Time Remains

Sean Gryb∗1 and Karim Thébault†2

1Institute for Mathematics, Astrophysics and Particle Physics, Radboud University, Huygens Building,
Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands

2Munich Center for Mathematical Philosophy, Ludwig Maximilians Universität, Ludwigstrasse 31,
D-80539, Munich, Germany

August 7, 2014

Abstract

On one popular view, the general covariance of gravity implies that change is relational
in a strong sense, such that all it is for a physical degree of freedom to change is for it to
vary with regard to a second physical degree of freedom. At a quantum level, this view
of change as relative variation leads to a fundamentally timeless formalism for quantum
gravity. Here, we will show how one may avoid this acute ‘problem of time’. Under our
view, duration is still regarded as relative, but temporal succession is taken to be absolute.
Following our approach, which is presented in more formal terms in (Gryb and Thébault
2014), it is possible to conceive of a genuinely dynamical theory of quantum gravity within
which time, in a substantive sense, remains.

Contents

1 Introduction 2
1.1 The Problem of Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Our Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Understanding Symmetry 5
2.1 Mechanics and Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Freedom by Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Voluntary Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Understanding Time 15
3.1 Change and Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Quantization and Succession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Time and Gravitation 21
4.1 The Two Faces of Classical Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Retaining Succession in Quantum Gravity . . . . . . . . . . . . . . . . . . . . . . 24

5 Conclusion / Discussion 26
5.1 Related Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.2 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

∗email: s.gryb@hef.ru.nl
†email: karim.thebault@gmail.com

1

mailto:s.gryb@hef.ru.nl
mailto:karim.thebault@gmail.com


We should not be able to tell the story of our relations with
another, however little we knew him, without registering
successive movements in our own life. Thus every individual
– and I myself am one of those individuals – measured
duration by the revolution he had accomplished not only round
himself but round others and notably by the positions he had
successively occupied with relation to myself.

Time Regained (Proust 1931)

1 Introduction

1.1 The Problem of Time

A key feature of Einstein’s theory of gravity is its invariance under arbitrary transformations of
the spacetime manifold. This diffeomorphism symmetry implies that only the coordinate-free
information contained in the geometry has a physical basis within the theory. Unfortunately, it
is not entirely clear how one should understand the implications of diffeomorphism invariance
for the specific role of time in the theory. In the Lagrangian formulation, where the theory is
expressed in terms of the Einstein-Hilbert action, this difficulty manifests itself in our inability
to find a representation of time in terms of an action of the real numbers implementing time
translations on the space of physical (i.e., diffeomorphism invariant) solutions.1 Similarly, in
the Hamiltonian formulation of the theory, which is the basis for many modern approaches to
the quantization of gravity, we find ourselves lacking a coordinate free means of representing
time.2

Imagine a loaf of bread that we can irregularly cut up into a sequence of slices. The loaf
is spacetime and the slices are instantaneous spatial surfaces. A foliation is then a parameter-
ization of a spacetime by a time ordered sequence of spatial slices. Such a parametrization is
local in the sense that it is defined for every point on every spatial slice. Diffeomorphism invari-
ance implies that spacetimes described by general relativity that are related by re-foliations are
physically equivalent. Within the Hamiltonian formulation, which dates back to (Dirac 1958),
we make the restriction to spacetimes that admit a foliation into such sequences of space-like
hypersurfaces (the globally hyperbolic spacetimes (Geroch 1970)). Spacetime diffeomorphism
invariance is implemented in two parts: i) spatial diffeomorphism invariance; and ii) spacetime
foliation invariance.3 We thus have within the theory an ability to re-slice a spacetime into an
infinite number of different decompositions of space and time without changing anything phys-
ical. It is the conceptual and technical complications involved in representing this symmetry
that leads to the acute ‘problem of time’ within the formalism.

Foliation symmetry further implies that any observable quantity within the theory must not
be dependent upon the local temporal labelling of spacetime.4 This leads us directly to the

1See (Belot 2007) for an extensive discussion of this and related points regarding the representation of time and
change within Lagrangian field theories, including general relativity. With regard to the Hamiltonian framework,
the present analysis differs on several key interpretational and formal points. See (Thébault 2012b) and (Pitts
2014) for critical discussion.

2This is one key aspect of the ‘problem of time’ in quantum gravity. See (Isham 1993) for a classic or (Anderson
2012) for an updated review.

3Note that this does not constitute the full group of spacetime diffeomorphisms since neither large diffeomor-
phisms nor diffeomorphisms that fail to preserve the space-like embedding of hypersurfaces are represented in
the Hamiltonian formalism.

4Formally, this is usually expressed in terms of the requirement that the functions that represent observable
quantities should commute with the Hamiltonian constraints, which are taken to implement foliation invariance
(Bergmann 1961). There are, however, many subtleties regarding both the role of Hamiltonian constraints and
the definition of observables in canonical general relativity (Anderson 2013; Pitts 2014).

2



question of how we should understand the change in physical quantities? In addition to not
having a representation of time, we seem also to have lost a clear methodology for representing
change! Our conceptual machinery appears in need of retooling.

According to the correlation or partial observables view of time in general relativity, the
radical moral one should draw from diffeomorphism invariance is that change is relational in a
strong sense, such that all that it is for a physical degree of freedom to change is for it to vary
with respect to a second physical degree of freedom; and there is no sense in which this variation
can be described in absolute, non-relative terms.5 This radical relationalist view of time implies
that there is no unique parameterization of the time slices within a spacetime, and also that
there is no unique temporal ordering of states. Furthermore, it implies a fundamentally different
view of what a degree of freedom actually is: such parameters no longer have distinct physical
significance since they can no longer be understood as being free to change and be measured
independently of any other degree of freedom. This means that all one-dimensional systems
must be understood as stationary since a relational notion of change cannot be constituted:
there is no degree of freedom for the system to change with respect to. A one-dimensional
pendulum is thus, under this understanding of dynamics, a stationary system with no genuine
degrees freedom. And a two-dimensional pendulum is to be understood as a one-dimensional
system, with the change in the (arbitrarily chosen) free variable expressed in terms of the other
‘clock’ variable.

On this view, it should be no great surprise that when the equations of a classically foliation-
invariant theory are quantized, one arrives at a timeless quantum gravity formalism6 – since, in
essence, this facet is already implicit within the classical theory. Both classically and quantum
mechanically, the functions that faithfully parametrize the true degrees of freedom of the theory
– the observables – are taken to be those which are completely independent of the local time
parametrization and, both classically and quantum mechanically, these perennials cannot, by
definition, vary along a dynamical trajectory. Thus, we see that this first response to the problem
of time in classical and quantum gravity is essentially one of capitulation. The definition of an
observable quantity within the correlation view is such that it cannot change along a dynamical
trajectory. Although, we can recover a weak sense of change as relative variation, there is no
scope for the basic one-dimensional ordering structure that, in our view, is constitutive of time.
To us this seems unsatisfactory as a solution, and in the remains of this paper we will articulate
an alternative.

1.2 Our Solution

The starting point of our approach is the conviction that the radical variant of relationalism
with regard to change and time discussed above has gone a little too far. The lessons for time
drawn from general covariance are more subtle, do not imply we should dispense with time alto-
gether. Rather, a consistent interpretation of the underdetermination of possibilities implied by
the temporal relabelling symmetry of the theory, leads to a view in which temporal succession
is understood as absolute. Temporal relabelling symmetries do not result in identification of
instantaneous states as identical physical possibilities and thus the formalism of theories with
these symmetries can be interpreted such that both the change in a given degree of freedom and
the ordering of such change along a dynamical history are fundamental structures in the theory.7

5The correlation view is most closely associated with the work of Carlo Rovelli (Rovelli 2002; Rovelli 2004) and
is put forward in slightly different ways by many physicists working on canonical quantum gravity, for example
see (Dittrich 2006; Dittrich 2007; Thiemann 2007; Bojowald, Höhn, and Tsobanjan 2011). For a more detailed
appraisal of the strengths and weaknesses of this view in the classical gravity context, see (Thébault 2012b). Also
see (Rovelli 2007) for discussion of some subtitles regarding the interpretation of the scheme.

6This is the Wheeler–DeWitt-type ‘frozen formalism’, endemic within both the old quantum geometrodynam-
ics approach (DeWitt 1967), and modern variants of canonical quantum gravity (Thiemann 2007).

7It is important to note here that by ‘time ordering’ we specifically do not mean anything like an arrow of time,
or objective difference between past and future. Rather, ‘time ordering’ here, and in what follows, merely implies

3



Given that, one can exploit the interpretational underdetermination present to take a further
step regarding the status of duration. One may insist that although change itself is taken to
absolute, the labelling of change in terms of a measure of duration, is something purely relative.
Such a Machian view of time is simple both to motivate and to realise within classical particle
models with global temporal re-labelling symmetry, but far less easily constituted within the
full general theory of relativity due to the need to define observables that respect foliation in-
variance. It is therefore understandable that the radical morals with regard to change and time
discussed above are often drawn. However, it is still true that within general relativity there
exist fundamental temporal structure relating to ordering in time. In canonical general relativ-
ity, such structure is encoded within the fact that the arbitrary slicings are always labelled in
terms of a monotonically increasing local time parameter (as implied by the positivity of the
lapse multiplier). This structure is also present in a more subtle sense within the Lagrangian
theory due to the form of the Einstein–Hilbert action. This is because the variation of the
Einstein–Hilbert action (subject to the appropriate boundary conditions)8 requires finding a
curve that minimizes the integral of the scalar curvature, and these curves, by definition, re-
quire parametrization by a monotonically increasing parameter. Thus, the formalism of general
relativity should not be seen as telling us to dispense with time ordering altogether.

Furthermore, on our view, that the (canonical) quantization of general relativity leads to
a timeless formalism should be understood as a consequence of an incorrect treatment of the
temporal symmetries of the classical theory. By treating local temporal labellings as entirely
unphysical, and change as entirely relational, we do not retain in the quantum formalism the
full classical dynamics or the implicit temporal ordering structure.9 The question remains, how-
ever, if conventional quantization techniques cannot preserve the essential temporal structure
of general relativity, how do we find a new methodology that can?

Our solution involves two non-trivial steps, the motivation for which will be outlined at
length below. The first relates to a fundamental re-description of gravity in terms of the shape
dynamics formalism originally advocated by Barbour and collaborators (Barbour 2003; Barbour
2012; Anderson, Barbour, Foster, and O’Murchadha 2003; Anderson, Barbour, Foster, Kelleher,
and O’Murchadha 2005) and then brought into modern form in (Gomes, Gryb, and Koslowski
2011). From the view of the current paper (see also the more detailed argument of (Gryb and
Thébault 2014)), the existence of shape dynamics as, in a precise sense, a dual to general relativ-
ity reveals classical gravity to be essentially Janus-faced. There exist an underdetermination of
symmetries that leave space for interpretation in terms of two distinct gravitational ontologies:
the traditional ‘Einstein ontology’ of spacetimes invariant under four dimensional coordinate
transformations; and a second, hitherto masked, ontology of sequences of scale-invariant three
dimensional spatial surfaces (i.e., spatial geometries invariant under re-scallings of lengths). As
shall be detailed later, this second ontology is closely related to a proposal for the interpretation
of the degrees of freedom of gravity made by James York (York 1986), and so we will call it the
‘York Ontology’.

the existence of a monotonically increasing parameterization of time slices which is, by definition, time-reversal
invariant.

