arXiv:gr-qc/9801029v1  9 Jan 1998
1
INTRODUCTION
1
Absolute Spacetime:
The Twentieth Century Ether
Carl H. Brans1
Received
All gauge theories need “something ﬁxed” even as “something changes.”
Underlying the implementation of these ideas all major physical theories
make indispensable use of an elaborately designed spacetime model as the
“something ﬁxed,” i.e., absolute. This model must provide at least the fol-
lowing sequence of structures: point set, topological space, smooth man-
ifold, geometric manifold, base for various bundles. The “ﬁne structure”
of spacetime inherent in this sequence is of course empirically unobserv-
able directly, certainly when quantum mechanics is taken into account.
This issue is at the basis of the diﬃculties in quantizing general relativity
and has been approached in many diﬀerent ways. Here we review an ap-
proach taking into account the non-Boolean properties of quantum logic
when forming a spacetime model. Finally, we recall how the fundamental
gauge of diﬀeomorphisms (the issue of general covariance vs coordinate
conditions) raised deep conceptual problems for Einstein in his early de-
velopment of general relativity. This is clearly illustrated in the notorious
“hole” argument. This scenario, which does not seem to be widely known
to practicing relativists, is nevertheless still interesting in terms of its
impact for fundamental gauge issues.
1
Introduction
Gauge theories, to which Friedrich Hehl has contributed so much, explore
the mysterious fundamental role which symmetries play in our understand-
ing of the physical world. To have a symmetry we need two parts: some-
thing ﬁxed while something else changes. Much of the progress of modern
physical theories has come as a result of studying this “ﬁxed/changing” di-
chotomy, analyzing it and suggesting new paradigms. For most of the history
1Loyola University, New Orleans, LA 70118, email:brans@loyno.edu

1
INTRODUCTION
2
of physics, space, and more recently spacetime, has in some sense or other
been the underlying ﬁxed object on which theories are written and in terms
of which experimental results are reported. The active, changing, part of the
symmetry, the gauge group, consists of coordinate changes, or to use more
contemporary terminology, diﬀeomorphisms. Yet, in spite of its uncontro-
vertible central role, a thorough understanding of spacetime models is still
one of the most elusive goals of modern physics.
In this paper, I would like to review questions related to these issues
using the now discredited ether models of the eighteenth and nineteenth
centuries for comparison.
In the earlier times, the ether was some not-
directly-observable-substratum thought to be needed by certain theories. For
example, in the Newtonian gravitational precursor to ﬁeld theory, Newton
thought that action at a distance was “... so great an absurdity , that...no
man, who has in philosophical matters a competent faculty for thinking,
can ever fall into it.” [1]. He speculated that in the case of gravitation,
the force may be produced by varying densities of the mechanical ether
in the presence of gravitating masses.
In later periods it played a more
passive role, providing a ﬁxed, preferred reference system relative to which
velocities should be measured for calculations of Lorentz forces and current
sources in Maxwell equations. We will not be concerned with the actual
details of these old ether models here, but only use them as a backdrop
to consider contemporary questions revolving around observability issues in
physical models. The interested reader is invited to consult the massive two
volume work on the ether by Whittaker [2].
Those of us who work day by day in theoretical physics and especially
relativity may tend to take for granted the huge package of assumptions that
we impose on our spacetime models, most of which cannot be supported by
direct experiment. It is this tacit acceptance of unobservable properties of
our model that motivates this paper. Of course, these assumptions have been
questioned by many workers from the time of Greek physics to the present.
Max Jammer has given us an excellent review of this subject [3]. Without
any claim to completeness we can also note more contemporary work, for
example, Connes [4], Rovelli [5], Madore and Saeger [6], Heller and Sasin [7],
Brans [8]. Other participants of this meeting have contributed to this ﬁeld,
including Rosenbaum [9] in his lecture to this meeting, and L¨ammerzahl and
Macias [10].
Many of these questions border on the philosophical, and philosophers
and historians of science have certainly made their contributions. Again the
literature is too huge to survey here. I mention only the work of Gr¨unbaum
[11] and Earman [12].

