# Spacetime Conventionalism Revisited

Ufuk I. Tasdan  University of Bristol
Bristol, BS8 1TH, United Kingdom<EMAIL_ADDRESS>and  Karim P. Y.
Thébault  University of Bristol
Bristol, BS8 1TH, United Kingdom<EMAIL_ADDRESS>
(Date: Draft of .)

###### Abstract.

We provide five rearticulations of the thesis that the structure of spacetime
is conventional, rather than empirically determined, based upon variation of
the structures that are empirically underdetermined and modal contexts in
which this underdetermination occurs. Three of the five formulations of
conventionalism will be found to fail. Two are found to open up new
interesting problems for researchers in the foundations of general relativity.
In all five cases, our analysis explores the interplay between geometric
identities, symmetry, conformal structure, and the dynamical content of
physical theories with the conventionalism dialectic deployed as a tool of
explication, clarification, and exploration.

###### Contents

  1. 1 Conventionalism About What?
    1. 1.1 Geometric Conventionalism and Universal Forces
    2. 1.2 Rearticulating Spacetime Conventionalism
  2. 2 Non-Conformal Structure and Universal Forces
  3. 3 Inertial Structure and Local Stress-Energy Flux
  4. 4 Conformal Structure and Tidal Deformation
  5. 5 Nomic Structure and Dynamical Fields
  6. 6 The Bach Conjecture
  7. 7 Summary
  8. A Conformal Tensors and the Bianchi Identities

##  1\. Conventionalism About What?

###  1.1. Geometric Conventionalism and Universal Forces

Geometric conventionalism is a form of underdetermination thesis regarding the
space or spacetime structure of the physical theory.111The thesis of
conventionalism regarding the geometry of space was first proposed by Poincaré
(1902), following in the tradition of Riemann (1854), and developed in a
variety of forms by Dingler (1911), Carnap (1922), and Reichenbach (1928). See
Ben-Menahem (2006) and Duerr and Ben-Menahem (2022) for an overview. Ivanova
(2015a, b) gives a helpful analysis of Poincaré’s conventionalism, and
Torretti (1978) of Dingler’s. A translation of the 1919-1921 doctoral
dissertation of Carnap (1922) can be found in Carnap (2019). Further analysis
with varying degrees of historical grounding can be found in (Grünbaum 1968;
Sklar 1974, 1985; Glymour 1977; Friedman 1983; Malament 1985; Norton 1994;
Weatherall and Manchak 2014; Dewar et al. 2022). One may demarcate varieties
of geometric conventionalism to arise from how one answers the following two
questions:

  1. (1)

Which geometric statements regarding a model of space (or spacetime) are not
empirically-determined truths?

  2. (2)

Which physical differences arise in the comparison of different conventions
about such statements?

Reichenbach (1928) famously responded to the first question that statements
regarding the spatial or spacetime metric tensor of any given model are not
empirically determined; and, to the second question that the physical
difference between two metrics (and thus two models) would manifest as an
apparent universal force. In particular, Reichenbach (1928), like Poincaré
(1902) in the disc experiment, compares the measuring rod of observers living
in two different worlds having flat spacetime. In this thought experiment,
inhabitants of one of the world have rulers under the effect of heat and
perceive the spacetime as curved. However, “(h)eat as a force can thus be
demonstrated directly” (p. 13) because heat affects different materials
differently. As a result, the geometry of spacetime is underdetermined if the
heat is taken as a universal force:

> “Thus the only distinguishing characteristic of a field of heat is the fact
> that it causes different effects on different materials. But we could very
> well imagine that the coefficients of heat expansion of all materials might
> be equal then no difference would exist between a field of heat and the
> geometry of space.” (Reichenbach 1928, p. 26)

What forms of conventionalism might we consider beyond the specific form
articulated in terms of universal forces? As just noted, in his original
argument Reichenbach took metrical relations of spacetime to be conventional
and made the specific assumption that the choice in convention could be
underdetermined by introducing universal forces. Significantly, Dieks (1987)
argues that it is in the Reichenbachian spirit to interpret gravitation as a
universal ‘force’, since any effect of the curvature of spacetime can be
explained by a suitable gravitational effect. Indeed, in Reichenbach (1928) we
find the following text:

> [The] universal effect of gravitation on all kinds of measuring instruments
> defines therefore a single geometry. In this respect we may say that
> gravitation is _geometrized_. (Reichenbach 1928, p. 256) (italics in
> original)

An explicit defence of this more liberal ‘universal effect’ reading can be
found in the later work of Carnap (1966):

> Reichenbach called them ‘forces’, but it is preferable here to speak […] in
> a more general way, as two kinds of ‘effects’. (Forces can be introduced
> later to explain the effects.) [p. 169]

Further discussion of this point in the context of the interpretation of
Reichenbach can be found in Sexl (1970) and (Duerr and Ben-Menahem 2022,
§2.4.2).

In what follows we will draw inspiration from these ideas for re-framing of
the debate via generalised forms of the conventionalist thesis that do not
rely on force-based representations. In particular, we will consider different
formulations of the spacetime conventionalist thesis in terms of physical
differences that are taken to result from different proposals for the relevant
universal effect.

###  1.2. Rearticulating Spacetime Conventionalism

We have just seen that one is able to historically situate a mildly
generalised form of geometric conventionalism based upon universal effects
rather than universal forces. For a foundational, as opposed to primarily
historical, perspective the next step is then to consider the salience of
wider forms of conventionalist thesis.222Our project is not primarily a
historical one and so we will not seek to defend the historical as opposed to
foundational salience of our generalisations. That said, our framework for
generalisation overlaps in several core respects with the reconstruction of
Poincaré’s conventionalism provided by (Duerr and Ben-Menahem 2022, p.167). To
do this productively, however, one must place some constraints upon the
relevant argument pattern. With this in mind, we will consider the following
generalised forms of our two questions:

  1. (1)

Which statements regarding the structure of a spacetime theory are empirically
underdetermined?

  2. (2)

Which physical differences arise in the comparison of different conventions
about such structure?

Setting the two questions up in this way allows us to confine ourselves within
a generalised spacetime conventionalist argument pattern whilst allowing for
rearticulation along various dimensions.

The first dimension of rearticulation we will pursue involves considering
various different ways of formulating the relevant physical differences in
(2). The second dimension of rearticulation is to vary the structure in
question in (2). Such structure should, of course, still be physically well-
motivated as the basic structure of a spacetime theory, but need not
necessarily be a geometric structure. Third, and finally, we can identify a
more subtle dimension of rearticulation in terms of the modal context of
empirical underdetermination thesis. The idea is that the context in which one
is answering the questions (1) and (2) is itself underdetermined to the degree
that it might be specified as obtaining between: i) two observers within a
single model of a theory; ii) two models within the same theory; iii) two
models of different theories; or iv) two ways of interpreting the sets of
models within a single theory. What is essential within our family of
generalised notions of spacetime conventionalism is that in each and every
case it is required that a basic structure of a spacetime theory is
empirically underdetermined, and this underdetermination leads to the
possibility for physical differences to arise between conventions regarding
how to break the underdetermination.

