/home/www/ftp/data/gr-qc/dir_0011050/0011050.dvi


Uniqueness of Simultaneity

Domenico Giulini
Institute for Theoretical Physics

University of Zürich
Winterthurerstrasse 190

CH-8057 Zürich, Switzerland

November 14. 2000

Abstract

We consider the problem of uniqueness of certain simultaneity struc-
tures in flat spacetime. Absolute simultaneity is specified to be a non-
trivial equivalence relation which is invariant under the automorphism
group Aut of spacetime. Aut is taken to be the identity-component of
either the inhomogeneous Galilei group or the inhomogeneous Lorentz
group. Uniqueness of standard simultaneity in the first, and absence of
any absolute simultaneity in the second case are demonstrated and related
to certain group theoretic properties. Relative simultaneity with respect
to an additional structure X on spacetime is specified to be a non-trivial
equivalence relation which is invariant under the subgroup in Aut that
stabilises X. Uniqueness of standard Einstein simultaneity is proven in
the Lorentzian case when X is an inertial frame. We end by discussing
the relation to previous work of others.

Introduction

Simultaneity is a relational structure on or of spacetime which helps to globally
organise events. It may or may not be thought of as intrinsic property of
spacetime, depending on the limitations on the amount of structure put on
the set of events. If it is definable solely by means of the structural elements
assigned to spacetime one usually speak of absolute simultaneity. In this case a
natural question is whether it is unique. If it is not definable in such a way, one
has to add some further structure with the help of which we may then define
some relative simultaneity, ‘relative’ to the added structure.

Adding sufficiently much structure will always allow to define some notion
of relative simultaneity, even though such a definition will generally be far from
unique. Uniqueness is no issue for practical applications, as many of the modern
global navigational systems – like GPS or LORAN-C – demonstrate. For them
to work it is sufficient that suitable and well defined methods for clock synchro-
nisation exist. Typically, such methods will heavily depend on the contingent

1



properties of the physical situation at hand, that is, abstractly speaking, on
the properties of specific solutions to the dynamical equations of many body
systems (earth, satellites etc.). Such solutions will clearly not respect any of the
fundamental spacetime symmetries of the underlying theory.

There is hence no question that abundant simultaneity structures can be
defined on spacetime. But this fact does not teach anything to the spacetime
theorist. He or she is interested in the structure of spacetime and whether
simultaneity may be regarded as (unique?) part of it. If spacetime does not
provide a sufficiently rich structure by itself, the natural question would be ‘how
much’ structure is missing. That is, whether we can add some minimal amount
of structure with respect to which a (unique?) characterisation of simultaneity
can be given. Here, ‘minimal’ could, for example, mean that the added struc-
ture should be invariant under as many of the original symmetries as possible.
This will be the intuitive idea behind the approach followed here. The added
structure should of course be physically interpretable and give rise to a physi-
cally ‘reasonable’ definition of simultaneity. For example, one may regard any
definition as inadmissible in which a physically realizable world-line is allowed
to intersect a set of mutually simultaneous events in more than one point. This
will also be a criterion which we adopt in this paper.

Our discussion is close in spirit to that of Malament (1977) and Sarkar &
Stachel (1999), and was in fact motivated by them. These authors are also
concerned with certain uniqueness properties of Einstein’s definition of syn-
chronisation. Their background is the debate about the ‘conventionality-of-
simultaneity-thesis’, an updated summary of which was given by Janis (1999).
Here the issue of uniqueness comes in because one adopted strategy to refute
this thesis is to first identify non-conventionality with uniqueness and then to
prove the latter. Clearly, this identification can be challenged upon the basis
that every proof of uniqueness rests upon some hypotheses which the simul-
taneity relation is supposed to satisfy and which may themselves be regarded
as conventional. Arguments of this kind were brought forward in particular by
Janis (1983) and Anderson et al. (1998, 123-126) and merely point towards a
certain ambiguity in the possible meaning and range of the word ‘convention’.1

For this reason we concentrate on the question of uniqueness, which seems to
be a much better behaved notion about which definite statements can be made
once conditions for simultaneity relation are specified.

There are various difficulties with the uniqueness arguments by Mala-
1Janis (1983, 101) characterises a simultaneity structure as free of conventions, if it is

‘singled out by facts about the physical universe’. But this is potentially also full of am-
biguities. For example, are solutions to equations of motion which are actually realised in
nature considered as ‘facts about the physical universe’ ? If yes, and this is hard to deny,
we can use them to define arbitrarily many (possibly practically very useful) simultaneities.
To choose between those clearly requires a convention. On the other hand, on the largest
observable scales our world is well described by a ‘cosmological’ solution in which the electro-
magnetic microwave background radiation singles out a preferred (irrotational) congruence of
local observers whose orthogonal spatial hypersurfaces define a preferred simultaneity struc-
ture. Does the ‘cosmological’ character sufficiently distinguish this solution to let us conclude
non-conventionality?

2



ment (1977) and Sarkar & Stachel (1999). Whereas the arguments given by
Malament appear mathematically correct, we agree with Sarkar & Stachel that
he puts physically unwarranted restrictions on the simultaneity relation.2 On
the other hand, the mathematical arguments given by Sarkar & Stachel are
mathematically incomplete. Our last section is devoted to a more detailed dis-
cussion of these issues. In this paper we wish to present a fresh and systematic
approach from first principles which, we believe, is free from the uncertainties
just mentioned.

Let us point out right at the beginning that our requirements on simul-
taneity differ slightly from the ones used by Malament and Sarkar & Stachel:
they require the simultaneity-defining equivalence relation to be invariant un-
der all causal automorphisms (explained in section 5), whereas we only require
invariance under spacetime automorphisms (to be defined below), which form
a proper (i.e. strictly smaller) subgroup of the former. This implies that a pri-
ori our invariance-requirement is weaker and will therefore generally allow for
more invariant equivalence relations. Uniqueness results in our setting should
therefore be considered as stronger. But our actual motivation for sticking with
spacetime automorphisms is simply that those causal automorphisms which
are not spacetime automorphisms, like constant scale transformations, are no
physical symmetries.3 For this reason we will also not include space- and time-
reflections in the group of spacetime automorphisms. The latter were also ex-
cluded by Sarkar & Stachel but not by Malament. Since our presentation aims
to be self-contained, it will also contain a fair amount of background material.