2
CONTEMPORARY SPACETIME MODELS
3
2
Contemporary Spacetime Models
In almost all theories a model of spacetime, say M, is required with at least
the following properties:
• Point set. That is, M contains atomic elements, p ∈M, representing
idealized point events. The existence and identity of this structure is
absolutely necessary for all of the following ones.
• Topological manifold. M must have a topology such that it is locally
Euclidean. That is, each p must lie in a neighborhood homeomorphic
to Rn for some integer n. Again, this property is needed for the others.
• Smooth manifold. M, in addition to being locally homeomorphic
to Rn, must be locally diﬀeomorphic to it. The local diﬀeomorphisms
constitute the local coordinates needed to express smooth functions
and to operate on them diﬀerentially.
• Geometric manifold. M must carry a smooth metric, connection
and perhaps, as Hehl has taught us, torsion.
• Bundle structures. Finally, additional gauge structures and ﬁelds
require local patching together of products of M with models of the
ﬁeld/gauge group space.
The advent of supersymmetry, etc.
surely makes this list incomplete,
but it does provide some idea of the extensive and detailed structure that
spacetime models use. Here we want to point out that this involved logical
construction is built from bricks that are as essentially unobservable as were
the vortices and atoms of the mechanistic ether of earlier times. It is in this
sense that the title of this talk compares spacetime to the ether. But before
getting into the details of these foundational lacunae, let us look at the basic
gauge group of spacetime relativity, starting from its historical roots.
3
The Classical Ether
Trautman [13] has suggested a helpful way to look at the progress of relativity
in spacetime physics using modern terminology. For more philosophical and
historical details along this path see Jammer [3]. Starting with Aristotelian
physics we can consider the basic spacetime model, M, as a simple direct
product of space with time, each having some intrinsic, absolute properties,
M = space × time = R3 × R1.
(1)
In this context, what passed for dynamics was deﬁned by ascribing of natural
places to speciﬁc types of matter, “earthy” things to the earth, etc.

3
THE CLASSICAL ETHER
4
Skipping centuries of interesting intellectual history, we arrive at the rela-
tivity of Galilei and Newton. In actuality, as Jammer points out, Newton was
strongly attached to the idea of some absolute nature for space, something
more along the lines of (1). Nevertheless the formalism of his mechanics
logically leads to a replacement of (1) with something like
M = Bundle =



space =R3
−→
M
↓p
R1=time
(2)
Thus, spacetime, M is a bundle over time, R1, with space as ﬁber, R3.
What distinguishes bundles from ordinary products as in (1) is the absence
of any natural identiﬁcation of the ﬁber over one base point with that over
some other base point. Each ﬁber, “space at a given time,” is isomorphic
to R3, but in no natural or canonical way. In classical mechanics, the ﬁber
group, relativity gauge group G, is Galilean group. This is essentially the
real linear aﬃne group of dimension three. We can complete this picture
to describe classical mechanics in modern terms by adding a preferred ﬂat
Euclidean metric on the ﬁber, three-space, and a preferred linear structure
on the base space, time. Finally, we need a corresponding bundle connection
whose geodesics provide the paths of free particles.
Thus, the formalism of Newtonian mechanics is more naturally associated
with a bundle structure as in (2), rather than with absolute space structure,
(1). Nevertheless, Newton felt strongly drawn to the latter, perhaps in part
because of his diﬃculties with action-at-a-distance, as illustrated in the quo-
tation in the Introduction. Thus, to Newton, space needed some mechanical
properties to enable it to transfer force and energy over distances in ﬁeld
theories.
This mechanical structure must be associated with some “sub-
stance,” the ether, which incidentally provides an absolute rest frame, or the
reduction of the bundle (2) to the trivial product, (1).
The attractiveness of such an absolute space model was reinforced with
the advent of the uniﬁed ﬁeld theory of electromagnetism. In this theory
velocity appears twice. First in the Lorentz force law,
F = q(E + v × B)
(3),
where v is the velocity of the charge q being acted on by the electromagnetic
ﬁeld, (E,B). The question left hanging is: “velocity relative to what reference
frame?” Secondly, the velocity v appears in the source ﬁeld equation
∇× B = µ0ρv + 1
c2
∂E
∂t ,
(4)
along with the new quantity of dimension speed, c ≡1/√ǫ0µ0. Again, the
question of what reference frame should be used to measure the source cur-
rent velocity arises. Furthermore the new, unexpected speed, c, reappears as

4
END OF THE CLASSICAL ETHER: SPECIAL RELATIVITY5
the speed of electromagnetic waves in one of the consequences of the vacuum
ﬁeld equations,
(∇2 −1
c2
∂2
∂t2 )
 E
B

