The Non-Equivalence of Einstein and Lorentz

Clara Bradley

Abstract

In this paper, I give a counterexample to a claim made in Norton
(2008) that empirically equivalent theories can often be regarded as the-
oretically equivalent by treating one as having surplus structure, thereby
overcoming the problem of underdetermination of theory choice. The case
I present is that of Lorentz's ether theory and Einstein's theory of spe-
cial relativity. I argue that Norton's suggestion that surplus structure is
present in Lorentz's theory in the form of the ether state of rest is based
on a misunderstanding of the role that the ether plays in Lorentz's theory,
and that in general, consideration of the conceptual framework in which
a theory is embedded is vital to understanding the relationship between
di�erent theories.

1 Introduction

The existence of distinct but empirically equivalent theories is seen as a threat
to the scienti�c realist. This is because empirically equivalent theories are of-
ten thought to be epistemically equivalent - that is, the evidence supports both
theories equally - which has the consequence that the choice between them is un-
derdetermined by the evidence. Many realists have responded to this challenge
by arguing that empirical equivalence is not su�cient for epistemic equivalence.
In particular, empirically equivalent theories may not be equally con�rmed by
the evidence. One notable advocate of such a position is Glymour (1977) who
argues that spacetime theories which are empirically equivalent can receive dif-
ferent degrees of con�rmation - if one spacetime theory has structure over and
above the other, then that theory will be less supported by the evidence because
it makes additional, unnecessary ontological claims. This avoids the problem
facing the realist since if one theory is epistemically preferred to other empiri-
cally equivalent theories, then there is no underdetermination of theory choice.

There is another route for the realist. This is to argue that the cases the
anti-realist gives as examples of underdetermined theories do not in fact corre-
spond to distinct theories, but to one theory formulated in di�erent ways, and
therefore the underdetermination is dissolved. Norton (2008) follows this line,
by arguing that instead of thinking, as per Glymour, that the theory with more
structure is con�rmed to a lesser degree, this extra structure should instead be
interpreted as surplus structure. In this paper, I �rst present Norton's argument
for why this allows us to treat empirically equivalent theories as theoretically

1



equivalent. I then argue that despite being a case that Norton uses to support
his argument, the case of Lorentz's ether theory and Einstein's theory of special
relativity does not �t his framework because of the particular role that the ether
plays in Lorentz's theory which means that it cannot be regarded as surplus, and
therefore the issue of underdetermination remains. Finally, I look at some con-
sequences of my argument for understanding the change from Lorentz's theory
to Einstein's as well as the comparison of theories more generally.

2 Norton's Argument

I take Norton's argument to be the following:

P1. Theories that are empirically equivalent are often structurally simi-
lar.

P2. Consider two empirically equivalent theories, T1 and T2. There are
two cases:

a) T1 and T2 are structurally equivalent.

b) T1 and T2 are structurally inequivalent, but one (say T1) has extra struc-
ture over the other (T2) which can be regarded as surplus.

P3. Theories are equivalent when they are structurally equivalent.1

C. Therefore, in either of the two cases above, the theories can be re-
garded as formulations of a single theory.

For case (a), the conclusion follows simply from P3. For case (b), the conclusion
follows because if we regard the extra structure in T1 as surplus - that is, it
plays no role of the theory of which it is part - and take in to account only
the physically signi�cant structure, then we have left structurally equivalent
theories.

P1 is not particularly controversial since it makes a weak claim and it doesn't
seem unreasonable to think that empirically equivalent theories will at least
have some structure in common. The identi�cation of theoretical equivalence
with structural equivalence given in P3 is more controversial. First, in order to
compare the structure of two theories, one �rst has to characterise the structure
of a theory. Norton is assuming that there is a natural way to do this. Second,

1There is a wide literature on theoretical equivalence and Norton is not clear where he
stands in this. In particular, Norton suggests that equivalence is both given by `intertrans-
latability' and by structural equivalence. However, these are often not thought of as compatible
views - the former is a syntactic relation between theories, using notions such as de�nitional
equivalence, which Glymour (1970, 1977) endorses, while the latter is a relation between the
mathematical structures of theories, which has been made more precise using the tools of
category theory (see Halvorson (2016); Weatherall (2016a,c, 2017)). Since Norton wants to
use comparison of mathematical structure to determine whether theories are equivalent, I take
him to be endorsing a form of structural, rather than de�nitional, equivalence.

2



there are di�erent ways of comparing the structure of theories and depending
which criteria one uses, one might reach di�erent conclusions regarding which
theories are equivalent. Finally, one might think that structural equivalence is
not a strong enough criterion for theoretical equivalence - one might also want to
preserve something over and above structure, such as representational content.2

While the task of characterising and comparing the structure of theories in
the general case is problematic, for the examples I look at in this paper I take
there to be a relatively natural way of doing so. In particular, I will look at
theories that are characterised in terms of their spacetime structure, which can
be compared by looking at their automorphism groups. Instead, I will argue that
the main problem with Norton's argument is that the two cases of empirically
equivalent theories he considers, (a) and (b) above, do not cover all cases of
empirically equivalent theories. I will present an example of two theories that
are empirically equivalent, where there is a clear sense in which one has more
structure than the other, but where it is not the case that one can think of
the theory with more structure as having surplus structure. This is the case of
Lorentz's ether theory and Einstein's theory of special relativity.

