clock. Because of that, it admits a
metrical interpretation’. (This quotation, on p. 160, is from Rovelli: who is one of
three distinguished interpreters of general relativity whom Brown quotes as kindred
spirits.)

Brown supports this position by reviewing some of the physics that underpins this
metrical interpretation: i.e. the physics that explains why gij is surveyed by rods and
clocks, and its null and timelike geodesics are the worldlines of light-rays and massive
non-rotating test-particles respectively. This review brings out that, with the exception
of this last case—the worldlines of test-particles—the metrical interpretation depends,

19



not only on general relativity’s field equations, but also on the strong equivalence prin-
ciple (SEP).

This dependence is important for this paper. For it is precisely by violating SEP
that Section 5’s effects will have light propagate outside the light-cones defined by gij.
So it will be worth first spelling out SEP, and seeing how the metrical interpretation
of gij uses it. The main point will be that SEP is a conjunction of two propositions;
and though both are part of “textbook general relativity”, only one of them (which I
will call Universality) is essential to general relativity—and it will be by violating the
other proposition (called Minimal Coupling) that Section 5’s effects violate SEP, and
thereby relativistic causality.

Agreed, what is essential to general relativity is partly a matter of interpretative
judgment, and partly a purely verbal matter. And unfortunately, there is no sharp
consensus about how to formulate the equivalence principle; nor about how to make
the distinction between the weak and strong principles. But I am sure that all general
relativists:

(i): would accept as reasonable the decomposition of SEP into Universality and
Minimal Coupling, which Brown articulates (pp. 169-172, citing Ehlers and Anderson);
and

(ii): would accept that only Universality is essential to general relativity; after
all, there are many articles discussing non-minimal coupling in what the article calls
‘general relativity’.

4.3.2 The equivalence principles, weak and strong

So let us start with the weak equivalence principle. Though I will not need to formu-
late it exactly,12 the basic idea is that local mechanical experiments cannot distinguish
gravity and inertia. A bit more precisely: they cannot distinguish a homogeneous grav-
itational field from the inertial effects of uniform acceleration. Nor can they distinguish
free fall, i.e. motion under gravity but under no other forces, from motion subject to
no force at all. So this is the idea of “Einstein’s elevator”: (which Einstein called “the
happiest thought of my life”). This means in particular that test-particles of different
masses move in the same way under gravity alone—i.e. move identically, given identi-
cal initial conditions, and if subject to no other force. (“Galileo’s law”: two different
masses dropped simultaneously from the Tower of Pisa fall in identical ways.) Hence
the idea of treating gravity as spacetime curvature, in the sense of taking freely-falling
test-particles to travel along geodesics of a curved connection.

On the other hand, the strong equivalence principle, SEP—our main concern—is
about how the various non-gravitational forces relate to gravity thus treated; and in
particular, how they relate to the connection induced by the metric tensor. Again, the
formulation varies from one author to another. But I will simply follow Brown (and so

12For Brown’s discussion, cf. pp 25-26 and 161-163. Cf. also Norton (1985) and Ghins and Budden
(2001).

20



Ehlers and Anderson), with slight modifications. The main point will be that SEP is
the conjunction of two propositions.

The first proposition, Universality, is that the physics of each of the non-gravitational
forces picks out the same affine connection. More precisely, we envisage that the theory
of any such force adopts the following framework:

(i): The theory is generally covariant. This means, roughly speaking, that it is pre-
sented in coordinate-independent differential equations for appropriate scalars, vectors
and tensors representing fields.

(ii): The theory invokes an affine connection on spacetime so as to have an appro-
priate coordinate-independent notion of differentiation on fields.
Given this framework, Universality then says that all the theories of the non-gravitational
forces are to invoke the same connection, ∇ say.

This assertion is similar to the basic idea above of the weak equivalence principle,
for the following reason. Suppose that each such theory asserts that a test-particle
that is free—i.e. a particle that is “small” enough not to affect what is influencing
it, and is subject to zero force of the kind in question—travels along a geodesic of
the common connection ∇. Indeed, in the light of the four-dimensional formulation
of Newtonian mechanics and special relativity, that is a natural assertion.13 Then
Universality makes it very natural to assert that in a theory of some or all of these
non-gravitational forces, a test-particle subject to none of these forces—a test-particle
that falls freely, i.e. subject only to gravity—should also travel along a geodesic of the
common connection. After all: if the particle did not do so, this would mean that the
absence of several forces yielded, in a combined theory of the forces, a motion different
from the common prescription of the ingredient theories. So “agreed votes” from the
ingredient theories would “cancel one another out” within the combined theory: which
would be distinctly odd.

The second proposition, Minimal Coupling, is in effect a bold generalization of
this last assertion, that in a theory of some or all of the non-gravitational forces, a
freely-falling test-particle travels along a geodesic of the common connection. This
generalization occurs in three ways:

(i): Minimal Coupling concerns all matter and fields, not just test-particles;
(ii): For Minimal Coupling, the matter and fields can be interacting, i.e. subject

to some or all the non-gravitational forces
(iii): Minimal Coupling makes a specific prescription about what laws govern the

matter and fields in the setting of a curved connection representing gravity. Namely:
the laws of the corresponding special relativistic theory are to be valid locally.

It is the third point, (iii), that is the heart of Minimal Coupling. To state it

13In both these theories, there is a distinction between timelike and spacelike curves, and so the
theory asserts the particle to travel along a timelike geodesic. The main difference is that in special
relativity, the connection is uniquely determined by (the requirement of compatibility with) the met-
ric. We need not consider now the issue whether this assertion needs a dynamical or ‘constructive’
explanation of the kind Brown favours. But I will discuss this issue in Section 4.3.3.1.

21



more precisely: differential geometry teaches us that the partial derivatives in the
differential equations of a special relativistic theory implicitly represent the standard
flat connection of IR4; and Minimal Coupling says that the general relativistic laws are
given simply by replacing these partial derivatives by the curved connection’s covariant
derivatives.14

Broadly speaking, this prescription means that the transition to the general rela-
tivistic laws is as simple as it could be, while reducing to the special relativistic laws in
the case of a flat connection. For a curved connection means that covariant derivatives
add “correction terms” to special relativity’s familiar partial derivatives. But Minimal
Coupling says that the general relativistic equations do not add anything else. In par-
ticular, they do not include terms proportional to any kind of curvature (whether the
scalar curvature, or one of the various curvature tensors): which ceteris paribus they
might well do, since any such terms would be zero in the setting of special relativity’s
flat connection, and so would not be refuted by the empirical success of the special
relativistic theory.

So Minimal Coupling represents a proposal for simplicity. And evidently, it is
fallible: nothing in the framework of general relativity forbids a matter field from
coupling to spacetime curvature, and so requiring a curvature term in the differential
equations that govern it. And as announced, we will see examples that violate it, in
Section 5.15

4.3.3 Motion along geodesics

So much by way of clarifying that the violation of SEP will involve violating Minimal
Coupling. Returning to Brown’s overall moral, we need to extract just two points from
his review of the physics that underpins this metrical significance of the tensor gij.
These concern: (a) motion along timelike geodesics and (b) light propagation along
null surfaces.

4.3.3.1 Timelike geodesics There is one aspect of the metrical significance of gij
that is well known to be independent of SEP: viz. the motion of massive test-particles.
(But again, this aspect will illustrate Brown’s moral.) Thus Einstein and other gen-
eral relativists initially took it as a postulate that freely-falling massive non-rotating
test-particles followed timelike geodesics of (the curved connection determined by) the
metric tensor. Brown remarks that this is the analogue for a theory with “geometrized
gravity”, of the interpretation of Newton’s first Law, for Newtonian mechanics and
special relativity, that he rejects: viz. the interpretation that these theories’ timelike

14Partial derivatives are usually represented by a comma, and covariant derivatives by a semi-colon.
So the semi-colon abbreviates the correction terms, and Minimal Coupling is sometimes called the
‘comma-to-semi-colon’ rule.

15Aficionados know several such: thanks to Steve Adler for mentioning the conformal massless
Klein-Gordon field.