In what follows we will explore a selection of sights of interest in the
conceptual space defined along these three dimensions of rearticulation. Some
of the pathways towards spacetime conventionalism that we will explore will
take us rather far from the original debates, and others will ultimately be
found to lead us into a cu-de-sac. We take this tour to be a valuable and
productive one for two principal reasons.

First, in what follows, we will deploy our generalised framework for varieties
of spacetime conventionalism as a tool of foundational analysis towards both
clarification of old conceptual problems, or indeed pseudo-problems, and the
identification of new problems. Articulation of new conceptual problems can,
in turn, be understood as a constitutive element of scientific progress
according to the framework articulated by Laudan (1977), c.f. Feyerabend
(1981). In this sense, our ultimate aim is to apply conventionalism as a lens
to advance the scientific understanding of spacetime theory.

Second, we take our project to provide important clarificatory insights into
the formal interplay between geometric identities, symmetry, conformal
structure, and dynamical equations within the foundations of general
relativity. The five forms of conventionalism articulated in what follows are
woven together by twin formal threads of conformal structure and the Bianchi
identities. Each of the forms of conventionalism we will consider, to at least
some degree, trades on a package of re-reading the interconnection between
parts of the formalism of general relativity based on one or both of these
aspects. We will consider putative conventionalisms relating to non-conformal
structure, local conformal structure and to the decomposition of the Einstein
equation into conformally invariant and non-conformally invariant parts.
Further, we will consider putative conventionalisms based upon understanding
the role of the Bianchi identities as formal geometric identities,
conservation equations, and dynamical field equations.

We start in Section 2 by considering a formulation of geometric
conventionalism due to Weatherall and Manchak (2014). We consider the
implications of the no-go result proved by the authors in the context of a
general framework for the analysis of the methodological implications of no-go
theorems due to Dardashti (2021). We will find that this analysis of the
Weatherall-Manchak no-go result further motivates the programme for
rearticulation along a different dimension that we have just set out.

In Section 3 we present our first attempt at rearticulation of conventionalism
in which we focus on inertial structure and putative universal effects that
result from differing representations of local stress-energy 4-current by
coincident observers following different world-lines. Ultimately, the view is
found to be unconvincing, since there are good reasons to reject the physical
salience of the energy 4-current and thus this prima facie plausible form of
conventionalism can be dissolved. In Section 4 we then consider the conformal
structure and the universal effect of tidal deformation. We argue that by
considering the invariant decomposition of the Riemann tensor into Weyl and
Ricci parts we can dissolve a further potential source of conventionalism with
regard to the deforming and non-deforming aspects of tidal effects. In each
case, although the new forms of conventionalism are found to fail, our
discussion serves to elucidate new perspectives on the representational roles
of different objects within the foundations of general relativity.

In Section 5, we pull our focus back to a wider form of interpretational
question regarding the dynamical understanding of the Weyl tensor fields and
the role of the Bianchi identities. In particular, we consider a putative
underdetermination with regard to the nomic structure of general relativity.
This analysis serves to open up the possibility for a very different form of
conventionalism in terms of how to isolate the dynamical content of general
relativity. In particular, we argue that underdetermination with regard to the
structure that specifies which relativistic spacetimes are dynamically
possible models leads to conventionalism with regard to which fields are
understood as dynamical and which non-dynamical. Furthermore, we find that
consideration of the initial value problem of general relativity serves to
further establish, rather than dissolve, the putative underdetermination with
regard to nomic structure that is at the heart of Spacetime Conventionalism 4.

Finally, in Section 6, we consider a proposal based upon the decomposition of
the Einstein tensor into a conformally invariant and non-conformally invariant
part. The potential for underdetermination of this decomposition is then taken
to be the hallmark of a novel form of conventionalism whose validity is
expressible in terms of (the failure of) a precise mathematical ‘Bach
conjecture’, whose truth is as yet unknown.

##  2\. Non-Conformal Structure and Universal Forces

Spacetime Conventionalism 1 (1) The non-conformal structure of relativistic
spacetime is not an empirically-determined truth. (2) Physical differences
between conformally equivalent models of relativistic spacetime with regard to
the existence of universal forces arise in the comparison of different
conventions about affine structure.

Any pair of conformal equivalent relativistic spacetimes are, by definition,
such that $(M,g_{\mu\nu})$ and $(M,\tilde{g}_{\mu\nu})$ where
$\tilde{g}_{\mu\nu}=\Omega^{2}g_{\mu\nu}$. Each of these spacetimes is
accompanied by a metric compatible, torsion-free, Levi-Civita derivative
operator, $\nabla$ and $\tilde{\nabla}$ respectively. Relativistic spacetime
conventionalism can then be formulated as the thesis that the same curve,
$\gamma$, can be a geodesic relative to $\nabla$ but be associated with an
acceleration relative to $\tilde{\nabla}$ given by a ‘universal force’ induced
by a tensor field $G_{\mu\nu}$ via the equation
$G^{\nu}_{\>\mu}\tilde{\xi}^{\mu}$ where $\tilde{\xi}$ is the tangent field to
$\gamma$ with unit length relative to $\tilde{g}_{\mu\nu}$. So formulated a
relativistic conventionalist thesis, equivalent to Spacetime Conventionalism 1
has a contradiction as a deductive consequence (Weatherall and Manchak 2014,
Proposition 2).

The result of Weatherall and Manchak constitutes a ‘no-go’ theorem for a
specific form of spacetime conventionalism and so understood, the theorem
might be taken to close off one of the principal avenues for the articulation
of such a view within the context of spacetime theory. It is worth noting in
this context that recent analysis due to Duerr and Ben-Menahem (2022) has
strongly contested whether the characterisation of conventionalism within the
antecedent conditions of the Weatherall and Manchak no-go result is
historically or conceptually adequate. Whilst our perspective on spacetime
conventionalism intersects in a number of respects with Duer and Ben-Menahem,
our attitude to the work of Weatherall and Manchak is rather different. This
difference can be understood by drawing upon the hugely insightful work on no-
go theorems in physics due to Dardashti (2021).

Within Dardashti’s framework, a no-go result has been established if and only
if an inconsistency arises between a derived consequence of a set of physical
assumptions, $P$, represented by a mathematical structure $M$, and a goal $G$
within a framework, $F$. By contraposition, the implications of the no-go
result can then be understood in terms of different ‘methodological pathways’
based upon a disjunct of the form $\neg P\vee\neg M\vee\neg F\vee\neg G$.
Presuming that we do not want to give up the goal, we may then reinterpret the
implication as a ‘go theorem’ with the implication $\neg P\vee\neg M\vee\neg
F$. That is, if we would like to achieve a certain goal, then the theorem
provides us with a formal basis to articulate the specific mathematical,
physical, or framework assumptions one must reject to achieve it.