1 Flat Spacetime and its Automorphisms

Throughout we deal with flat spacetime which we denote by M. It consists
of a manifold together with certain geometric structures. The manifold is as-
sumed to be diffeomorphic to R4 with its natural differentiable structure. We
think of R4 as being endowed with a basis which is fixed once and for all, un-
less explicitly stated otherwise. Linear transformations are then interpreted as
diffeomorphisms (active point transformations), not as changes of bases.

An effective method to (implicitly) specify geometric structures is via the
choice of an automorphism group Aut, which is a subgroup of the group of
bijections of R4. In most cases, like in ours, it will turn out to be a finite
dimensional Lie group which acts by diffeomorphisms, but a priori this need
not necessarily be so. Once the choice of Aut is made, a geometric structure is

2For fairness one should say that Malament had a different motivation, namely to prove
that the standard simultaneity relation of special relativity is uniquely definable in terms
of the relation of causal connectibility. Since the latter is invariant under a strictly larger
group than the group of physical symmetries of spacetime, he had indeed good reasons to put
the stronger invariance requirement. Note that this makes existence less and uniqueness –
provided existence holds – more likely.

3We argue on the level of modern classical and quantum field theories in flat space, thereby
ignoring General Relativity.

3



said to exist on, or be a property of, spacetime M iff4 this structure is invariant
under Aut.5 Invariant structures are sometimes called ‘absolute’ (like absolute
simultaneity), but one should keep in mind that this notion of absolute depends
on, and is hence relative to, the choice of Aut. That choice should really be
considered as a physical input.6

In this paper Aut is either the inhomogeneous Galilei or the inhomogeneous
Lorentz group. Let us briefly recall the essential requirements which lead to
these groups.

• Elements of Aut should be bijections of R4; that is, they should be maps
which are injective (same as ‘into’) and surjective (same as ‘onto’). Hence
each transformation has an inverse and no point (event) ‘gets lost’ in a
transformation. Note that a priori we do not require transformations to
be continuous or even smooth, which – if you think about it – would be
hard to justify physically. In fact, smoothness will be implied by this and
the next condition.

• We assume we are given ‘forceless point-particles’, that is, elementary
point objects whose inertial trajectories define an affine structure on M
with respect to which the trajectories become ‘straight lines’. Aut is now
required to preserve this affine structure, i.e., transformations in Aut must
map straight lines to straight lines.

It is the main result of real affine geometry that bijections of Rn (n ≥ 2) which
map straight lines to straight lines must be affine maps: x 7→ Ax + a, where
A is an invertible n × n-matrix and a ∈ Rn. A proof of this fact is given
in sections 2.6.3-4 of (Berger 1987).7 Hence Aut must be a subgroup of the
real affine group in 4-dimensions, called Aff(4, R), which is given by the semi-
direct product R4 o Gl(4, R) of the group of translations (R4) with the group
of general linear transformations (Gl(4, R)). For (a′, L′) and (a, L) in Aff(4, R)
their multiplication law reads:

(a′, L′)(a, L) = (a′ + L′a, L′L). (1)

4Throughout we use ‘iff’ as abbreviation for ‘if and only if’.
5This is the central idea of the ‘Erlanger Programm’ of Felix Klein (1893): to characterise

a geometry (in a generalised sense) by its automorphism group (Klein calls it ‘Hauptgruppe’).
Relations belong to that geometry (are ‘objective’ according to Weyl (1949, Chap. III.13)) iff
they are invariant under Aut. This statement stands independent of the logical question of
whether any such ‘objective’ relation is actually derivable or definable within a given axiomatic
setting (Weyl 1949, 73). This issue has recently been raised again by Rynasiewicz (2000) in
the context of the ‘conventionality of simultaneity’ debate.

6Eventually it depends on the fundamental dynamical laws of quantum field theory (with-
out gravitation), which denies the notion of empty space even locally. What we call the
automorphisms of spacetime is the stabiliser of the ground state (‘vacuum’) within the group
of dynamical symmetries of the theory. Note that this point of view eventually also denies
that there exists a fundamental distinction between kinematical and dynamical symmetries.

7To complete Berger’s (1987) argument one needs to supply a proof of his proposition
2.6.4, which states that there are no non-trivial automorphisms of the real numbers. This well
known fact can be shown in an elementary fashion.

4



• We assume that all spacetime-translations are part of Aut (‘homogeneity of
spacetime’). Hence Aut is of the form R4oAut∗, where Aut∗ ⊆ Gl(4, R). Of
Aut∗ it is further assumed that it contains the spatial rotations (‘isotropy
of space’) as matrices of the form:8

R(D) =
(

1 ~0>
~0 D

)
, (2)

where D ∈ SO(3). Note that we did not include space- and time-reflections.

• We assume the relativity principle to hold, which says that velocity trans-
formations B(~v) (called boosts) are part of Aut∗. The boosts are assumed
to be continuously and faithfully labelled by ~v ∈ V ⊆ R3, where V is
connected. Finally, let R(D) be as in (2), then we assume the following
equivariance condition9, which should be regarded as part of the require-
ment of ‘isotropy of space’:

R(D)B(~v)[R(D)]−1 = B(D~v) . (3)

Given these conditions marked with •, one can rigorously show that Aut is
either the inhomogeneous Galilei or the inhomogeneous Lorentz group for some
yet undetermined velocity parameter c. The identification of c with the velocity
of light is a logically independent step which need not concern us here. The
idea to just use the relativity principle and not the invariance of the velocity
of light in order to arrive at (something close to) the Lorentz group was first
consistently spelled out by Frank & Rothe (1911). The way sketched here is
mathematically more complete and partly based on the work of Berzi & Gorini
(1969).