However, before I do so, I am going to present an example where Norton's
argument does seem to apply neatly. This is the case of Newtonian gravita-
tion - Newton's laws plus Newton's law of gravitation - set in Newtonian and
Galilean spacetime respectively. Galilean spacetime consists of the structure
(M,∇, tab,hab), a manifold isomorphic to R4 with a derivative operator, a tem-
poral and spatial metric.3 This corresponds to a foliation of spacetime into
simultaneity hypersurfaces which each have the structure of Euclidean space.
Newtonian spacetime consists of (M,∇, tab,hab,ξa), which adds to Galilean
spacetime a privileged observer represented by a timelike vector with �xed 4-
velocity ξa, which we can interpret as what Newton thought of as `absolute
space', giving a state of absolute rest, and therefore a notion of absolute veloc-
ity. From the way we have presented these spacetime structures, it is clear that
Newtonian spacetime simply consists of Galilean spacetime + X, where X is
some structure - namely, the structure corresponding to ξa. Therefore, if we re-
gard ξa as surplus structure then we can regard the two theories as structurally,
and thus according to Norton, theoretically equivalent.

A more rigorous way of seeing that Newtonian spacetime has more struc-
ture than Galilean spacetime is by comparing the automorphism groups of both
structures. In this, we follow Barrett (2015), who suggests the following struc-
tural comparison:

(SYM*) A mathematical object X has more structure than a mathematical
object Y if Aut(X)⊂Aut(Y ).

2See Nguyen (2017) and Co�ey (2014) for views on theoretical equivalence that aim to go
beyond formal accounts.

3The operator ∇ is a �at derivative operator on R4 compatible with tab and hab. Both tab
and hab are smooth, symmetric �elds on M, where tab has signature (1,0,0,0) and h

ab has
signature (0,1,1,1), and they obey the orthogonality condition habtbc = 0. For details, see
Malament (2012, 4.1).

3



The symbol ⊂ is taken to be non-inclusive.
The automorphisms of a spacetime are di�eomorphisms which preserve the

structure on the spacetime. So, for example, the automorphisms of Galilean
spacetime, (M,∇, tab,hab), are di�eomorphisms which preserve tab, hab, and
∇. These turn out, not surprisingly, to be the Galilean transformations, given
by spatial and time translations, rotations, re�ections, and Galilean velocity
boosts.

The automorphisms of Newtonian spacetime, (M,∇, tab,hab,ξa) are then
di�eomorphisms which preserve the Galilean spacetime structure, but also pre-
serve ξa. This means that the transformations which are automorphisms of
Newtonian spacetime will be more restrictive than those of Galilean spacetime.
In fact, they correspond to spatial and time translations, rotations and re�ec-
tions4, but not Galilean velocity boosts. So we see that it is true that:

Aut(Newt) ⊂Aut(Gal)

Here, Newt and Gal correspond to Newtonian and Galilean spacetime re-
spectively. Newtonian spacetime therefore has more structure than Galilean
spacetime by (SYM*). And this `more' structure clearly corresponds to that
structure ξa, picking out a privileged frame of reference. Therefore, if we regard
ξa as surplus structure, we are in e�ect equivocating between models in New-
tonian spacetime related by a Galilean velocity boost, and return something
which is struturally equivalent to Galilean spacetime.

I will now turn to the case of Lorentz and Einstein, and see how it di�ers to
this case.

3 Background to Lorentz's and Einstein's Theory

In order to understand Lorentz's and Einstein's theory, I will consider how
Maxwell's equations can be interpreted in di�erent spacetime structures. First,
Minkowski spacetime is given by (M,gab), where M is the manifold R4 and gab
is a complete, �at and smooth metric with signature (1,3).5 The metric gab gives
Minkowski spacetime `lightcone structure'. The automorphisms of Minkowski
spacetime are the Lorentz transformations, which include the analagous trans-
lations, rotations and re�ections to the Galilean case, as well as Lorentz boosts,
which are analagous to Galilean boosts, except they do not preserve the spacelike
vectors as Lorentz boosts do.6 In Minkowski spacetime, Maxwell's equations ad-
mit the following, observer independent formulation, where the electromagnetic
�eld is given by the smooth, antisymmetric tensor �eld Fab, and the charge-
current density is given by the vector �eld Ja:7

4As long as these rotations and re�ections are about ξa.
5For details, see Malament (2012, 2.1).
6Sometimes the term `Lorentz transformations' is used to refer just to Lorentz boosts, but

I will use the phrase to refer to all transformations that are automorphisms of Minkowski
spacetime.

7The demonstration in this section follows that of Weatherall (2016b) and Malament (2012,
2.6).

4



∇[aFbc] = 0 (1)

∇nFna = Ja (2)

To understand how the electric and magnetic �elds are related to Fab , we
must consider the way in which they appear to di�erent observers. In Minkowski
spacetime, the electric �eld relative to a observer O at a point p with four-
velocity va is given by:

Ea = Fanv
n (3)

The magnetic �eld is given by:

Ba =
1

2
�abcdvbFcd (4)

�a1...anis an orientation tensor, which gives a choice of `handedness' or right-
hand rule. It follows from (3) and (4) that the electric and magnetic �elds are
always orthogonal to va. We also need to de�ne the electric charge density µ,
and 3-current density ja relative to O. These are given by:

µ = Java (5)

ja = h
a
bJ

b (6)

hab is the spatial projection tensor at the point determined by v
a. We can

also de�ne Fab in terms of the electric and magnetic �elds, as:

Fab = 2E[avb] + �abcdv
cBd (7)

Relative to a choice of constant timelike vector �eld va (∇avb = 0), Maxwell's
equations can then be given in the following form, where D is the derivative
induced on hypersurfaces orthogonal to va (see Malament (2012, 1.10)):

DbB
b = 0

�abcDbEc = −vb∇bBa
(8)

DbE
b = µ

�abcDbBc = v
b∇bEa + ja

(9)

Equations (8) and (9) correspond to Equations (1) and (2) respectively.
It is important to note that if one considers a di�erent observer in Minkowski

spacetime, represented by a di�erent choice of timelike vector, Maxwell's equa-
tions in the form of (8) and (9) still hold. This is a consequence of the fact
that the equations are invariant under the Lorentz transformations, which are
automorphisms of Minkowski spacetime. Einstein's theory of special relativity
takes spacetime to have the structure of Minkowski spacetime, and hence this
is the interpretation of Maxwell's equations in Einstein's theory.

5



If we try to interpret these equations in Galilean spacetime (M,∇, tab,hab),
problems arise because the spacetime symmetries are given by the Galilean
transformations, where Galiean boosts preserve the spacelike vectors and not
the timelike vectors, rather than the Lorentz transformations, where Lorentz
boosts preserve neither spacelike nor timelike vectors. This means that if we
consider a new observer, related to the �rst by a Galilean boost, Maxwell's
equations in the form of (8) and (9) do not hold except for the case of constant
electric and magnetic �elds, with no sources.8 Therefore, we cannot successfully
interpret Maxwell's equations in Galilean spacetime.

In order to overcome this problem, we can adopt a privileged state of motion
in the framework of Galilean spacetime and state that Maxwell's equations
hold only in this frame of reference. But we can understand the adoption of a
privileged state of motion as corresponding to moving to Newtonian spacetime,
(M,∇, tab,hab,ξa). In this setting, ξa can be taken to represent the `ether state
of rest'.9

We can now understand Maxwell's equations by substituting ξa for va in
equations (8) and (9), and de�ne Fab as:

Fab = 2E[atb] + �abcdξ
cBd (10)

Here, Ea = h
abEb. We then recover Maxwell's equations in the following

form:

∇[aFbc] = 0 (11)

(ξaξn −han)(ξbξc −hbc)∇bFcn = Ja (12)

We can see explicitly that the ether state of rest, represented by ξa, is required
to make sense of Maxwell's equations in Newtonian spacetime.

Using the resources of Newtonian spacetime, we can also de�ne an inverse
Minkowski metric:

gab = ξaξb −hab (13)

In Minkowski spacetime, Fab and F
ab are related as follows:

Fab = gacgbdFcd (14)

and so equation (12) can be written in Minkowski spacetime as:

gangbc∇bFcn = Ja (15)
8For proof, see Weatherall (2016b).
9Although the ether state of rest plays an analogous role to absolute space in Newtonian

Gravitation, it should not be identi�ed with it. This is because the ether need not be said to
be `absolutely' at rest, it merely needs to be one state of motion in which Maxwell's equations
hold.

6



This makes explicit the way in which the metric appears the same way in both
formulations of Maxwell's equations. This gives the sense in which they `say the
same thing'.

I take Lorentz's theory to correspond to Maxwell's equations interpreted in
Newtonian spacetime. Lorentz of course would not have thought of his theory
in this way, since he did not have the resources to do so. However, it allows us
to see clearly the connection between the introduction of the ether state of rest
with the endorsement of the Newtonian picture of space and time.

This gives the basis of both theories. But we need to know more than
Maxwell's equations to adjudicate between them. In particular, we need to
consider the behaviour of matter. In Einstein's theory, the laws governing matter
are Lorentz invariant. But Lorentz was born in the Newtonian tradition, where
Galilean invariant Newton's Laws govern matter. So the reason we would want
to endorse Lorentz's theory over Einstein's is if we characterize inertial frames
of reference by the Galilean frames of reference - and this in turn would be
if we believe that Newton's Laws hold in all frames.10 If inertial frames of
reference are Galilean one's then, by the argument above, we are forced to adopt
Newtonian spacetime. But this theory on it's own - a combination of Newton's
Laws and Maxwell's equations with moving observers charcterized by Galilean
frames of reference - is inconsistent with experimental results, since it implies
that we would be able to detect motion with respect to the ether, which several
experiments failed to detect. What Lorentz did was construct a theory that
maintained the Newtonian conception of space and time but was consistent with
these experiments. He introduced a set of coordinate transformations, under
which Maxwell's equations remained invariant. This is called his `theorem of
corresponding states':

If there is a solution of the source free Maxwell equations in which the
real �elds E and B are certain functions of x0 and t0, the coordinates
of S0 and the real Newtonian time, then there is another solution
of the source free Maxwell equations in which the �ctitious �elds E′

and B′ are those exact same functions of x′ and t′, the coordinates
of S and the local time in S. - Janssen (1995, ch.3, p.48)

The frame S0 is the rest frame of the ether, the frame S one moving through
the ether. What this theorem says is just that Maxwell's equations are invariant
under some coordinate transformations, given by:11

x′ = γ(x0 −vt0) y′ = y z′ = z t′ = γ[t0 − (v/c2)x0]

The factor γ is de�ned as
√

1−v2/c2
−1
. This is just a mathematical theo-

rem. In order to have any empirical consequences, Lorentz had to add a physical

10It might also be if we think that there is a notion of absolute simultaneity, but it seems
the reason we would want to hold this is if it is implied by the laws.