= 0.
(5)
So, Maxwell’s uniﬁed ﬁeld theory of electromagnetism leaves us with three
speeds, that of the source of the ﬁelds, (4), that of the object on which the
ﬁelds act, (3), and the ﬁeld waves themselves, (5). The Galilean relativity
in (2) must then break down, since the presence of such speeds breaks its
invariance, and we return to some absolute space model, (1), where space is
now the “luminiferous” ether, with spacetime
M = ether × time.
(6)
Of course, there were strong voices in opposition to the notion of absolute
space, most notable Bishop Berkeley [14] and later Mach who referred to
the notion of absolute space as a “conceptual monstrosity” [15]. Einstein
claimed that such ideas were instrumental in the evolution of his thinking
about relativity.
At this point it might be appropriate to recall all of the eﬀort that was put
into the design of mechanical or pseudo-mechanical models of such an ether
[2]. It is natural to wonder how all of the work of contemporary physics
involving elaborate spacetime structures and superstructures may likewise
appear in the next century.
4
End of the Classical Ether: Special Relativity
But of course some hundred years ago Michelson and Morley results forced
serious rethinking of the classical ether-space model, (6). While Lorentz and
others attempted to preserve the ether by proposing length contractions and
clock dilations as a result of motion through it, Einstein cut to the heart
of the matter in his principle of special relativity, closely tied to the prin-
ciple of operationalism which informally claims that if something conspires
successfully against its observation, then its existence should not be used as
part of a physical theory. Thus, if the ether’s only claim to existence is as
an absolute rest reference frame, and its properties make motion through it
unobservable, then from the viewpoint of physics it doesn’t exist.
In fact, Einstein taught us to think in terms of a uniﬁed spacetime model,
with no preferred a priori splitting (apart from the qualitative space-like,
time-like, light like ones) of space from time. The transformation group pre-
serving these spacetime properties, the gauge group of special relativity is of
course the Poincar´e group, that is, the homogenous Lorentz group plus trans-
lations. Friedrich Hehl and his colleagues have been leaders in emphasizing
the importance of this group especially in the context of general relativity
and its generalizations, [16],[17],[18].

5
GENERAL COVARIANCE: EINSTEIN’S RELATIVITY 6
With the many successes of special relativity, it seems that the ether has
ﬁnally been put to rest. Indeed it has in this classical sense. If you can’t ob-
serve it, it doesn’t exist is a standard motto. Or to paraphrase an old axiom:“
No stuﬀhas existence until it is observed to have existence.” But, should
we not apply this to “stuﬀ”=manifold properties? So, is spacetime the new
ether? Clearly, it does not play the same mechanical role of “transmitter
of forces,” as the vortex constituted stuﬀof the old mechanical one. Also,
it clearly does not provide an “absolute rest reference frame.” But, it does
have other, similar properties.
It provides, in operationally unobservable
ways, the substratum to carry the many structures used by modern theories.
and it is the point of this paper that spacetime structures in modern theories
comprise a replacement for it and so have become a “new ether.”
5
General Covariance: Einstein’s Relativity
In addition to the Principle of Equivalence, which we will not consider, the
Principle of General Relativity and Mach’s Principle, are generally taken
as foundations in models of how Einstein arrived at General Relativity. Of
course, the actual history is more complicated and interesting, and the reader
can consult the volume one of the Einstein Studies, [19], for a deep and
accurate account of the story.
For our purposes, it is suﬃcient to point out that Einstein was aware of
the rigid structure still remaining on the spacetime of special relativity by the
Lorentz metric and the associated preferred set of inertial reference frames.
Mach’s Principle addresses the issue of why the ﬁxed stars have constant
velocity in the inertial frames, while the Principle of General Relativity pro-
poses extending the physically acceptable frames beyond this restricted set.
In other words, while special relativity had weakened the assumption of a
preferred (zero) absolute-velocity-deﬁning ether, it replaced it by a preferred
(zero) absolute-acceleration-deﬁning one. So, in the spirit of this paper, the
next step toward generally covariant theories was a result of re-examining
and loosening previous rigid structures. John Norton [20] has given us an
thorough and highly interesting analysis of how Einstein arrived at his equa-
tions of General Relativity. Here we will only skim over the issue of the
identity of spacetime points as illustrated in Einstein’s hole dilemma. (See
also [5]).
Consider a model universe, with matter and metric ﬁelds, T, g on a man-
ifold containing a region U, which will be the “hole.” Einstein was thinking
in coordinates, so let T (x), g(x) be expression of solution ﬁeld equations in
terms of coordinates (global) x.
Now re-coordinate, x →x′, with x = x′ in U, but not everywhere. Then
T ′(x′), g′(x′) is also a solution, with
g′(x′) = g′(x) = g(x),
(7)