2 Simultaneity

Simultaneity, S, is a relation on M, that is, a subset of M × M. If (p, q)
belongs to this subset we write S(p, q), which stands for the statement: ‘the
point (event) p on M is in relation (later called ‘simultaneous’) to the point q’.
More precisely, we require S to be an equivalence relation, which means that it
ought to satisfy the following three conditions:

S(p, p) ∀p ∈ M (reflexivity) , (4)
S(p, q) ⇒ S(q, p) ∀p, q ∈ M (symmetry) , (5)
S(p, q) and S(q, r) ⇒ S(p, r) ∀p, q, r ∈ M (transitivity) . (6)

8Vectors in R3 carry an arrow overhead and are considered as 1 × 3-matrices. The super-
script > denotes matrix-conjugation. 4 × 4-matrices are written in time ⊕ space - form.

9Condition (3) is usually not stated explicitly, but tacitly assumed in statements to the
effect that one may w.l.o.g. (sic) restrict attention to boosts in a preferred direction (Berzi &
Gorini 1969, 1519), and that rotations about the ~v-axis necessarily (sic) commute with B(~v)
(Torretti 1996, 80). On the other hand, Berzi & Gorini and Torretti explicitly make use of (3)
but with R being a spatial reflection which reverses the boost direction (Torretti 1996, 79),
which unnecessarily involves reflection transformations (which we exclude), whereas the same
can be achieved by choosing for R a π-rotation about an axis ⊥ to the boost direction.

5



It is hard to see how one could do without the first two conditions, but tran-
sitivity is certainly not needed in order to talk about the simultaneity of pairs
of events. For example, it already allows to synchronise each member of a set
of clocks with a preferred ‘master-clock’, which is indeed sufficient for certain
practical purposes. However, transitivity is needed in order to consistently talk
about mutually simultaneous events in sets of more than two.

2.1 Equivalence Relations

Let us recall a few general properties of equivalence relations which we will fre-
quently use. An equivalence relation S on a set M is the same thing as a par-
tition of M. Recall that a ‘partition’ is defined to be a covering by non-empty,
mutually disjoint sets. In the present context such sets are called equivalence
classes. The equivalence class in which p lies is called p’s equivalence class or
[p], and given by

[p] := {q | S(p, q)} . (7)
This definition makes sense since [p] and [q] are either disjoint or identical.
Before showing this, we first note that reflexivity implies p ∈ [p]. Hence no
[p] is empty and each p lies in some equivalence class. Now, if S(p, q) then
[p] = [q] since symmetry and transitivity immediately imply that S(p, r) iff
S(q, r). Moreover, in the same way we see that r ∈ [p] ∩ [q] implies S(p, q)
and consequently [p] = [q], which proves the claim. Conversely, a partition
M =

⋃
i Ui defines an equivalence relation through S(p, q) ⇔ p and q lie in

the same Ui. The conditions of reflexivity, symmetry, and transitivity are easily
checked. Hence we have shown that an equivalence relation on M is the same
as a partition of M.

Two particularly boring equivalence relations are: 1) [p] = [q] ∀p, q ∈ M
(just one equivalence class), and 2) [p] 6= [q] ∀p, q ∈ M where p 6= q (each point
is a different class). We call an equivalence relation non-trivial if it is different
from these two.

2.2 Invariant Equivalence Relations

Suppose M carries an action of a Group G: (g, p) 7→ g · p. We say that the
equivalence relation S is invariant under this action iff

S(p, q) ⇔ S(g · p, g · q) , ∀g ∈ G , ∀p, q ∈ M . (8)

Expressed in terms of the equivalence classes (7) this is the same as10

[g · p] = g · [p] , ∀g ∈ G , ∀p ∈ M . (9)

Proof. That (8) implies (9) is seen as follows:

[g · p] = {q | S(g · p, q)}
10For U ⊆ M or H ⊆ G we write: g · U := {g · p | p ∈ U} and H · p := {g · p | g ∈ H}.

6



= {q | S(p, g−1 · q)}
= {g · r | S(p, r)}
= g · {r | S(p, r)} = g · [p] .

Conversely, if S(p, q) then q ∈ [p] and (9) implies g·q ∈ [g·p] so that S(g·p, g·q).
This proves the equivalence of (8) and (9).

As already mentioned in the introduction, we regard a G = Aut–invariant
equivalence relation as a physical property of spacetime. Our central require-
ment on ‘absolute simultaneity’ then reads as follows:

Requirement 1 Absolute Simultaneity is a non-trivial Aut-invariant equiva-
lence relation on M each equivalence class of which intersects any physically
realizable trajectory in at most one point.

If no such absolute structure exists, we need to add some further structural
elements X to M. X could be a subset of M, like a single wordline which
models an individual observer, as in Malament (1977), or a whole 3-dimensional
family of such observers which define a reference frame, as in Sarkar & Stachel
(1999). Straight lines or families of straight lines correspond to inertial observers
and inertial reference frames respectively. Now, let AutX be the subgroup of Aut
that preserves (stabilises) X. For example, if X is a subset of M, this means
that AutX should map points of X to points of X (pointwise X need not be
fixed), and if X is a partition of M by subsets, like a foliation by straight lines,
it means that AutX should preserve this partition, i.e., map lines to lines. A
relation is then said to exist on M relative to X, or be a property of (M, X), iff it
is invariant under AutX .11 In other words, the relation is required to break none
of the residual spacetime symmetries which still exist relative to the structure
X. Our central requirement on ‘relative simultaneity’ then reads as follows:

Requirement 2 Simultaneity relative to X is a non-trivial AutX -invariant
equivalence relation on M each equivalence class of which intersects any physi-
cally realizable trajectory in at most one point.

We see that in order to classify simultaneity-structures we essentially need
to classify G-invariant equivalence relations, where G is Aut or some subgroup
thereof. This will be done to some extent in the following sections. There we
will make extensive use of the following simple observations: Let Gp denote the
stabiliser subgroup of p ∈ M in G, that is, Gp := {g ∈ G | g · p = p}. If S(p, q)
then (8) immediately implies S(p, g ·q) for all g ∈ Gp. Hence the whole Gp-orbit
of q, denoted by Gp · q, lies in [p]. Moreover, suppose S(p, q) and that for some
g, with p′ = g · p and q′ = g · q, we have

Gp · q ∩ Gp′ · q′ 6= ∅ , (10)

then [p] = [p′]. The proof is simple: since the orbits Gp · q and Gp′ · q′ lie in [p]
and [p′] respectively, [p] and [p′] intersect and must hence be equal.