8See (York 1986) for a discussion of this variational principle in the context relevant here.
9Equivalently, in more formal language, if Hamiltonian constraints are treated as generating purely unphysical

transformations one does not, in the quantum theory, retain their role in generating dynamics of the quantum
state, nor in providing a temporal ordering.

4



Figure 1. Janus was the Roman god of gateways, transitions and time, and is usually depicted as
having two, non-identical faces pointing in opposite directions as above. Thus, figuratively (and in fact
geometrically) the two faces of Janus are an apt representation for the two faces of gravity.

Given this understanding of gravity as having two dual faces, when confronted with the
problem of understanding the role of time in classical gravitation, one has the option of choosing
whichever formalism – shape dynamics or general relativity – is formally and conceptually easier
to work with. Here, we choose to use shape dynamics, and assume the particular characterization
of the theory is given by the York ontology. It is from this basis that the second step in our
proposal can be made. Originally in (Gryb and Thébault 2012) and then more recently in
(Gryb and Thébault 2014), a procedure for the relational quantization of theories with temporal
relabelling symmetries was outlined. Whereas the first of these two papers served to offer a
range of conceptual arguments leading to the need for relational quantization, the second served
to place relational quantization within a formal framework for understanding symmetries in
physical theory in general.

One of the principal motivations of the current paper is to explicate further the philosophical
foundations of this approach to symmetries and time. In particular, in what follows we will:
First, introduce a general methodology for the classification of symmetries and symmetry-related
variables according to physically motivated criteria, see §2.1-2.2; Second, demonstrate that our
classification scheme leads naturally to a procedure for quantization via the introduction of
dummy variables, see §2.3; Third, provide philosophical motivation for the Machian view of time
discussed above, see §3.1; Fourth, show how our philosophical motivations for the treatment of
time symmetries mesh with our general prescriptions for symmetry and lead to the procedure for
relational quantization, see §3.2. These arguments establish a framework sufficient to motivate
the relational quantization of gravity from the perspective of the York ontology, see §4.1 and
§4.2, and in doing so provide a demonstration that, given suitable starting assumptions, time
can remain in quantum gravity.

2 Understanding Symmetry

2.1 Mechanics and Representation

In a classical physical theory the relationship between a mechanical system and its theoretical
representation can be given through the specification of three pairs of structures, one of each
pair formal and one of each pair physical. These pairs relate in turn to: degrees of freedom,
dynamical laws, and the specifiable initial (or boundary) data. Explicitly, for a finite dimensional
classical system, we can consider a physical system as being represented formally by: i) a

5



configuration space with n-configuration variables and some pre-determined metric structure;
ii) a nomological restriction on curves in the configuration space that selects only curves that
are geodesic with respect to the pre-determined metric structure; iii) a set of further initial (or
end-point) conditions on the curves.10 Such a specification serves as a representation of: a) the
physical degrees of freedom; b) the dynamical law which determines the evolution of the system;
and c) the physical conditions on the preparation of the system. The representational pairings
are then i-a, ii-b, iii-c. We will designate the formal side of these pairings the formalism of the
system and the physical side the characteristic behaviour of the system.

One can be more explicit with regard to the formalism side of the set up by defining the
Lagrangian formalism. For an n dimensional system, the configuration space is a manifold,
C, with elements qi, i = 1, ...,n. At a given point q ∈ C we can define a tangent space TqC.
The disjoint union of all the tangent spaces of C is the tangent bundle TC. The elements of
the tangent bundle are pairs (q, q̇) of configuration variables q and vectors tangent to those
variables q̇. A curve within the tangent bundle, γ : R → TC0, will correspond to a history of a
system: a sequence of configurations and velocities. The parameterisation of the curve is given
by some monotonically increasing parameter, t. We then define the Lagrangian, L : TC → R,
and the action, I[γ] =

∫
γ
L[qi, q̇i]dt. Given the specification of a curve (including its endpoints),

the extremisation of the action, δI[γ] = 0, according to the principle of least action leads

to the Euler–Lagrange equations, d
dt

(
∂L
∂q̇i

)
= ∂L

∂qi
, that specify a set of solutions, γS, which

uniquely determine the possible classical histories of the system given an initial point in TC.
We now have a formal representation of physical degrees of freedom in terms of the velocity
and configuration variables; the relevant nomological restrictions, in terms of the variational
principle and Lagrangian; and the preparation conditions, in terms of the specification of the
end points of the curves in the variation.

An alternative, but generally equivalent, Hamiltonian formalism can then be derived by
defining a cotangent bundle of our configuration manifold: the phase space Γ = T∗C. This is
the disjoint union of all the cotangent spaces T∗qC (these are dual to the tangent spaces – i.e.,
elements of the cotangent space are linear functionals on the tangent space). A point in phase
space, (q,p), consists of a point in our original configuration space, q ∈C, paired with a covector
at q, p ∈ T∗qC. These covectors, which we call the conjugate momenta, are given by, pi =

∂L
∂q̇i

.
We can equip phase space with a symplectic structure by defining the symplectic potential
θ = p ∧ dq and the (closed) symplectic 2-form ω = dθ. If ω is non-degenerate, then one can
equip phase space with a Poisson bracket such that

{
qi,pj

}
= δij. Using this Poisson structure,

we can define a Hamilton vector field, vf (g), for any phase space functions (f,g) : Γ → R, via
vf (g) = {g,f}. We can then define the flow of some function f by the integral curves of its
Hamilton vector field.

To fix the dynamics, we introduce the Hamiltonian functional, H[qi,pi] = p
iqi − L, and

derive Hamilton’s equations, ṗi = vH(p) and q̇i = vH(q). The classical trajectories, γ̄S, are,
thus, uniquely defined by the flow of the Hamiltonian functional on phase space. For the
purposes of this formalism, the preparation conditions are represented by specification of initial
values of the positions and momenta variables.

All this is familiar to anyone with basic knowledge of mathematical physics. It is important,
however, to be clear regarding the relevant representational relationships. For a given system
we have, on the one hand, the physical characteristics of the system contained in the nature
of the degrees of freedom, the physical preparation of the system, and the law-like regularities
in behaviour; and, on the other, we have the formal description of the system contained in
the formal variables, the formal initial (or end-point) conditions, and nomological restrictions
on these. The connection between these two triples is precisely what a physical theory gives
us. In some very simple cases, such connections can be understood unambiguously; however,

10It can be shown, see for instance (Lanczos 1970), that these three requirements (which are collectively known
as Jacobi’s principle) are equivalent to Hamilton’s variational principle for mechanics.

6



invariably, when we want to understand realistic systems, we often run into situations where the
relationships between the formal and physical aspects is no longer one-to-one. In such situations,
we can say that some form of representative redundancy is present within our formal-physical
set up. One of the main aims of this paper is to detail a new scheme for understanding this
representative redundancy. In particular, we aim to provide a general and physically motivated
classification of different types of symmetry transformations. This classification will entail which
formal differences have the capability of representing distinct physical possibilities and which
correspond to pure redundancy.

Such a symmetry classification scheme has direct implications for the ontology that can be
associated with a physical theory. At a classical level, this is because, by placing constraints on
which formal differences can correspond to different physical possibilities, our understanding of
symmetry also places constraints on the consistent interpretations of a theory. These constraints
will invariably still leave the ontology underdetermined: the same possibility counting can
be consistent with very different ontologies for a physical theory. However, in some specific
circumstances – including the case of time parametrization symmetries that is our main focus
– difference as to the symmetry classification scheme can have crucial ontological importance.
The ontological importance of symmetry classification is even greater considered in the context
of quantization. As we shall see in detail later, the way in which we treat symmetries classically
determines which possibilities will be realised at the quantum level.

Clearly, then, a symmetry classification scheme is something of great importance, and should
be built upon a chain of sound physical and formal inferences. What is needed are general
principles that are based upon physical reasoning but lead to precise mathematical diagnoses.
Unfortunately, from our perspective, what is provided by most existing approaches are precise
mathematical principles leading to a diagnostic schema that is neither physically well motivated
nor formally rich enough.

The standard classification of symmetries follows a scheme where the existence of local or
‘gauge’ symmetries – i.e. transformations that depend on space or time and under which the
action is invariant – indicates that identical physical possibilities are being represented in terms
of distinct instantaneous states in the formalism.11 The representative redundancy inherent in
such symmetries is directly connected to otiose degrees of freedom which are then eliminated
during (or before) quantization. In this standard scheme, we can remain agnostic regarding
the status of redundancies associated with global symmetries – i.e. transformations that do not
depend on space or time and under which the action is invariant. The representative redundancy
related to such symmetries need not be eliminated during (or before) quantization. Thus, within
this scheme, the difference between local and global symmetries has huge potential impact at
the level of both classical and quantum mechanical interpretation.

In our view, the weight placed upon the local vs. global distinction represents a first major
problem with the scheme. There are theories which display local symmetry but do not feature
identical physical possibilities represented in terms of distinct instantaneous states. In such the-
ories, there are in fact no excess degrees of freedom and, thus, the application of standard ‘local’
symmetry quantization techniques will lead to the elimination of genuine physical differences.
Moreover, the loss of representative machinery will have direct interpretational consequences:
there will be ontologies which are excluded from the cast of possible interpretations for no good
reason.

A second problem with the standard scheme is that it is ill-suited for dealing with a recently

11The most comprehensive modern book on gauge theory and its quantization is (Henneaux and Teitelboim
1992), the original classic is (Dirac 1964). See (Belot and Earman 2001; Belot 2003; Earman 2003; Rickles
2004; Rickles 2007) for philosophical analysis of the connection between possibility spaces and this notion of
gauge symmetry. The crucial factor in most of these accounts is the ‘indeterminism’ that can result from not
eliminating the excess possibilities associated with local symmetries. See (Thébault 2012a) for details on why such
arguments are not decisive in general. See (Pons 2005) argument that the standard Dirac analysis of canonical
quantization is incomplete.

7



discovered form of symmetry: the class of hidden symmetries. These symmetries are of a
peculiar form, in that they are derived from implicit rather then explicit redundancies in the
formal-physical relationship. When hidden symmetries occur, we are able to ‘trade’ a certain
symmetry of the theory for a second set of transformations which was originally not a symmetry.
Such symmetries, thus, lead to a different form of interpretational underdetermination since we
have a choice as to different formulations of a theory with different symmetries – over and above
the way we understand the implications of these symmetries.

Each of these problems with the standard scheme is directly related to issues regarding
time and gravity. With regard to the first problem, time reparametrization symmetries are an
example of local symmetries which should not be associated with excess degrees of freedom.
The standard scheme leads to an erroneous categorisation of dynamically related instantaneous
states as constituting the same physical possibility. This rules out an interpretation of the
temporal ontology of the theory in hand in terms of anything other than radical relationalism.
With regard to the second problem, the identification of hidden symmetries provides the basis
of the duality between General Relativity and Shape Dynamics mentioned above. It is thus
only through the understanding of hidden symmetries that the ‘dual faces’ of gravity can be
identified. We will return to these points in detail when we enter into the specific discussions
of time (§3) and gravity (§4). Before then, we would like to construct our own scheme for the
categorisations of symmetries which does not suffer from the identified defects. In order to do
this, we need to find general physical principles for distinguishing types of redundancy.