11Originally, Lorentz included a factor on the right hand side of these equations to stand for
a possible transverse deformation, but this was set equal to 1 in 1904 for dynamical reasons.
These coordinate transformations are what we recognise now as a Lorentz boost.

7



assumption to the e�ect that the �elds E′ and B′ are those �elds that are pro-
duced in the frame S. This is obtained with what is called the `generalised
contraction hypothesis':

If a material system, with a charge distribution that generates a
particular electromagnetic �eld con�guration in S0, a frame at rest
in the ether, is given the velocity v of a Galilean frame S in uniform
motion through the ether, it will rearrange itself so as to produce the
con�guration of particles with a charge distribution that generates
the electromagnetic �eld con�guration in S that is the correspond-
ing state of the original electromagnetic �eld con�guration in S0. -
Janssen (1995, ch.3, p.47)

In other words, corresponding states - states related by the coordinate trans-
formations given in the theorem of corresponding states - physically transform
in to one another. It is important to emphasise that this is a physical (dynam-
ical) transformation applied to Galilean frames, since Lorentz held on to the
Galilean transformations as being the true coordinate transformations between
observers. The coordinates x′ and t′ are auxiliary coordinates that represent
the coordinates in which Maxwell's equations hold, not the true spacetime co-
ordinates x and t of the moving frame. From these two posits, Lorentz was able
to account for the negative result of any ether drift experiment.

It turns out that what is needed to ensure that the generalised contraction
hypothesis holds is that all laws, including those governing matter, are Lorentz
invariant. Of course, Lorentz needed to derive that this was true, or at least
justify that it was the case in order to have a compelling theory - I refer to
Janssen (1995) for such dynamical arguments that Lorentz gave. The important
point is that for Lorentz, this fact about the Lorentz invariance of all laws is a
coincidence - it is because of the behaviour of matter through the ether that it
holds. To ensure full empirical equivalence with Einstein's theory, we also need
that the rods and clocks of a moving observer measure the auxiliary coordinates
and local time that is made reference to in the theorem of corresponding states.
In other words, moving Galilean frames must appear `as if' they are Lorentzian
frames.

In what follows, when I speak of Lorentz's theory I will take it to be that
which is empirically equivalent to Einstein's. Lorentz himself did not endorse
this theory until after 1905, when Einstein published his theory.12And as men-
tioned before, this way of looking at Lorentz's theory in terms of Newtonian
spacetime is an anachronistic way of doing so. However, I think that it captures
the essential features of Lorentz's theory, and it is in this form that is particu-
larly philosophically interesting, since we can compare Lorentz's and Einstein's
theory directly in terms of their spacetime structure.

12Poincaré earlier suggested that observers will register the `local time' of Lorentz's theory
rather than the true Newtonian time, but Lorentz only accepted this later (see Janssen (1995,
3.5.4)).

8



4 Lorentz-Einstein as Counterexample

Barrett (2015) shows that under his standard of comparison (SYM*), Minkowski
spacetime has strictly less structure than Newtonian spacetime. That is, the
automorphism group of Newtonian spacetime is a strict subset of those of
Minkowski spacetime. We therefore seem to have a case where Norton's ar-
gument should apply, and Newtonian spacetime can be interpreted as having
surplus structure resulting in the equivalence of Lorentz and Einstein's theory.
Norton himself introduces this example, suggesting that the surplus structure
in Lorentz's theory is the ether state of rest. This is clear in the following quote:

The mainstream of physics found Lorentz's view implausible and
decided that his ether state of rest was physically super�uous struc-
ture, so that the two theories were really just variants of the same
theory - Norton (2008, p.36)

However, by comparing the structure of both theories, it is clear that the ether
state of rest is not the surplus structure that Norton is looking for. This is
because the ether state of rest corresponds to that privileged reference frame
ξa, which if removed gives back Galilean and not Minkowski spacetime, in which
(as shown in Section 3) the laws of electromagnetism can not be formulated.
In other words, simply removing the ether state of rest from Lorentz's theory
neither results in a sensible theory, nor one that is structurally equivalent to
Einstein's theory.

It is not clear what one could remove from Newtonian spacetime in order
to recover Minkowski spacetime. What one is doing is removing the elements
(tab,h

ab,ξa) and replacing them with gab, which, as we have seen, is a combina-
tion of the elements ξa and hab. We can understand it physically as removing
the splitting in to separate notions of space and time given by the elements of
Newtonian spacetime, and replacing them with some other notion where simul-
taneity becomes relative.

Another way of seeing this is that Galilean boosts map timelike vectors
to other timelike ones while preserving the spacelike vectors, whereas Lorentz
boosts map both timelike and spacelike vectors to di�erent timelike and space-
like vectors. So Galilean boosts preserve the simultaneity structure of Newto-
nian spacetime, while Lorentz boosts do not. Consequently, while both Galilean
and Minkowski spacetime are ways of removing the preferred state of rest from
Newtonian spacetime and therefore the notion of absolute velocity, Galilean
spacetime corresponds to doing just that, whereas Minkowski spacetime cor-
responds to something di�erent - it removes the preferred state of rest, while
also modifying the notion of simultaneity. This results in a completely di�erent
spacetime structure with a di�erent interpretation.