11Again this notion of relative existence of geometric relations may be found in Klein’s
‘Erlanger Programm’ (Klein 1893, §2).

7



2.2.1 Existence

One may ask for general criteria for when G-invariant equivalence relations
exist. For example, assume G’s action on M to be 2-point-transitive, which
means that for any set of four mutually distinct points p1, p2, q1, q2 there exists
a g ∈ G such that g · p1 = q1 and g · p2 = q2. This is equivalent to saying that
the stabiliser subgroups Gp act transitively. Then, obviously, the only invariant
equivalence relation is the trivial one where [p] = M for all p. On the other
hand, if G’s action is not transitive but still non-trivial, we can, for example, just
set [p] := G · p to define a non-trivial G-invariant equivalence relation. Hence
the mathematically most interesting situations arise when G acts transitively
but not 2-point-transitively. This is precisely the situation we are dealing with.
Due to the spacetime translations, Aut clearly acts transitively on M, but the
stabiliser subgroup of, say, the origin, Aut∗, does not. Its orbits consist of 3-
dimensional submanifolds which are planes in the Galilean and hyperbola or
light-cones in the Lorentzian case.

A general criterion for the existence of G-invariant equivalence relations, or
equivalently, G-invariant partitions, does indeed exist. Before we state it, recall
that a subgroup K of G is called ‘maximal’ iff there is no proper subgroup H
of G which properly contains K. We have

Theorem 1 Let G act transitively on M. There exists a non-trivial G-
invariant equivalence relation on (equivalently: partition of) M iff the stabiliser
subgroups Gp are not maximal.

Note that maximality either applies to all or none of the stabiliser subgroups,
since for a transitively acting G they are all conjugate: Gp = g · Gq · g−1 if
p = g · q. A proof of Theorem 1 may be found as proof of Theorem 1.12
in Jacobson (1974). We will not make essential use of this theorem because we
prefer to give direct arguments. But it is still useful to know since it highlights a
group theoretic property (maximality of stabiliser subgroups) that distinguishes
the inhomogeneous Galilei from the inhomogeneous Lorentz group and which
pinpoints the mathematical origin of their different behaviour regarding the
existence of absolute simultaneity-structures.

3 Galilean Relativity

We speak of Galilean relativity if Aut is the inhomogeneous (including transla-
tions), proper (no space reflections), orthochronous (no time reflection) Galilei
group, which we denote by IGal. According to the general results given above
we only need to specify its homogeneous part Aut∗. It is given by the homo-
geneous Galilei group Gal, which is the semi-direct product of spatial rotations
(R ∈ SO(3)) and boosts (~v ∈ R3). Hence we have

IGal = R4 o (R3 o SO(3)) , (11)

where the first o on the right side comes from (1) and the second corresponds
similarly to the law (~v′, R′)(~v, R) = (~v′ +R′~v, R′R). It is implemented by letting

8



Aut∗ ⊂ Gl(4, R) be the subgroup of 4 × 4 – matrices of the form:
(

1 ~0>

~v R

)
. (12)

Note that the first semi-direct product in (11) is such that the action of Aut∗

on R4 is not irreducible: it leaves invariant the subgroup R3 of spatial transla-
tions (due to the zero-vector in the upper right corner of (12)). Consequently,
R

3
o Aut∗ is a subgroup of Aut. Note also that the same is not true for time

translations. With respect to Theorem 1 this implies that Aut∗ – the stabiliser
subgroup of the origin in R4 ∼= M – is not a maximal subgroup of Aut, since we
can still adjoin the group R3 of spatial translations. Hence Theorem 1 guaran-
tees the existence of a non-trivial Aut-invariant equivalence relation. But this
will be proven directly below.

IGal is parameterised by ten real numbers: three for R, three for a boost-
vector ~v, three for a spatial translation vector ~a and one for a time-translation
b. A general element g ∈ IGal can then be uniquely labelled by (R, ~v, ~a, b). The
law for multiplication and inversion then simply read12

g′′ = (R′R , ~v′ + R′~v , ~a′ + R′~a + b′~v , b′ + b) , (14)
g−1 = (R−1 , −R−1~v , −R−1(~a − b~v) , −b) . (15)

Writing p in M as (t, ~x), the action of g on p reads:

~x 7→ ~x′ = R~x + ~vt + ~a , (16)
t 7→ t′ = t + b . (17)

The subgroup Euc ⊂ IGal of Euclidean motions consists of spatial rotations
and translations. It is given by all elements where ~v = ~0 and b = 0. Its
orbits are the planes of constant t which we denote by Σt. Note that boosts
act like translations in each Σt separately, but scaled with a factor t. Only
time-translations permute the planes Σt. The general law is g · Σt = Σt+b.

3.1 Galilean Simultaneity

Let S be a G = IGal–invariant, non-trivial equivalence relation. We choose a
hyperplane Σt and a point p ∈ Σt.

First we assume that a point q 6= p exists on Σt such that S(p, q). For the
moment we forget about M and restrict attention to Σt which we regard as R3
with standard inner product and norm ‖·‖. Euc ⊂ IGal acts transitively on Σt

12IGal can be embedded in Gl(5,R) as follows:

g(R, ~v, ~a, b) −→



R ~v ~a
0 1 b
0 0 1


 ∈ Gl(5,R) (13)

which also gives (14,15). In this picture M is identified with the 4-dimensional hyperplane
x5 = 1, which then leads to (16,17).

9



by standard Euclidean motions. The stabiliser subgroup Eucp of p consists of all
rotations centred at p. The orbit Eucp ·q is a 2-sphere around p of radius ‖p−q‖.
Now, let g ∈ Euc be a translation by a vector of norm less then 2‖p−q‖. Clearly
S(p′, q′) for p′ = g·p and q′ = g·q. Moreover, since the distance between p and p′
is less then 2‖p−q‖, the Eucp-orbit of q and the Eucp′ -orbit of q′ intersect, which
implies [p] = [p′]; compare discussion surrounding (10). Since any point on Σt
can be reached by a finite number of translations of norm less than 2‖p−q‖, we
only need to iterate this argument to show that all points of Σt lie in the same
equivalence class.