2.2 Freedom by Degrees

The first physical principle for distinguishing types of redundancy we can identify relies upon
the notion of action, I[γ]. As noted above, the action is directly connected to the nomological
restrictions that allow the formalism of a theory to pick out physical dynamics. If we consider
the infinitesimal variation of a curve in a particular direction in the configuration space (by
‘direction’ we will mean along an infinitesimal segment of the flow of a particular phase space
function) and find that the action is invariant up to a total derivative, then the degree of freedom
associated with that direction (i.e., the degree of freedom identified with the particular phase
space function whose flow is in the relevant direction) is related to a manifest form of symmetry.
It is important to point out, for our considerations, that a manifest symmetry can either be
a local symmetry or a global symmetry. This criteria relates to a property of the action itself
and has nothing to do any additional variational principles one would like to further impose on
the action (for example, to extract classical equations of motion). To fully classify a degree of
freedom, we will need a second criteria, described below, which leads to a richer set of physically
distinct cases than what is explicitly considered by standard textbook definitions of symmetry.

Physically speaking, a manifest symmetry implies that there will be multiple possible se-
quences of configurations (representing histories in the configuration space) which are either
physically indistinguishable or correspond to different values of a conserved charge. In the first
case, the symmetry corresponds to a mathematical transformation within the formalism that
has no effect on physically measurable quantities. Thus, it can apply to the Universe as a whole.
In the second case however, the transformation has an experimental implication: it changes the
value of the conserved charge. The only way to attribute meaning to this is to have an emergent
structure within the formalism that allows one to measure changes of the variable conjugate
to the relevant conserved charge. Thus, the second case is only relevant to sub-systems of the
Universe that are dynamically isolated from the rest of the Universe, which serves as an emer-
gent background. For a more detailed discussion of how these backgrounds emerge, see (Giulini
2013).

Furthermore, the physical indistinguishability of the histories could be due either to practical
limitations within the particular experimental set-up being considered (as is nicely illustrated,
for example, in (Wharton 2009)) or to fundamental limitations within the system. On its own,

8



the presence of a manifest symmetry does not indicate that identical physical possibilities are
being represented in terms of distinct instantaneous states in the formalism.

Given an orbit on configuration space, generated by the flow of some phase space function,
in which the action is not invariant, there are two further possibilities for the relevant degree
of freedom identified with the phase space function in question: it might be the case that there
are no symmetries associated with it, or it might be the case that there are hidden symmetries
associated with it. This third case has not been explored until recently, but will prove important
for gravity. We will return to its detailed consideration later.

The second principle we can identify relates specifically to the nature of the variational
principle used to vary the relevant variables in the action. As was noted above, it is essential
to remember that the abstract initial (or end-point) conditions used in the variational principle
are part of a formalism which stands in a possibly non-unique representative relationship with
a class of physical systems defined in terms of their characteristic behaviour (i.e., physical
observables within the system, physical preparations of the system, and dynamical laws obeyed
by the system). Let us again consider the variation of the action based upon the variation of
a configuration space curve in a direction associated with a particular degree of freedom (as
defined more precisely above). In this case, let us focus upon the details of the variational
principle used along the entire history of the system in question, including the boundary (in
space and time).

If the characteristic behaviour – which is fixed by the actual physical degrees of freedom,
the physical preparation of the system, and the dynamical law – places no restriction upon such
a variation, then we say that it is a free variation. Under our proposed scheme, this type of
variation implies that the relevant degree of freedom is an otiose formal artefact since nothing in
‘the world’ places a restriction upon the relevant variable’s value. Nothing in the characteristic
behaviour of the system fixes anything in the mathematical formalism, and so it must be a
facet of redundancy within our representation. The alternative is that, for a given direction and
associated degree of freedom, restrictions are placed on the variation. In such a situation, we
say that we have a fixed endpoint variation (because the end points of the variation are fixed
in configuration space), or fixed variation for short, and we expect that the relevant degree of
freedom has some representative relationship to something physical.

It is important to note why one would expect that it is the second, and not the first principle,
which is decisive in the categorisation of a symmetry-related degree of freedom as inherently
redundant or not. Although the first principle does derive from conditions on the action, it does
not derive from conditions specifically on the variation principle. It is precisely the variational
principle that fixes the full characteristic behaviour of the system. Thus, it is only the variation
principle that can ultimately be sensitive to the difference between redundancies that are linked
to dynamical conservation properties, and those that are entirely due to our use of excessive
coordinates within the instantaneous representation of a physical state.

Keeping this important point in mind, and given our two physically motivated principles,
we can set about categorising types of redundancy according to a physically motivated diag-
nosis. One would then hope that the mathematical exactitude of standard techniques will be
recoverable where these techniques have proven physically reliable. This indeed proves to be
the case if we consider the most blatantly unphysical form of redundancies: those connected
to manifest symmetries and free variations. In such situations, the relevant degree of freedom
encodes no dynamical information, and defines a direction which is, by definition, superfluous to
the representation of the world. Such variables are gauge variables and the relevant symmetries
are gauge symmetries. Thus, we see the precise reasons why gauge symmetries (as classified by
our scheme) will lead to identical physical possibilities being represented in terms of distinct
instantaneous states in the formalism.

Our categorisation is sufficient, although not necessary, to recover the conventional math-
ematical categorisations of gauge symmetries (Henneaux and Teitelboim 1992). As discussed

9



above, the standard account of gauge symmetries is usually in terms of a local (i.e., functionally
dependent on space and time) transformation that leaves the action (or Lagrangian) invariant.
Our categorisation of gauge symmetry is sufficient (but not necessary) for this locality condi-
tion. It is also sufficient (but not necessary) to recover the other standard notions of gauge
symmetry in terms of: i) the failure for the the Legendre transformation between the velocity-
configuration and phase space to be invertible, and; ii) existence of first class primary constraints
(Dirac 1964). This means that for all the standard cases where there is no perceived ambiguity
about the cause and interpretation of redundancy – e.g., electromagnetism, Yang-Mills theories,
the standard model (all in the presence of no spatial boundaries) – our definition will coincide
with the standard definition. However, as we shall see, for the case of time labelling symmetries,
and indeed many other symmetries which will not be discuss at length here,12 our scheme still
allows for an alternative, physically motivated categorisation.

Here, we should note an important point for the purposes of our discussion: those degrees
of freedom identified as gauge within the classical theory are always (in some sense) eliminated
within the process of constructing the quantum theory. Since these degrees of freedom are not
representing anything about the physics of the system, the quantum correlates of these degrees
of freedom must not be associated with observable operators in the quantum formalism. In
essence, the methodology for ensuring faithful treatment of gauge degrees of freedom is always
the same: treat the direction associated with the degree of freedom as non-physical.

A further, more subtle form of redundancy derives from the presence of a degree of freedom
associated with a manifest symmetry, but a fixed variation. Since the variation is fixed we know
that the relevant variable is connected to something physical. In general, we can understand this
‘something physical’ as the conservation of some empirically determinable quantity throughout
the system’s evolution. For this reason, we call the relevant symmetries conservation symme-
tries.13 There is no necessary connection between the existence of conservation symmetries and
identical physical possibilities being represented in terms of distinct instantaneous states (or
sequences of states) in the formalism. It is a further interpretational step, needing further mo-
tivation, to eliminate such potential redundancy. We can thus understand the interpretational
implication of the existence of a conservation symmetry in terms of an underdetermination of
possibilities. One has the freedom to classify the transformations related by the symmetry as
connecting the same or different physical possibility. Our scheme does not provide means to
adjudicate between these options.

Again, our definition allows us to recover the parts of the standard scheme that are phys-
ically well motivated: all symmetries associated with Noether’s first theorem are, under our
definition, conservation symmetries.14 An example particularly relevant to the considerations
of this paper is that our classification scheme non-standardly directs us to categorise symmetries
associated with temporal relabelling as conservation symmetries and, as well shall discuss later,
this proves absolutely pivotal for understanding the role of time in relational quantum theories,

12Note that asymptotically flat GR and Yang–Mills theory in the presence of spatial boundaries which break
gauge invariance are examples of theories that also do not fall into the category of pure gauge theories by our
classification, although they would by the standard treatments (Henneaux and Teitelboim 1992). We believe that
our classification scheme is more appropriate for these cases because a notion of background is introduced by the
relevant boundary conditions.

13By this terminology, we do not wish to imply that all conserved quantities are related to conservation
symmetries (e.g., electric charge is a conserved quantity arising in gauge theories and is not a conservation
symmetry by our definition), although conservation symmetries necessarily have conserved charges associated
with them. Their distinguishing feature, which is relevant for us here, is that, in the quantum theory, conservation
symmetries allow for superpositions of states with different values of the conserved charge.

14Our scheme also allows us to capture classical symmetries exhibiting conserved charges that would not
normally fall under the treatment of Noether’s theorem. For example, General Relativity with asymptotically
flat boundary conditions is locally invariant under spacetime diffeomorphisms. Nevertheless, it still has conserved
charges associated to it (the ADM momenta (Arnowitt, Deser, and Misner 1962)) because the boundary variation
is performed in a fixed way, using our terminology, due to how the asymptotic boundary conditions are imposed.
Thus, the Poincaré invariance on the asymptotic boundary is a conservation symmetry by our definition.

10



including prospective theories of quantum gravity.15 The key point is that, in general, conser-
vation symmetries, since they are tied into the characteristic behaviour of the system, must be
treated entirely differently from gauge symmetries when constructing the quantum formalism.
They are associated with physical directions on phase space and should correspond to classical
observables. In the quantum theory, there should, thus, be self-adjoint operators associated
with the classical observables which act on the physical Hilbert space. One should therefore
expect that general quantum states are formed by taking superpositions of the eigenstates of
these operators. This is entirely unlike gauge symmetries, where the eigenvalues of the associ-
ated operators (i.e., the quantized generators of the classical symmetries) must be set to zero
in the quantum formalism (following the Dirac quantization algorithm).

A simple case of a conservation symmetry will illustrate this point well. Let us consider
a standard Newtonian point particle system treated as an isolated sub-system of the Universe
consisting of three particles starting in different positions which evolve under the force of gravity.
Now consider what happens if one performs time-dependent spatial translations to this system
of particles, under which Newton’s equations are manifestly invariant.