So why might Norton have thought that the ether state of rest is surplus
structure? One reason is that, like absolute space, it is in principle undetectable
in Lorentz's theory. However, it is undetectable for a di�erent reason than
absolute space, and this di�erence corresponds to the reason why absolute space,
and not the ether state of rest, can be regarded as surplus structure.

9



This can be seen by considering the symmetry argument that one can give
for why we ought to regard absolute space as being surplus. Given a model
of Newtonian Gravitation in which the laws hold, a transformation that adds
a constant velocity to all particles gives another model in which the laws also
hold and which is observationally equivalent to the original. But under such
a velocity boost in Newtonian spacetime, the element which corresponds to
absolute space - the distinguished vector ξa- is not invariant, and therefore
solutions to Newton's laws related by a constant velocity boost correspond to
distinct, but observationally indistinguishable, state of a�airs. In particular,
what is undetectable is precisely the state of rest represented by ξa. Therefore,
the state of rest plays no role in the empirical consequences of the theory, and
ought to be regarded as super�uous. And indeed, we can equivocate between
models related by a Galilean velocity boost by removing the state of rest, which
takes us to Galilean spacetime.

In Lorentz's theory, the situation is more subtle. In the rest frame of the
ether, the laws governing matter are Newton's laws. Since Newton's laws are
invariant under the Galilean transformations, this implies that inertial observers
are related by Galilean boosts. But under a Galilean velocity boost, we do not
get a solution to Maxwell's equations as a function of the Galilean coordinate x
and t. So such a transformation does not map solutions to solutions, as it did
in the previous example. But we do get an observationally equivalent state of
a�airs in the following sense - as a result of dynamical e�ects (the interaction
of matter with the ether), Maxwell's equations (and all other laws) hold as a
function of x′ and t′ which are the coordinates that moving observer measures,
and so observers are unable to tell whether they are at rest or moving with
respect to the ether.13

Therefore, although we can conclude that the ether is undetectable, this is
not for the same reason as absolute space is in Newtonian gravitation, since
Galilean velocity boosts in Lorentz's theory do not take solutions to solutions
(as they do in Newtonian gravitation). Importantly, in Lorentz's theory, the
ether is actually required in order to ensure that it is undetectable, since it is
the interaction with the ether that provides the physical underpinning for the
generalized contraction hypothesis, and thus the claim that moving observers
can never detect the presence of the ether.14

This demonstrates that it is not undetectability on its own that makes some-
thing surplus. For some structure to be surplus, it must be that it can be re-
moved to result in an empirically adequate theory. This depends not only on

13One might respond that if we consider a Lorentz velocity boost rather than a Galilean
velocity boost, then we do get an analogous argument - such a boost maps solutions of
Maxwell's equations to solutions and yet the ether state of rest di�ers. However, it is not
clear that we can say that such models are observationally equivalent, since in Lorentz's
theory such a transformation does not have physical meaning, other than when coupled to
Galilean inertial observers.

14Lorentz tried to not make any speci�c assumptions about the makeup of the ether. One
might argue therefore that the ether is not actually important to Lorentz's theory. However,
the fact that he doesn't make any assumptions about how the ether is made up doesn't mean
he doesn't think the ether is what justi�es the dynamical e�ects.

10



which objects are undetectable, but what role the objects in a theory play, which
depends on the conceptual framework of the theory as a whole. The ether is in
principle undetectable in Lorentz's theory but it plays an ineliminable role in
the theory, and therefore it cannot be thought of as surplus.

Another way we could have seen that the ether state of rest is not surplus in
Lorentz's theory is by noticing that Maxwell's equations, written in Newtonian
spacetime, make explicit reference to the structure corresponding to the ether
state of rest (Equation 12). This is unlike Newton's laws written in Newtonian
spacetime, where the laws only make explicit reference to structure found in
Galilean spacetime. For example, Newton's second law in Newtonian spacetime
can be expressed as:

Fa = mvn∇nva

We do not need a notion of spatial length for timelike vectors (which the
structure ξa gives) to make sense of this equation. Of course, we have shown
that we can reinterpret a quantity in Maxwell's equations written in Newtonian
spacetime to give Minkowski spacetime, and in this sense Maxwell's equations
don't require this bit of structure, but we don't need to reinterpret, or change
the form of, anything in Newton's second law to recover Galilean spacetime.
In the conceptual framework of Lorentz's theory, the ether state of rest is an
essential piece of structure.

I have argued that Norton was wrong to think that the ether state of rest
is surplus structure in Lorentz's theory. This does not rule out that one can
still think of Lorentz's theory as having surplus structure. One might argue
that if we think of Minkowski spacetime as a substructure of Newtonian space-
time, which we can do on the basis that Aut(Newt) ⊂Aut(Mink), then we
can think of Minkowski spacetime as resulting from equivocating between those
models related by a Lorentzian velocity boost. The di�erence between the case
of Galilean/Newtonian spacetime and the case of Minkowski/Newtonian space-
time - the fact that in the former, Newtonian spacetime is obtained by simply
removing the state of rest, whereas in the latter, Newtonian spacetime is ob-
tained by removing the state of rest but also modifying other structure - might
just be due to the way we wrote down the spacetime structures.

The problem with this argument is that although one might be able to
think of Newtonian spacetime as Minkowski spacetime + X (i.e. that New-
tonian spacetime is structurally equivalent to something that corresponds to
Minkowski spacetime + X), this does not say anything about how one interprets
the spacetime structure. What Lorentz's theory is, under the characterization
given above, is that theory of Maxwell's equations corresponding to a preferred
state of rest in the framework of Galilean spacetime.