5
GENERAL COVARIANCE: EINSTEIN’S RELATIVITY 7
within U. But matter and ﬁelds are diﬀerent outside of U. So, Einstein was
deﬂected from seeking a generally covariant theory since the following fact
would seem to be paradoxical: matter outside of U does not determine the
ﬁelds inside of U uniquely (or vice versa). At this point we must be cautious
about treating this as trivial, since we are so accustomed to accepting general
covariance as an obvious desideratum. From the viewpoint of development
of the theory, there is more here than confusion about coordination.
In fact, it does seem on ﬁrst glance that Einstein and Grossman were con-
fused about the expression of the same metric merely displayed in diﬀerent
coordinates. For example
ds2 = dx2 + dy2,
(8)
or
ds′2 = cosh2(x′)dx′2 + dy′2.
(9)
Clearly (8) and (9) represent the same metric, and Einstein was aware of
this. However changing the notation in (9) results in
ds′2 = cosh2(x)dx2 + dy2.
(10)
If we then identify the points of the manifold with the pair (x, y), then
(8) and (10) are truly diﬀerent metrics in some sense, although they are
diﬀeomorphic (isometric).
In fact it is possible to deﬁne point=“ordered
pair of numbers,” not “diﬀeomorphism equivalence class of ordered pair of
numbers in each coordinate system.”
This discussion highlights the diﬀerence between the active and passive
interpretations of the transformation (diﬀeomorphism).
Actually, as dis-
cussed in detail by Norton [20], Einstein’s fourth presentation of this ar-
gument shows that rather than being confused at the diﬀerence between
(8) and (9), he was laying the groundwork for the modern interpretation of
diﬀeomorphisms as physical gauge transformations.
What may be surprising about this is that it seems to rob the individ-
ual points of their identity, in the absence of a metric. In other words, if
P1, P2 are two points in a manifold, some diﬀeomorphism maps one into the
other. The geometry and ﬁelds around P1 become those around P2, in physi-
cally equivalent geometric manifolds, so P1 cannot be distinguished from P2.
Rovelli, [5] discusses this subject in some detail, distinguishing spacetime
models as ML, “local,” with a particular smoothness and “atlas” as opposed
to MN, “non-local,” which is equivalence class of all ML under diﬀeomor-
phisms (gauge transformations).2 However, let us recall that this discussion
concerns the mathematical model which is mapped by some assumed “dif-
feomorphism” onto an absolute point set, spacetime. In fact, without some
underlying pointset, there can be no notion of diﬀeomorphism.
2Corresponding to this, mathematicians distinguish “smooth structure” from “smooth
manifold.” We will discuss this later.

6
ABSOLUTE SPACETIME: QUANTUM THEORY 8
Nevertheless, the idea remains that the use of diﬀeomorphism as phys-
ically unobservable gauge “wipes out” the individual identity of points. In
fact, in discussing his ﬁnal generally covariant ﬁeld equations Einstein said
in a letter to Schlick in 1915, “ thereby time and space lose the last remnant
of physical reality. All that remains is that the world is to be conceived as a
four-dimensional (hyperbolic) continuum of 4 dimensions.” [21]. Our point
here is that this continuum carries at least as much structure as the replaced
ether.
6
Absolute Spacetime: Quantum Theory
There still remains the list of absolute spacetime properties described in the
introduction such as topology, smoothness, etc., which seem to be arbitrar-
ily chosen. This leads to the question of regarding the role of spacetime
as object or scratch pad. This is a question certainly bordering on philos-
ophy philosophy, but also closely related to the operational foundations of
quantum theory.
The principal distinguishing characteristic of quantum logic as opposed to
classical logic is that in quantum theory questions correspond to projection
operators in Hilbert space. The logical operators, “or,” ∨, corresponds to
“span of vector space union,” while “and,” ∧, corresponds to intersection.
This results in a non-Boolean algebra,
a ∧(b ∨c) ̸= (a ∧b) ∨(a ∧c),
(11)
By contrast classical questions concern “set-inclusion,” so “or,” ∨, becomes
set union, ∪, and “and,” ∧, becomes ∩, set intersection. Thus, for point sets,
a ∩(b ∪c) = (a ∩b) ∪(a ∩c).
(12)
In other words, quantum logic is not in general consistent with the logic of
set-inclusion, which is fundamental to point-set questions.
For further discussion of these questions, see Brans[8], and especially
Marlow [22]. Maybe there is not enough in this bare-bones quantum logic
approach to work with (produce a theory), but, many others, too many
to mention here, have also looked into the inﬂuence of quantum theory on
spacetime point set properties in diﬀerent ways. However the contributions
of Connes [4] and Madore [23] stand out as leading to much current work in
this area.
7
Absolute Spacetime: Choice of Smoothness
Until recently this was thought to be trivially determined by the topology, at
least for relatively simple manifolds such as R4. However, it is not trivial. In
fact, some of the most exciting developments of diﬀerential topology recently