In our terms, such a translation is precisely a manifest symmetry since it corresponds to a
global variation of configuration space curves in a direction along which the action is invariant.
We can further classify the symmetry as manifest-fixed, because, for a Newtonian system,
we are not free to vary the data on the end points – the physics of the system places definite
restrictions such that only some spatially translated variations are equivalent. In this case, these
restrictions are just the conservation of linear momentum and the relevant constant of motion
is just the total linear momentum of the system. Thus, we have a symmetry which is fixed and
manifest – a conservation symmetry in our terminology. Standard quantum mechanics, which
is the quantum theory defined based upon Newtonian mechanics, is then such that we can have
superpositions of momentum eigenstates, as one would expect from our general prescription.
If, on the other hand, one has misclassified the spatial translations as gauge symmetries (i.e.,
manifest-free), then the quantum formalism that resulted would be a quantum theory of a single
momentum eigenstate – which is clearly not a faithful quantization of Newtonian theory. What
is lost in this analysis is the ability to treat the centre-of-mass velocity of the system as an
operator in the quantum theory, since forcing the system to a momentum eigenstate forces this
observable to be precisely zero. However, if the three particle system is an isolated sub-system
of the Universe, then the behaviour of the centre-of-mass velocity clearly has meaning as part
of the characteristic behaviour of the system, and such a misclassification would fail to capture
the full behaviour of the system. It is precisely this form of categorisation error that we hold
to be behind the idea that ‘time disappears in quantum gravity’, in complete analogy to how
‘centre-of-mass velocity disappears’ in the example above.

The simplest case in our classification scheme is where there is no redundancy in the relevant
representative relationship between a physical degree of freedom and its formal correlate. This
is the case where there is no manifest symmetry and the variation is fixed. This case corresponds
to a conventional dynamical degree of freedom. Its initial or boundary conditions are specified
by the variational principle and, in a phase space formalism, it simply evolves according to
the flow of the energy function or Hamiltonian. The different possibilities corresponding to the
direction associated with the degree of freedom are physically distinguishable, and so must be
counted as distinct possibilities. Quantum mechanically, these degrees of freedom correspond
to observable operators that can exist in the appropriate superpositions.

The most non-trivial case is if there is no manifest symmetry and the variation is free. In this
case, it is possible that there is a hidden symmetry in the system. This can only happen if there
is another manifest symmetry in the theory that has a particular type of formal relationship
with the one at hand (it is second class with respect to it – i.e., the Poisson brackets of the

15The conserved quantity associated with relabelling symmetry is the Hamiltonian function itself, which, as
one can easily show, is a conserved quantity of the classical evolution.

11



constraints generating these symmetries is not weakly zero). If this is the case, the elements of
the formalism can be modified (without changing the physical predictions of the theory) in such
a way that the first symmetry becomes manifest. This is called symmetry trading and has been
used to construct the shape dynamics formalism introduced in (Gomes, Gryb, and Koslowski
2011). The general theory of symmetry trading is developed in (Gomes and Koslowski 2012).
The main formal result behind the symmetry trading formalism is that the two symmetries in
questions can gauge fix each other, since they are generated by constraints that are second class
with respect to each other. Thus, one has a choice to interpret one of the constraints to be the
generator of a symmetry while the other is the gauge fixing of it, but the opposite interpretive
choice can equally be made. The two theories are equivalent because there is a special choice
of gauge in both theories where the evolution on phase space is identical, given some initial
data that solve the initial value constraints. We will give an more detailed description of this
idea in the context of general relativity further down. For the time being, we can merely point
out the clear interpretational implication of the existence of a hidden symmetry in terms of
an underdetermination of symmetries. One has the freedom to reformulate a theory such that
different sets of transformations become symmetries. These new symmetries can be of either
the gauge or conservation type. The scheme does not provide means to adjudicate between
these options.

The quantum mechanical implications of trading hidden for non-hidden symmetries are sub-
tle yet potentially very powerful. As shall be outlined below, one of the major possible benefits
of symmetry trading is that it allows us to exchange one symmetry – which we are unsure how
to quantize, for another for which there are available techniques. Thus, the underdetermination
of the symmetries of a theory at the classical level can give benefits regarding quantization: it
gives one more options.

Finally, if there is no manifest symmetry, the variation is free and there is no symmetry
which can be traded, then the situation is more complicated because extra constraints need to be
imposed in order for the symmetry to be preserved by the dynamics. These new constraints often
lead, for the same reasons, to even further constraints leading to an infinite regress of constraints.
This is a sign of an inconsistent system (i.e., a mathematically poorly posed formalism). We can
now collect together all the possible types of degrees of freedom classified as distinct within our
scheme in Figure 3. The appropriate interpretation implications and prescription for treating
the quantum mechanical equivalents are then given in Figure 4.

Variation Symmetry in Action Classification

Free Manifest Gauge Symmetry

Fixed Manifest Conservation Symmetry

Fixed None Physical Direction

Free Hidden Tradable Symmetry

Free None Possible Inconsistency

Figure 3. New symmetry classification defined using physical principles.

The tables make clear both the generality and potential physical importance of our scheme.
If a degree of freedom is misclassified then not only will the interpretation of its role within
the classical formalism be incorrect, the quantum formalism derived will fail to capture the
characteristic behaviour of the classical system in the appropriate limit. A mistake at this stage
will lead to an incorrect quantum theory. The key claim that will be defended later in this
paper is that precisely such a misclassification has been made for the case of gravity, and that
the so-called timelessness of quantum gravity is actually a manifestation of this mistake, and
not the absence of basic temporal structure in the relevant system class.

12



Classification Interpretational Implication Quantum Degree of Freedom

Gauge Symmetry Eliminate otiose variables Operator annihilates wavefunction

Conservation Symmetry Underdetermination of possibilities Operator obeying identity

Physical Direction Fixed possibilities Operator with no restriction

Tradable Symmetry Underdetermination of symmetries Represented in new way?

Possible Inconsistency Formalism inconsistent? Quantization blocked

Figure 4. Implications of new scheme. Figure 4

2.3 Voluntary Redundancy

In the previous section, we outlined a classification scheme for the redundancies that can oc-
cur within the representative relationship between classical mechanical formalisms and classical
physical systems. One of the most important applications for this scheme is to ensure a phys-
ically well-motivated quantization of the theory in question – i.e., one leading to a quantum
formalism where the relevant quantum mechanical analogues to the classical degrees of free-
dom are faithfully represented. Unfortunately, for some classical systems, standard techniques
for quantization do not lead to quantum formalisms with such properties.16 Thus, even if we
correctly classify the symmetries in the classical formalism, we may lose track of them during
the process of quantization. Furthermore, while, for simple systems and symmetries, it can be
a straightforward task to isolate the degree of freedom associated with a particular symmetry,
the general case is famously problematic when explicit systems are considered (as an example,
compare electromagnetism to Yang–Mills theory). In order to prevent such problems, we rec-
ommend a general formal procedure that provides a concrete method for explicitly isolating the
degree of freedom associated to the symmetry in question. The description we will give here
for this procedure will be rather more intuitive than explicit. More technically inclined readers
may refer to §3 of the companion paper (Gryb and Thébault 2014) for details.

The procedure we recommend for dealing with this problem involves introducing even more
redundancy into the formalism by introducing auxiliary fields that artificially parametrize the
symmetry in question. These auxiliary fields, which are so named because they can be integrated
out by inserting their classical equations of motion, are introduced into the theory in such a way
that they shift all the degrees of freedom of the theory along the orbit of some gauge group in a
way that we will describe in more detail below. Formally, these auxiliary fields ‘compensate’ for
the symmetry group in question, giving them a role very similar to that of a gauge compensator
field. These auxiliary degree of freedom can then either be varied in a free way or fixed way
by either imposing or not imposing a specific functional restriction onto the corresponding
compensator field. This restriction will be referred to as the best-matching constraint because
it originally appeared in the context of a procedure called best matching developed by Barbour
(for a nice introduction to best matching see (Barbour 2010)). Although the use of these
compensator fields is often a matter of finding a convenient way to mathematically isolate a
degree of freedom associated with some symmetry, for the case of reparametrization symmetry
– which is the primary case of interest to us here – the introduction of a compensator field is a
mandatory technical step in being able to faithfully represent the symmetry.

To illustrate what the best-matching conditions achieve, we will briefly describe the role of
the compensator fields in the formalism. These fields are simply symmetry group parameters
that represent an active transformation of the configuration variables of the system. The group
G itself is determined upon specification of one’s formalism. In other words, once one chooses: i)
the set of configuration variables one would like to use to describe a particular system and ii) a
particular parametrization of the physical observables of the theory in terms of these variables,

16This issue is over-and-above the occurrence of anomalies – which we will not discuss here.

13



then the choice of group is given by the quotient of the former by the latter. In the case
where G can be represented as a Lie group, the formalism becomes straightforward to describe.
The active transformation of the configuration variables, q, can be written by exponentiating
the contraction of the compensator fields, θ, with the generators, t ∈ g, of the the relevant
Lie algebra. This gives a set of actively transformed quantities q̄ = eθ· tq that depend on the
compensator fields θ. It is clear from this definition what these barred coordinates represent: the
difference between them and the original q’s is just given by motion in the direction associated
with the symmetry group orbit (i.e., along the flow of the constraints generating the symmetry
in question). Thus, they can be used to absorb the symmetric degree of freedom we are looking
for.

For a simple example of how this works, consider the case of N Newtonian particles in 1d
with coordinates qi and suppose we want to consider the dilatations as a symmetry group. The
action of this group on these coordinates is simply qi → eθ(t)qi, where θ here is the compensator
field parameterizing the dilatations. The effect this has on the theory is to perform time-
dependent rescalings of the coordinates. We can ensure that this new field θ does not change
the predictions of the theory (and is, thus, a genuine auxiliary field), by imposing an additional
phase space constraint πθ −

∑
i q
i = 0 onto the theory, which is the infinitesimal generator of

the gauge symmetry

qi → e−φqi θ → θ + φ (1)

under which the quantity q̄i = eθqi is clearly invariant. The net effect of this procedure is to
mix up the part of the qi associated with the global scale with the new field θ. The additional
gauge symmetry above shows how this auxiliary field acts to compensate the (at this moment)
artificial dilatational symmetry just introduced.

The compensator fields are then used to implement either a fixed or free variation in a two-
step process. First, we define the momenta p, conjugate to q, and πθ, conjugate to θ, and perform
a canonical transformation (i.e., a transformation that preserves the symplectic 2–form on the
extended phase space) from the original coordinates (q,θ; p,πθ) to a set of barred coordinates
(q̄, θ̄; p̄, π̄θ). This canonical transformation is completely determined by the definition of q̄ and
the requirement that θ̄ = θ (for details see (Gryb and Thébault 2014)). The effect of this
transformation is to effectively mix the compensator field with the symmetric degree of freedom
in such a way that the barred momentum, π̄θ, represents the momentum conjugate to the
symmetric degree of freedom, as expressed in terms of the new variables.

The second step is to perform the actual variation. How we do this depends crucially upon
the free vs. fixed distinction. If the variation is fixed, we know that the degrees of freedom in
the original phase space, (q,p), were all physical. We should therefore treat only the directions
associated with the introduction of the compensator fields (and their momenta) as corresponding
to surplus representative structure. This equates to making the variation independent of two
phase space degrees of freedom17 for each compensator field, and can be done explicitly by
enforcing the canonical restrictions πθ = 0. This condition can be treated using standard gauge
theory methods developed by Dirac (Dirac 1964). Although the details are not important here,
the main result is that the restriction πθ = 0 eliminates the extra redundancy introduced into
the theory when adding the compensator fields.18

If the variation is free, then an additional step must be taken since the original theory already
had non-physical redundancies. This step involves adding an additional constraint to the system
that guarantees that the action is independent of the velocities of the θ̄’s. This will ensure that

17For the remainder of the text, we will count phase space – as opposed to configuration space – degrees of
freedom as is conventional in canonical approaches to quantum gravity.