Returning to the Weatherall-Manchak result. Since the theorem contained in
their Proposition 2 is both valid and non-trivial, we take there to be good
cause to explore its implications as a ‘go theorem’ in the context of the
negation of the various physical, mathematical and framework
assumptions.333See Duerr and Ben-Menahem (2022) for specific arguments that
involve questioning the Weatherall-Manchak result on the basis of both its
mathematical assumptions, (NORM) that $\tilde{\xi}$ is a unit norm and (RIEM)
that $(M,\tilde{g}_{\mu\nu})$ is a Riemannian geometry, and physical
assumptions, (CONF) that the old and new metrics are conformally related
$\tilde{g}_{\mu\nu}=\Omega^{2}g_{\mu\nu}$ and (FORCE) that the ‘universal
force’ is induced by a tensor field $G_{\mu\nu}$ via the equation
$G^{\nu}_{\>\mu}\tilde{\xi}^{\mu}$. That is, we take the result to motivate
‘methodological pathways’ in which we seek to articulate different forms of
the generalised spacetime conventionalist goal based upon different physical,
mathematical and framework assumptions. Our specific approach is based upon
articulating a set of such pathways in terms of the different dimensions of
rearticulation of conventionalism as per the analysis of the introduction.
Thus, our project is further motivated by a ‘go theorem’ interpretation of the
Weatherall-Manchak result. We start by considering a form of spacetime
conventionalism founded upon inertial structure and an intra-model modal
context.

##  3\. Inertial Structure and Local Stress-Energy Flux

Spacetime Conventionalism 2 (1) The inertial structure of a particular
relativistic spacetime is not an empirically-determined truth. (2) Physical
differences between coincident observers with regard to local stress-energy
flux arise in the comparison of different conventions about which time-like
vector field is chosen as an observer’s inertial frame.

The second form of the conventionality of spacetime conventionality we will
consider relates to the interpretation of different observers of the same
model, and thus the same geometric facts, in terms of different local
dynamical facts. The idea is that within a spacetime there can be observers
that are coincident and yet will disagree about the dynamical interpretation
of the same local geometric facts. The formal property that we will isolate in
the privileged class of observers who have the analogue of a vanishing
universal force is that they are Killing observers and thus the answer to the
first of our two conventionalism questions is inertial structure. The physical
differences that feature in the second question are then with regard to local
stress-energy flux.

We can better understand the motivation for Conventionalism 2 by briefly
considering the problem of the local definition of conservation of energy in
general relativity.444See Pitts (2010, 2016, 2021); Lam (2011); Read (2020);
Dewar and Weatherall (2018); Duerr (2019b, a) for discussions of the problem
of defining gravitational energy. A further noteworthy paper, along lightly
different lines, is Lehmkuhl (2011). There it is argued that in the context of
general relativity energy-momentum density as expressed by $T_{\mu\nu}$,
cannot, in fact, be regarded as an intrinsic property of matter, but rather
should be understood as a relational property that matter possesses only in
virtue of its relation to spacetime structure. For an outstanding general
discussion of energy conditions in general relativity see Curiel (2014). The
problem can be straightforwardly stated as follows. For a vanishing
cosmological constant, the Einstein equations take the familiar form:

(1) $G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi T_{\mu\nu}$

where $T_{\mu\nu}$ is the energy-momentum tensor associated with all matter
fields and their interactions, and $G_{\mu\nu}$ is the Einstein tensor which
is determined by the metric $g_{\mu\nu}$ via the Ricci curvature $R_{\mu\nu}$
and Ricci scalar $R$.

The Levi-Civita derivative, $\nabla_{\mu}$, can be explicitly constructed from
the metric and an arbitrarily chosen derivative operator. Given such a
derivative operator on a manifold $M$, it can be proved that there exists a
unique smooth tensor field on $M$ such that for all smooth fields
$\xi^{\sigma}$:

(2)
$R^{\rho}_{\sigma\mu\nu}\xi^{\sigma}=-2\nabla_{[\mu}\nabla_{\nu]}\xi^{\rho}$

(Malament 2012, Lemma 1.8.1). $R^{\rho}_{\sigma\mu\nu}$ is then the unique
Riemann curvature tensor field associated with $\nabla_{\mu}$. The Ricci
curvature tensor is defined via the contraction of the Riemann curvature as:

(3) $R_{\sigma\mu}=R^{\rho}_{\sigma\mu\rho}$

An important identity that holds for any Riemann tensor is the second Bianchi
identity. For the torsion-free case, this can be expressed in terms of the
Levi-Civita derivative as:

(4) $\nabla_{[\alpha}R^{\rho}_{\mid\sigma\mid\mu\nu]}=0$

This equation follows from the basic mathematical properties of the Riemann
tensor and is thus naturally understood in this context as a formal identity
without physical content.555See (Malament 2012, p.69-79) and (Landsman 2021,
p.61) for proofs. Interestingly, however, when combined with the definition of
the Einstein tensor the Equation (4) can be shown to imply:666See (Malament
2012, p.322). N.b. one also uses further symmetry properties of the Riemann
tensor in this short derivation.

(5) $\nabla_{\mu}G^{\mu\nu}=\nabla_{\mu}T^{\mu\nu}=0$

At this point, it is very tempting to read the equation
$\nabla_{\mu}T^{\mu\nu}=0$ as an expression of the local conservation of
energy (or stress-energy) and it seems like we have derived a dynamical
conservation equation from a formal identity. However, such a naïve way of
thinking about energy conservation in GR immediately runs into trouble.

Consider an observer moving along the integral curve of an arbitrary time-like
vector field $\xi^{\mu}$. Let us define the stress-energy 4-current along the
direction defined by this vector as $j[\xi^{\mu}]=T_{\>\nu}^{\mu}\xi^{\nu}$.
This 4-current does not in general vanish, so we cannot be guaranteed that
$\nabla_{\mu}j[\xi^{\mu}]=0$ by the fact that $\nabla_{\mu}T^{\mu\nu}=0$. This
might be interpreted to mean that we no longer have ‘strict’ conservation of
energy according to the curved spacetime version of Gauss’ law (Wald 1984, pp.
69-70). Moreover, we might plausibly interpret $j[\xi^{\mu}]$ as the conserved
current associated with the observers’ time translations and thus understand
$\nabla_{\mu}j[\xi^{\mu}]=0$ to imply a failure of conservation of locally
measured energy. Next, recall that the class of Killing vector fields are
given by the satisfaction of the Killing equation:

(6) $\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu}=0$

This is equivalent to the statement that the metric is invariant along the
integral curves of $\xi_{\nu}$ or, in terms of the Lie derivative, that we
have that $\mathfrak{L}_{\xi}g_{\mu\nu}=0$. The existence of a time-like
Killing vector field is then (in this sense) equivalent to the symmetry of
time translation invariance holding along the relevant time-like curves. The
existence of a time-like Killing vector field is then necessary and sufficient
for the vanishing of the stress-energy 4-current. This, in turn, implies that
for the class of Killing observers whose, worldlines are integral curves of
the vector fields solving (6), the vanishing of the stress-energy 4-current
and thus ‘strict’ conservation of energy is guaranteed.

We can then formulate Spacetime Conventionalism 2 by interpreting the stress-
energy flux for the non-Killing observers as playing the functional role of a
Carnapian universal effect. Killing and non-Killing observers can be
coincident within the same spacetime patch but will attribute different
interpretations to the same geometric facts in terms of the presence or
absence of this additional associated flux. Putative physical differences with
regard to local stress-energy flux thus can be understood to arise in the
comparison of different conventions about which timelike vector field is
chosen as an observer’s rest frame. The existence or not of such effects
depends upon a convention as to the choice of timelike vector field as a rest
frame.