22



geodesics ‘form ruts or grooves in spacetime which somehow guide the free particles
along their way’; (2005, p. 24; here, ‘free’ of course also excludes gravity; cf. also
pp. 139-143 for references to advocates of this interpretation). But from about 1918
onwards, a succession of theorems by Eddington, Einstein and others made it clear
that this postulate is unnecessary: a massive test-particle must follow such a geodesic,
because of the conservation of energy-momentum (more precisely: the vanishing of the
covariant divergence of the body’s stress-energy tensor).16

So for Brown, these theorems are like the dynamical explanations of length con-
traction and time dilation in special relativity that he favours. All are genuine physical
“constructive” explanations of the chrono-geometric significance of the metric tensor—
as against the pseudo-explanations that just make a postulate, for example that time-
like geodesics “form ruts” for test-particles. Brown also points out that these theorems
are limited, but in a way that supports his interpretation. Namely, extended free-
falling bodies will in general experience tidal gravitational forces, and so will not follow
geodesics: underlining the point that it is not “in the nature” of freely-falling bodies
to follow the alleged ruts.

4.3.3.2 Null surfaces and geodesics The propagation of light, or more gener-
ally electromagnetic radiation, along null surfaces provides an interesting comparison
with massive test-particles, Section 4.3.3.1 above. The situation is similar in that it
again illustrates Brown’s moral. Thus one can prove from Maxwell’s equations for
electromagnetism on a general relativistic spacetime that electromagnetic radiation
will propagate along null surfaces. So one does not need to postulate that these sur-
faces “form ruts” for light to follow: there are theorems that it must do so. But the
situation is also dissimilar from Section 4.3.3.1, in that the theorems invoke SEP: for
it dictates the form that Maxwell’s equations take in general relativity.

We can see both the similarity and the dissimilarity in the simplest of this kind of
theorem: viz. where we simplify the description of the light wave, by taking the limit
of short wavelengths. In this limit, light is described as consisting of rays, with each
ray being characterized by a curve in spacetime; (and at each point along the curve,
an intensity of the light, corresponding to the amplitude of the wave—but we need not
consider intensities). This is called the ‘geometric optics limit’, since geometric optics
(as vs. wave optics) describes light as composed of such rays.

When we take this limit, the direction of the ray is given by the wave-vector (i.e.
covector, 1-form) k which is the gradient of the phase: the tangent vector to the ray
is the corresponding contravariant vector ki ≡ gijkj. It is straghtforward to show from
Maxwell’s equations, as dictated by the SEP, that:

(i): k is a null vector, i.e. k2 = 0; and

16For details, cf. e.g. Misner Thorne and Wheeler (1973, pp. 471-480), Wald (1984, p. 73), and
Geroch and Jang (1975). Brown notes various subtleties about these theorems. In particular, although
the form of the field equations determines gij to be a (0,2) symmetric tensor, nothing in the equations
dictates that gij should have Lorentzian signature.

23



(ii): the ray is a geodesic, i.e. the ray parallel-transports its own tangent vector:
ki∇ikj = 0; (Misner, Thorne and Wheeler 1973, pp. 568-583).

For the purposes of Section 5, I need to take note of how these theorems can also
be generalized; i.e. they hold good away from the geometric optics limit. The first
point is that the propagation of waves is a complex subject, and textbooks of optics,
or more generally wave physics, sport several inequivalent notions of the velocity of
a wave. For example, one (to which we will return) is the phase velocity: writing
the wave-vector k = (ω, k) as usual, vph =

ω
|k|. But for our concerns with causality,

it seems to be agreed that the relevant notion is wavefront velocity (also known as:
‘signal velocity’); which is essentially the velocity of the boundary between the regions
of zero and non-zero excitation of the field concerned. Mathematically, the wavefront
is given by the characteristics of the wave equation describing the field. So the gist
of the general theorems is that the characteristics of Maxwell’s equations on a general
relativistic spacetime are the null surfaces defined by the tensor gij; (cf. Friedlander
1975: Theorem 3.2.1).

So to sum up: the moral is as before. It is by a theorem, not by an interpretive
postulate, that gij earns the name of ‘metric’. In particular, SEP implies that Maxwell’s
equations take a form that makes the “physical light-cones” that are defined by the
(wavefront velocity of/characteristics for) the propagation of light coincide exactly with
the “geometric cones” defined by gij(X

i, Xj) = 0.

5 Two effects

5.1 The overall shape of the examples

The overall shape of the promised examples violating relativistic causality is now clear.
At the end of Section 4, we have seen the conceptual distinction between:

(i): the geometric light-cones defined by the tensor gij, and
(ii): the physical light-cones that are traversed by electromagnetic radiation; (or

mathematically, and generalizing from electromagnetism: the characteristics for the
wave equation governing the field concerned).
And we have seen how for electromagnetism in classical general relativity, SEP makes
(i) and (ii) coincide.

So we naturally look for a theory that has a regime in which SEP fails, in such a
way that the physical light-cones turn out to be wider, rather than narrower, than the
geometric light-cones. That is: the vectors tangent to the physical light-cones are to be
spacelike, rather than timelike—understanding ‘spacelike’ and ‘timelike’ with respect
to gij. (Here ‘regime’ is physicists’ jargon for a certain set of ranges of values of a
theory’s parameters. For example, in fluid mechanics one could speak of the regime of
high density and low viscosity. And sometimes, the regime is specified, wholly or in
part, by specifying a state of the system concerned.)

24



More specifically, in terms of the decomposition of SEP (in Section 4.3.2): we
look for regimes where Minimal Coupling fails, and curvature-dependent terms enter
the action and equations of motion for the electromagnetic field in such a way that
electromagnetic propagation is described in terms of an effective metric whose light-
cones are wider than those of gij. Hence, one talks of ‘superluminal light’: which is
not a contradiction in terms, since ‘superluminal’ is to be understood as greater than
the speed c defined by gij.

We consider two such regimes, both arising in quantum electrodynamics (QED): so
they concern superluminal photons. In fact, they are but two of a family of effects in
which vacuum polarization affects the propagation of photons, owing to the vacuum
being modified by one or another external environment.17 But I shall confine myself
to these two. The first is the Drummond-Hathrell effect, which concerns QED in a
general relativistic spacetime, i.e. photon propagation in an external gravitational
field (Section 5.2). The second is the Scharnhorst effect (Section 5.3), which concerns
photon propagation in the flat spacetime between two perfectly conducting planar
plates.18

5.1.1 Limitations and alternatives

Before going into details, I should make three other general points about the overall
shape of these examples. The first two are, I admit, limitations of the examples: about
observability, and approximations; subsequent Sections will give more details. The
third is a pointer towards other similar examples, which I will not discuss further.

5.1.1.1 Observability? The word ‘effect’ carries the connotation that one could
observe it. No such luck, I’m afraid! Both effects are so tiny as to be well beyond
present observation—and perhaps all future observation. Incidentally: here, ‘tiny’
does not mean that all photons travel at a speed greater than c by a tiny amount.
Rather (as one would expect for a quantum theory), it means that:

(i): there is a probability distribution for finding the photon to have travelled at
various speeds; and

(ii): the probability for a photon to be found to have travelled at a speed that is
greater than c, by a large enough margin to be observationally distinguished from c,
is tiny. (In other words: the probability for a photon to be found a measurably large

17A detailed study of photon dispersion and birefringence (polarization-dependent phenomena) is
Adler (1971). And in recent years, a framework for unifying these results has begun to emerge: e.g.
Dittrich and Gies (1998).

18Brown’s own discussion is on p. 165-172. Among other references, he cites Shore (2003) and
Liberati et al (2002). These and Shore (2003a, 2007), and some of their references, have been my
sources for what follows. I again stress, as in Sections 1 and 3.3, that our present understanding
of these effects, is undoubtedly incomplete—there are plenty of open questions hereabouts, for both
physicists and philosophers. A vivid illustration of this is the recent results about the Drummond-
Hathrell effect (Hollowood and Shore 2007, 2007a), mentioned in Section 5.2.1.2.

25



distance outside the geometric light-cone is tiny.)

5.1.1.2 An artefact of approximations? QED is an extraordinarily accurate
theory, and a long-established one. But it is very complicated, so that calculations
within it often have to adopt various approximation schemes: and these effects are
no exception. We shall see that the currently feasible calculations involve various
approximations; of which one is perhaps especially dubious, as regards the prediction
of superluminal photons. Of course, our authors stress this; (e.g. Brown 2005, p. 168;
Shore 2003, pp. 508, 513, 2003a, Sections 3.2, 4.3).

