1. Conventionalism about what?

2. Non-conformal structure and universal forces

Any pair of conformal equivalent relativistic spacetimes are, by definition, such that $\left( {M,{g_{\mu \nu }}} \right)$ and $\left( {M,{{\tilde g}_{\mu \nu }}} \right)$ where ${\tilde g_{\mu \nu }} = {{\rm{\Omega }}^2}{g_{\mu \nu }}$ . 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 expression $G_{{\rm{\;}}\mu }^\nu {\tilde \xi ^\mu }$ 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, Reference Weatherall and Manchak2014, 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 (Reference Duerr and Ben-Menahem2022) has strongly contested whether the characterization 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 (Reference Dardashti2021).

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. ^{Footnote 3} That is, we take the result to motivate “methodological pathways” in which we seek to articulate different forms of the generalized 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

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. ^{Footnote 4} The problem can be straightforwardly stated as follows. For a vanishing cosmological constant, the Einstein equations take the familiar form

(1) $${G_{\mu \nu }} = {R_{\mu \nu }} - {1 \over 2}R{g_{\mu \nu }} = 8\pi {T_{\mu \nu }},$$

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) $$R_{\sigma \mu \nu }^\rho {\xi ^\sigma } = - 2{\nabla _{[\mu }}{\nabla _{\nu ]}}{\xi ^\rho }$$

(Malament, Reference Malament2012, Lemma 1.8.1). $R_{\sigma \mu \nu }^\rho $ 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) $${R_{\sigma \mu }} = R_{\sigma \mu \rho }^\rho .$$

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) $${\nabla _{[\alpha }}R_{\left| \sigma \right|\mu \nu ]}^\rho = 0.$$

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. ^{Footnote 5} Interestingly, however, when combined with the definition of the Einstein tensor, equation (4) can be shown to imply ^{Footnote 6}

(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 general relativity 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 $j\left[ {{\xi ^\mu }} \right] = T_{{\rm{\;}}\nu }^\mu {\xi ^\nu }$ . This 4-current does not in general vanish, so we cannot be guaranteed that ${\nabla _\mu }j\left[ {{\xi ^\mu }} \right] = 0$ 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, Reference Wald1984, 69–70). Moreover, we might plausibly interpret $j\left[ {{\xi ^\mu }} \right]$ as the conserved current associated with the observers’ time translations and thus understand ${\nabla _\mu }j\left[ {{\xi ^\mu }} \right] = 0$ to imply a failure of conservation of locally measured energy. Next, recall that the class of Killing vector fields is 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 ${\mathfrak{L}_\xi }{g_{\mu \nu }} = 0$ . 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. ^{Footnote 7} 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. ^{Footnote 8}

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 or not of Killing vector fields. In this context, one could consider introduction of a gravitational-energy–momentum tensor ${t^{\mu \nu }}\left[ {{g_{\alpha \beta }},{\nabla _{\alpha \beta }}} \right]$ which is such that a total energy–momentum complex is conserved with respect to the flat metric’s torsion-free derivative operator (Pitts, Reference Pitts2010, Reference Pitts2016, Reference Pitts2021). 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 localized 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 characterization of the causal basis of genuine universal gravitational effects due to curvature without the need for background structure: this is to reframe 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

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) $${a^\mu } = - R_{\beta \nu \alpha }^\mu {\xi ^\alpha }{\xi ^\beta }{\chi ^\nu }.$$

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 rescaling, 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 non-trivial 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 rescaling 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 coupling 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) $${C_{\rho \sigma \mu \nu }} = {R_{\rho \sigma \mu \nu }} - {1 \over 2}\left( {{g_{\rho [\nu }}{R_{\mu ]\sigma }} + {g_{\sigma [\mu }}{R_{\nu ]\rho }}} \right) - {R \over 6}\left( {{g_{\rho [\mu }}{g_{\nu ]\sigma }}} \right)$$

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 ${\phi ^*}\left( A \right) = {\phi ^*}\left( {B + C} \right) = {\phi ^*}B + {\phi ^*}C$ . 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 antisymmetric parts of a decomposition of the Riemann tensor in terms of SO(3,1) irreducible tensors. ^{Footnote 9} 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. ^{Footnote 10} Let us first interpret the Weyl tensor as the free gravitational field, and the metric tensor as its (second-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

(9) $${E_{\rho \sigma }} = {C_{\rho \sigma \mu \nu }}{\xi ^\mu }{\xi ^\nu },\;\;\;\;\;\;\;\;{H_{\rho \sigma }} = {1 \over 2}{\varepsilon _{\rho \nu \alpha }}C_{\sigma \mu }^{\nu \alpha }{\xi ^\mu },$$

where ${\varepsilon _{\rho \sigma \mu }}$ is the effective volume element in the rest space of the comoving observer and is explicitly given by

(10) $${\varepsilon _{\rho \sigma \mu }} = \sqrt {\left| {{\rm{det\ g}}} \right|} \delta _{[\rho }^0\delta _\sigma ^1\delta _\mu ^2\delta _{\nu ]}^3{\xi ^\nu }$$