Three clear lines of response to this argument for Spacetime Conventionalism 2
are available. The first is simply to note that in the general class of
Einstein spacetimes Killing observers are not guaranteed to exist at all. The
existence or not of Killing vector fields depends upon there being non-trivial
continuous isometries of the metric, and there are good reasons to believe
that such transformations do not exist in general for physically realistic
spacetimes.777Following the results of Fischer (1970), what can be proved
explicitly is that for globally hyperbolic Einstein spacetimes which admit an
initial value formulation, the topology of the space of physically
distinguishable initial states (i.e. the ‘superspace’ composed of distinct
Riemannian three-geometries) is such that the geometries admitting continuous
isometries are singular points. Furthermore, initial data without non-trivial
Killing vector fields is generic in the sense that ‘geometries of high
symmetry are completely contained in the boundary of geometries of lower
symmetry’ (p. 303). There is thus a good formal basis to expect that a generic
(globally hyperbolic) Einstein spacetime will lack Killing vector fields. At a
simpler, more intuitive, level it is not hard to convince oneself that any
physically realistic spacetime will be too ‘messy’ in terms of inhomogeneities
and anisotropies to contain Killing vector fields. In a generic spacetime we
can therefore expect that there will not be any set of ‘freely falling’
observers who measure locally vanishing stress-energy 4-current. Spacetime
Conventionalism 2 would then be best understood as a viable but highly limited
conventionalist position.888It is true that, in general, local conservation of
stress-energy will approximately hold for ‘small enough’ patches of spacetime,
with how small depending on the relevant curvature (Wald 1984, pp. 70). Thus,
there will, in general, exist ‘approximate Killing observers’ for whom within
some ‘small enough’ region there are no tidal forces and gravitational stress-
energy 4-current vanishes. Operationally, however, observers cannot be
arbitrarily small, and thus making such a move would violate the spirit of the
Reichenbachian argument.

The second response relies upon the introduction of further structure into the
theory. The first step is to note that the magnitude of the stress-energy
4-current depends upon the affine properties of the world-line of the
observer. This means that the quantity given the flux into a region is still
conventional in the sense that it will be different for different observers
moving within the same spacetime patch irrespective of the existence of not of
Killing vector fields. In this context, one could consider introduction of a
gravitational-energy-momentum tensor
$t^{\mu\nu}[g_{\alpha\beta},\nabla_{\alpha\beta}]$ which is such that a total
energy-momentum complex is conserved with respect to the flat metric’s torsion
free derivative operator (Pitts 2010, 2016, 2021). This would eliminate any
potential for conventionalism with regard to total energy conservation but at
the cost of introducing extra background or auxiliary structures into the
theory.

This brings us to our third ultimately more satisfactory response to Killing
conventionalism: the rejection of the physical salience of the energy
4-current. The key observation is that it is far from clear that focusing on
localised energy fluxes is germane to the context of relativistic physics;
arguably the ambiguities we have encountered are a product of focusing on
concepts and quantities that are a relic of the bygone, pre-relativistic era.
Consider in particular the fact that the failure of the vanishing of the
stress-energy 4-current can occur for non-Killing observers even in flat
spacetimes. This surely indicates that we should not think of it as encoding
the results of additional universal effect resulting in the failure of energy
conservation. Rather it motivates us to look for a completely general
characterisation of the causal basis of genuine universal gravitational
effects due to curvature without the need for background structure: this is to
re-frame the discussion to focus on tidal forces and their explicit geometric
representation, rather than energy conservation. To do this it will prove
essential to consider the physical significance of conformal structure within
general relativity.

##  4\. Conformal Structure and Tidal Deformation

Spacetime Conventionalism 3 (1) The local conformal structure of a particular
relativistic spacetime is not an empirically-determined truth. (2) Physical
differences between coincident observers regard to mechanical stress due to
tidal forces arise in the comparison of different conventions about the
decomposition of the Riemann tensor into deforming and non-deforming parts.

Consider a smooth one-parameter family of geodesics. Define two vector fields:
a timelike vector field $\xi^{\mu}$ tangent to the family of geodesics and a
second vector field $\chi^{\nu}$ that represents the infinitesimal
displacement to an infinitesimally nearby geodesic. For a given Riemann
curvature tensor, the acceleration due to geodesic deviation is given by:

(7) $a^{\mu}=-R^{\mu}_{\beta\nu\alpha}\xi^{\alpha}\xi^{\beta}\chi^{\nu}$

In general terms, tidal ‘forces’ should be understood as the effects of
geodesic deviation as induced by Riemann curvature. There are two physically
distinct senses in which a geodesic can undergo deviation. The first is
through conformal re-scaling, which would change local spatial scales but
preserve local shape. The second is through deformation which would preserve
local spatial scales but change local shape.

Each effect can have nontrivial physical consequences. However, there are
compelling physical reasons to expect the manifestations of the effects for
local observers to be rather different. Since geodesic deviation through
conformal re-scaling does not change the shape of a material body it will not
induce mechanical stress upon that body. Rather, physical manifestations will
be indirect and contingent on the material content and relevant scales and
couplings constants. For example, redshifts or thermal effects due to
expansion. By contrast, geodesic deviation through deformation will change
local shape and thus manifest as a ‘universal effect’ which induces mechanical
stress on a material body. We have direct empirical evidence that such
stresses are able to overwhelm internal forces in realistic physical
situations through the observed phenomena of spaghettification in which the
deforming geodesic deviation induces an extreme form of tidal shear in the
region outside a black hole event horizon that rips stars into thin strips.

Spacetime Conventionalism 3 is then the view that seeks to exploit
underdetermination in the representation of tidal forces. The proposal would
be that different coincident observers may decompose the consequences of the
geodesic deviation given by Equation (7) differently, and so adopt different
conventions with regard to the degree of tidal deformation and associated
mechanical stress. This would not only underdetermine the tidal ‘forces’ in a
physically significant sense but also underdetermine the local conformal
structure of spacetime.

The proposed new formulation of spacetime conventionalism has, however, been
set up to fail. It is precisely in the relevant respect that the Riemann
curvature tensor has a unique decomposition in terms of its Ricci and Weyl
parts. This decomposition equates to exactly the decomposition of geodesic
deviation into deforming and non-deforming parts. We can see this as follows.
The Weyl curvature tensor, $C_{\rho\sigma\mu\nu}$ can be defined via the
Riemann and Ricci curvature tensors as the traceless tensor:

(8)
$C_{\rho\sigma\mu\nu}=R_{\rho\sigma\mu\nu}-\frac{1}{2}\left(g_{\rho[\nu}R_{\mu]\sigma}+g_{\sigma[\mu}R_{\nu]\rho}\right)-\frac{R}{6}\left(g_{\rho[\mu}g_{\mu]\sigma}\right)$