5.1.1.3 The cat out of the bag: bi-metric theories In both these effects, the
idea of two metrics is “modest”, in that the second metric is “merely” effective. That
is: it is a structure that helps describe a certain regime of the theory, rather than
occurring in its fundamental equations describing all regimes. Nor does the structure
“occur implicitly” in the fundamental equations in the sense of being mathematically
determined by them, say by being a function of quantities that occur explicitly—and
the same function regardless of the regime, or a choice of state.19

But once the idea of a theory having two metrics is out of the bag, one naturally
speculates! Thus a theory might postulate ab initio two metrics, in the sense of two
(0,2) symmetric tensors, neither of which mathematically determines the other (so
neither is a function of the other); and then the theory might divide between these
tensors the various roles—or something like the roles—played in general relativity by
just the one tensor gij.

A second possibility for such a bi-metric theory is a theory with the following three
features.

(1): It postulates a (0,2) symmetric tensor, call it again gij, that is fundamental in
that it occurs in the theory’s basic equations.

(2): But another tensor g̃ij is a function of the first, g̃ij = g̃ij(gi′j′), where the
function concerned is: (i) universal; (ii) exact not approximate; (iii) not invertible,
or at least not usefully invertible, so that we cannot equivalently rewrite the basic
equations using g̃ij.

(3): And yet our “critters”—rods, clocks, test-particles and light-rays—“display” g̃
rather than the fundamental tensor g.

Indeed, there is a long tradition of such bi-metric theories. Weinstein (1996) is
a fine philosopher’s introduction to this tradition: launched, like Brown’s discussion,
from consideration of non-minimal coupling (cf. footnote 9). A recent example of the
first possibility above is Drummond’s re-formulation of variable speed of light theories
(2001). But I shall follow Brown (2005, pp. 172-175) in discussing a recent example

19Incidentally, the emergence in relativistic pilot-wave theories of Lorentz-invariance at the observ-
able quantum level from the non-Lorentz-invariant sub-quantum level is an example of such “implicit
occurrence”.

26



of the second possibility: the TeVeS theory of Bekenstein. This is a relativistic theory
of gravity which postulates, in addition to a fundamental (0,2) symmetric tensor g, a
vector field U (which is dynamically constrained to be timelike) and a scalar field φ;
and these together define a tensor g̃. So ‘TeVeS’ stands for ‘tensor-vector-scalar’.

The motivation for the theory lies in the tradition of modified Newtonian dynamics
(called ‘MOND’) begun by Milgrom in the 1980s, to account for the anomalously fast
rotation of galaxies and clusters without having to invoke dark matter, viz. by making
gravity decrease more slowly than inverse-square for very large distances. Thus in the
TeVeS theory, the scalar field φ makes gravity stronger at large distances; and the
vector field U enhances gravitational lensing, making the theory empirically adequate
to the observations that gravitational lensing is stronger than would be expected from
the lensing galaxy’s visible matter: observations which are usually taken to require
dark matter.

But I will not need more details of the theory’s motivation. What matters for us is
that the theory defines another tensor g̃ as a function of all three of g, U and φ. Roughly
speaking, one obtains g̃ by multiplying g in the spacelike directions orthogonal to U by
a function of φ, and by dividing g parallel to U by the same function. From the theory’s
postulated action, and the resulting equations of motion, one shows that the “critters”
listed above, rods etc., survey g̃, not g. So again we see Brown’s moral: that it is by
a detailed physical argument that a tensor—here g̃, not g—earns its chrono-geometric
significance.

So much by way of discussing a heterodox, though so far unrefuted, relativistic
theory of gravity. Now I turn to proposals for superluminal light propagation in QED
in an otherwise orthodox relativistic setting: general relativistic for the Drummond-
Hathrell effect, and special relativistic, though with a preferred rest, for the Scharnhorst
effect.

5.2 The Drummond-Hathrell effect

Drummond and Hathrell (1980) studied the effect on light propagation of vacuum
polarization in a curved spacetime. Vacuum polarization gives a photon an effective
size characterized by the electron’s Compton wavelength λC = h̄/mc (with m the
electron mass). This suggests that if the photon propagates in an anisotropic spacetime
with typical curvature length-scale L ∼ λC , its motion might be affected. Here we can
already see the point made in Section 5.1.1.1, that such an effect on the motion would
be tiny; since for ordinary astrophysical objects—even the event-horizon of a black
hole—the gravitational field is weak enough that the curvature length-scale L is vastly
larger than λC .

20

Drummond and Hathrell’s analysis confirms (subject to three main approximations)
the suggestion that the photon’s motion, in particular its speed, is affected by gravity.

20Section 5.2.1 will give a numerical estimate for a solar mass black hole.

27



That is: they deduce an effective action, and corresponding (non-linear) generalizations
of Maxwell equations (i.e. equations of motion for light), which display an interaction
between quantized electromagnetism and spacetime curvature. So both the action
and the equations contain curvature-dependent terms, and violate SEP. Besides, the
equations of motion imply—again, subject to approximations—that in some scenarios
(in particular the Friedmann-Robertson-Walker and Schwarzschild spacetimes) light
propagates superluminally. I will first summarize these results of Drummond and
Hathrell, and mention some work by later authors (Section 5.2.1). Then I will discuss
how despite this superluminal propagation, causal contradictions can apparently be
avoided (Section 5.2.2).21

5.2.1 Faster than light?

5.2.1.1 Three approximations Drummond and Hathrell showed that vacuum
polarization implies an effective action for the electromagnetic field in a curved space-
time that contains curvature terms and so violates SEP. But their derivation is subject
to three approximations. They are:

(i): Only one-loop Feynman diagrams are considered.
(ii): Gravity is assumed to be weak, in the sense that the effective action keeps only

terms of first order in the curvature tensors R (scalar), Rij (Ricci) and Rijkl (Riemann).
In a bit more detail: this restriction implies that results are valid only to the lowest
order in the parameter λ2C /L

2, where L is the typical curvature length scale; so the
results are more accurate, the larger L i.e. the weaker is the gravitational field.

(iii): The photons are assumed to be low frequency, in the sense that the effective
action neglects terms involving higher orders in derivatives of the fields.

These approximations are in ascending order of ‘importance’, in the sense of ‘recal-
citrance’ ! That is:

As to (i): One-loop diagrams contribute to the action terms proportional to the
fine structure constant α ∼ 1/137, and higher-loop diagrams would contribute terms
proportional to powers of α—which are smaller. So we can expect the one-loop ap-
proximation to give the dominant effects.

As to (ii): There is a trade-off here. As mentioned, the effect depends on L being
comparable with λC ; but our results are more accurate the larger L is.

As to (iii): There are two reasons why we need to consider high-frequency photons;
one theoretical and one experimental. The theoretical reason develops the remarks at
the end of Section 4.3.3.2. There I reported that for questions about causality, the
relevant notion of the velocity of a wave (of the several available!) is the wavefront
velocity vwf : it represents the velocity of the boundary of the region of excitation
of the field, and is given mathematically by the characteristics of the field’s wave
equation. I also reported that with SEP, vwf is c: i.e. the characteristics for the
orthodox Maxwell’s equations in general relativity are the null hypersurfaces defined

21My main sources are Shore (2003, 2003a); cf. footnote 16.

28



by gij. But now SEP, and these orthodox equations, are gone, and so we have to
ask: what is vwf for Drummond and Hathrell’s non-linear generalization of Maxwell’s
equations? Fortunately, a theorem of Leontovich (from 1972) states that for a large
class of partial differential equations (including Drummond and Hathrell’s equations),
the wavefront velocity is the infinite frequency limit of the phase velocity. That is:

vwf = limω→∞ vph(ω) = limω→∞
ω

| k | . (5.1)

(The proof is sketched in Shore (2003a, Section 3.2) and (2007, p. 10-11).) So vwf is
independent of frequency; but we need to know the high-frequency limit of vph(ω).

The experimental reason relates to the fact that on Drummond and Hathrell’s
analysis, the correction to the photon’s speed is O(αλ2C /L

2), with L the typical curva-
ture length scale, as before. Experimentally, this is a ratio of a square of a quantum
scale λC to an astrophysical scale, and is therefore minuscule. (In a black hole example
below, it will be O(10−34).) In order to assess whether we could observe this correction,
Drummond and Hathrell suggest (1980, p. 354) that we make two assumptions:

(a): The typical time over which propagation can be followed is characterized by
L, so that the length difference to be observed is given by αλ2C /L.