8
CONCLUSIONS
9
have come as a result of what can be termed “exotic smoothness” on spaces
of relatively trivial topology, for example R4. In many respects, the devel-
opment of this subject parallels that of non-Euclidean and then diﬀerential
geometry. Thus, for many years there were conjectures about the uniqueness
of Euclidean geometry, both mathematically and as physics. Similarly, but
more recently, there have been conjectures that there is essentially only one
way to do calculus globally on topologically simple manifolds. The phrase
“how to do calculus globally” corresponds to what mathematicians call a dif-
ferentiable or smoothness structure. Physically such a structure is a global
system of reference frame patches covering all of spacetime smoothly, that
is, with smooth (C∞) coordinate transformations in their overlaps.
The
phrase, “essentially only one” means only one equivalence class under dif-
feomorphisms of the manifold. This diﬀerence between diﬀerent smoothness
structures and non diﬀeomorphic ones can be a slippery concept to master,
but is central to an understanding of diﬀerential topology. In a way, it is
parallel to that involved in Einstein’s hole argument discussed above. Just as
the choice of diﬀerent coordinates may make the metric look diﬀerent when
the underlying geometry is actually the same, so will a recoordination make
the diﬀerential structure appear to be diﬀerent, when in fact it is equiv-
alent (diﬀeomorphic). This equivalence inducing class of diﬀeomorphisms
corresponds to the underlying principle of general relativity.
By direct calculation, it is possible to show that, up to diﬀeomorphisms,
there is only one smoothness structure on each Rn for n = 1, 2, 3. For n > 4
the same result was obtained later making use of cobordism techniques.
However, the case n = 4 remained an open one. Because of the topological
triviality it was natural to conjecture that it too would turn out to be trivial
with respect to diﬀerential topology. Thus, it was a tremendous surprise
when as a result of the work of Donaldson, Freedman and others it was
established that
Theorem; (Donaldson, Freedman, et al.)There are an inﬁnity of
smooth manifolds of topology R4, none of which are in the diﬀeomorphism
class of the any other (including the standard one).
Thus, the diﬀeomorphism gauge does not cover the entire range of physics
on topologically trivial R4! Do these provide new structures for new physics?
See Brans [24] for a general review of these topics, and Asselmeyer [25] for a
speciﬁc suggestion of physical content.
8
Conclusions
In this paper we tried to survey some of the extensive structures used on
contemporary spacetime models, noting their direct physical unobservability
and reﬂect on this rigid ﬁne structure in the light of the historical paral-
lel of the luminiferous ether. A hundred or so years ago, it was generally
thought (apart perhaps from a few people like Gauss) that geometry was

9
ACKNOWLEDGMENTS
10
“pre-physics,” a natural given. Now, we have take it for granted that geo-
metric structures carry physical ﬁelds.
Perhaps it is now appropriate to speculate that the mathematical struc-
tures “point set” (e.g., Boolean, or non-Boolean...), “topological” (e.g., non-
Euclidean...), “smoothness” (e.g., exotic...), etc., might also serve to carry
physical properties in a manner analogous to that of “geometry.”
9
Acknowledgments
I am indebted to John Norton and Carlo Rovelli for very helpful comments,
suggestions and insights. Also, this work was partially supported by a grant,
LaSpace, R150253.
Finally, of course, we are all grateful to Friedrich Hehl for his persistent
clariﬁcation of the role of various spacetime gauge structures.
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