2.1. Ambiguities in derivation

We begin with a brief review of how to arrive at GR as the only consistent way of including self-interaction in the dynamics of a classical spin-2 field, formulated on a Lorentzian manifold $\left( {M,\eta } \right)$ (where $\eta $ is the Minkowski metric), in order to pinpoint several ambiguities.

1. 1. Stipulate a classical spin-2 field $h$ , i.e., a tensor field representing exactly two polarizations, together with a free equation of motion.

2. 2. Include universal coupling to matter: the energy–momentum tensor of all fields other than the spin-2 field itself (denoted by ${\rm{T}}$ ) source the free spin-2 equation of motion:

   (EOM (0)) $$G_{\beta \gamma }^{\left( 0 \right)} = {T_{\beta \gamma }}$$

   Assuming that $T$ is conserved on-shell by virtue of the matter equations of motion alone, a consistency problem arises: a coupling between the matter fields and the $h$ field means that $T$ will no longer be conserved on-shell, i.e., $\partial T\mathop = \limits^{{\rm{on}} - {\rm{shell}}} 0$ no longer holds for the novel matter equations of motion that include coupling to $h$ . Yet $G_{\beta \gamma }^{\left( 0 \right)}$ in the free equation of motion is identically zero when hit with the partial derivative which, via (EOM(0)), implies $\partial T = 0$ (not just on-shell)

3. 3. Attempt to remedy the consistency problem by including a contribution to the energy–momentum tensor from $h$ itself (denoted by ${t^{\left( 0 \right)}}$ ).

4. 4. Adding an energy–momentum tensor contribution from $h$ , ${t^{\left( 0 \right)}}$ , leads to a new spin-2 equation of motion sourced by the matter term $T$ :

   (EOM (1)) $$G_{\beta \gamma }^{\left( 1 \right)}: = G_{\beta \gamma }^{\left( 0 \right)} - t_{\beta \gamma }^{\left( 0 \right)} = {T_{\beta \gamma }}.$$

But it would be inconsistent to stop at this point: we can iterate the argument above, which leads to adding the energy–momentum tensor for $h$ associated to the left-hand side of EOM(1), which we will denote ${t^{\left( 1 \right)}}$ .

This problem arises for all steps $n \gt 0$ : Let $G_{\beta \gamma }^{\left( n \right)}$ be the $n$ -EOM-term for $h$ , sourced by ${T_{\beta \gamma }}$ . Once corrected by $t_{\beta \gamma }^{\left( n \right)}$ , the ( $n + 1$ )-EOM is:

(EOM (n + 1)) $$G_{\beta \gamma }^{\left( {n + 1} \right)}: = G_{\beta \gamma }^{\left( n \right)} - t_{\beta \gamma }^{\left( n \right)} = {T_{\beta \gamma }}.$$

1. 5. Upon taking the limit ${\rm{n}} \to \infty $ , and redefining ${\rm{\eta }} + {\rm{h}}$ as ${\rm{g}}$ , the resulting field equation containing all required self-interactions is the Einstein field equation.

The ambiguities of this procedure are so problematic as to undermine any plausible claim of a derivation of GR from classical spin-2 theory.

Step 1 is ambiguous in choice of spin-2 field representation: A classical spin-2 field can be represented in various fashions (Barceló et al. Reference Barceló, Carballo-Rubio and Garay2014, section 2). This is noteworthy in that the choice of spin-2 field representation demonstrably makes a difference in the self-interaction approach even if various representations may be physically equivalent in the standard flat spacetime context. Most proposals, with the exception of Barceló et al. (Reference Barceló, Carballo-Rubio and Garay2014), tacitly presuppose a Fierz–Pauli representation—the $h$ field is taken as a second-rank, symmetric Lorentz tensor field. An alternative, however, is the spin-2 field representation that involves a trace-free condition on $h$ ( $h_\mu ^\mu = 0$ ). This can be realized as a specific (partial) gauge-fix of the Fierz–Pauli spin-2 representation. The trace-free representation but not the Fierz–Pauli representation always implies that the volume element linked to a composite metric $g: = \eta + h$ is that of Minkowski spacetime (cf. Barceló et al. Reference Barceló, Carballo-Rubio and Garay2014, section 5). Moreover, step 1 is ambiguous in the choice of spin-2 equation of motion: The Fierz–Pauli spin-2 equation of motion is a contingent but arguably preferable choice in so far as it is the unique gauge-invariant second-order equation for a free spin-2 field in the Fierz–Pauli representation. ^{Footnote 3} We will assume the Fierz–Pauli equation in the following.

With step 2 we face the choice of matter coupling: on what grounds is $h$ taken to be sourced by $T$ and not in a matter-field-specific manner?