(b): Observability requires this to be O(λ) where λ is the wavelength of the light.
These assumptions suggest that observability requires that αλ2C /λL > 1, while our
approximations (ii) and (iii) above required respectively that L À λC and λ À λC .
Agreed, assumption (b) is unduly pessimistic, since modern spectroscopy enables it to
be weakened by some six, or even eight, orders of magnitude. That is: observability
might require only that, say, αλ2C /λL > 10

−8. Nevertheless, to observe the effect we
would obviously like λ as small as possible.

So much by way of caveats about the approximations. Turning now to reporting the
results, there is the proverbial “good news and bad news”: results giving superluminal
propagation, and results suggesting that it does not occur. Following Shore, I will
report these in order.

5.2.1.2 Good news and bad So far as I know, the main way in which superlu-
minal propagation is derived is by applying to Drummond and Hathrell’s “Maxwell’s
equations” (derived from their effective action) the geometric optics (short wavelength)
limit; (cf. Section 4.3.3.2). When we do this, the previous result, that k is null, k2 = 0,
is replaced by a more complicated equation. We will not need it; but for completeness
it is

k2 − 2a
m2

Rijk
ikj +

8b

m2
Rijmnk

ikjaman = 0 , (5.2)

where a and b are real constants (of magnitude about 1% and 0.1% of α), Rij is the
Ricci tensor, Rijmn is the Riemann tensor and a is the polarization vector (a spacelike
4-vector normalized to unity). For us the important point about eq. 5.2 is that it is
homogeneous and quadratic in k, and so we can write it in the form

g̃ijkikj = 0 ; with g̃
ij ≡ g̃ij(Rklmn, ap) . (5.3)

29



This (frequency-independent) function g̃ of curvature and polarization represents an
effective metric, as follows. The tangent vector to a light ray is now given by pi := g̃ijkj;
i.e. light rays are curves xi(s) with dxi/ds = pi. This definition of pi implies

(g̃−1)ijp
ipj = g̃ijkikj = 0 . (5.4)

So g̃−1 defines an effective metric: (though we still raise and lower indices with gij).
We will from now on write G for g̃−1. So our question is whether in some solutions,
G’s light cones—in Section 5.1’s terms: the physical light cones—are wider than the
geometric light cones defined by gij. In other words: a solution exhibits superluminal
light if p is spacelike (with respect to gij)—are there such solutions?

Indeed, Drummond and Hathrell (1980, p. 354; and later authors including Shore)
show that the Friedmann-Robertson-Walker spacetime, and the Schwarzschild space-
time, are such solutions. But as we would expect from the discussion above, the effect
is numerically tiny. In particular, for classical electromagnetism in the Schwarzschild
spacetime, there is a geometric optics solution (i.e. a solution of k2 = 0, ki∇ikj = 0)
describing a light ray in a circular orbit with radius r = 3GM (Misner, Thorne and
Wheeler (1973) pp. 672-677); and for one solar mass at the singularity, the quantum
correction to the classical speed c is O(10−34).

There are also results suggestive of superluminal propagation, for generalizations
of the Drummond and Hathrell action. In particular, Shore has derived an action
containing all orders in derivatives (cf. approximation (iii) above): though like Drum-
mond and Hathrell, he retains only terms of O(RF F ) in curvature R and field strength
F . And he has shown that this action, applied to Bondi-Sachs spacetime (describing
gravitational radiation far from its source), implies that vwf ≡ vph(∞) is superluminal.

On the other hand, the “bad news”: there are indications that these results (even
Shore’s about Bondi-Sachs) are artefacts of the approximations used in deriving them.
For a complete analysis of high frequency propagation requires one to consider higher-
order terms in curvature and field strength, not just in derivatives. And there is some
mathematical evidence—just recently, strong evidence—that these terms will, for high
frequencies ω, drive vph(ω), and so vwf , to c.

I will not need details about this evidence. But for completeness: the evidence
concerns the correction to the classical light cone condition k2 = 0 being an integral
whose integrand contains as a factor a phase roughly of the form

exp[−is2Ω2(R, ω)P (R, s)] ; (5.5)
where s is the integration variable, R is a generic curvature component and Ω(R, ω) and
P (R, s) are functions (not exactly known) of the gravitational field, and also of ω and
s respectively; and where Ω2 ∼ Rω2

m4
. So it seems likely that whatever the behaviour

of P and the other factors in the integrand, high frequencies ω → ∞, Ω → ∞ and
therefore rapid variation in the exponent, will drive the integral, i.e. the correction to
the classical condition k2 = 0, to zero. (For an introduction to these details, cf. Shore
(2003: pp. 516, 518; 2003a, Section 4.3; 2007, pp. 28-29).)

30



Just recently, this evidence has been much strengthened. Hollowood and Shore
(2007, 2007a) have shown by means of new techniques that vwf ≡ vph(∞) is always
c. But they also show that micro-causality fails, i.e. commutators of fields at space-
like separations do not vanish, but fall off exponentially; so that there is violation of
relativistic causality in sense (ii) of Section 3.2.1.

So much by way of reviewing the prospects for superluminal propagation in the
Drummond-Hathrell effect. Of course, whether (and in what sense) it occurs, our
main philosophical point—viz. Brown’s moral that a tensor earns its chrono-geometric
significance by dint of detailed physical arguments—is vividly illustrated. For in this
effect, the light-cones, and more generally geometric structure, defined by gij are at one
remove (though a numerically minuscule one!) from the physical behaviour of light:
which is instead described by the effective metric Gij.

5.2.2 Avoiding contradictions

Assuming now that there is superluminal propagation, I turn to how causal contradic-
tions can be avoided. For as announced in (B) of Section 3.3, the argument against
causal contradictions is not the usual philosophical one, that a closed “causal loop” im-
plies a severe consistency condition which one just presumes the solutions in question
satisfy. The argument is rather that the kind of superluminal propagation envisaged
could not be exploited to produce a causal loop, i.e. a zig-zag, there and back, into
the causal past of an initial event.

More exactly, Shore makes two points. The first is general (and also made by
Drummond and Hathrell 1980, p. 353); the second is specific. First, he suggests a
zig-zag process from an event A to a spacelike event B, and then from B to an event C
that is spacelike to B, but in the causal past of A, may be expected to require ‘that the
laws of physics should be identical in the local frames at different points of spacetime,
and that they should reduce to their special relativistic forms at the origin of each
local frame’ (2003, p. 511; cf. 2003a, Section 2.1).22 But this requirement is just SEP
(cf. Section 4.3.2): which of course fails in the framework of the Drummond-Hathrell
effect.

Second, Shore points out that one can investigate the causal structure given by the
effective metric Gij by using general notions and results that have been developed for
classical general relativity, i.e. for the causal structure fixed by the usual metric gij.
Thus there is a well-known spectrum of properties of causal “good behaviour”—one of
which, though strong, is especially relevant to assessing the causal structure fixed by
Gij. This is the concept of stable causality: I shall not give the exact definition of this;
(cf. e.g. Hawking and Ellis 1973, p. 198; Geroch and Horowitz 1979, p. 241; Wald

22I presume his idea is that only with this can we be sure that there can be a process from B to
C which is like that from A to B. But the threatened zig-zag might exploit different processes in its
two legs. But nevermind: we will see, now and in Section 5.3.2, stronger reasons to doubt that there
can be such zig-zags.

31



1984, p. 198). We only need the idea: that a spacetime (M, gij) (M the manifold)
is stably causal if not only does it lack closed timelike curves, but also the spacetime
resulting from it by a slight opening out of gij’s light-cones at every point does not have
any such curves. The idea is of course motivated by wanting causal good behaviour
to be robust to perturbations. Stable causality also follows from global hyperbolicity,
which we saw in Section 3.2.2 to be usually assumed in quantum field theory.

Clearly, stable causality can be applied to our effective metric Gij in two ways:
(i): If one knows (M, gij) is stably causal, one can expect that (M, Gij) has no

closed timelike curves, since Gij differs from gij only by terms of O(α);
(ii): One can ask, more ambitiously, whether (M, Gij) is itself stably causal; so that

its causal good behaviour is itself robust to perturbations (in particular to overcoming
Section 5.2.1’s approximations!). In some cases, this question can be answered posi-
tively by invoking the following characterization of stable causality. As usually stated
for (M, gij), it is that a spacetime is stably causal iff it has a “global time function”,
i.e. a smooth function f : M → IR whose gradient is everywhere timelike (Hawking
and Ellis 1973, Prop. 6.4.9, p. 198; Wald 1984, Theorem 8.2.2, p. 198). But this
equivalence of course remains valid for (M, Gij) with its definition of ‘timelike’. So
one can be assured that (M, Gij) is stably causal by finding a global time function f .
And indeed, Shore remarks (2003, p. 513; 2003a, Section 2.4; 2007, pp. 12, 29) that in
the Friedmann-Robertson-Walker spacetime discussed in Section 5.2.1, the usual global
time coordinate is such a function, for Gij no less than for gij. So here is a clear case
where superluminal light, i.e. the physical light cones being wider than the geometric
ones, harbours no causal anomalies.