18More explicitly, the condition πθ = 0 is a first class constraint which can be gauge-fixed by the condition
θ = θ̄ = 0, thus eliminating two phase space degrees of freedom, as outlined in, for example, (Henneaux and
Teitelboim 1992).

14



the theory is independent of any freely specifiable information associated with the symmetric
degree of freedom, independently of when the end points of the variation are specified. Formally,
we can express the relevant requirement as the disappearance of the transformed momentum
variable to the compensator field, i.e., via imposing the best-matching constraint equation:
π̄θ = 0.

Now, let us try and understand more clearly the role played by the compensator fields
after the transformation has mixed them with the symmetric degrees of freedom. The manifest
symmetry requirement states that the action is invariant under the symmetry in question. In
terms of the transformed compensator fields, this property, together with the Euler–Lagrange
equations of the system, implies that π̄θ = constant. This means that, in the manifest case, we
always have a conserved quantity. This quantity is called the Noether charge associated to a
global symmetry.

In a fixed variation, this charge is determined by the initial conditions of the system. For
example, if the symmetry in question is represented by linear translations in space, the momenta
of the compensator fields correspond to the total linear momentum in each of the three spatial
directions. Such quantities are, of course, conserved in the evolution of any isolated system and
we thus see that the conservation of the Noether charge relevant to linear translations simply
is the expression of conservation of linear momentum. Classically, the value of a conserved
charge is something definitely determined for once and for all time by the initial state of the
relevant system. Quantum mechanically however, things are more flexible. Systems can, and
generally do, exist in superpositions of different values of the relevant charge. This is a direct
manifestation of the fact that conservation symmetries are rooted in physical dynamics rather
than redundant representative structure.

In the free case, however, there is no way to fix the corresponding charge. Instead, this is
done by the best-matching conditions, which force the charge to vanish.19 This has important
implications for the quantum theory. The application of the quantum analogue of the best-
matching conditions forbids the existence of superpositions of eigenstates of that charge, since
only the zero eigenvalue is allowed. Below, we will see how this seemingly innocuous point
becomes essential to understanding time and its denial in the context of quantum theories of
gravity.

3 Understanding Time

3.1 Change and Order

Things change, and time – whatever it may be – certainly is, at a minimum, a means for
describing this change. Newton was father to a notion of time that gives us much more than
just a measure of change, since his absolute time ‘of itself, and from its own nature, flows
equably without relation to anything external’ (Newton 1962). Such a Newtonian time can be
taken to constitute an absolute temporal background against which both the temporal order of,
and the temporal distance between, events is defined irrespective of any changes that may take
place. Thus, in a universe of no change, it makes sense to distinguish both the order of and
time between events that are, other than their absolutely determined temporal position, entirely
identical. More precisely, we can say that on the Newtonian view, both the metric (distance)
and topological (ordering) structure of time are fixed absolutely irrespective of changes in the
material universe.

At the other end of the spectrum from a Newtonian notion of time, we can conceive of a
radically relationalist conception along the lines discussed in Section 1. Recall that, on this
correlation view of time, that is most closely associated with the work of Rovelli (Rovelli 2002;

19In electrodynamics, the best-matching constraint reduces to the usual Gauss constraint, so the terminology
is a bit confusing: this ‘charge’ does not correspond to the usual electrodynamic ‘charge’.

15



Rovelli 2004), all that it is for a physical degree of freedom to change is for it to vary with
regard to a second physical degree of freedom – and there is no sense in which this variation
can be described in absolute, non-relative terms. Within this picture of the world, we explicitly
deny both time’s metric and topological structures. The notion of time one may recover from
change is an inherently arbitrary and approximate one. We are free to choose any degree of
freedom as an internal clock, and such an arbitrarily chosen clock may, for some finite interval,
give us a useful means of marking both the duration between and order of events as defined
by correlations between the other degrees of freedom. However, such an internal clock can
only reliably give us an approximation of a temporal ordering, since it may always start to run
backwards after some finite interval, and so generally will not give us an ordering of events
in ‘time’ that is globally defined in terms of a linear sequence: the parametrization is not
monotonically increasing. Moreover, such a method of distinguishing one variable as ‘the clock’
inevitably involves neglecting the dynamics of precisely that variable: once the internal clock
choice is made we are no longer able to describe the change in that clock degree of freedom.
This is, in effect, to reduce the dimensionality of our system by one. Thus, in the context of this
strong relationalism about time: a one-dimensional system is always static, a two dimensional
system is really a one dimensional system, and so on.

Under a radical relationalist view, the ontology of time is constrained to be a very sparse
one. Only relative variation, exists and there is no fundamental time ordering structure – only
ordering relative to an arbitrary and approximate internal clock. Adoption of such a view
has direct implications for the ‘metaphysics of time’ in that it is inconsistent with all three
of the notions in the famous McTaggart schema (McTaggart 1908). Recall that McTaggart
distinguishes between, the dynamic time of the ‘A-series’, the directed, but non-dynamic B-
series, and the ordered but un-directed ‘C-series’. The C-series should reasonably be taken to
be the minimal position since without such structure the richer metaphysics of time cannot be
defined.20 However, the ordering structure required for the C-series is precisely what is denied
within the radical relationalist view. Thus, we see that there is a precise sense that radical
relationalism involves giving up time, rather than simply relativising it. If ordering structure
(C-series change) is taken as an essential feature of the time concept, then since such a feature
is inconsistent with radical relationalism, the view can be categorised as an essentially timeless
one.

Let us now consider a third ontology of time. This viewpoint constitutes a ‘middle way’, with
less absolute time structure than the Newtonian, but more than the Radical Relationalist. Our
starting point is set by the views of Ernst Mach. Mach both criticised absolute notions of time
on epistemic grounds and put forward a positive account of the kind of temporal structure we
are presented with. According to Mach it is ‘utterly beyond our power to measure the changes
of things by time...quite the contrary, time is an abstraction at which we arrive through the
changes of things.’ (Mach 1883). Thus, on the Machian view, a consistent notion of time can
be abstracted from change such that the inherently interconnected nature of every possible
internal measure of time is accounted for. According to the Mittelstaedt–Barbour (Mittelstaedt
1976; Barbour and Pfister 1995) interpretation of Mach, we can understand this second Mach’s
principle as motivating a relational notion of time that is not merely internal but also equitable,
in that it can be derived uniquely from the motions of the entire system taken together. Thus,
any isolated system – and, in fact, the universe as a whole – should have its own natural clock
emergent from the dynamics. The Machian view of time thus involves a relative notion of
duration as abstracted from change. There is then a clear sense in which we can think of the
Machian relationalist view of time as connected to the radical relationalist view: each involves a
relational view of duration as derived from relative change. However, for a notion of time to be

20We are very grateful to Matt Farr for making us aware of this connection. See (Farr 2012a; Farr 2012b; Farr
UP) for insightful comments regarding McTaggart’s C-series, its relation to the A and B Series, and details on
the similar conception of time that appears in the work of (Black 1959) and (Reichenbach 1956).

16



relational in the Machian sense, it is not enough to be merely relational – it must also be unique
and equitable. We cannot, therefore, merely identify an isolated subsystem as our relational
clock, since to do so is not only non-unique but would also lead to an inequitable measure,
insensitive to the dynamics of the clock system itself. This subtle difference regarding the
Machian and radical notions of relational duration, encodes a more extreme difference regarding
the status of time ordering. In assuming that there is a unique method for abstracting duration
from change, the Machian view is also assuming that there is an absolute ordering within the
change – otherwise the abstraction process would be underdetermined. This means that the
Machian relationalist view involves the assumption of absolute temporal ordering structure.
The ordering of events is fixed absolutely irrespective of changes in the material universe. This
is entirely different to the radical relationalist view where such structure is explicitly forbidden.
Although the metric structure of time is relativised, under the Machian view the topological
structure is presumed to remain absolute.

We thus have three internally consistent views of time, each containing a different level
of absolute time structure: i) Newtonian (absolute duration, absolute ordering structure); ii)
Machian relationalist (relative duration, absolute ordering structure); iii) radical relationalist
(relative duration, relative ordering). Given a physical theory with time re-labelling symmetry,
is there any restriction as to which of the three ontologies we can consistently interpret the
theory in terms of? Clearly this depends upon the symmetry categorisation scheme which one
adopts.

According to one influential view21 for theories which feature temporal relabelling symme-
tries, radical relationalism is forced upon us by the mathematical structure of the theories in
question. Since the time symmetries present in these theories are judged, according to the
standard classification scheme, to be gauge symmetries, we are compelled to take the interpre-
tational step of strong relationalism about time. As indicated above, to us the logic of such
arguments seems a strange one. Reliance upon a purely mathematical prescription to determine
our interpretation of the ontology associated with a physical formalism seems to put the cart
before the horse. Moreover, it can actually be proved that the scheme in question does not apply
to at least the case of theories where global temporal relabelling is a symmetry (Barbour and
Foster 2008).22 Our suggestion is that one should remain agnostic as to the ‘true nature of time’
implied by a physical theory until after one has carefully considered the relationship between
the formalism of that theory and the class of systems which it is taken to represent. In particu-
lar, one must have an understanding of the physical basis of the relevant temporal symmetries
in terms of how the characteristic behaviour of the system class are related to the abstract
degrees of freedom, the boundary and initial (or end point) conditions, and the nomological
restrictions on the evolution. Thus, on our view, it is only after applying something like our
physical prescription for classifying symmetries that one is able to consider the interpretation
of the formalism in terms of a temporal ontology.

Let us do this explicitly for the case of a simple finite dimensional system where global
temporal relabelling is a symmetry – we will consider the more difficult case of infinite dimen-
sional systems with local temporal relabelling symmetry, i.e., general relativity, in the following
section. The situation is then this. We have a class of physical systems which have a finite
number of physical degrees of freedom and for which there is no physical difference between the
system passing through a sequence of physical states at different rates. Such a system may be
represented via slightly adapted version of Newtonian mechanics called Jacobi’s Theory.23 This
theory can be constructed in terms of a configuration space with an identical number of degrees

21On this, see, in particular,(Rovelli 2004, §3.2.4).
22This proof is an addition to independent arguments questioning the role of the Hamiltonian constraint as the

generator of a gauge symmetry (Pons, Salisbury, and Shepley 1997; Pons and Salisbury 2005; Pons, Salisbury,
and Sundermeyer 2010; Pitts 2014). See §5.1.

23This is after the great German mathematician Carl Jacobi. See (Lanczos 1970) for details of both the
formalism and its historical context. Jacobi’s theory is called ‘relativistic mechanics’ in (Rovelli 2004).

17



of freedom as a the corresponding Newtonian configuration space but with an action which
displays an extra symmetry. This symmetry is reparametrization invariance, and is equivalent
to redefining the time parameter used to mark change between states within the theory.