The most severe ambiguity arises in step 3, the choice of energy–momentum tensor(s) associated to a spin-2 equation of motion. ^{Footnote 4} There is just no good heuristic available to guide this choice, nor can it easily be regarded as a starting assumption (by contrast with the first two steps). We will defer discussion until section 3.1. Step 3 suffers from a further ambiguity in choice of $h$ -self-energy–momentum tensor interaction in the sense that it is not clear why no forms of coupling of $h$ to matter in a different sense of universality (for instance, as not mediated via the energy–momentum tensor) or of no universal nature at all (interaction with $h$ differs in form from matter type to matter type) should be considered (in analogy with the ambiguity of step 2). Arguably more severely though, and specific to the self-interaction term, it is unclear whether the interaction of $h$ with an energy–momentum tensor is to be primarily described as self-sourcing at the level of the equations of motion in analogy to the sourcing notion in electrodynamics, or as self-coupling at the level of action (schematically: $h \cdot T$ ) in reference to modern particle physics parlance (unlike for a conventional energy–momentum tensor which does not contain $h$ itself, self-interaction and self-coupling are not equivalent in the context of a self-energy–momentum tensor for $h$ ).

Even setting aside all these ambiguities in setting up the self-interaction problem, the spin-2 construction of GR faces yet further hurdles in actually delivering GR. Most importantly, the construction is only really feasible if the self-sourcing relation leads to the Einstein field equations (or to an action equivalent to the Einstein–Hilbert action). Butcher (Reference Butcher2012) obtains, through a procedure similar to that above, that the gravitational action at each respective step $n$ is given by $\mathop \sum \nolimits_{i = 1}^n {S_i}$ , where ${S_1}\left[ {\gamma, h} \right] = {1\over\lambda} \int {d^4}x\sqrt { - \gamma } {G_{\mu \nu }}{h^{\mu \nu }}$ , ${S_2}\left[ {\gamma, h} \right] = {1\over2} \int {d^4}x{h^{\rho \sigma }} {{\delta {S_1}}\over{\delta {\gamma ^{\rho \sigma }}} }$ , and ${{\delta S_i\left[ {\gamma, h} \right]}\over{\delta {h^{\mu \nu }}}} = {{\delta {S_{i - 1}}\left[ {\gamma, h} \right]}\over{\delta {\gamma ^{\mu \nu }}}}$ (recursively). Now, ${S_1}$ , ${S_2}$ , ${S_i}$ are the first-, second-, $i$ th-order terms obtained from (formally) expanding the Einstein–Hilbert action $S = {1\over\lambda} \int {d^4}x\sqrt { - g} R$ with respect to $g$ around $\eta $ in terms of orders of $h$ . But from only knowing the sequence prescription and the starting terms, can one indeed find that $\mathop \sum_{i = 1}^{\infty} {S_i}$ converges to $S$ ?

We refrain from claiming to see how convergence to the Einstein–Hilbert action (with $\eta + h$ to be replaced by $g$ ) can in any sense be guessed rather than just tested; in particular, a direct guess based on visual inspection of a finite number of elements of the sequence fails. Even if it were for instance discovered that the “sequence within the series”, i.e. ${({S_i})_{i \in \mathbb{N}_1}}$ , is explicitly given by ${S_i}\left[ {\gamma, h} \right] = {{2}\over{i!}}{\left[ {\int {d^4}x{h^{\mu \nu }} {{\delta}\over{\delta {\gamma ^{\mu \nu }}}}} \right]^{i - 2}}{S_2}\left[ {\gamma, h} \right]$ , it is not at all clear from the mere sequence rule what the overall series $\mathop \sum \nolimits_{i = 1}^n {S_i}$ would converge to. A common feature of this and other derivation attempts is that background knowledge of GR provides guidance at crucial steps; none are able to uniquely recover GR starting from only concepts available in special relativistic field theory. Notably, the exact details of the attempted derivation varies in the literature, even in the most recent attempts (e.g., Butcher Reference Butcher2012; Barceló et al. Reference Barceló, Carballo-Rubio and Garay2014)—an inconsistency which in itself points to the imprecise nature of the current state of the derivation. The ambiguities discussed in this section stand in sharp contrast to any claim that one can uniquely derive GR from spin-2 theory.

2.2. Limited sector

How much of GR can we expect to recover through the self-interaction approach? It is understandably appealing to treat gravitons as perturbations around a flat spacetime as a prelude to quantization, and to expect curved spacetime geometry to emerge from interactions among the gravitons. From the point of view of heuristics and theory construction, it is natural to start from a case of limited scope and hope that it generates insight into the full theory, leaving global issues, and the complexities of full non-linear interactions, to be treated at a later stage. However, if we take seriously the idea that the spin-2 approach gives us a kind of theoretical reduction of GR (cf. Salimkhani Reference Salimkhani2020), the question of how much can be reproduced is much more pressing. Here we highlight two well-known limitations of the self-interaction approach that hold even if the ambiguities above were resolved.