5.3 The Scharnhorst Effect

I turn to superluminal light propagation in QED in a special relativistic setting: the
Scharnhorst effect (Scharnhorst 1990; Barton 1990). Again I will first summarize results
(Section 5.3.1). Then I will discuss how causal contradictions can apparently be avoided
(Section 5.3.2).23

5.3.1 Faster than light?

In 1990, Scharnhorst and Barton showed that for the vacuum between two infinite
perfectly conducting plates, QED predicted that photons would have a wavefront ve-
locity vwf larger than c. More precisely, vwf is enhanced in the direction orthogonal to
the plane of the plates. Though these derivations were based on approximations, in
particular on assuming low-frequency photons (cf. (iii) in Section 5.2.1.1), there is ev-
idence (Barton and Scharnhorst 1993, pp. 2040-2044; Scharnhorst 1998, pp. 706-707)
that the result is not an artefact of the approximations, but a genuine effect, albeit an
unobservably small one. (Cf. below for the small size).

23My main source is Liberati et al (2002); cf. footnote 16.

32



But unlike Section 5.2.1.1, I shall not go into details about the approximations.
After all (as Barton and Scharnhorst stress: 1993, p. 2044), QED in flat spacetime is
much better understood than QED in curved spacetime, and the system of two plates
has been much studied for another striking effect, the Casimir effect: according to
which there is a force between the plates, even when the quantum electromagnetic
field between them is in the vacuum state.24 I will only stress (as Barton, Scharnhorst
and Liberati et al. do) that the superluminal propagation does not violate Lorentz-
invariance of the theory (the action and equations of motion which imply the propa-
gation), but reflects only the non-Lorentz-invariance of the vacuum state.

There are two points here.
(i): Suppose a theory obeys a symmetry in the sense that a certain transformation,

e.g. a spatial rotation or a boost, maps any dynamical solution to another solution.
This by no means implies that every solution should be invariant, i.e. mapped onto
itself, under the transformation: after all, not every solution of Newtonian mechanics
is spherically symmetric!

(ii): Agreed, the vacuum state for empty Minkowski spacetime is required to be
Lorentz-invariant since it should “look the same” in a translated, rotated or boosted
frame. But the presence of the plates breaks this symmetry, just as a pervasive
inertially-moving medium would do: licensing a non-Lorentz-invariant vacuum state.

A simple Newtonian example illustrates both these points (given by Liberati et al.
Section 2.2). A wavefront of sound spreading from a point source in a fluid at rest in
a Newtonian spacetime is described as spherical in the rest frame of the fluid, but in a
Galilean-boosted frame it is described as blunted (due to reduced relative speed) in the
direction of the boost, and as elongated (increased speed) in the opposite direction.

I turn to numerical details of QED’s predictions. Writing gij for the Minkowski
metric (with signature (-,+,+,+)), the effective metric between the plates, i.e. defining
(to order α2) the physical light-cones of photon propagation, is

Gij = gij −
ξ

1 + ξ
ninj ; (5.6)

where ni is the unit spacelike vector orthogonal to the plates; (and we again raise and
lower indices with gij). If a is the distance between the plates, we have

ξ = 4.36 × 10−32
(

10−6m
a

)4
. (5.7)

This is minuscule. But since it is positive, the light-cones of Gij are slightly, albeit
unobservably, wider in the direction orthogonal to the plates than those of gij: super-
luminal propagation.

24For the history and philosophy of this effect, cf. Rugh, Zinkernagel and Cao (1999).

33



5.3.2 Avoiding contradictions

There is of course a large literature about superluminal propagation in a special rela-
tivistic setting, as part of the yet larger literature on the foundations of special rela-
tivistic kinematics. Fortunately, Liberati et al. (2002) connect the Scharnhorst effect
with this literature, of which they give a judicious and detailed discussion; (as of course
does Brown: 2005, Chapters 2-6). So I am happy to do no more than report some of
their main points: certainly, I could not do better! I shall report four points: two are
general and correspond roughly to the first point of Shore’s in Section 5.2.2; two are
specific to the Scharnhorst effect, and correspond to Shore’s second point, about stable
causality.

First, Liberati et al. emphasise that Lorentz-invariance does not preclude superlu-
minal propagation: a speed c can be invariant without being a maximum signal speed.
One sees this clearly in the style of derivation of an invariant speed, and of the Lorentz
transformation, pioneered by von Ignatowsky in 1910, and repeatedly rediscovered and
developed since then. These derivations neither assume, nor deduce, that a signal can-
not travel faster than c. (Rather, one deduces that there cannot be a reference frame,
or a coordinate system, with a relative speed greater than c; cf. Liberati et al. 2002,
Sections 2.1, 2.3.)

Liberati et al. also emphasise that superluminal propagation does not imply that
there can be a causal zig-zag from a cause A to a spacelike effect B, which is itself the
cause of a second effect C lying in the causal past of A. To sustain this implication, one
would presumably have to require that ‘in any given reference frame, the only criterion
for saying that an event is the cause of another one [is] the time ordering in that frame’.
Which is surely false. Although ‘there are no precise definitions of [cause and effect]
... the criteria used to establish that e1 is a cause of e2 are based on considerations ...
about the so-called “arrow of time” ’(2002, Section 3.1).

But whatever the vagaries of the notions of cause and effect, there is obviously no
threat of a contradiction provided that with respect to one particular reference frame,
all superluminal propagation is forward in time.25 And turning now to the Scharnhorst
effect, it is straightforward to check that this is so, with respect to the rest frame of
the plates. Indeed, here we connect with the notions of causal good behaviour, in
particular stable causality, studied in classical general relativity and invoked by Shore,
as we saw in Section 5.2.2. Thus we need only consider the spacetime IR4 containing
the two infinite plates, distance a apart, and equipped with the usual Minkowski metric
outside the plates, and the effective metric eq. 5.6 inside. It is obvious that the time
coordinate t of the rest frame of the plates has ∇t everywhere non-zero and timelike.
So the spacetime is stably causal: not only are there no closed timelike curves; but
also none could arise by a slight widening of the cones throughout the spacetime (i.e.
between the plates, a further widening from Gij). Obviously, a similar argument will

25Drummond and Hathrell also make this point (1980, p. 353). One might add: consider such
propagation, or even instantaneous action-at-a-distance, in a Newtonian spacetime.

34



secure stable causality for scenarios with more than one pair of plates, at least if they
do not move relative to one another.

Finally, what about pairs of plates in relative motion? Liberati et al. consider
various scenarios, arguing that a causal contradiction will not arise. Then they end by
suggesting that the general threat of contradiction should be analyzed using Hawking’s
Chronological Protection conjecture. The idea of the conjecture is that if a spacetime
is causally well-behaved “early on”, it cannot become badly-behaved later. More pre-
cisely: a region of closed timelike curves which does not extend indefinitely into the
past must have a “first” closed null curve, at which—Hawking argues—uncontrollable
singularities will occur, implying the breakdown of quantum field theory on curved
spacetime, and the need for some sort of quantum theory of gravity; (cf. Earman 1995,
pp. 188-193). Applying this idea to a spacetime in which early on, several pairs of
plates are well separated, and individually, stably causal (cf. above), we infer that if
some scheme for the plates’ later motion seems to yield a causal contradiction, then,
as Liberati et al. put it: ‘causal paradoxes are the least of your worries since you are
automatically driven into a regime where Planck-scale quantum gravity holds sway’
(2002, Section 3.2.4).

6 Conclusion

By way of concluding this paper, let me briefly list three of my main claims:—
(i): In some QED scenarios, relativistic causality is apparently violated.
(ii): These scenarios raise open questions, not just about how to define relativistic

causality, and how to avoid causal contradictions (in more interesting ways than by
saying ‘there is a severe consistency constraint’), but about the much wider question,
of the future of relativistic quantum physics.