(Goswami and Ellis, Reference Goswami and Ellis2021, appendix A). The equation for geodesic deviation due to the Weyl curvature is then simply

(11) $${a^\mu } = - C_{\beta \nu \alpha }^\mu {\xi ^\alpha }{\xi ^\beta }{\chi ^\nu } = E_\nu ^\mu {\chi ^\nu }.$$

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. ^{Footnote 11} 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

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 potential 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 Friedmann–Lemaître–Robertson–Walker 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 rewritten as

(12) $${\nabla _{[\alpha }}{R_{\mu \nu ]\rho \sigma }} = 0.$$

This can be expanded via the Weyl–Ricci decomposition (8) as

(13) $${\nabla _\nu }{C^{\rho \sigma \mu \nu }} = {\nabla ^{[\sigma }}{R^{\rho ]\mu }} + {1 \over 2}{g^{\mu [\sigma }}{\nabla ^{\rho ]}}R$$

In the context of the further decomposition of the Weyl tensor as per equation (9), the contracted Bianchi identities then lead directly to the temporal and spatial derivatives of the electric and magnetic parts of the Weyl tensor, and these equations then lead in vacuo to the propagation equations for gravitational waves (Goswami and Ellis, Reference Goswami and Ellis2021). ^{Footnote 12}

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

(14) $${\nabla _\sigma }{F^{\rho \sigma }} = {J^\rho },$$

where ${F^{\rho \sigma }}$ is the electromagnetic field tensor and ${J^\rho }$ is the source current, and note that we can rewrite the equations (13) such that they take an analogous form:

(15) $${\nabla _\nu }{C^{\rho \sigma \mu \nu }} = {J^{\rho \sigma \mu }},$$

where ${J^{\rho \sigma \mu }} = {\nabla ^{[\sigma }}{R^{\rho ]\mu }} + {1 \over 2}{g^{\mu [\sigma }}{\nabla ^{\rho ]}}R$ (Hawking and Ellis, Reference Hawking and Ellis1973, 85). This striking analogy suggests we might think of equation (13) 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 rewrite the source current directly in terms of the stress–energy tensor such that we have an expression of the form

(16) $${\nabla _\nu }{C^{\rho \sigma \mu \nu }} = {\nabla ^{[\sigma }}{T^{\rho ]\mu }} + {1 \over 2}{g^{\mu [\sigma }}{\nabla ^{\rho ]}}T_\nu ^\nu $$

We would then seem to have a dynamical equation for the Weyl curvature as sourced by the stress–energy tensor (Danehkar, Reference Danehkar2009). 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 the 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. ^{Footnote 13} 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 (13) 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 (Reference Lehmkuhl2011, 469):

Following this formulation, we would then have that if the Bianchi identities (13) 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 (13) 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. ^{Footnote 14}

What would appear to be the best strategy for breaking this interpretational underdetermination is to consider the initial value problem. ^{Footnote 15} 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. ^{Footnote 16}

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. ^{Footnote 17} 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 (Reference Friedrich1996) 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

(17) $$R_{\nu \lambda \rho }^\mu = C_{\nu \lambda \rho }^\mu, {\rm{\;\;\;\;\;\;\;\;}}{\nabla _\mu }C_{\nu \lambda \rho }^\mu = 0,$$

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. ^{Footnote 18}

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. ^{Footnote 19} Consider a “ ${\rm{\Lambda }}$ -vacuum” spacetime of the form

(18) $${G_{\mu \nu }} + {g_{\mu \nu }}{\rm{\Lambda }} = 0.$$

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

(19) $${G_{\mu \nu }} = - {g_{\mu \nu }}{\rm{\Lambda }}.$$

In this context, it is natural to think of ${\rm{\Lambda }}$ 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 ${\rm{\Lambda }}$ . 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, Reference Peebles, James and Ratra2003; Huterer, Reference Huterer and Goodstein2011).

This requirement for reference to external physical theories 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. ^{Footnote 20} 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 and 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. ^{Footnote 21} 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

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 (Reference Weatherall and Manchak2014) 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

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

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 characterize 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 characterize a matter–energy field and a further “universal effect,” restrictively. ^{Footnote 22} Clearly, by the theorem of Weatherall and Manchak (Reference Weatherall and Manchak2014), ${A_{\mu \nu }}$ will not be expressible as a Newtonian force.

The reformulation (20) we are looking for is indeed possible, by defining ${B_{\mu \nu }}$ to be the Bach tensor ^{Footnote 23} which can be expressed without introducing further structure in terms of the Weyl and Ricci tensors as

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

The Bach tensor is conformally invariant in $D = 4$ (Bach, Reference Bach1921). Thus, defining ${A_{\mu \nu }}: = {B_\mu } - {G_{\mu \nu }}$ , we have that for $D = 4$ equation (20) holds if and only if the Einstein equation does, as desired. ^{Footnote 24}

The idea of a “pure geometry” interpretation of the Bach tensor is noted in the mathematical physics literature. In particular, (Bergman, Reference Bergman2004, 18) 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.”

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 and Graham, Reference Fefferman and Robin Graham2012, 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-two 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 characterized 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 research.