Next we assume that a point p′ 6∈ Σt exists such that S(p, p′). Let p′ be
a member of, say, Σt′ , where t 6= t′. Consider the stabiliser subgroup IGalp′
of p′. Besides certain rotations (which need not concern us at the moment),
it contains the 3-dimensional subgroup which is given by the combinations of
translations and boosts for which ~a = −t′~v. By construction this subgroup
fixes Σt′ pointwise, but acts transitively on Σt via translations of the form
~x 7→ ~x+(t−t′)~v. This already proves that [p](= [p′]) contains both hyperplanes,
Σt and Σt′ . The latter is seen by just reversing the rôles of p and p′ in the
argument.

So far our arguments show that, for any p ∈ M, [p] is a union of planes Σt
one of which contains p. It is consistent with IGal-invariance to choose for [p]
just the single plane containing p since g · Σt = Σt+b implies (9). We may thus
call this the minimal or finest non-trivial IGal-invariant equivalence relation on
M. If [p] contains more than one plane, say Σt and Σt′ , then for IGal invariance
(here only time-translations matter) it is necessary that all planes Σt+n(t′−t) for
n ∈ Z are also contained in [p]. Therefore, if λ denotes the infimum of all time-
differences of planes contained in [p], then [p] is the union of all planes Σt+nλ
for n ∈ Z, where p ∈ Σt. If this infimum were zero we would obtain the trivial
equivalence relation where all of M is a single class. Hence we have

Theorem 2 Let S be a non-trivial Aut = IGal – invariant equivalence relation
on M. Then the possible equivalence classes [p] are given by:
(i) the plane Σt containing p;
(ii) the union over n ∈ Z of planes Σt+nλ, where 0 < λ ∈ R and p ∈ Σt.

Next to the mathematical space M that represents physical spacetime, one
may also associate a mathematical space T that simply represents time, namely
the set of equivalence classes given by the quotient

T := M/S . (18)

In case the equivalence classes in M just consist of single hypersurfaces Σt, T
is isomorphic to R. The action of IGal on T is just (g, t) 7→ t + b. In case
there are more than one Σt in each equivalence class, T is isomorphic to the
circle S1 = R/{identification mod λ}. In this sense time is periodic with period
λ < ∞. Note however that this does not mean that we may make periodic
identifications in M and represent spacetime by Mλ := M/Z, where Z ⊂ IGal
is represented by the discrete time translations t 7→ t + nλ , n ∈ Z. The point

10



being that this space (homeomorphic to S1 × R3) would not support an action
of IGal, since the boost transformations (t, ~x) 7→ (t, ~x + t~v) are incompatible
with such an identification. This is due to the group-theoretic fact that time-
translations do not form a normal subgroup in IGal, in contrast to spatial
translations. For example, periodic spatial identifications by some integer lattice
Z

3 ⊂ R3, which certainly does form a normal subgroup, results in a closed spatial
space whose topology is that of the 3-torus, T 3, and a spacetime M′ = R × T 3
which still carries an action of IGal, though not an effective one, since Z3-valued
spatial translations now act trivially. We summarise the results of this section
in

Theorem 3 Standard Galilean simultaneity is the unique absolute simultaneity
satisfying Requirement 1 for Aut = IGal. It is also the unique non-trivial IGal-
invariant equivalence relation on M for which time is non-cyclic, or for which
the equivalence classes in M are connected.

4 Lorentzian Relativity

We now wish to explore the consequences of replacing IGal by the inhomoge-
neous Lorentz group ILor (also known as Poincaré group). This group has not
only quite a different group-structure than IGal, but also very different orbits
in M. This results essentially from the way boost-transformations are imple-
mented, which are now not allowed to boost beyond a finite limit-velocity c,
usually taken to be the velocity of light, but the value of c is unimportant for
us as long as 0 < c < ∞. In the following we choose units such that c = 1.

ILor is obtained by choosing for Aut∗ ⊂ Gl(4, R) the homogeneous (proper,
orthochronous) Lorentz group, Lor. To define it, consider the real 4 × 4-
matrices L which leave the diagonal-matrix – called the Minkowski metric –
η := diag(1, −1, −1, −1) invariant in the following sense:

LηL> = η . (19)

Those L with determinant +1 form the proper Lorentz group which is also
denoted by SO(1, 3), a notation with an obvious meaning in view of (19). This
group has two components: one where the time-time component L00 ≥ 1, the
other where it is ≤ −1. The former is the identity component and leads to our
group Lor of proper orthochronous Lorentz transformations. As for the Galilei
group we excluded reflections of space or time.

To see the group-theoretic difference between Gal and Lor, recall that Gal
is a semi direct product {boosts} o {rotations}, which implies that boosts and
rotations separately form subgroups, and that the boost-subgroup is normal
(i.e., invariant under conjugations in Gal; compare (3)). Lor, on the other hand,
is a simple group, that is, it does not contain any normal subgroups other than
the identity and the whole group. Rotations still form a subgroup, but boosts
do not. In general, a boost multiplied by a boost is a boost times a non-trivial
rotation (the latter being the origin of ‘Thomas-Precession’). Requirement (3)

11



still holds, of course, and merely means that boosts form an invariant set under
conjugation with rotations but not under all conjugations in Lor.

The translation subgroup in ILor acts on M just in the same way the
translations in IGal do. This is also true for spatial rotations, which together
with boosts make up Lor. The former correspond to matrices of the form

(
1 ~0>
~0 R

)
, (20)

where R ∈ SO(3). On the other hand, boost with velocity ~v = v~n, where
v := ‖~v‖ < 1, correspond to matrices of the form (γ := 1/

√
1 − v2):

(
γ γ~v>

γ~v 1 + (γ − 1)~n ⊗ ~n>
)

, (21)

which act like

~x 7→ ~x′ = ~x + γ~vt + (γ − 1)(~n · ~x)~n , (22)
t 7→ t′ = γ(t + ~v · ~x) . (23)

Whereas (22) is merely a deformation of the boost action in (16), (23) differs
significantly from (17). Thinking of M as R4, it means that the family of parallel
planes t = const., which can be characterised by their normal-direction (here
w.r.t. standard Euclidean metric of R4) given by (1,~0), will be transformed into
the family of tilted planes with normal direction given by (1, ~v), i.e. tilted by
the angle tan α = v. Hence any two planes from the first and second family
respectively intersect.