The tensorial nature of this expression of course means that the decomposition
of Riemann curvature into Ricci and Weyl parts is invariant under the push-
forward of diffeomorphisms, $\phi$ of the spacetime manifold $M$. That is, for
any three tensors which are related such that $A=B+C$ we will have that
$\phi^{\star}(A)=\phi^{\star}(B+C)=\phi^{\star}B+\phi^{\star}C$. Observers
using different coordinate systems in a given spacetime will necessarily agree
on the decomposition. Moreover, the decomposition of the Riemann curvature
tensor into Ricci and Weyl components has its origin in the symmetry
transformation properties of the Riemann tensor. In particular, we can
explicitly derive the decomposition by considering the symmetric and anti-
symmetric parts of a decomposition of the Riemann tensor in terms of SO(3,1)
irreducible tensors.999Following Ramond (1997) we can decompose
$R^{\rho}_{\sigma\mu\nu}$ in terms of $SU(2)\otimes SU(2)$ as it is locally
isomorphic to $SO(3,1)$. We decompose the Riemann curvature tensor into
symmetric parts  (9) $\left(3\otimes 3\right)\oplus\left(5\otimes
1\right)\oplus\left(1\otimes 5\right)\oplus\left(1\otimes
1\right)\oplus\left(1\otimes 1\right)$ and antisymmetric parts
$\left(3\otimes 3\right)\oplus\left(3\otimes 1\right)\oplus\left(1\otimes
3\right)$ We then find that the Weyl tensor $C_{\rho\sigma\mu\nu}$ transforms
as $\left(5\otimes 1\right)\oplus\left(1\otimes 5\right)$ and it is
conformally symmetric. The object that transforms as $\left(3\otimes 3\right)$
in the symmetric part the corresponds to traceless part of Ricci tensor $R$.
We thus find that there is a solid mathematical foundation for taking the
distinction as observer-independent and intrinsic to the structure of a given
spacetime.

A further decomposition of the Weyl tensor then allows an explicit and
insightful connection to tidal deformation.101010Here we are following Goswami
and Ellis (2021). The origin of this decomposition is Matte (1953). Further
discussion can be found in Bel (2000). Thanks to Juliusz Doboszewski for help
with these historical sources. Let us first interpret the Weyl tensor as the
free gravitational field, and the metric tensor as its (2nd order) potential
field and consider a timelike unit vector field $\xi^{\nu}\xi_{\nu}=-1$
representing a family of observers. Next, parallel to the way one can split
the electromagnetic field into electric and magnetic parts in the rest frame
of $\xi^{\nu}$, we can split the Weyl curvature tensor,
$C_{\rho\sigma\mu\nu}$, into electric and magnetic parts constructed as
symmetric traceless tensors orthogonal to $\xi^{\nu}$. The explicit
expressions are:

(10) $\displaystyle E_{\rho\sigma}=C_{\rho\sigma\mu\nu}\xi^{\mu}\xi^{\nu}$
(11) $\displaystyle
H_{\rho\sigma}=\frac{1}{2}\epsilon_{\rho\nu\alpha}C^{\nu\alpha}_{\sigma\mu}\xi^{\mu}$

where $\epsilon_{\rho\sigma\mu}$ is the effective volume element in the rest
space of the co-moving observer and is explicitly given by:

(12) $\epsilon_{\rho\sigma\mu}=\sqrt{|\text{det
g}|}\delta^{0}_{[\rho}\delta^{1}_{\sigma}\delta^{2}_{\mu}\delta^{3}_{\nu]}\xi^{\nu}$

(Goswami and Ellis 2021, Appendix A). The equation for geodesic deviation due
to the Weyl curvature is then simply:

(13) $\displaystyle a^{\mu}$ $\displaystyle=$
$\displaystyle-C^{\mu}_{\beta\nu\alpha}\xi^{\alpha}\xi^{\beta}\chi^{\nu}$ (14)
$\displaystyle=$ $\displaystyle E^{\mu}_{\nu}\chi^{\nu}$

We can thus understand the electric part of the Weyl tensor as uniquely
responsible for the mechanical stress associated with the tidal deformation
effect – that is, the geodesic deviation that changes the shape of bodies in
geodesic motion.111111There is also an important connection between the
electric part of the Weyl tensor and gravitational waves - see Goswami and
Ellis (2021) for more details on the formal relation. A recent, insightful
discussion of gravitational waves and gravitational energy in the context of
the analogy with electromagnetic waves can be found in Gomes and Rovelli
(2023). Tidal deformation is a non-conventional effect which can be associated
with a specific part of the Weyl curvature tensor of any given spacetime.
Spacetime Conventionalism 3 is a resounding failure.

##  5\. Nomic Structure and Dynamical Fields

Spacetime Conventionalism 4 (1) The nomic structure that specifies which
relativistic spacetimes are dynamically possible models is not an empirically-
determined truth. (2) Physical differences with regard to dynamical and non-
dynamical interpretations of field equations arise in the comparison of
different conventions about whether certain fields are understood to couple or
interact within equations.

Whilst the dynamical role of the Weyl tensor forecloses the possibility for a
putative conventionalism about tidal deformation, it opens up an opportunity
for a more interesting, and ultimately more tenable, form of conventionalism
with regard to interpretations of the equations that enforce the connection
between Weyl and Ricci curvature. In this section, we will consider a novel
form of conventionalism that is built upon underdetermination with regard to
the nomic or law-like structure that specifies which relativistic spacetimes
are dynamically possible models. In particular, an underdetermination with
regard to which equations are taken to encode nomic structure that directly
restricts dynamically possible models and which equations are taken to be
gauge identities or constraints that restrict kinematically possible models.
This underdetermination then, in turn, leads to conventionalism with regard to
which fields are understood to couple or interact which in turn implies
physical differences. We can articulate the line of thought behind this form
of conventionalism as follows.

First, recall that solving the Einstein equation (1) does not uniquely
determine the Riemann curvature. This can be seen most obviously in the
example of vacuum Einstein spacetimes which are defined as the class of Ricci
flat Lorentzian manifolds – i.e. Einstein spacetimes in which the Ricci
curvature tensor $R_{\mu\nu}$ is zero. The Riemann curvature of a Ricci flat
Lorentzian manifold is entirely determined by the Weyl curvature tensor
$C_{\mu\nu\alpha\beta}$. In contrast, we might also consider Weyl flat
spacetimes, in which the Ricci curvature encodes all geometric degrees of
freedom. Weyl flat spacetime cannot admit any local deformations of shape and
are illustrated most vividly by the FLRW spacetimes that approximately
describe our universe. The intersection of the Ricci flat and Weyl flat cases
is then the unique Riemann flat spacetime: Minkowski spacetime. In general, a
spacetime will have non-zero Ricci and Weyl curvature and the obvious question
is then how these two forms of curvature can be related if the Einstein
equation only relates to the Ricci part.

The answer is found in the Bianchi identities. Equation (4) can be re-written
as:

(15) $\nabla_{[\alpha}R_{\mu\nu]\rho\sigma}=0$

this can be expanded via the Weyl-Ricci decomposition (8) as:

(16)
$\nabla_{\nu}C^{\rho\sigma\mu\nu}=\nabla^{[\sigma}R^{\rho]\mu}+\frac{1}{2}g^{\mu[\sigma}\nabla^{\rho]}R$

In the context of the further decomposition of the Weyl tensor as per Equation
(10), the contracted Bianchi identities then lead directly to the temporal and
spatial derivatives of the electric and magnetic part of the Weyl tensor, and
these equations then lead in vacuo to the propagation equations for
gravitational waves (Goswami and Ellis 2021).121212For the original treatments
along related lines see Newman and Penrose (1962); Newman and Unti (1962);
Hawking (1966).