The key question is then: what kind of symmetry is reparametrization invariance – is it
manifest? and is it free or fixed? The answer to the first question is fairly straightforward.
Since, by construction, reparametrization invariance is a symmetry of the action, it can only be
considered a manifest symmetry. What is interesting to note, however, is that the phase space
direction associated with the symmetry is in fact given by the energy function or Hamiltonian.
Thus the ‘symmetry direction’ is also the ‘dynamics direction’.

The next step is to determine whether we are dealing with a conservation or gauge symme-
try by distinguishing whether we have a free or fixed variation. Again this is a fairly simple
question to answer since, again by construction, we have a configuration space where all the
abstract degrees of freedom directly correspond to physical degrees of freedom – those of the
corresponding Newtonian system. Explicitly, since infinitesimal change to the endpoints of the
variational principle is in every direction parametrized by physical degrees of freedom, such
variation cannot be done freely. It is fixed by the characteristic behaviour of the system, in
particular the preparation conditions. Thus, reparametrization invariance, or temporal rela-
belling symmetry, is a conservation and not a gauge symmetry.24 This is precisely as one would
expect since, as we have just noted, the phase space direction associated with the symmetry is
precisely the direction of dynamical change within the theory – thus, if the symmetry were a
gauge symmetry, then we should not expect to have any physical dynamics whatsoever since
the ‘dynamics direction’ would be an otiose representative structure. As we discussed above, in
the case of a gauge symmetry we find identical physical possibilities being represented in terms
of distinct instantaneous states. In this case, the supposedly identical physical possibilities are
dynamically related states. In interpreting time relabelling symmetry as a gauge symmetry we
would be interpreting dynamical change as unphysical. In that eventuality, the only option for
an interpretation of the relevant temporal ontology would be the radical relationalism men-
tioned above. However, by the lights of our scheme, such a move is formally unjustified. We
do not see temporal relabelling as a gauge symmetry and so can license an interpretation of
theories with such a symmetry in terms of a more substantive notion of time.

One important point regarding the conservation symmetry classification of reparametriza-
tion invariant theories relates to the interpretation of the role of energy. As noted above, all
conservation symmetries have associated conserved quantities or charges. For the case in hand,
the charge will be equivalent to the energy of the system. This means that the energy of the
universe is interpreted as a constant of motion. This is in contrast to many existing views
whereby, in a reparametrization invariant theory, energy is a constant of nature. Classically,
this difference has no empirical implications, but, quantum mechanically, it implies that we
should expect superpositions of energy eigenstates – as we would usually for conserved charges
in the quantum version of a theory with conservation symmetry. We will return to this feature
of what we call ‘relational quantum theories’ in the following subsection.25

Before then, we must first consider the interpretational implications of our conservation
symmetry approach to classical reparametrization invariant theory. Under our symmetry clas-
sification scheme, global reparametrization invariance is a conservation symmetry. The exis-
tence of global reparametrization invariance does not, therefore, motivate us to classify sets of
instantaneous states as identical. Thus, the interpretation of a theory which displays global

24 The conserved quantity associated with it varies in interpretation depending on the system in question.
For timelike geodesics in spacetime, it corresponds to the mass of the test particle. For Jacobi mechanics, it
corresponds to the energy of the system. In all cases, it is associated with the constrained Hamiltonian of the
system.

25One might wonder why one shouldn’t expect the energy to simply be zero. At the moment, we have no physical
principle to impose this condition. Indeed, in gravity, the analogue of the total energy (i.e., the homogeneous part
of the Hamiltonian constraint) is the cosmological constant, which has been directly observed to be non-zero.

18



reparametrization invariance in terms of radical relationalism is not well motivated by our sym-
metry classification scheme. Such an interpretation would involve counting as identical possible
instantaneous states which are not, by our scheme, related by gauge symmetry transforma-
tions. Thus, our symmetry classification scheme renders an interpretation of the given class of
reparametrization invariant theories in terms of radical relationalism highly implausible.

Our scheme does not, however, give us means to adjudicate between the two remaining
options: as is typical for a conservation symmetry we have an underdetermination of possibil-
ities, and this underdetermination reflects precisely the difference between the Newtonian and
Machian views. Under our scheme, physical theories which display global reparametrization
invariance can be interpreted in terms of the Newtonian concept of time. Such an ontology
of time involves counting as distinct possibilities histories which are related by a conservation
symmetry.26 On the other hand, under our scheme, physical theories which display global
reparametrization invariance can also be interpreted in terms of Machian relationalism. Such a
view involves counting as identical histories which are related by conservation symmetries.

In the end, we are left with an interpretative choice regarding the status of temporal duration.
To make this choice, further arguments in addition to the categorisation of symmetries are
need. Here we will adopt the Machian view. This is because we find very plausible the Machian
arguments in favour of an internal and equitable notions of time. Such arguments are essentially
epistemological, and rely on the fact that the notion of duration we have available is always
derived from change, and since there is no way for us to ever have access to an absolute duration
measure, we would be better off doing without it. However, by relying on such motivations to
do away with absolute duration, are we not opening the door to a charge of double standards.
Is there not a parallel epistemological worry regarding ordering structure: there seems to be
no possible way for us to ever gain access to this ‘absolute time ordering’ and so its adoption
also seems to involve a rather strong metaphysical act of faith. There are two obvious ways of
addressing this worry:

First, rather pragmatically, we can simply note that time ordering does appear to corre-
spond to part of our physical formalism, and so, unless we can find an empirically adequate
re-formulation without it, we have no good cause to question its status. There is, of course,
a rather impressive precedent for the use of background structures to solve pressing theoret-
ical problems, namely Newton’s use of absolute space to give a coherent formulation of the
principle of inertia (Rynasiewicz 1995a; Rynasiewicz 1995b; Pooley FC). And, in any case, if
metaphysical minimalism is taken to be the main motivation for the elimination of non-empirical
backgrounds, then accepting the fairly thin notion of a time ordering background is far more
palatable than a full-strength Newtonian-style notion of time.

More ambitiously, we could accept that time ordering should be founded in accessible fea-
tures of our theory but that perhaps the arena for doing this is quantum and not classical
theory. It is possible that the process by which temporal ordering in classical physics emerges
is connected to the classical limit of a fundamental quantum theory or even a broken symmetry
in such a theory. A (rather technical) example that suggests the plausibility of this idea is in
quantum field theory where a monotonic ordering is naturally encoded in the renormalization
group (RG) flow near a conformal fixed point. Thus, the RG flow equation of a such a field
theory could be reinterpreted as a time evolution equation in a shape dynamics theory. A sim-
ple toy model featuring such behaviour was studied in (Barbour, Lostaglio, and Mercati 2013).
Furthermore, there are exciting indications that a similar scenario could be used to reproduce
certain models of inflation (McFadden and Skenderis 2010). However, many interesting open
questions remain and such suggestions are still very tentative.

Given these two prospective justificatory strategies, let us accept, for the time being at least,

26This is unless of course one where to adopt some sophisticated form of ‘temporal substantivalism’ where,
via the introduction of anti-Haecceitist reasoning about temporal points or otherwise, the possibility counting
matches that of the Machian. See (Thébault 2012a; Thébault 2012b).

19



that our middle-way, succession-as-absolute-and-duration-as-relative, ontology of time can be
coherently philosophically defended. What fruits can it bear when brought back into the domain
of physical theory? Can it give us new insights into the nature of time in relational quantum
theories?

3.2 Quantization and Succession

In the previous section, we defended both the classification of temporal relabelling symmetries
as conservation symmetries and the interpretation of theories with such symmetries in terms of
a temporal ontology within which duration is relative but succession absolute. Each of these
moves gains significance for the future development of physical theory when seen in the con-
text of quantization. This is particularly clear when considering the so-called problem of time
that arises within attempts to quantize general relativity, but is also the case for simple glob-
ally reparametrization invariant models. The temporal relabelling symmetries of such models
should, according to conventional classification schemes, be understood as gauge symmetries.
This means that quantization, whether achieved via the voluntary redundancy route detailed
above or otherwise, leads to a quantum theory in which the phase space directions defined by the
flow of the Hamiltonian vector field (i.e. the null vector field associated with the Hamiltonian
constraint) are treated as unphysical and, correspondingly, the quantum system is restricted
to a zero eigenstate of the relevant charge. For the case of globally reparametrization invari-
ant models, this is explicitly equivalent to treating dynamical directions as unphysical and to
restricting the system to a single zero energy eigenstate. Thus, the classification of global tem-
poral relabelling symmetries as gauge symmetries leads directly to a frozen quantum formalism.
The only ontology we can associate to such a picture is that of radical relationalism – and we
are left without time.

The alternative, for which we are arguing, is to treat global temporal relabelling as a con-
servation symmetry. The problem is then how to quantize the theory such that both the
reparametrization symmetry and absolute temporal succession structure is retained. This is
where the methodology of voluntary redundancy comes into its own. To our knowledge, the
only way to achieve quantization of a classical model such that time remains in the sense de-
sired, is to use this method. Explicitly what we do (see companion paper (Gryb and Thébault
2014) for more details) is choose a particular extension of the phase space such that our single
configuration compensator field, τ, has a canonical conjugate, πτ , proportional to the energy
of the system. Our single constraint is then constructed by the combination of the original
Hamiltonian plus the new momentum variable, so we have H(qi,pi) + πτ = 0. As discussed
in §2.3 above, imposition of this constraint leads to a variational principle dependent upon all
but two of the degrees of freedom – i.e., precisely the number we started with. The quantum
theory we reach by applying standard methods then preserves these physical degrees of freedom
faithfully in that it allows their quantum analogues to change independently of each other.

Furthermore, as expected, the quantum theory we arrive at is such that we can represent
states in superpositions of eigenvalues of energy since total energy is the Noether charge asso-
ciated with our conservation symmetry. Our quantum formalism can therefore accommodate
fundamental temporal structure associated with succession via the parametrization chosen to
distinguish the distinct energy eigenstates. However, since this parametrization is arbitrary, un-
like in conventional quantum theory, there is no preferred classical temporal background which
fixes a notion of duration. Time remains, but only in the form of succession. Because time is
retained in a relational sense, we call this procedure for the quantization of theories with global
reparamterization symmetries Relational Quantization. From our perspective it is one of the
crucial ingredients in the construction of a genuinely dynamic theory of quantum gravity as is
consistent with the Machian view of time.

20



4 Time and Gravitation

4.1 The Two Faces of Classical Gravity

Our description of the universe is replete with different scales – from the unimaginably small
distances of particle physics, up to the unimaginably big distances involved in modern astronomy
and cosmology. The theories relevant to these domains have one important feature in common:
they treat such scales as an absolute background structure. Thus, in almost all modern physical
theory, if we uniformly double the lengths involved, the phenomena will change.27 Such scale
dependence is also part of our best theory of gravity: general relativity. Although the theory
incorporates a huge amount of descriptive freedom, it still privileges length scales. Interestingly
however, the type of argument that drove Einstein to try and eliminate coordinate dependence
from Newton’s theory of gravity also motivates us to eliminate scale dependence from Einstein’s.
Just as he argued that we have no empirical access to absolute coordinate structures – only
relative ones; we may argue that we have no empirical access to absolute scale structures – only
relative ones.