With respect to Theorem 1 we also remark that in R4 o Aut∗ the action
of Aut∗ = Lor on R4 is now irreducible, hence no subspace of translations is
left invariant, as it was the case for spatial translations when Aut∗ = Gal. This
implies that Aut∗ = Lor and all its conjugations by translations – which make up
the stabiliser subgroups Autp – are maximal subgroups of Aut. From Theorem 1
we can therefore anticipate that there will be no Aut = ILor-invariant non-
trivial equivalence relation on M. Below we prefer to give a simple direct proof
of this fact.

4.1 Lorentzian Simultaneity

Since spacetime translations and spatial rotations in IGal and ILor act identi-
cally on M, all the arguments which were given in the framework of Galilean
relativity and which did not use boost do also apply in the present case. In par-
ticular, each equivalence classes of any non-trivial ILor-invariant equivalence
relation S contains one or more of the planes Σt. But now comes the point:
since boosts transform the family of planes Σt to a tilted family Σ′t′ , any mem-
ber of which intersects any member of the former, all the planes must be in the
same equivalence class. This implies

12



Theorem 4 The only ILor-invariant equivalence relation on M is the trivial
one where M is the only equivalence class. Hence absolute simultaneity satis-
fying Requirement 1 does not exist in Lorentzian relativity.

The proof is almost trivial: since Σt lies within a single class, and since boosts
in ILor map classes to classes, the image Σ′t of Σt under a boost also lies within
one class. But it intersects all Σt and hence all other classes, which implies that
there is only one class.

4.2 Lorentzian Simultaneity Relative to an Inertial Frame

What we have just learned is that Aut∗ = SO(1, 3) does not leave spacetime with
enough structure to be able to define absolute simultaneity. But what about
relative simultaneity? For this we have to add some further structure X. Clearly,
if we choose X so that Aut gets broken down completely, AutX -invariance will
be an empty requirement and any equivalence relation will do. As outlined in
the introduction, the task is to choose X rich enough to ensure existence but
otherwise as symmetry-preserving as possible. If this leads to a sufficiently big
AutX , the residual invariance requirement may ensure uniqueness.

The structure X we wish to consider here is an inertial reference frame,
which in our case (flat geometry) corresponds to a foliation of M by (necessar-
ily parallel) timelike straight lines. Let now X stand for such a foliation by lines
which are all parallel to the four-vector v = (1, ~v). The foliation X is obviously
invariant (meaning lines are transformed to lines) under all spacetime transla-
tions, and obviously not invariant under any non-trivial boost. In fact, we can
w.l.o.g. assume ~v = ~0, for otherwise let B(~v) denote the boost which maps the
t-axis to a line in X, which we call the t′-axis, and refer the whole situation to t′

and the planes Σ′t′ perpendicular (w.r.t. Minkowski metric) to the t
′-axis. The

full stabiliser group AutX = ILorX is now seen to consist of time translations
and the group Euc of spatial Euclidean motions: AutX = R × Euc.

First, we can now use the spatial rotations in Euc to argue exactly as in
Section 3.1: if in some Σt there exist two different points for which S(p, q), then
the argument there shows that [p] contains Σt. Since time translations can map
Σt to any other hyperplane in this family, each hypersurface Σt is contained in
some equivalence class.

Next suppose that for a p ∈ Σt some p′ ∈ Σt′ for t′ 6= t exists so that S(p, p′).
Now we cannot proceed as in Section 3.1 since AutX contains no boosts. Instead
we argue as follows: let `p′ denote the straight line in M through p′ which is
parallel to the t-axis, and let r be its point of intersection with Σt. The stabiliser
Eucp′ of p′ in Euc consists of rotations which in Σt′ rotate about p′ and in Σt
rotate about r.

If p 6= r we can move p by an element of Eucp′ to get another point q in Σt
for which S(p, q). Hence we are back to the case above which now shows that
[p](= [p′]) contains Σt and Σt′ , the latter again by reversing the rôles of p and
p′ in the argument.

13



This conclusion is avoided iff r = p, i.e., iff p lies on `p′ . The conclusion that
each equivalence class contains some hyperplanes is avoided iff any two different
p, p′ for which S(p, p′) lie on the same straight line parallel to the t-axis. There
clearly are non-trivial AutX -invariant equivalence relations whose classes are
contained in the straight lines parallel to the t-axis. These are readily classified:
S(p, q) iff either p, q are on the same such line, or τ nλ · p = q for some n ∈ Z,
where τλ ∈ AutX is the time translation τ 7→ t + λ. This leads to the following
classification of all possible equivalence relations:

Theorem 5 Let X be a foliation of M by timelike straight lines and S a non-
trivial AutX = ILorX – invariant equivalence relation on M. Then the possible
equivalence classes [p] are given by:
(i) the plane Σt 3 p perpendicular (Minkowski metric) to the timelike lines X;
(ii) the union over n ∈ Z of planes Σt+nλ, where 0 < λ ∈ R and p ∈ Σt;
(iii) the line in X through p;
(iv) the union over n ∈ Z of points τ nλ · p, where τλ is the translation by an
amount λ > 0 along the line in X through p.

In case (ii) each straight timelike line intersects each equivalence class a
countably infinite number of times. In case (iii) each timelike line in X is its own
equivalence class and hence intersects an equivalence class in uncountably many
points. In case (iv) the same is true for countably many points. Therefore, these
cases do not define a notion of relative simultaneity satisfying Requirement 2,
since ‘physically realizable trajectories’ will certainly include all timelike straight
lines (inertial motion). On the other hand, case (i) does satisfy the condition
that each equivalence class is cut at most (in fact: exactly) in one point by each
physically realizable trajectory, which here, for definiteness, we may e.g. specify
to be all timelike piecewise differentiable curves. Hence we have

Theorem 6 Einstein simultaneity is the unique relative simultaneity satisfying
Requirement 2 for AutX = ILorX , where X denotes an inertial frame (=folia-
tion of M by timelike straight lines).