11



5 Consequences

5.1 Earman's Principle

Earman (1989, ch.3) argues that there is an important methodological principle
that ought to determine what the structure of spacetime is:

The symmetries of the dynamical laws should be the same as the
symmetries of spacetime.

In order to understand this principle, we must know what it means to be a
symmetry of the dynamical laws and a symmetry of spacetime. Symmetries
of spacetime correspond to the automorphisms of the spacetime structure in
question. The dynamical symmetries are not so easy to specify precisely - they
correspond to transformations between solutions of the laws of motion. How-
ever, it can't be any such transformation, since this would imply that a map
between arbitrary solutions counts as a symmetry.15 It must be a transforma-
tion that preserves the physically signi�cant structure, but the problem is: how
do we know what the physically signi�cant structure is, other than through the
spacetime symmetries? In the context of spacetime theories, some people have
argued that there is a relatively natural way of picking out the dynamical sym-
metries - the idea is to distinguish between the �elds that represent spacetime
structure from the �elds which represent the matter content of spacetime, and
then de�ne a dynamical symmetry in terms of di�eomorphisms which act only
on the matter �elds and which maps solutions to solutions (See Earman (1989)
and Pooley (2013)). Whether this will work for all spacetime theories is not
clear, but it su�ces for the simple cases at hand.16

The argument Earman (1989) gives for all dynamical symmetries being
spacetime symmetries is that a dynamical symmetry without a spacetime sym-
metry would indicate some surplus structure in the structure of spacetime. This
is because there would be two models of the theory which correspond to dis-
tinct spacetime possibilities (in other words, they are non-isomorphic) which
nonetheless are observationally equivalent. For example, the theory of New-
tonian gravitation in Newtonian spacetime has Galilean velocity boosts as a
dynamical symmetry but not a spacetime symmetry. This indicates that New-
tonian spacetime has surplus structure.

The argument for all spacetime symmetries being dynamical symmetries is
that if the laws are generally covariant, then it follows from a transformation
being a spacetime symmetry that it is also a dynamical symmetry. As we saw
in Section 3, it is not possible to interpret the generally covariant version of
Maxwell's equations in a spacetime structure - Galilean spacetime - which has
a spacetime symmetry which is not a dynamical symmetry.

15See Belot (2013) for a more detailed argument against such a notion of symmetry.
16There have also been attempts to generalise this argument to cover all kinds of theories,

for example by Dasgupta (2015) and North (2009). However, it becomes even more di�cult
to specify the dynamical symmetries in the general case.

12



Applying this to the case of Lorentz's and Einstein's theory, we �nd the
following. In Einstein's theory, there is a perfect match between spacetime and
dynamical symmetries - both are the Lorentz symmetry transformations.

In Lorentz's theory, the spacetime symmetries are the Newtonian ones while
the dynamical symmetries are the Lorentz transformations, since the laws in
Lorentz's theory are invariant under these transformations. Hence all spacetime
symmetries are dynamical symmetries, but the group of dynamical symmetries
is larger than the group of spacetime symmetries. If we follow Earman, this
would imply that Lorentz's theory has surplus structure. The problem with
Earman's argument in this case is that it doesn't take in to account how the
dynamical symmetries are interpreted in Lorentz's theory. This is similar to the
reasoning that we gave in the last section for why the ether is undetectable but
not surplus in Lorentz's theory: Lorentz took moving bodies to be characterised
by the Galilean transformations, since Newton's Laws, which were held to govern
matter, are Galilean invariant. But since Maxwell's equations do not hold in
these frames, a dynamical explanation was needed to explain their observations.
The interaction between moving bodies and the ether was meant to provide this
explanation. The resulting laws that did hold in these frames of reference were
forced (in accordance with negative results from ether drift experiments) to be
Lorentz invariant ones.

So while it is true that the laws are invariant under the Lorentz transforma-
tions, this is a result of dynamical e�ects in Galilean frames of reference, and not
because Lorentzian frames correspond to the frames of reference for moving ob-
servers. Consequently, Lorentz's theory does fail the test of symmetry matching
in the sense that the laws are invariant under a wider range of transformations
than the spacetime structure. However, this is not due to Lorentz's theory hav-
ing surplus structure, it is due to the peculiarity of Lorentz's theory that all laws
transform in the same way in Galilean frames of reference. In other words, the
Lorentz transformations only have meaning in Lorentz's theory when combined
with Galilean frames of reference and the dynamical e�ects of the ether, and
therefore one needs to maintain the structure of Newtonian spacetime in order
to make sense of this dynamical symmetry in the context of Lorentz's theory.

This example shows how Earman's principle, by making reference only to
formalism and not to the interpretation of theories, misses subtleties in compar-
ing theories. While Lorentz's theory does fail the test of symmetry matching,
this has to do with the whole conceptual framework on which the theory rests,
and therefore one can't say that the problem with Lorentz's theory is simply
that the spacetime and dynamical symmetries don't match, and that this means
that the theory has surplus structure. If we want to maintain that Einstein's
theory is superior because it satis�es Earman's principle, we have to say what
it is about the matching of symmetries in this theory that makes it superior.
We will look at some arguments in the following section.