There at least two distinct ways in which scale can enter a theory: the first is through the
presence of dimensionful couplings, while the second is through the conformal factor of the
metric (since this carries information about the absolute lengths of vectors). The former notion
of scale invariance is intimately related to the renormalization group (RG) flow of a particular
theory because it is the fixed points of the RG flow that are characterized by only dimensionless
numbers. The latter is tied to geometry and whether or not lengths are preserved under parallel
transport. In particular, invariance under local length scales can be expressed by requiring that
gauge-invariant observables be invariant under the local symmetry

gab(x) → eφ(x)gab(x), (2)

which is called a local Weyl or often conformal transformation. Although both notions of scale-
invariance should be required in a truly scale-invariant theory, we will mostly be concerned with
the more modest goal of implementing local Weyl invariance within our framework, leaving the
more technically challenging problem of understanding the UV properties of our framework for
the future.

What remains to be seen is to what extent local scale invariance can be implemented within
a physically viable theory. Weyl’s early attempts at a four dimensionally scale-invariant theory
of gravity (Weyl 1922) were ultimately unsuccessful, in terms of leading to a theory of quantum
gravity, because of strong indications that the theory is unstable. Although work in this vein is
ongoing (Mannheim 2012; Hooft 2010) there are still significant issues to be overcome.

Here we will outline a proposal, different from Weyl’s, which seeks to implement both a
Machian notion of time, and a three dimensional version of local scale invariance. The two
ideas are in fact naturally connected. As has been argued in (Barbour 2012), Mach’s general
stance of epistemic scepticism with regard to non-relational concepts, should lead us to the
conclusion that it is the local scale invariant ‘shapes’ of instantaneous configurations of the uni-
verse that should be taken as fundamental. These shapes can be determined by local observers
through measurements of angles, which, on this view, are what is taken to be fundamental.
The fundamental character attributed to angles can be justified through a simple epistemolog-
ical observation: all measurements of lengths are, necessarily, local comparisons. Thus, real
experiments only ever measure ratios of lengths in some local region, then use these local mea-
surements to deduce lengths for distant objects. Such arguments can be used to justify the
expectation that real experiments should be insensitive to local spatial Weyl transformations of
the form (2), which do not change the result of local measurements of ratios of lengths. What
is left after removing the information about absolute local lengths are simply local angles.

27More specifically, the Higgs and gravitational sectors are not conformally invariant.

21



In order for our definition of angles to be meaningful, there must exist a preferred notion
of global time through which the instantaneous configurations can be defined. This is for the
simple reason that spatial angles are clearly not invariant under local boosts or, more generally,
under foliation-changing spacetime diffeomorphisms. Thus, we can see intuitively that the local
temporal relabelling symmetry (which we understand to be explicitly foliation changing) is in
conflict with three dimensional scale invariance. Just as observers using the spacetime picture
of gravity must assume the information about scale for distant objects, observers using the
scale-invariant picture must assume information about time for distance clocks by choosing a
preferred time foliation.

Here we will examine this apparent conflict and offer conceptual foundations for its resolution
(following the more formal arguments of (Gomes, Gryb, and Koslowski 2011)) by showing that
these are really complimentary, and not conflicting, pictures of reality. The first step in our
reasoning relies upon the notion of hidden symmetry which has already been discussed briefly
above. To recap, the idea is to identify, for a particular theory, a direction in which there
is no manifest symmetry and the variation is free. In this case, it is possible that there is a
hidden symmetry in the system. As discussed above, this can only happen if there is another
manifest symmetry in the theory that has a particular type of formal relationship with the
one at hand.28 If this is the case, the elements of the formalism can be modified (without
changing the physical predictions of the theory) in such a way that the first symmetry becomes
manifest. Remarkably, it has been proved for the case of canonical gravity that, given certain
reasonable simplicity assumptions, the unique set of hidden symmetries can be identified as
three dimensional Weyl transformations which preserve the total volume of space (in the case of
spatially closed topologies) (Gomes 2013). Intuitively, one can think of these as transformation
that redistribute scale from one region to another in a way that is very similar to what happens
to the 2d surface of a balloon when the balloon is squashed or deformed. By definition, hidden
symmetries can only be identified when there exists another symmetry which is manifest and
has the required formal relationship such that the two sets can be understood as dual to each
other. For the case in hand, the relevant dual to the scale symmetry is almost all of the foliation
symmetry. This means that if we symmetry trade such that the hidden volume preserving scale
transformation symmetry becomes manifest, we simultaneously switch to a theory with merely
global, rather than local, time relabelling symmetry (this corresponds to a single, non-local
Hamiltonian constraint).

Specifically, what was proven in (Gomes, Gryb, and Koslowski 2011) is that there exits
two theories on the phase space of General Relativity that are physically equivalent but have
different symmetries: one is the standard ADM theory, which is foliation invariant, and the other
is Shape Dynamics, which is invariant under (volume preserving) conformal transformations.
The physical equivalence of the formalisms is expressed by the fact that there is a special gauge
choice in both theories where the dynamical trajectories on phase space are identical, given some
valid initial data. It is possible, however, for these theories to differ if, for whatever reason, there
are global obstructions to imposing the special gauges in both frameworks. These possibilities
have been explored, for instance, in the case of black holes (Gomes 2014).

Once symmetry trading is completed, the theory we get has a neatly divided set of sym-
metries: volume preserving conformal transformations and spatial diffeomorphism symmetries
can be classified as gauge symmetries; and global time relabelling can be classified as a con-
servation symmetry. This package of symmetries provides a clear constraint on the possible
interpretations since it entails that spatial diffeomorphism and three dimensional conformal
transformations connect physically identical instantaneous states, and leaves upon the interpre-
tation of the global time relabelling symmetry. This is in contrast to the symmetry package of
general relativity. As discussed in §1, there we have i) spatial diffeomorphism symmetry; and ii)

28The formal requirement is that the constraint surfaces corresponding to the symmetries are ‘orthogonal’, i.e.,
second class, on phase space. For the general theory behind symmetry trading, see (Gomes and Koslowski 2012).

22



local time relabelling symmetry. The first of these corresponds to a manifest free variation and
thus, under our definition, is a gauge symmetry and therefore involves transformations between
physically identical instantaneous states. This point is fairly uncontroversial (however see (Pitts
2013; Pitts 2014)). Local time relabelling symmetry or re-foliation symmetry, on the other hand,
in our scheme corresponds to a (very complicated) manifest fixed variation. This means that
there is scope for a much greater degree of underdetermination regarding the implications of
the symmetry for possibility counting: the group of local time relabelling symmetries has an
infinite number of parameters, the group of global relabelling symmetries only has one. Our
categorisation is in contrast to the standard scheme where re-foliation is classified as a gauge
symmetry, and thus, supposedly, gives us a means to identify physically identical instantaneous
states. However, the sense in which re-foliations can be understood as connecting physically
identical instantaneous states is a notoriously subtle and heavily qualified one (Thébault 2012b;
Pitts 2014). Thus, the strength of our classification scheme in this instance is that it leaves open
ambiguity regarding the implications of the symmetry for possibility counting, exactly where it
exists in practice.

Let us recapitulate: there exists a unique formal move that allows us to re-describe gravita-
tional systems in a fundamentally different way. In the language introduced earlier, we have an
underdetermination of symmetries. This first level of underdetermination is regarding a choice
between two packages of symmetries. According to our understanding of symmetries there is
then also an underdetermination of possibilities – depending upon how we interpret the con-
servation symmetries within the two packages. If we wish to break such underdetermination
further motivations are needed.

We saw earlier that there a good Machian motivations for breaking the underdetermination
of possibilities as related to global time-relabelling. Thus, as far as the shape dynamics package
of symmetries is concerned we have a means to fix the interpretation of gravity fairly precisely.
Let us consider a strategy for breaking the underdetermination of symmetries and see if we can
motivate passage to the shape dynamics formalism. Above we argued that, in order to classify
the symmetries relevant to a class of systems consistently, we must first specify precisely what
the relevant physical degrees of freedom and preparation conditions are. This translates into
asking what degrees of freedom are independently specifiable on the boundary of the variation
or, in the case in hand, ‘What is fixed on the boundary in the action principles of General
Relativity?’ This question was posed by Wheeler to York, and is addressed in (York 1986). Our
view corresponds to that taken in Section 4 of that paper: what is fixed on the boundary is: i)
a three geometry invariant under scale and coordinate labelling symmetries; and ii) the mean of
the ‘York time’ variable (which is canonical conjugate to the spatial volume). We will refer to
this identification of the independent degrees of freedom of gravity as York’s ontology. We take
these variables to faithfully parametrize the characteristic behaviour of gravity and, thus, take
their variation to be of the fixed kind, while variation with respect to all other variables is free.
York’s identification of a locally scale-invariant three geometry as a variable to be fixed in the
variational principle of gravity is consistent with the principle of scale-invariance just argued
for on the basis of Mach’s principles. However, one might note that York’s second requirement:
to keep the variable conjugate to the spatial volume fixed, is in direct conflict with the global
principle of scale-invariance. Our view on this will be rather pragmatic at this stage. The fixing
of this ‘York time’ can be motivated by the directly observable red-shift, which is undeniably
part of the characteristic behaviour of gravity. However, on Machian grounds, one might expect
the red-shift to result as an emergent phenomenon from a fully scale-invariant theory. In this
eventuality, we would still expect York’s proposal to be valid in some effective limit in the
quantum regime. However, since a concrete proposal where such a scenario is realized has not
yet be developed, we will consider York’s ontology directly.

From the York perspective, canonical general relativity has the rather undesirable feature
of neither having manifest invariance under volume preserving conformal transformations, nor

23



being a conventional gauge theory with respect to diffeomorphisms, nor varying the York time
in fixed manner. The first two difficulties can be resolved by noting that the volume preserving
scale transformations are a hidden symmetry of the canonical version of general relativity (as
we have just noted). The last difficulty can be dealt with using our proposal for Relational
Quantization procedure detailed in the previous section.

Before we embark on the final phase of our analysis and detail our specific proposal for time
in a substantive sense, to remain within a theory of quantum gravity, let us take a moment to
consider the philosophical consequences of symmetry trading on the way we should think of the
ontology of classical gravity. The traditional understanding of general relativity (in canonical
terms or otherwise) is as a theory of spacetimes invariant under spacetime diffeomorphisms.
This is essentially the ontology for gravity that was implied by Einstein’s seminal work in the
early part of last century, and which is still one key pillar of the scientific understanding of the
universe. That this theory has a unique and robust formal correspondence, or duality, to another
theory of gravity, which (under certain restrictions)29 has the same physical consequences is
highly non-trivial. As mentioned in the introduction, on our view, it is taken to imply that
gravity is essentially Janus-faced. From this perspective, we should see the Einstein ontology of
diffeomorphism invariant four dimensional spacetime geometries as only one face of gravity. The
other, newly unveiled face being constituted by the ontology proposed by York: sequences of
three dimensional spatial geometries accompanied by the specification of the York time variable
and invariant under both diffeomorphism and scale transformations.