(iii): Philosophically, these scenarios illustrate Brown’s moral that the mathemat-
ical representatives of geometry get their geometrical significance by dint of detailed
physical arguments.

Acknowledgements:– For comments, conversation and correspondence, I am very
grateful to: audiences in Dubrovnik, Oxford and Cambridge; two referees, and Bill
Demopoulos, Ian Drummond, Gordon Fleming, John Norton, Brian Pitts, Graham
Shore; and especially to Steve Adler, Harvey Brown, Dennis Lehmkuhl, Bob Wald and
Steven Weinstein.

7 Appendix: Two familiar examples

This Appendix reports the two examples of quantum-theoretic violations of relativistic
causality which are most familiar to philosophers of physics: the pilot-wave approach,

35



and the Newton-Wigner representation. I shall say much more about the former.

Both examples concern Minkowski spacetime. So one naturally asks which of Sec-
tion 3.2.1’s precise formulations are violated. But this is a subtle, and even contro-
versial, question, since the relations of these examples’ formalisms to the local field
operators that are those formulations’ topic, are indirect: and in some respects, un-
known or controversial. So here again there are questions I cannot pursue: it must
suffice that the references below are the place to begin finding the answers.

7.1 The pilot-wave

There are various pilot-wave approaches to relativistic quantum theory. But I shall
sketch one well-developed approach which is strongly analogous to the best-known
pilot-wave approach to non-relativistic quantum theory. As an example of violating
relativistic causality, it is in a sense only an example “by courtesy”: for it takes the rel-
ativistic light-cone structure as a “merely emergent” or phenomenological description.
For this reason, and also because of its being analogous to Newtonian action-at-a-
distance, this example provides an interesting comparison with the others. I shall:
begin with the pilot-wave approach to non-relativistic quantum theory (Section 7.1.1);
then discuss action-at-a-distance within it, and the Newtonian analogy (Section 7.1.2);
and finally turn to the relativistic case (Section 7.1.3).

7.1.1 The non-relativistic case

Recall that the system comprises both a wave and one or more point-particles. Let us
begin with the wave and how it evolves over time. The wave is a complex-valued func-
tion ψ on configuration space Q (e.g. Q = IR3N for N spinless particles in euclidean
space), which always evolves by the Schrödinger equation. The Schrödinger equation
is local in the mathematical sense: roughly, the evolution depends on ψ and its spatial
derivatives but not on differences of ψ at different points in Q. But it is non-local in
the physical senses that:

(i): it is defined on configuration space, not real space; and
(ii): wave-functions of even a single particle propagate instantaneously: if at time

t = 0 a wave-function ψ(0) has compact spatial support (i.e. is non-zero only in a
compact spatial region), then at all later times t, no matter how small, ψ(t) is non-zero
throughout all space.

Despite the continuous deterministic Schrödinger evolution, an analysis of measure-
ment processes demonstrates an effective collapse of the wave function, nowadays often
called ‘decoherence’: which explains the instrumental success of orthodox textbooks’
notorious projection postulate.26

26More precisely, it explains it, once allied to the pilot-wave theory’s invoking particle positions to
provide definite events. That decoherence alone is not enough to solve the measurement problem is
argued by e.g. Bub (1997, pp. 221-223, 231-232, 236) and Adler (2003).

36



Each point-particle has a continuous trajectory which is determined by the wave-
function according to the guidance equation. We need to note three features of the
guidance equation:—

(i): Classical analogues: This equation is natural. Indeed, it follows from the ortho-
dox probability current for the Schrödinger equation found (for the one-particle case) in
most textbooks. It is also the obvious wave-mechanical analogue of a central equation of
classical Hamilton-Jacobi theory, viz. p = ∂S

∂q
. More generally, much of the pilot-wave

approach bears comparison with Hamilton-Jacobi theory. In particular, many quan-
tum effects are due to the presence (in the analogue of the classical Hamilton-Jacobi
equation) of an extra potential term dependent on the wave-function, viz. the quantum

potential U . In the simplest one-particle case, U := − h̄2
2m

∇2R
R

; where R is given by the
polar decomposition of the wave-function, i.e. ψ(q, t) = R(q, t) exp(iS(q, t)/h̄).

(ii): Probability and equivariance: The pilot-wave approach recovers the orthodox
quantum probabilities by averaging over particle position using | ψ |2 as the probability
density. This is the Born rule, understood non-instrumentalistically; i.e. understood
with probabilities like those of classical statistical mechanics. Besides: taken together
with the Schrödinger equation, the guidance equation implies that this probabilistic
interpretation of the wave-function, viz. that | ψ |2 is the position probability density,
is preserved over time.

(iii): Non-locality: The guidance equation also implies that in a multi-particle sys-
tem, the motion of each particle is determined in part by the simultaneous actual
positions of the other particles. For the guidance equation says that the particle’s
momentum is the gradient of the phase S of the wave-function at the point in config-
uration space corresponding to all these actual positions. To be precise, for the ith
particle, and the N actual positions qj, j = 1, . . . , N : pi ≡ mq̇i = ∇iS |q1,..., qN .

Non-locality is also evident in the quantum potential for many particle systems.
Thus for two spinless particles, so that ψ(q1, q2, t) = R(q1, q2, t) exp(iS(q1, q2, t)/h̄),

the quantum potential is U (q1, q2, t) = − h̄
2

2m
1
R
(∇21R + ∇22R); and only for the special

case of product states ψ(q1, q2, t) = ψ1(q1, t)ψ2(q2, t), is the quantum potential a sum,
U = U1(q1, t) + U2(q2, t).

On the other hand, this non-locality cannot be exploited to send a signal in
the sense of affecting the statistics of distant experiments—provided the system is in
“quantum equilibrium”, i.e. provided that | ψ |2 is indeed the position probability den-
sity. That is, the orthodox quantum no-signalling theorem is recovered by averaging
over particle positions with | ψ |2.

7.1.2 Action-at-a-distance?

In Section 7.1.1’s summary, two features look like action-at-a-distance: the instan-
taneous spreading of wave-functions, and the non-local guidance equation. (In this
Subsection, I set aside the quantum potential, discussion of which would be similar to
that of the guidance equation.)

The first is undeniable, but also unsurprising. For this is a feature of orthodox non-

37



relativistic quantum theory which the pilot-wave approach simply inherits—and which
one naturally expects to disappear in a relativistic theory, not least because in a rela-
tivistic setting such superluminal propagation seems to threaten causal contradictions.
But cf. Section 7.1.3 and Section 7.2.

But we need to pause here over the second, the guidance equation. I agree that
it is natural to take it as asserting action-at-a-distance; (though as just noted, this
“action” in an individual process could only be used to send a signal if the system was
in quantum dis-equilibrium). But I should register that this can be (and has been)
resisted—for reasons that apply equally to the more familiar case of Newton’s law of
universal gravitation, F = G m1m2

R2
, mentioned in (2) of Section 2. So I shall explain

both the natural construal and the reasons for resistance, for both cases.

For both the pilot-wave and Newton’s law, the value of a quantity is a function of
the simultaneous value of another quantity. For gravity, the force is determined by the
simultaneous value of distance (or position of the other mass); for the pilot-wave, the
momentum (or velocity) is determined by the simultaneous value of many positions.
(Agreed, there are other differences which are crucial for physical calculations and
explanations: above all, that the size of the effects of Newtonian gravity drops off with
distance, while that of the pilot-wave need not.) In both cases, it is very natural to take
the mathematical statement of functional dependence as a causal statement, asserting
action-at-a-distance. And it is natural to support this causal construal by appealing to
counterfactuals. Thus it is natural to say that in a Newtonian world, a counterfactual
like ‘If the centre of the Sun were now, say, 10,000 km away from where it actually is,
the Sun’s gravitational pull on the Earth would now be in a slightly different direction’
is true. And the second occurrence of ‘now’ (rather than ‘eight minutes from now’)
suggests instantaneous causation. In particular, within contemporary philosophy of
causation: an advocate of the counterfactual analysis of causation (Lewis 1973) would
take this counterfactual to indicate instantaneous causation.

And similarly for the guidance equation. For example, for the two particles in an
EPR-Bell experiment: it is natural to say that a counterfactual like ‘If the L-particle
were now in a different position, the velocity of the R-particle would now be different’
is true, and indicates causation.