5 Relation to Work of Others

Here we shall basically focus on the work of Malament (1977) and Sarkar &
Stachel (1999). Both are directly concerned with uniqueness issues, but neither
gives a systematic classification of invariant equivalence relations. Malament
proves uniqueness of Lorentzian simultaneity relative to a single observer. In this
case X is a single timelike straight line. But instead of spacetime automorphisms
Aut he takes all causal automorphisms, which we denote by Autc, by which he
understands all bijections f of spacetime such that p − q is non spacelike iff
f (p) − f (q) is non spacelike. It has been proven by Alexandrov (1975) that any
such transformation is a combination of transformations in ILor, time reflections
(t, ~x) 7→ (−t, ~x), space reflections (t, ~x) 7→ (t, −~c), and dilatations p 7→ λp

14



with λ ∈ R+ (positive real numbers).13 More precisely, Malament proved the
following

Theorem 7 (Malament 1977) Let X be an initial observer, i.e., a timelike
straight line. Let S be an AutcX -invariant non-trivial equivalence relation, which
also satisfies the following condition: there exists a point p ∈ X and a point
q 6∈ X such that S(p, q). Then S is given by standard Einstein simultaneity.

In a recent review, Anderson et al. (1998, 124-125) claim Malament’s proof
to be technically incorrect. We disagree, as do Sarkar & Stachel (1999) and
apparently also Janis (1999). However, it is true that the proof presented by
Malament (1977) leaves out some details. To settle this technical issue, we
present an alternative and somewhat more detailed proof in the Appendix which
uses the language developed in previous sections.

Let us look at the same situation from our point of view, where instead of
Autc we take Aut = ILor. We may w.l.o.g. take X to be the time axis; oth-
erwise we boost and translate the selected observer to rest at the origin and
take the conjugate of all subgroups to be mentioned by that combination of a
boost and a translation. Then AutX = R × SO(3), where R consist of pure time
translations and SO(3) are the rotations in Euc fixing the t-axis. Picking a single
inertial observer out of an inertial reference frame eliminates the space trans-
lations in Euc. This distinguishes the present case from that discussed above
and makes a big difference concerning the question of uniqueness. Consider the
two-parameter family of subsets of M:

σ(τ, r) := {(t, ~x) ∈ M | t = τ, ‖~x‖ = r}, (24)

given by the points (for r = 0) of the t-axis and all concentric 2-spheres about
the spatial origin in each t = const. plane. This is already an AutX -invariant
partition of M so that taking the σ(τ, r) as equivalence classes would define
a notion (though not a very reasonable one) of relative simultaneity satisfying
Requirement 2. However, it obviously violates the condition in Malament’s
theorem, since no point on X is related to a point off X. But this can be easily
cured: just take AutX -invariant unions of sets σ(τ, r) which connect points on
X with points off X, and define these unions as new equivalence classes. For
example, in each t = const. hypersurface, we can take a central ball and spherical
shells of radius 1:

σ′(τ, n) :=
⋃

r∈[n−1,n)
σ(τ, r) , (25)

where n is a positive integer. Another possibility would be to unite sets σ(τ, r)
in different hyperplanes t = const. We just have to take care that no two sets

13The same is true for any bijection which in both directions preserves just ‘lightlike’ or
just ‘timelike’ separations. Moreover, in the time oriented case, the same statements hold if
one restricts to just future (or past) oriented separations and if time reflections are eliminated
from the list of possible transformations. Note that, mathematically speaking, the particular
non-trivial aspect of these results lies in the lack of any initial continuity requirement for the
bijective maps; the listed requirements suffice to imply continuity.

15



which are causally related are in the same equivalence class. For example, as
slight modification of (25), we may take:

σ′′(τ, n) :=
⋃

r∈[n−1,n)
σ(τ + mr, r) , (26)

which also consist of an inner ball and concentric spherical shells, but now
taken from the half-cones C±(τ, α) with vertex on X at time τ and opening
angle α = | tan−1(1/m)|. They open to the future (+ sign) for m > 0 and to
the past (− sign) for m < 0. For the half-cones to be acausal, i.e. spacelike
hypersurfaces, we need opening angles bigger than π/2, i.e., |m| < 1. m = 1
gives future light-cones, m = −1 past light-cones.

This abundance of possibilities does not violate Malament’s theorem, since
the onion-like partitions of the spatial hypersurfaces Στ or C(τ, α) is not invari-
ant under scale transformations; only the partition of M into the Στ or C(τ, α)
(α fixed) is. But, in turn, the latter is not invariant under time reflections. This
is why Malament’s theorem works. (See the appendix for more details.)

Sarkar & Stachel pointed out that time reflections had to be included in
order to prove Malament’s theorem, and that without them one could still have
lightlike half-cones as equivalence classes (half-cones of other opening angles are
also possible, or course). They assert – and we agree with this – that there
is no physical reason to require invariance under time reflections. But likewise
is there no physical reason to include dilatations, which they do include in
their definitions, although in footnote 11 of their paper Sarkar & Stachel (1999)
explicitly state that their ‘considerations are independent of a requirement of
invariance under scale transformations’ (i.e. dilatations). They do not mention
that dropping dilatations adds a plethora of new invariant equivalence relations,
like those in (25)(26) (where e.g. the shell thickness is totally arbitrary and may,
in addition, depend on r).

Finally Sarkar & Stachel (1999) consider the case where X is an inertial
frame and try to show uniqueness of standard Einstein simultaneity. Expressed
in our terminology they argue as follows: if X is an inertial frame, AutX contains
spatial translations perpendicular (w.r.t. Minkowski metric) to the lines in X.
The orbit of any point p under these translations is the plane Σt containing p.
Since ‘they [the spatial translations] are not to affect the simultaneity relation,
these translations must take each simultaneity hypersurface to itself’ (Sarkar &
Stachel 1999, 217), which are therefore given by the Σt’s. But this does not
provide a proof, since the underlined part does not follow. It is not true that
the equivalence classes of a G-invariant equivalence relation must separately
be G-invariant sets; only the partition must be G-invariant, but G may well
permute the equivalence classes. The precise statement is given in equation (9).
From our Theorem 5 it is also clear that classifying invariant equivalence classes
is not quite sufficient, one also needs to impose some condition which eliminates
those relations whose classes contain timelike related points. No such condition
is mentioned by Sarkar & Stachel (1999).