13



5.2 Why did Einstein Supersede Lorentz?

If we want to maintain that Einstein's theory is a distinct, superior theory to
Lorentz's, then we have to provide reasons for why Einstein's theory is to be
preferred epistemically. The arguments given in this paper imply that it cannot
be because Einstein's theory doesn't have surplus structure.17

One reason one might give is that Einstein's theory has less structure,
since Minkowski spacetime has strictly less structure than Newtonian space-
time. Therefore, by Occam's razor, we ought to prefer Einstein's theory. How-
ever, there are reasons to be unsatis�ed with this argument. Firstly, it doesn't
explain historically why Einstein's theory was preferred - the comparison of
spacetime structure has only been made relatively recently. Additionally, sim-
plicity of spacetime structure is not all that we might care about in a theory -
there are other theoretical virtues such as uni�cation and explanation. It is also
only one source of simplicity in a theory.

It might be that the simpler structure of Minkowski spacetime means that
other theoretical virtues are met, which could be recognised historically as
important. Michel Janssen argues that one virtue of Einstein's theory over
Lorentz's is explanatory virtue18 - two observationally equivalent states of a�airs
which receive very di�erent explanations in Lorentz's theory receive a unifying
explanation in Einstein's theory. For example, consider the case of a moving
magnet inducing a current in a stationary wire. This receives a di�erent expla-
nation in Lorentz's theory from the case of the same wire moving through the
magnetic �eld of the magnet when stationary, even though the current produced
is the same. This is because in one case the magnet is stationary with respect
to the ether and in the other the wire is, so they correspond to distinct situa-
tions. On the other hand, in Einstein's theory, the two cases are understood as
the same situation just looked at from di�erent perspectives.19 Additionally, in
Lorentz's theory the Lorentz invariance of all dynamical laws is in need of expla-
nation on a case by case basis - it is just a coincidence that all matter behaves
in the same way when they move through the ether. In Einstein's theory, on the
other hand, the Lorentz invariance of the laws is explained by the Minkowski
metric - Lorentz invariance is a symmetry of the spacetime, so the fact that the
observations are the same in Lorentzian frames of reference can just be thought
of as re�ecting `normal spatio-temporal behaviour' (Janssen (1995, ch.4, p.7)).
In other words, inertial reference frames naturally become the Lorentzian one's.
This argument suggests that the matching of symmetries in Einstein's theory
means that the dynamics can now be explained by the spacetime structure,

17The question of why Einstein superseded Lorentz was raised in Zahar (1973a,b). He argues
that while Special Relativity seems to have greater heuristic power than Lorentz's theory, the
crucial reason is that Special Relativity continued into General Relativity, which had novel
empirical success. While I do not want to deny that this is a reason for why Einstein's
theory was eventually accepted, I am concerned here with the question of why the heuristic
of Einstein's theory is superior to Lorentz's.

18See Janssen (2002a,b, 2009).
19This example is taken from Janssen (2002b), but it was also the example Einstein used

to demonstrate the unsatisfactory consequences of Lorentz's theory.

14



which it couldn't be in Lorentz's theory.
The problem with this argument is that it takes the arrow of explanation

to be from spacetime structure to dynamical laws. Some people have argued
(see, for example, Brown and Pooley (2006)) that this gets things the wrong way
round - dynamics should explain spacetime structure. On this view, it is not the
case that Einstein's view is better because it allows the dynamics to be explained
in terms of spacetime structure. However, this view also takes Einstein's theory
having better explanatory power, since in Lorentz's theory the mismatch be-
tween dynamical and spacetime symmetries means that the spacetime structure
is not explained by the dynamics.20 Therefore, we can maintain the explana-
tory virtue of Einstein's theory over Lorentz's while remaining agnostic as to
the arrow of explanation between dynamics and spacetime structure.

There is also an epistemological virtue to Einstein's theory - in Lorentz's
theory, we are unable to observe the ether state of rest, whereas in Einstein's
theory, we have no analogous undetectable structure. This is not su�cient
on it's own since, as argued before, the ether is unobservable but not surplus.
However, both the epistemological and explanatory bene�ts of Einstein's theory
give reason to think that Einstein's theory is not only theoretically simpler
(because it has less structure), but also conceptually simpler.

This is contrary to what Acuna (2014) argues - that these pragmatic virtues
are tenuous, and that Earman's defence of symmetry principles in terms of
physical and ontological simplicity is a better way of arguing for the superiority
of Einstein's theory. However, he takes the violation of Earman's principle to
re�ect that the ether is super�uous. For example, he says:

The violation of the symmetry principles in Lorentz's theory re-
sults in that the ether - in connection with the privileged ether rest
frame and the Newtonian structure of space-time - becomes suspi-
cious of representing nothing physical. The fact that special rela-
tivity does not postulate entities or structures that are dubious in
this sense makes it a physically and ontologically simpler theory than
Lorentz's. That is, the comparative simplicity of Einstein's theory is
not a merely pragmatic virtue, but it re�ects more solid ontological
foundations. - Acuna (2014, p.294)

I have argued that this view is mistaken, and therefore that we need to incorpo-
rate pragmatic reasons which look not only to the formal structure of theories,
but also how they are interpreted, in order to say why Einstein's theory is
preferred.

20In fact, on this view a theory such a Lorentz's is not even coherent, since it maintains
that the dynamical symmetries underwrite the spacetime symmetries and so there cannot be
a mismatch between them. I think that this goes too far and that Lorentz's theory should be
treated as a plausible theory, but I will not discuss this further here.