But I admit that one can resist this; in fact Dickson (1998, Section 9.4, pp. 196-
208) does so. His overall position is reminiscent of Norton’s causal anti-fundamentalism
(Section 2). He is cautious about causal judgments, and suspicious of the counterfactual
analysis of causation. Like Norton, he suggests that under determinism, the most we
can safely say is that the total present state is the effect of earlier states: he is wary of
logically weaker, localized, facts or events as causal relata.27

27Beware of a false move. We could not really avoid action-at-a-distance, by (i) allowing such
localized facts or events as causal relata, and then (ii) saying that the various present localized facts
or events (e.g. in the Newtonian case: m1’s position, and the force on m2) are joint effects of a common
cause, viz. the total state at any earlier time. For one can consider a very recent time, and the state
at that time of a very distant region—say, the positions and velocities of bodies in that region; and so
get “as near as makes no difference” to instantaneous causation, viz. the causal “contribution” from

38



In more detail: Dickson agrees that in everyday life we tend to interpret counter-
factuals as non-backtracking, i.e. to take a counterfactual supposition about a state
of affairs at time t to preserve most of the past of t. And this makes us endorse the
counterfactuals above, for we think along the following lines: if the Sun were now in
a different place, nevertheless its past and the Earth’s past would be as it actually
was until very recently, and so the Earth would be very nearly where it actually is,
and so would indeed feel the Sun’s pull in a slightly different direction. But, says
Dickson, it is unclear what this interpretative tendency—apparently conventional and
so alterable—has to do with causation. Thus he and the counterfactual analysis agree
that if instead we take counterfactuals as backtracking, then for a deterministic theory
like Newtonian gravity, a counterfactual supposition about the present implies differ-
ences from actuality indefinitely far into the past: differences which could in general
grow as we go further into the past (Lewis 1973a: pp. 75). But Dickson sees this as a
sign—not of how backtracking counterfactuals are irrelevant to causation—but of how
poorly we understand causation, in a deterministic world no less than (and perhaps
even more than!) in a indeterministic one.

So much by way of:
(i) reporting the traditional, natural construal of the guidance equation (and New-

ton’s law of gravitation) as involving instantaneous causation; and
(ii) registering that one can nevertheless resist this.

My own view is that the traditional construal is right: but as I admitted in (2) of
Section 2, to defend this view would be a large project in the philosophy of causation,
which I duck out of. (And as I said there, I also do not have a general criterion of
when spacelike functional dependence amounts to spacelike causation (a violation of
relativistic causality), rather than merely reflecting the joint effects of a common cause.
But I think it is clear that all my examples involve the former, not merely the latter.)

So to assess whether the pilot-wave approach violates relativistic causality, I turn
to ...

7.1.3 The relativistic case

I turn to the adaptation of the pilot-wave approach in Section 7.1.1 to relativity. There
have been various obstacles, various significant achievements—and there are important
projects yet to be done. I shall confine myself to reporting (from Holland 1993, Bohm
and Hiley 1993) some salient points of work on:

(i): single-particle relativistic wave equations: where I will emphasise the question
whether sub-luminal trajectories can be defined from the current as usually defined;
and

(ii): quantum field theory: where I will give more details, and emphasise the use of
a preferred frame and the non-locality of the quantum potential (Section 7.1.3.1).

So in various respects, what follows just scratches the surface of a large subject.

the distant bodies to the nearby present fact or event.

39



For example, one topic which I will ignore is the pilot-wave account of many-particle
wave equations; (for the many-particle Dirac equation, cf. Bohm and Hiley 1993, pp.
214-225, 272-286; and briefly, Holland 1993, p. 509). So I will not discuss guidance
equations for particles that are non-local on analogy with that in (iii) of Section 7.1.1,
viz. by taking a gradient of S in configuration space at a point determined by all the
particles’ positions. But we will still see non-locality in Section 7.1.3.1 below, viz. in
the behaviour of the quantum potential.

So first, I consider (i): single-particle wave equations. To give a pilot-wave interpreta-
tion of such equations, we naturally ask whether the formalism allows the definition of
a 4-vector current jµ with the properties that:

(a): jµ is conserved, ∂µj
µ = 0;

(b): jµ’s time-component j0 is positive, and so might be interpreted as a probability
density;

(c): jµ is always timelike, so that its integral curves can be the worldlines of the
particle concerned, which therefore travels subluminally.

It turns out that the Klein-Gordon equation, describing a spin 0 particle, resists a
pilot-wave interpretation. Its 4-vector current jµ enjoys property (a), but violates (b)
and (c). On the other hand, the Dirac equation has a current satisfying all of (a)-(c);
(Holland 1993, pp. 498-509; Bohm and Hiley 1993, pp. 232-238 for the Klein-Gordon
equation, and pp. 214-222 for the Dirac equation).

7.1.3.1 Quantum field theory The pilot-wave approach in Section 7.1.1 carries
over successfully to the relativistic quantum field theory of bosons, such as a spin 0
particle or the photon. The general idea is threefold:

(a): We first formulate the theory in the Schrödinger picture, using the space
representation of the field coordinate ψ. This formulation is entirely orthodox, though
not widely presented by the textbooks.

(b): Then we make the polar decomposition of the wave-function, obtaining (as
in Section 7.1.1) a conservation equation, and a quantum analogue of the Hamilton-
Jacobi equation containing a new quantum potential term.

(c): Then we interpret (a) and (b), much as we did in Section 7.1.1. Namely:
the orthodox wave-function Ψ is a physically real (though of course mathematically
complex) field on the configuration space of the field ψ: i.e. it is a functional of the
field ψ. Ψ always evolves by the Schrödinger equation. So far, again so orthodox; (at
least as regards formalism—orthodoxy might cavil at calling a wave-function physically
real). But there is also at all times an actual field configuration, which evolves by a
guidance equation which is natural, derived from the formalism, and a generalization
of that in Section 7.1.1. And we recover the orthodox probabilistic results by averaging
over field configurations, using the Born rule understood non-instrumentalistically.

I shall say a bit more about (a)-(c), concentrating on the interpretative aspects in
(c); and for simplicity, on a neutral, spin 0, massless particle, described classically by
a real scalar function ψ(q, t). (For many more details, cf. Holland 1993, pp. 519-537;

40



Bohm and Hiley 1993, pp. 238-247, 286-295.)

(a): Space representation:— As usual in the transition from a quantum theory of a
fixed number of particles to a quantum field theory:

(i): the role of the coordinate q in the former, viz. being the value of a degree of
freedom, is taken over in the latter, by the field ψ; and:

(ii): the role of the former’s label i, viz. labelling the degrees of freedom, is taken
over by the continuous index x ≡ q.
So the configuration space is the infinite-dimensional space of possible configurations
ψ : IR3 → IR; and the wave-function is Ψ[ψ(x), t] = 〈 ψ(x) | Ψ(t) 〉 . That is: the
wave-function Ψ is a functional of the real scalar field ψ, and a function of the time t:
it is not a point-function of x.

Ψ obeys the Schrödinger equation ih̄ ∂Ψ/∂t = ĤΨ; where the Hamiltonian oper-
ator Ĥ is derived from the classical Hamiltonian by the usual canonical quantization
heuristic that “Poisson brackets become commutators”; so that in a representation in
which ψ(x) is diagonal, the momentum is given by the functional derivative −ih̄ δ/δψ.
(b): Quantum potential:— We make a polar decomposition of Ψ as Ψ = R exp(iS/h̄),
with both R = R[ψ(x), t] and S = S[ψ(x), t] being real functionals of the field. Then
the Schrödinger equation yields: a conservation law suggesting that R2 is a probability
density for the field; and a quantum analogue of the classical field’s Hamilton-Jacobi
equation. This quantum analogue adds to the classical equation a new potential term,
U [ψ, t] := − 1

2R

∫
d3x δ

2R
δψ2

. This is the field-theoretic quantum potential. Recalling

the transition to quantum field theory summarized in (a) above, we see that it is
analogous to the elementary quantum potential U (qi) for several particles labelled i:

U (qi) = − h̄
2

2m
1
R

(Σi ∇2i R); cf. (iii) of Section 7.1.1. We also see that the field-theoretic
quantum potential is inherently non-local.