The next crucial step towards Spacetime Conventionalism 4 is to consider the
form of Maxwell’s equations:

(17) $\nabla_{\sigma}F^{\rho\sigma}=J^{\rho}$

where $F^{\rho\sigma}$ is the electromagnetic field tensor and $J^{\rho}$ is
the source current, and note that we can re-write the equations (16) such that
they take an analogous form:

(18) $\nabla_{\nu}C^{\rho\sigma\mu\nu}=J^{\rho\sigma\mu}$

where
$J^{\rho\sigma\mu}=\nabla^{[\sigma}R^{\rho]\mu}+\frac{1}{2}g^{\mu[\sigma}\nabla^{\rho]}R$
(Hawking and Ellis 1973, p.85). This striking analogy suggests we might think
of the Equation (16) as field equations for the Weyl curvature, just as the
Einstein equations provide field equations for the Ricci curvature.
Furthermore, we can use the Einstein equation to re-write the source current
directly in terms of stress-energy tensor such that we have an expression of
the form:

(19)
$\nabla_{\nu}C^{\rho\sigma\mu\nu}=\nabla^{[\sigma}T^{\rho]\mu}+\frac{1}{2}g^{\mu[\sigma}\nabla^{\rho]}T_{\nu}^{\nu}$

We would then seem to have a dynamical equation for the Weyl curvature as
sourced by the stress-energy tensor (Danehkar 2009). In such circumstances,
the difference between Weyl and Ricci curvature appears to have disappeared in
the sense that both can be seen as dynamical fields encoding geometric degrees
of freedom that are sourced by stress-energy tensor.

Notwithstanding the argument just given, we can also arrive at a non-dynamical
interpretation of the Weyl tensor by starting with a different reading of the
modal status of the Bianchi identities. That is, we understand the identities
as encoding restrictions on kinematical rather than dynamical possibilities. A
Bianchi identity is, in the general case, defined as the consequence of the
‘gauge’ symmetry properties of a theory via Noether’s second theorem.131313For
a detailed discussion of Noether’s second theorem in a historical context see
Kosmann-Schwarzbach (2010). For a rigorous formal overview see Olver (1993).
For discussion in the context of gauge theories and quantization see Henneaux
and Teitelboim (1992). The Noether derivation of Bianchi identities does not
require stationarity of the relevant action. Then, in the context of general
relativity, as was discussed in Section 3, the second Bianchi identity follows
from the basic mathematical properties of the Riemann tensor and is thus
naturally understood as a formal identity without physical content. As such,
Equation (16) is derivable completely independently of the action and would
hold for any spacetime theory formulated on Riemannian geometries with a
torsion-free connection.

These considerations lead us to a second non-dynamical interpretational option
in which we treat the Bianchi identities as encoding kinematic rather than
nomic structure. In the context of general relativity, this means that the
space of kinematically possible models of the theory is preselected such that
the identities are obeyed. This, in turn, means that the Ricci and Weyl
curvatures are kinematically constrained to be coordinated such that the
Riemann curvature obeys the relevant identity. Dynamics is then encoded solely
within the Einstein equation which then fixes the dynamically allowed Ricci
curvature. Residual freedom within the Weyl curvature can be understood as
fixed via the choice of initial conditions. Under such an interpretation the
relation between the Ricci and Weyl curvature is a product of a pre-
established kinematical harmony, not a substantive dynamical relationship. The
Weyl tensor is a non-dynamical field.

A useful way of understanding the connection between the interpretation of
equations and the interpretation of fields has been suggested by Lehmkuhl
(2011):

> [A]n interaction demands that all fields present are dynamical fields […] it
> seems sensible to make a distinction between speaking of two fields
> interacting and two fields coupling. For a dynamical field can couple to a
> non-dynamical field […] but we would not speak of an interaction if only one
> of the two fields was dynamical: a non-dynamical field acts without being
> acted upon if it couples to a dynamical field. Hence, two fields interacting
> should be seen as sufficient but not necessary for the fields to couple,
> whereas two fields coupling is necessary but not sufficient for the two
> fields to interact. (p.469)

Following this formulation, we would then have it that if the Bianchi
identities (16) describe an interaction then the Weyl tensor is a dynamical
field of the same status as the Ricci tensor; they each act whilst
simultaneously being acted upon. However, if the Bianchi identities (16)
describe a coupling, then we should offer a non-dynamical interpretation of
the Weyl curvature tensor, it merely couples to the Ricci tensor, the two
fields do not interact; the Ricci tensor acts upon the Weyl tensor without
being acted upon.141414This idea parallels in certain key respects the
discussion of the distinction between matter and spacetime found in Martens
and Lehmkuhl (2020).

What would appear to be the best strategy for breaking this interpretational
underdetermination is to consider the initial value problem.151515Thanks to
Henrique Gomes for suggesting this idea to us. In particular, if the initial
value problem of general relativity were found to be fully specifiable
independently of the Bianchi identities, one could argue that only the
Einstein equations encode nomic structure, and the non-dynamical
interpretation is supported. Conversely, if the Bianchi identities play an
explicit role in encoding interactions between degrees of freedom within the
initial value problem then they would encode nomic structure and the dynamical
interpretation would be supported.161616The idea that (well-posed) initial
value problems could play a privileged role in picking out the ontic structure
of physical theories is explored in a rather different context by Callender
(2017). See also Baron and Evans (2021); James (2022).

Approaches to the explicit solution of the initial value problem in general
relativity proceed via hyperbolic reductions based upon a particular gauge
choice. The most famous of these is the original ‘harmonic’ or ‘wave’ gauge
treatment due to Choqouet-Bruhat.171717Here we are following (Choquet-Bruhat
2008, §VI) and (Landsman 2021, 7.5). In this approach, although the Bianchi
identities do play a substantive role, they do not form part of the basic
system of hyperbolic dynamical equations that are obtained as a reduction of
the Einstein equations. This would seem to support the non-dynamical
interpretation. However, at least in vacuum, there exists an alternative
approach to the hyperbolic reduction of general relativity due to Friedrich
(1996) in which the Bianchi identities are explicitly understood as hyperbolic
propagation equations for the Weyl tensor. Explicitly, and considering the
vacuum case, within this approach, the Weyl tensor is treated as one of the
fundamental dynamical variables in a system of equations given by:

(20) $\displaystyle R^{\mu}_{\nu\lambda\rho}$ $\displaystyle=$ $\displaystyle
C^{\mu}_{\nu\lambda\rho}$ (21)
$\displaystyle\nabla_{\mu}C^{\mu}_{\nu\lambda\rho}$ $\displaystyle=$
$\displaystyle 0$

which are equivalent to the vacuum forms of the Einstein equation and Bianchi
identity respectively. This is precisely to understand the Einstein equations
and Bianchi identities as interaction equations of equal status and, moreover,
to explicitly treat the Weyl tensor as a dynamical field.181818For a more
general application of the idea of treating the Bianchi identities as
alternative representations of the field equations in the context of matter
systems see Friedrich and Rendall (2000).