15



6 Conclusion

In this paper, I have given an example of two empirically equivalent theories
- Lorentz's ether theory and Einstein's theory of special relativity - where the
former has strictly more structure than the latter, but where one cannot re-
gard this theory as having surplus structure in the sense that it is inessential to
that theory and can be excised to give a theory structurally equivalent to the
other. The reason for this is that removing the candidate for surplus structure
- that corresponding to the ether state of rest - does not give something struc-
turally equivalent to Minkowski spacetime, where Einstein's theory takes place.
Therefore, contra Norton, this is not an example of one theory formulated in
two di�erent ways, where one formulation simply has surplus structure that the
other doesn't have.

This highlighted the following. First, one cannot infer from something being
unobservable, or absent in an empirically equivalent theory, to being surplus -
one has to recognise the role that object/structure plays in the theory of which
it is part. Second, one cannot simply look at the formal structure of theories
to tell us what is problematic or which theory is to be preferred - one has
to consider how the theories at hand can be interpreted and the conceptual
framework in which they are embedded. In the case of Lorentz's and Einstein's
theory, ignoring these points can lead to mistakenly thinking that the source of
problems in Lorentz's theory is the presence of surplus structure.

Acknowledgements

I would like to thank James Ladyman and Karim Thébault for helpful discus-
sions, and I am especially grateful to Jim Weatherall who not only prompted
the ideas in this paper, but whose detailed comments on earlier drafts have been
invaluable.

References

Acuna, P., 2014. On the empirical equivalence between special relativity and
lorentz's ether theory. Studies in History and Philosophy of Science Part B:
Studies in History and Philosophy of Modern Physics 46 (2), 283�302.

Barrett, T. W., 2015. Spacetime structre. Studies in History and Philosophy
of Science Part B: Studies in History and Philosophy of Modern Physics 51,
37�43.

Belot, G., 2013. Symmetry and equivalence. In: Batterman, R. (Ed.), The Ox-
ford Handbook of the Philosophy of Physics. Oxford University Press.

Brown, H., Pooley, O., 2006. Minksowski space-time: a glorious non-entity. In:
Dieks, D. (Ed.), The Ontology of Spacetime. Elsevier, pp. 67�89.

16



Co�ey, K., 2014. Theoretical equivalence as interpretive equivalence. The British
Journal for the Philosophy of Science 65 (4), 821�844.

Dasgupta, S., 2015. Symmetry as an epistemic notion (twice over). The British
Journal for the Philosophy of Science 67 (3), 837�878.

Earman, J., 1989. World Enough and Space-Time. MIT Press.

Glymour, C., 1970. Theoretical equivalence and theoretical realism. PSA: Pro-
ceedings of the Biennial Meeting of the Philosophy of Science Association,
275�288.

Glymour, C., 1977. The epistemology of geometry. Nous 11 (3), 227�251.

Halvorson, H., 2016. Scienti�c theories. In: Humphreys, P. (Ed.), The Oxford
Handbook of Philosophy of Science. Oxford University Press.

Janssen, M., 1995. A comparison between lorentz's ether theory and special
relativity in the light of the experiments of trouton and noble. Ph.D. thesis,
University of Pittsburgh.

Janssen, M., 2002a. Coi stories: Explanation and evidence in the history of
science. Perspectives on Science 10, 457�522.

Janssen, M., 2002b. Reconsidering a scienti�c revolution: the case of lorentz
versus einstein. Physics in Perspective 4, 421�446.

Janssen, M., 2009. Drawing the line between kinematis and dynamics in special
relativity. Studies in History and Philosophy of Modern Physics 40, 26�52.

Malament, D. B., 2012. Topics in the Foundations of General Relativity and
Newtonian Gravitation Theory. University of Chicago Press.

Nguyen, J., 2017. Scienti�c representation and theoretical equivalence. Philoso-
phy of Science 84 (5), 982�995.

North, J., 2009. The "structure" of physics: A case study. Journal of Philosophy
106, 57�88.

Norton, J. D., 2008. Must evidence underdetermine theory? In: M. Carrier,
D. H., Kourany, J. (Eds.), The Challenge of the Social and the Pressure of
Practice: Science and Values Revisited. Pittsburgh: University of Pittsburgh
Press, pp. 17�44.

Pooley, O., 2013. Substantivalist and relationalist approaches to spacetime. In:
Batterman, R. (Ed.), The Oxford Handbook of Philosophy of Physics. Oxford
University Press.

Weatherall, J., 2016a. Are newtonian gravitation and geometrized newtonian
gravitation theoretically equivalent? Erkenntnis 81 (5), 1073�1091.

17



Weatherall, J., 2016b. Space, time, and geometry from newton to einstein, feat.
maxwell. Unpublished Lecture Notes.

Weatherall, J., 2016c. Understanding gauge. Philosophy of Science 83 (5), 1039�
1049.

Weatherall, J., 2017. Category theory and the foundations of classical �eld the-
ories. In: Landry, E. (Ed.), Categories for the Working Philosopher. Oxford
University Press.

Zahar, E., 1973a. Why did einstein's programme supersede lorentz's? (i). The
British Jounal for The Philosophy of Science 24 (2), 95�123.

Zahar, E., 1973b. Why did einstein's programme supersede lorentz's? (ii). The
British Jounal for The Philosophy of Science 24 (3), 223�262.

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