(c): Guidance equation, interpretation:— The field-theoretic guidance equation, deter-
mining the motion of the postulated actual field configuration, takes the general form
∂ψ
∂t

=
δS[ψ(x),t]

δψ(x)
. Recalling again the summary in (a) above, this is clearly analogous to

Section 7.1.1’s guidance equation q̇i =
1
m
∇iS. For a neutral, spin 0, massless particle,

which would classically be governed by the wave equation 2ψ(x, t) = 0, the guidance

equation takes the form 2ψ(x, t) = −δU [ψ(x),t]
δψ(x)

.

This guidance equation, and the Schrödinger equation ih̄ ∂Ψ/∂t = ĤΨ from (a),
are the fundamental equations of motion for the total system which comprises both an
actual classical field configuration, and a wave-functional Ψ on the space of such con-
figurations. For our purposes, we need only two points about these two equations and
their interpretation. The first concerns the approximate status of Lorentz symmetry;
the second concerns the classical limit. (For more details about these points, cf. e.g.
Holland (1993, pp. 522-524).)

(i): Our use, from (a) onwards, of the parameter t amounts to postulating an
absolute simultaneity structure like that in Newtonian mechanics, Galilean quantum
mechanics and the non-relativistic pilot-wave approach. In particular, the constructions

41



summarized in (b) and (c) do not reveal t to have been “free up to” the rotation
of hyperplanes associated with a Lorentz boost: as happens for the time-evolution
in orthodox relativistic quantum theories. Indeed, the guidance equation and the
Schrödinger equation are not Lorentz-covariant, and the actual field ψ(x, t) is not a
Lorentz scalar. But just as in the non-relativistic case, we recovered orthodox quantum
mechanical results by averaging over possessed positions using | ψ |2 as the probability
density: so also here, one recovers the results of orthodox relativistic quantum theory
by averaging. In particular, Lorentz-covariance is an emergent symmetry: it fails at
the sub-quantum level of the field configurations, but holds good once we average over
field configurations.

(ii): For the pilot-wave approach, it is the right hand side term in the guidance
equation, i.e. the “quantum force” (generalized gradient of a potential), which is
responsible for the characteristic differences of the quantum theory from the classical
theory. Accordingly, we expect to obtain the classical limit when the magnitude and
gradient of the quantum potential are both negligible. And indeed, when this is so,
the guidance equation reduces to the classical wave equation 2ψ(x, t) = 0. But away
from the classical limit, the evolution of the field ψ(x, t) is in general highly non-local
(and non-linear). That is: how the field changes here and now depends on the present
value of the field arbitrarily far away.

We can now (at last!) summarize how this pilot-wave approach to relativistic quan-
tum theories violates relativistic causality. The situation is broadly like the action-at-
a-distance in the non-relativistic theory, due to the guidance equation and the quantum
potential. Namely, some fundamental equations of the theory are non-local with re-
spect to the theory’s absolute simultaneity structure; so that, setting aside doubts of
Dickson’s and Norton’s sort (cf. Section 7.1.2), an individual process involves instan-
taneous causation as in Newtonian gravitation. Or in terms of the light-cone structure
(which for the pilot-wave approach is emergent): individual processes involve spacelike
causation. On the other hand, when we average over these individual processes, we
recover the orthodox Lorentz-covariant formalism: in particular, micro-causality (Hol-
land 1993, p. 523) and Section 3.2.1’s two other formulations of relativistic causality—
but now interpreted as “mere” ensemble statements.

Finally, Section 3.3 raised the question how violations of relativistic causality avoid
contradictions, and announced that my examples would do so by forbidding a spacelike
zig-zag, “there and back”, into the causal past of an initial event—so that the usual
“bilking argument” for a contradiction could not get started.

For the pilot-wave approach, the situation is straightforward. In view of the un-
derlying absolute simultaneity structure, there is obviously no threat of a zig-zag, or
of a contradiction: no more than with the action-at-a-distance in Newtonian gravity.
This point is made (along of course with other points about non-locality) by e.g. Hol-
land, and Bohm and Hiley; (cf. Holland 1993, pp. 483-487, 494-495, 531-537; Bohm
and Hiley 1993 pp. 286-287; Kaloyerou 1993, p. 337). And in Section 5’s discussion
of the Drummond-Hathrell and Scharnhorst effects, the same point was made in a

42



somewhat generalized form: namely, that for superluminal propagation to be free of
contradictions, it is obviously enough that there be one frame of reference in which all
the propagation is forward in time.

7.2 The Newton-Wigner representation

I turn to my second example, which I will treat much more briefly. In effect, it develops
a point mentioned at the start of Section 7.1.1: that in orthodox non-relativistic quan-
tum theory, wave-functions propagate instantaneously. The point now is: this sort of
propagation also occurs in a part of conventional relativistic quantum theory—namely
in the Newton-Wigner representation.

The Newton-Wigner representation applies equally well to a relativistic quantum
theory of a fixed number of particles, and to a quantum field theory. It provides a basis
of the (pure) state space (the wave-functions) consisting of strictly localized states. At
first sight, this seems analogous to the Dirac delta-functions, or more generally wave-
functions with compact spatial support, of the non-relativistic theory. But these states,
and the spectral projectors of the corresponding position quantity, have some striking
features.

(i): The states propagate superluminally. Indeed: although there is no absolute
simultaneity structure, they propagate instantaneously in the following sense. If at
time t = 0 in some inertial frame, a Newton-Wigner wave-function ψ(0) has compact
spatial support (i.e. is non-zero only in a compact spacelike patch, Σ say, of the t = 0
hyperplane), then at all later times t > 0, no matter how small, ψ(t) is non-zero
throughout all space. Agreed, the great majority of the amplitude lies in the future
light-cone of Σ; and as t grows, the percentage of this majority rapidly tends to 100
percent. Nevertheless there is a “superluminal tail”.

(ii): Two spectral projectors associated with spacelike related spatial regions on
two different spacelike hyperplanes will not commute.

These features seem to imply, respectively, that:
(i’): One could signal superluminally by “releasing” a Newton-Wigner wave-packet

initially confined to a compact region of space. (And here ‘signalling’ could be taken
in a strong sense, viz. as a non-vanishing probability of a detection, triggering some
pre-arranged event, at spacelike separation.)

(ii’): A (sharp, von-Neumann-style) non-selective measurement of one such projec-
tor can influence that statistics of the measurement of the other.

These features have been analysed, and the Newton-Wigner formalism much devel-
oped, especially by Fleming. Most recently, his (2003, 2004) include replies to recent
literature; (the features are also surveyed in Fleming and Butterfield (1999, pp. 108-
130, 153-162)). His work emphasises (among many other points) that:

(a): Though these features, and the Newton-Wigner formalism, are little known,
they are not heterodoxies: they form part of the conventional framework of relativistic
quantum theories.

43



(b): Various arguments can be given that (i’) and (ii’) are in fact not implied (and
so causal loops are avoided). I will not go into these arguments: (Fleming and Butter-
field (1999, p. 157) gives some references). Admittedly, they are partial, reflecting our
lack of a developed theory of measurement for relativistic quantum theories. So here
it must suffice to make three points.

First: the general theme, that violations of relativistic causality may well be indica-
tive of future physics, also occurred in Section 5. Second: I stress that the ingredients
of these partial arguments are very different from those in Sections 7.1.3.1 and 5. For
example, one ingredient is to associate signalling with the group velocity of a wave
(there being no superluminal group velocities in the Newton-Wigner representation);
and another is to relativize the notion of localization to a hyperplane.

Third: in view of Section 3.2’s formulation of relativistic causality as micro-causality,
i.e. commutativity of spacelike operators, I should stress two general points of “reas-
surance”, about the non-commutation in (ii) above.

(1): If the spatial regions associated with the projectors are on the same hyper-
plane, then the projectors are of course orthogonal, representing the fact that a single
particle localized in one region has zero probability to be found in the other—and so
commute.

(2): The non-commutation in (ii) is consistent with micro-causality. For the ex-
pression of the Newton-Wigner operators, in terms of the operators that are the topic
of micro-causality, involves an integral over an entire spacelike hyperplane. The non-
commutation is thus “explained” by the existence of timelike paths connecting portions
of the integrands in the integrals occurring in the definitions of both spectral projec-
tors. (Of course, the fact that the Newton-Wigner representation and the conventional
one are related by an unbounded integral means that the senses in which the Newton-
Wigner operators, and the conventional ones, are “associated” with regions are distinct.
Accordingly, elucidating these different senses is a main theme of the recent literature:
cf. Fleming (2004, Sections 3c, 4d and 5b) and references therein.)

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