Study of the initial value problem of general relativity thus serves to
further establish, rather than dissolve, the putative underdetermination with
regard to nomic structure that is at the heart of Spacetime Conventionalism 4.
If we appeal to the initial value problem to identify the nomic structure of
general relativity then we find that such a specification depends upon a
choice as to which approach to hyperbolic reduction we apply. At least in the
vacuum case, this then explicitly allows for different options with regard to
the attribution of a dynamical or non-dynamical role to the Weyl tensor.

We can draw an analogy between this situation and the role of the cosmological
constant.191919See Martens and Lehmkuhl (2020) for a related discussion
regarding modified gravity and dark matter. Consider a ‘$\Lambda$-vacuum’
spacetime of the form:

(22) $G_{\mu\nu}+g_{\mu\nu}\Lambda=0$

Writing the equation in this way makes it natural to think of $\Lambda$ as a
non-dynamical constant of nature. We might, of course, alternatively re-write
the same equation as:

(23) $G_{\mu\nu}=-g_{\mu\nu}\Lambda$

In this context, it is natural to think of $\Lambda$ as a dynamical source for
the Ricci degrees of freedom. These are completely trivial rewritings of the
same equation, so it is not clear what could hang on the dynamical vs. non-
dynamical interpretation of Lambda. However, there are good reasons coming
from outside the theory to treat the cosmological constant vs. dark energy
interpretational question as a substantive physical issue (Peebles and Ratra
2003; Huterer 2011).

This requirement for reference to external physical or physics principles
could also be taken to be required for the case of our two interpretations of
the Ricci-Weyl relationship. Within general relativity, the equations
themselves do not give priority to the differing dynamical and non-dynamical
interpretations. However, in a wider theoretical context, and considering, in
particular, the relevance to quantization, the distinction between the
interpretation may in fact ground genuine physical differences. For example,
whereas it is generally the case that kinematical restrictions are converted
to super-selection rules in quantization, dynamical restrictions, such as
conservation laws, are applied as quantum nomological restrictions, which
allows for their violation subject to the uncertainty principle.202020See Gryb
and Thébault (2016) for more discussion in the context of the problem of time
and the cosmological constant. Spacetime Conventionalism 4 might thus offer
insight into alternative heuristic strategies for theory extension.

Finally, an alternative strategy for breaking the underdetermination is to
consider the deeper mathematical structure behind the tensor fields in
question. Appeal to the primacy of such structure as the basis upon which to
designate which fields are dynamical and which non-dynamical. In particular,
one can understand the Weyl and Ricci tensors as composite objects built out
of the metric tensor, spinors or connection coefficients, and then argue that
since the Bianchi identities but not the Einstein equations, follow from the
mathematical definition of these objects, only the latter can be understood as
nomic structure.212121Thanks to Henrique Gomes for suggesting this argument to
us. In the context of such a view, Spacetime Conventionalism 4 would, in its
failure, again provide a valuable heuristic, in this case towards a
particular, non-trivial, mathematical ontology of relativity theory. Such an
avenue of investigation warrants further consideration.

##  6\. The Bach Conjecture

Spacetime Conventionalism 5 (1) The geometric structure of spacetime is not an
empirically-determined truth. (2) Physical differences with regard to which
tensor fields play the role of geometric structure, arise through different
conventions in the decomposition of the Einstein tensor into conformally
invariant and non-conformally invariant parts.

The final form of conventionalism we will consider is, in a sense, a marriage
of the formulation of geometric conventionalism in the spirit of Weatherall
and Manchak (2014) with the idea discussed above of understanding conformally
invariant tensors as representing the geometric structure of spacetime. The
view is, however, to our knowledge entirely novel. Furthermore, it leads to a
precise mathematical conjecture the truth of which is as yet unproven.

The basic starting point for this particular strategy is to look to re-express
the Einstein equation in a conformally invariant form. That is, seek to show
that Einstein’s equation is true if and only if,

(24) $B_{\mu\nu}=8\pi T_{\mu\nu}+A_{\mu\nu},$

where $T_{\mu\nu}$ is the matter-energy tensor appearing in Einstein’s
equation, $B_{\mu\nu}$ is a conformally invariant tensor, and $A_{\mu\nu}$ is
a further tensor that is not conformally invariant. So far this is purely a
formal exercise in analysis. The interpretation move is then to take
$B_{\mu\nu}$ to characterise facts about geometric spacetime structure, which
is, by assumption, understood to be conformally invariant. By contrast, the
non-conformally invariant objects $T_{\mu\nu}$ and $A_{\mu\nu}$ are taken to
characterise a matter-energy field and a further ‘universal effect’,
restrictively.222222The interpretation of $T_{\mu\nu}$ as matter is equivalent
to matter criterion G of Martens and Lehmkuhl (2020). Clearly, by the theorem
of Weatherall and Manchak (2014), $A_{\mu\nu}$ will not be expressible as a
Newtonian force.

The re-formulation (24) we are looking for is indeed possible, by defining
$B_{\mu\nu}$ to be the _Bach tensor_ 232323Named for Rudolf Bach, not the
Baroque German composer. Although, the former was in fact a pseudonym for
Rudolf Förster. See Appendix A for further formal details. which can be
expressed without introducing further structure in terms of the Weyl and Ricci
tensors as:

(25)
$B_{\mu\nu}:=\nabla^{\sigma}\nabla^{\rho}C_{\rho\mu\nu\sigma}+\tfrac{1}{2}C_{\rho\mu\nu\sigma}R^{\rho\sigma}.$

The Bach tensor is conformally invariant in $D=4$ (Bach 1921). Thus, defining
$A_{\mu\nu}:=B_{\mu\nu}-G_{\mu\nu}$, we have that for $D=4$ the Equation (24)
holds if and only if the Einstein equation does, as desired.242424For $D=2$,
all relativistic spacetimes are trivial vacuum solutions to Einstein’s
equation since it can be proved both that the Einstein tensor is identically
zero and that the natural candidate for a two-dimensional Weyl tensor is
identically zero (Fletcher et al. 2018). For $D=3$, the Riemann curvature
tensor is completely determined by the local matter distribution (Giddings et
al. 1984), so although there are interesting conformally invariant tensors in
three dimensions, such as the Weyl tensor and Cotton tensor, the dynamics of
three-dimensional gravity is such that these are constrained to be zero. Thus,
Conventionalism for 5 for $D<4$ is uninteresting.

The idea of a ‘pure geometry’ interpretation of the Bach tensor is noted in
the mathematical physics literature. In particular, Bergman (2004) offers the
observation that:

> The Bach tensor is a tensor built up from pure geometry, and thereby
> captures necessary features of a space being conformally Einstein in an
> intrinsic way. (Bergman 2004, p. 18)

The pivotal issue in the context of Spacetime Conventionalism 5, is then
whether the Bach tensor $B_{\mu\nu}$ is the _unique_ conformally invariant
two-place tensor, at least up to a multiplicative constant. Adopting the
interpretation of geometric structure and matter-energy above: uniqueness
would imply that this decomposition allows one to distinguish uniquely between
geometric structure ($B_{\mu\nu}$) and matter-energy ($T_{\mu\nu}$) and the
‘universal effect’ ($A_{\mu\nu}$); and, _non_ -uniqueness would amount to a
sense in which the distinction between geometric structure and matter-energy
is conventional.