16



Appendix: Proof of Theorem 7

By hypothesis there exist p ∈ X and q 6∈ X such that S(p, q). W.l.o.g. we
may take X to be the t-axis and p to be the origin; otherwise we can boost the
observer to rest and translate p to the origin, and then consider the conjugates of
all subgroups to be mentioned by that combination of a boost and a translation.
We set q = (t′, ~x′).

From Alexandrov’s (1975) results we know that Autc is as specified above
and can hence infer that the stabiliser subgroup G := AutcX consist of the fol-
lowing transformations and combinations thereof: 1) time-translations, 2) time
reflections about any moment in time, 3) the group O(3) of all orthogonal spatial
transformations (i.e. including reflections), and 4) all dilatations about points
on X. Consider the subgroup Gp ⊂ G that fixes p (the origin). It consists
of (and combinations thereof): 2’) the single time reflections about time zero:
t 7→ −t, 3’) all of O(3), and 4’) dilatations about p: r 7→ λr , ∀r ∈ M and λ > 0.

We know that Gp · q ⊆ [p], hence we are interested in the Gp-orbit of q.
The orbit of q under transformations 4’), including the point p, consists of the
half-line L : λ 7→ λq, λ ≥ 0. The shape of the orbit of this half-line under all
remaining transformations in Gp crucially depends on whether t′ 6= 0 or t′ = 0
(i.e. whether or not p and q lie in the same hyperplane perpendicular to X).

Case 1: t′ 6= 0. The angle between X and the half line L ending on X is
α = tan−1(‖~x′‖/|t′|) with 0 < α < π/2. Hence the orbit of L under all trans-
formations in O(3) – which is the same as the orbit under SO(3), so spatial
reflections do not add anything new – is a half-cone with vertex p and opening
angle α, which opens to the future if t′ > 0 and to the past if t′ < 0. Acting
with the remaining time reflection 2’) results in a (full) cone with same vertex
and opening angle, which we call C(p, α).

Case 2: t = 0. Now L is perpendicular to X (α = π/2). The orbit of L under
O(3) is the plane Σt=0. The main difference to Case 1 is that now the time
reflection 2’) adds nothing new since it leaves Σ0 (pointwise) fixed. Hence the
full Gp-orbit is still Σ0.

Consider Case 1; then [p] ⊇ C(p, α) and hence by (9) [g · p] ⊇ g · C(p, α) =
C(g · p, α) for all g ∈ Autc. (The last equality is obvious if one writes g as a
linear transformation in Autc, which leaves the light cone at the origin invariant,
followed by a translation. But any g can be so written since the translations form
a normal subgroup.) Taking all spacetime translations for g shows [p] ⊇ C(p, α)
for all p ∈ M. But for 0 < α < π/2 the cones C(p, α) and C(p′, α) for any two
p, p′ necessarily intersect. This is indeed easy to see and needs not be proven
here. Hence all equivalence classes intersect and S is trivial.

Finally consider Case 2; then [p] contains the hypersurface Σ0. If it contains
any other point we are back to case 1 and S is trivial. Hence a non-trivial S
would have [p] = Σ0. Using time translations in (9) then shows that [p] would
likewise be given by the hyperplane perpendicular to X containing p. But this
proves the theorem since the partition of M into the hyperplanes Σt is indeed
Autc-invariant and hence defines a non-trivial equivalence relation.

17



References

[1] Alexandrov, A.D. (1975): ‘Mappings of spaces with families of cones and
space-time transformations’. Annali di Matematica (Bologna) 103, 229-257

[2] Anderson, R., Vetharaniam, I., and Stedman, G. (1998): ‘Conventional-
ity of synchronization, gauge dependence and test theories of relativity’.
Physics Reports 295, 93-180

[3] Berger, M. (1987): Geometry I (Springer-Verlag, Berlin)

[4] Berzi V., Gorini, V. (1969): ‘Reciprocity principle and the Lorentz trans-
formations’. Journal of Mathematical Physics 10, 1518-1524

[5] Frank P., Rothe, H. (1911): ‘Über die Transformation der Raumzeitko-
ordinaten von ruhenden auf bewegte Systeme’. Annalen der Physik 34,
825-855

[6] Jacobson, N. (1974): Basic Algebra 1 (W.H. Freedman and Comp., San
Francisco)

[7] Janis, A.I. (1983): ‘Simultaneity and conventionality’. In Physics, Philos-
ophy and Psychoanalysis (D. Reidel Publ. Comp, Dordrecht 1983), eds.
R.S. Cohen and L. Laudan

[8] Janis, A.I. (1999): ‘Conventionality of simultaneity’. The Stanford Ency-
clopedia of Philosophy (Fall 2000 Edition), Edward N. Zelta (ed.);
URL=http://plato.stanford.edu/entries/spacetime-convensimul/

[9] Klein, F. (1893): ‘Vergleichende Betrachtungen über neuere geometrische
Forschungen’. Mathematische Annalen (Leipzig) 43, 63-100

[10] Malament, D. (1977): ‘Causal theories of time and the conventionality of
simultaneity’. Noûs 11, 293-300

[11] Rynasiewicz, R. (2000): ‘Definition, convention, and simultaneity: Mala-
ment’s result and its alleged refutation by Sarkar and Stachel’. Talk pre-
sented at the Seventeenth Biennial Meeting of the Philosophy of Science
Association, Vancouver, Nov. 2.-5. 2000. Available online at
http://hypatia.ss.uci.edu/lps/psa2k/program.html

[12] Sarkar, S. and Stachel, J. (1999): ‘Did Malament prove the non-
conventionality of simultaneity in the special theory of relativity?’ Phi-
losophy of Science 66, 208-220

[13] Torretti, R. (1996): Relativity and Geometry (Dover Publications, New
York)

[14] Weyl, H. (1949): Philosopy of Mathematics and Natural Science (Princeton
University Press, Princeton)

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