The situation of dual theories which are empirically equivalent, yet ontologically radically
different is one of great interest within the philosophy of science since it seems to imply a par-
ticularly pernicious species of underdetermination.30 Interestingly, the case which has garnered
the most interest in the recent literature (Dawid 2007; Rickles 2011; Matsubara 2013), that of
the AdS/CFT correspondence in string theory/conformal field theory, is also one in which the
two duals theories are respectively a diffeomorphsism invariant theory of gravity and a theory
invariant under conformal transformations.31 However, one would not have expected that such
striking, and perhaps worrying, underdetermination scenarios could crop up in one of our most
established physical theories. Yet, for our purposes, this seeming theoretical vice will prove a
virtue. It is only by recognizing the second, scale-invariant face of gravity, that we can forge a
new path towards quantization without sacrificing Time.

4.2 Retaining Succession in Quantum Gravity

As was argued for extensively in Section 2, correct identification and treatment of the physical
degrees of freedom, and in particular their boundary variation behaviour, is essential to a
faithful quantization of any physical theory. One of the major impediments to the quantization
of gravity has been that, in canonical form, the symmetry and dynamics of the theory are
‘deeply entangled’.32 In the language of Section 2, this corresponds to the ‘fixed’ and ‘free’
aspects of the manifest symmetry being mixed together. Here we make no attempt to tackle

29These restrictions amount to global foliability conditions of the spacetime by foliations where the trace of
the extrinsic curvature of the hypersurfaces, as embedded into the spacetime, is constant.

30See (French 2011) for a excellent overview and analysis of various notions of ‘metaphysical underdetermina-
tion’ leading towards a motivation of the philosophical viewpoint of ‘ontological structural realism’. See (Pooley
2006) and (Thebault 2012, §19) for discussion of the problems that the radically different ontologies found within
theories of gravity may pose for the position.

31Indeed, it has recently been argued (Gomes, Gryb, Koslowski, Mercati, and Smolin 2013) that the ‘bulk-bulk’
equivalence of shape dynamics and general relativity can be used as an explanation for certain limiting regimes
of the AdS/CFT correspondence.

32Formally, this facet is encoded within the non-trivial structure of the Dirac–Bergmann constraint algebra, in
particular that the bracket between two Hamiltonian constraints only closes with structure functions is indicative
of the dual dynamics-symmetry aspect of these constraints. The constraint algebra of shape dynamics (and
the relationally quantized theory) on the other hand is a genuine Lie algebra, and so a clear formal distinction
between symmetry and dynamics can be made.

24



the fearsome, and as yet unsolved, problem of a disentangling the physical from gauge variations
within the theory. Rather, given the ‘gravity is Janus-faced’ revelation detailed above, we may
simply turn to the alternative scale-invariant formalism for classical gravity and hope for a
simpler representation of the relevant symmetries more amenable to quantization.

Starting with the canonical, or ADM, formulation of general relativity, symmetry trading
yields a theory with the following manifest symmetries: a) three dimensional spatial coordinate
invariance; b) three dimensional (spatial volume33 persevering) spatial conformal invariance;
and c) invariance under one dimensional global time relabelling (or reparametrization). If we
interpret this formalism in York’s terms as discussed above, then we arrive at a unique and
unambiguous specification of which of these symmetries correspond to free variations of the
physical degrees of freedom at the boundary. By our symmetry classification scheme, such
degrees of freedom are unphysical otiose variables. By construction, the York specification of
boundary data is insensitive to variation of the (volume persevering) conformal and coordinate
modes of the three dimensional metric tensor that characterises three dimensional spatial ge-
ometries. Thus, symmetries a) and b) are identified as gauge symmetries. Then we find, in
correspondence with the discussion of Section 3, that the reparametrization symmetry c), is
of the fixed kind. Thus, our prescription for quantization implies that we should introduce
two global degrees of voluntary redundancy that parametrize the direction associated with the
function that generates the reparametrization symmetry. As for the simple particle case, this
function is a global Hamiltonian, and this means that our extra variables are a time-ordering
label and a conserved charge associated with the ‘total energy’. We then append the second of
these to the Hamiltonian to get an extended, but physically equivalent, formalism, which then
can be quantized (at least in formal terms) via standard methods.

What we have gained in this rather circuitous route of symmetry trading, arbitrary extension
and quantization, is simple to state. We are now equipped to represent the state of the universe
at different times. This is because, as for the particle case, we are able to take superpositions of
energy eigenstates, and consider the independent evolution of observable operators. And yet,
we have not introduced a Newtonian-style background time into the theory. Time labellings are
encoded in the arbitrary parameter which was introduced into the theory during the extension
procedure, and so are neither fundamental nor observable. Rather, the temporal structure which
this formalism for quantum gravity contains is precisely the temporal succession structure we
associated with the ideas of Mach above. As was noted in the opening section, we should not
think of such a ‘temporal topological background’ as being entirely alien to general relativity
since it is in fact implicit within the canonical formalism at least (in terms of the positivity
requirement on the lapse multiplier). Moreover, it is only by retaining such structure that, in
the case of gravity, we can hope to preserve genuine change and avoid radical relationalism with
regard to time, as we have been advocating.

Our proposal for the quantization of gravity thus involves two substantive interpretative
moves. Firstly, the switch from the Einstein ontology (as implied by the general relativity
formalism for gravity) to the York ontology (as implied by the shape dynamics formalism of
gravity). Secondly, the promotion of time ordering (or topological) structure from an implicit
formal feature to an explicit background. Both individually, and as a package, we can provide
a range of motivations for these non-trivial steps.

Most straightforwardly, there is the motivation from pragmatism: by following our prescrip-
tion one opens up new strategies for theory development, and this, in the end, might be argued
to be the true goal of foundational research. It remains to be seen precisely what lasting value
the much vaunted ‘spirit of general covariance’ will prove to have as a heuristic for future the-
ory construction. It may prove pivotal, or it may prove to have been misleading. Thus, if new
and viable theoretical avenues can be opened up by reinterpreting symmetry in the context of

33For the case of open spatial topology, this global restriction turns into a specification of asymptotic boundary
conditions (Gomes and Koslowski 2012).

25



gravity, we would be churlish to entirely ignore them since they do not sit conformably with
an exciting abstract principle – no matter how fundamental it may currently appear to be. We
should not let fetishism for four-dimensional spacetime diffeomorphism invariance be a bar to
potential progress.

Furthermore, over and above the conceptual novelty of our proposal, it provides several
notable formal advantages. In traditional approaches to the quantization of gravity (i.e. the
‘Wheeler-DeWitt-type’ approaches), the resultant quantum formalism is such that only one
energy eigenvalue is allowed. Evolution of the quantum states can then only be obtained by
deparametrizing with respect to a degree of freedom, in the choice of which one must make an
arbitrary decision. The definition of the functions used to represent observable quantities in the
theory depend on this choice and, even for simple models, can lead to extremely complicated
expressions. Through our approach, we arrive at a formalism where there can be superpositions
of energy eigenstates, and the evolution of the full state can be given with respect to the auxiliary
time label. Thus, the evolution does not depend on any arbitrary choice of auxiliary time label.
The identification of the relevant observables is then also non-arbitrary, and is technically much
easier (because of the time-independence of the Hamiltonian).

Finally, in addition to conceptual novelty and formal tractability, there is the simplest
motivation of all: the motivation from time. It seems to be a basic requirement that, in one
way or another, we are able to abstract some concept of time from our physical formalism –
without it our physics would simply fail to be descriptively adequate. If the only approach to the
quantization of gravity were via a timeless formalism, then it would perhaps be fair to insist that
we must make do with the conceptual paucity of time merely as relative variation. However,
given that there is a viable alternative route towards quantization, via symmetry exchange and
relational quantization, the conceptual cost incurred by taking it should be counted as naught,
next to the benefit of retaining a minimal, yet substantive, concept of time.

5 Conclusion / Discussion

5.1 Related Arguments

Here we would like to highlight related arguments appearing in recent work (Pitts 2014), in
addition to closely connected earlier observations (Pons, Salisbury, and Shepley 1997; Pons and
Salisbury 2005; Pons, Salisbury, and Sundermeyer 2010). In certain key respects these authors
reach conclusions regarding time in general relativity that are closely related to our own. In
particular, for them, as for us, the Hamiltonian constraint of GR is not understood to generate
a gauge transformation, in the sense described by a local symmetry of the Lagrangian. The ar-
guments of these authors focus upon the idea that, to get the Hamiltonian framework to match
up with the Lagrangian notion of gauge symmetry, primary constraints must work in tandem
with their associated secondary constraints to produce a genuine notion of gauge transforma-
tion. Although in many respects such arguments support essentially the same conclusions as
those presented here, the chain of reasoning involved is largely independent. Contrasting and
comparing the two approaches in a more detailed way would be an interesting topic for further
study.

5.2 Concluding Remarks

The aim of this paper was to argue that there are strong formal and philosophical reasons to
expect time to remain within any theory of quantum gravity. Although the temporal symmetries
of classical gravity are subtle, such that the redundant and physical aspects of the formalism
are entangled, there does exist a precise formal recipe for making the unambiguous distinction
needed for a faithful quantization. This recipe relies not just upon the technical notions of
symmetry trading and relational quantization discussed here (and presented more formally

26



elsewhere), but also upon two quite general and simple philosophical morals. First, that physics
is not mathematics: it is our understanding of how the physical formalism relates to the world
that should govern our interpretation of its mathematical structures and not vice-versa. Second,
that, at base, time has two aspects: metric and topological. While the first does seem in conflict
with the relational, and ‘background free’ aspects of time in general relativity, the second appears
implicitly even within the Einstein formalism. Furthermore, when seen in the context of our
shape dynamics plus relational quantization proposal, topological or time ordering structure
plays an important and unambiguous role: it is The Remains of Time in quantum gravity. As
such, the virtues of a program towards its conservation are self-evident.

Funding

K.T. would like to thank the Alexander von Humboldt Foundation and the Munich Center
for Mathematical Philosophy (Ludwig-Maximilians-Universität München) for support. S.G.
would like to thank the The Netherlands Organisation for Scientific Research (NWO) and the
National Sciences and Engineering Research Council of Canada (NSERC) for financial support
and Utrecht University for accommodations.

Acknowledgements

We are appreciative to Oliver Pooley and Hans Westman for valuable discussion of relevant
points over the past years, to Seamus Bradley and Luca Moretti for comments on an earlier
draft, and, in particular, to Julian Barbour for detailed critical comment. Detailed comments
from two referees were also of great help. Thanks also to Eilwyn Lim for producing the Janus
illustration above.

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30


	Introduction
	The Problem of Time
	Our Solution

	Understanding Symmetry
	Mechanics and Representation
	Freedom by Degrees
	Voluntary Redundancy

	Understanding Time
	Change and Order
	Quantization and Succession

	Time and Gravitation
	The Two Faces of Classical Gravity
	Retaining Succession in Quantum Gravity

	Conclusion / Discussion
	Related Arguments
	Concluding Remarks