Fascinatingly, although this question is both mathematically precise and
foundationally important, its status appears, as yet, unsettled. The issue at
hand is one which can be more generally associated with finding the conformal
invariants in the theory of Lorentzian manifolds:

> “Classically known conformally invariant tensors include the Weyl conformal
> curvature tensor, which plays the role of the Riemann curvature tensor, its
> three-dimensional analogue the Cotton tensor, and the Bach tensor in
> dimension four. Further examples are not so easy to come by.” (Fefferman
> 2012, p.1)

Thus, although the review of the literature does not reveal any other known
conformal invariants in four dimensions, there is no proof of the absence of
such alternatives either.

One might thus propose that the debate over conventionality and non-
conventionality, as described above, is not just a matter of philosophical
debate, but of settling a precise mathematical conjecture, whose truth is not
yet known:

######  Bach Conjecture.

The Bach tensor is the unique (up to a multiplicative constant) conformally
invariant rank 2 tensor field on a four-dimensional Lorentzian manifold.

If the Bach Conjecture is false, then Spacetime Conventionalism 5 is true; if
the Bach Conjecture is true, then Spacetime Conventionalism 5 is false. We
thus take the proof or refutation of the Bach conjecture to be an important
problem in the foundations of general relativity that warrants considered
attention.

##  7\. Summary

One might reasonably judge proposals for the conventionality of spacetime
structure in terms of their respective novelty, interestingness, and
plausibility. Our discussion has characterised and critically evaluated five
formulations of the conventionality thesis and we hope the reader will agree
that all five have proved worthy of consideration. The first three options
considered have been argued to be implausible on grounds that vary from direct
refutation (Spacetime Conventionalism 3), to provable falsity under certain
assumptions (Spacetime Conventionalism 1), to rejection on the grounds of the
lack of physical salience of the central undetermined object (Spacetime
Conventionalism 2). Nevertheless, each of the three views illustrates
important morals regarding the interplay between empirical and mathematical
structures within the foundations of spacetime theory. The final two forms of
conventionalism we considered to be novel, interesting, and plausible.
Moreover, in terms of analysis of the dynamical role of the Bianchi identities
(Spacetime Conventionalism 4) and proof or refutation of the Bach conjecture
(Spacetime Conventionalism 5), they each open up a new direction of enquiry
that we may hope to be profitably pursued in future researches.

## Acknowledgements

The form, scope, and content of this paper were transformed through hugely
productive interactions with Bryan Roberts. In particular, both the framing of
varieties of conventionalism via the two questions and the formulation of the
Bach conjecture is due directly to Bryan Roberts. We further greatly profited
from detailed written comments and lengthy discussions with Bryan without
which the present paper would be a shadow of what it is. The paper has also
profited from a number of other detailed discussions that KT has had over the
last years with (among others) Erik Curiel, Juliusz Doboszewsk, Sam Fletcher,
Henrique Gomes, Sean Gryb, James Ladyman, Dennis Lehmkuhl, Oliver Pooley,
James Read, Simon Saunders, Dave Sloan, Jim Weatherall. We are also grateful
to online audiences at the DPG conference Heidelberg and the Philosophy of
Physics and Chemistry group in Buenos Aires, Argentina, and to Patrick Düerr
and two anonymous referees for helpful comments. The authors take full
responsibility for any errors, omissions or inaccuracies.

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A

##  Appendix A Conformal Tensors and the Bianchi Identities

The Bianchi identities can also be related to the Cotton Tensor
$\mathcal{C}_{\rho\sigma\nu}$ (Cotton; 1899) and Bach Tensor $B_{ab}$ as
follows. Following García et al. (2004), we can decompose curvature into
irreducible representations with respect to the pseudo-orthogonal group for
certain dimensions, $n$, as follows:

(26) $\displaystyle n=1$ $\displaystyle\rightarrow R_{\alpha\beta}=0$ (27)
$\displaystyle n=2$ $\displaystyle\rightarrow
R_{\alpha\beta}=Scalar_{\alpha\beta}$ (28) $\displaystyle n=3$
$\displaystyle\rightarrow
R_{\alpha\beta}=Scalar_{\alpha\beta}+\widetilde{R}_{\alpha\beta}$ (29)
$\displaystyle n\geq 4$ $\displaystyle\rightarrow
R_{\alpha\beta}=Scalar_{\alpha\beta}+\widetilde{R}_{\alpha\beta}+C_{\alpha\beta}$

where $\widetilde{R}$ denotes traceless Ricci part of the decomposition and
every tensor is in its Weyl 2-form. For $n=3$, the Cotton tensor paper which
is given by
$DR_{\alpha\beta}=DC_{\alpha\beta}+\frac{2}{n-2}\upsilon_{[\alpha}\land\mathcal{C}_{\beta]}=0$
where $D$ denotes exterior covariant derivative and $\upsilon$ denotes coframe
of a Riemannian space of $n$ dimensions.

(30)
$\mathcal{C}_{\rho\sigma\nu}=2\left(\nabla_{[\alpha}R_{\beta]\gamma}-\frac{1}{2\left(n-1\right)}\nabla_{[\alpha}Rg_{\beta]\gamma}\right)$

and for $n=4$, the Bach tensor can be defined as

(31)
$B_{\alpha\beta}:=\nabla^{\mu}\mathcal{C}_{\alpha\mu\beta}+L^{\mu\nu}C_{\alpha\mu\beta\nu}$

where $L^{\mu\nu}$ is called Schouten tensor and given by

(32) $L_{\mu\nu}=R_{\mu\nu}-\frac{1}{2\left(n-1\right)}Rg_{\mu\nu}.$

Although the Bach and Weyl tensors appear to be independent in tensorial form,
they can be directly associated with one another via a variational principle.
In particular, we can drive the Bach tensor from a Lagrangian composed of Weyl
tensors that takes the form:252525see Fiedler and Schimming (1980) for the
exact solutions.

(33) $L=C_{\rho\mu\nu\sigma}C^{\rho\mu\nu\sigma}+kg^{\mu\nu}T_{\mu\nu}$

where $T_{\mu\nu}$ is the relevant matter field with a non-zero constant. The
relevant Euler-Lagrange equations are then Bach’s equation of motion:

(34) $B_{\mu\nu}=kT_{\mu\nu}$

where $T_{\mu\nu}$ is traceless due to conformal invariance of $B_{\mu\nu}$.

Additionally, considering the vacuum case, the Bach tensor exhibits parallel
behaviour to the Weyl tensor in relation to the Bianchi identities given in
(20)262626see (Bergman; 2004, p. 39) for the discussion.:

(35) $\displaystyle\nabla^{d}C_{abcd}$ $\displaystyle=$ $\displaystyle
K^{d}C_{abcd}$ (36) $\displaystyle B_{\mu\nu}$ $\displaystyle=$ $\displaystyle
0$