First, there are certain surface terms which a spin-2 derivation (no matter which energy–momentum tensor scheme is chosen) can never reproduce (Padmanabhan Reference Padmanabhan2008, section III). ^{Footnote 5}

The second, and more pressing, concern regards how to make a transition from gravitons propagating against a background metric $\eta $ to a generic curved spacetime metric $g$ (e.g., Barceló et al. Reference Barceló, Carballo-Rubio and Garay2014). In general, $\eta $ and $g$ are not definable on homeomorphic manifolds, and the local symmetries and other structures of Minkowski spacetime (or other spacetimes with fixed curvature) cannot be extrapolated globally. It is also not clear when local perturbative treatments can be patched together. Riemannian normal coordinates can be applied in “local” patches on a manifold, clarifying the sense in which a solution “looks locally flat,” but the requirement that these patches can be stitched together imposes limits on the topology and global features of the manifold.

3.1. The first horn

The self-sourcing procedure rests on the identification of a concrete spin-2 energy–momentum tensor. The choice of an energy–momentum tensor is ambiguous in several senses. First of all, the choice of an energy–momentum tensor is generally considered to be ambiguous with respect to the addition of a superpotential ${\partial _\alpha }{{\rm{\Psi }}^{\left[ {\rho \alpha } \right]\sigma }}$ , where ${{\rm{\Psi }}^{\left[ {\rho \alpha } \right]\sigma }}$ is a third-rank tensor that is anti-symmetric in its first two indices (thereby, ${\partial _\rho }{\partial _\alpha }{{\rm{\Psi }}^{\left[ {\rho \alpha } \right]\sigma }} = 0$ ). The most general possible energy–momentum tensor found from the superpotential method for the case of the Fierz–Pauli action was derived in Baker (Reference Baker2021), which demonstrated that there are not only infinitely many possible superpotential additions, but that even for specific superpotentials there are infinitely many off-shell possibilities; therefore the superpotential method further complicates the selection of a unique energy–momentum tensor for self coupling.

Secondly, there is an ambiguity linked to the fact that the standard definition usually adhered to in special relativity to obtain the energy–momentum tensor relative to an action $A$ /Langrangian density $L$ —the canonical Noether energy–momentum tensor $T_C^{\mu \nu }$ —does not lead to a symmetric energy–momentum tensor, which is, however, required for the self-interaction approach. ^{Footnote 6} Given the unique energy–momentum definition for the canonical Noether energy–momentum tensor

$${T_C^{\mu \nu } = {\eta ^{\mu \nu }}{L} - {{\partial {L}} \over {\partial \left( {{\partial _\mu }{h_{\alpha \beta }}} \right)}}{\partial ^\nu }{h_{\alpha \beta }} - {{\partial {L}} \over {\partial \left( {{\partial _\mu }{\partial _\omega }{h_{\alpha \beta }}} \right)}}{\partial _\omega }{\partial ^\nu }{h_{\alpha \beta }} + \left( {{\partial _\omega }{{\partial {L}} \over {\partial \left( {{\partial _\mu }{\partial _\omega }{h_{\alpha \beta }}} \right)}}} \right){\partial ^\nu }{h_{\alpha \beta }} + \cdots} $$

for a spin-2 field ${h_{\alpha \beta }}$ , one obtains $T_C^{\rho \sigma } = {\eta ^{\rho \sigma }}{{L}_{FP}} - {\partial {{L}_{FP}}\over\partial \left( {{\partial _\rho }{h_{\mu \nu }}} \right)}{\partial ^\sigma }{h_{\mu \nu }}$ for the Fierz–Pauli Lagrangian density ${{L}_{{\rm{FP}}}}$ . Using the freedom to add a superpotential ${\partial _\alpha }{{\rm{\Psi }}^{\left[ {\rho \alpha } \right]\sigma }}$ , the symmetric energy–momentum tensor for the spin-2 field will be given (on-shell) by the canonical Noether $T_C^{\mu \nu }$ on the Fierz–Pauli Lagrangian ${{L}_{{\rm{FP}}}}$ and the divergence of a superpotential as

(1) $${T^{\rho \sigma }} = T_C^{\rho \sigma } + {\partial _\alpha }{{\rm{\Psi }}^{\left[ {\rho \alpha } \right]\sigma }}.$$

As shown in Baker (Reference Baker2021), however, there are infinitely many solutions to the specific problem of finding a symmetric energy–momentum tensor using this procedure. This approach alone fails to deliver a unique answer. The usual proposal to determine a unique answer employed in the spin-2-to-GR literature is to pick out a preferred symmetric energy–momentum tensor through the Hilbert definition:

$${T^{\gamma \rho }} = {2 \over {\sqrt { - g} }}{{\delta {L}} \over {\delta {g_{\gamma \rho }}}}{\biggr\vert_{g = \eta }}.$$

Notably, having some prescriptive definition for energy–momentum tensor like the Hilbert one is required to make the self-interaction spin-2 approach feasible to begin with: at each iterative step it is required to actually pick an energy–momentum tensor relative to the spin-2 field equation obtained in the previous step. ^{Footnote 7} This needs to be done in a systematic fashion if the self-sourcing is really to count as a performable procedure, which in turn seems to make a unique definition necessary—otherwise we would have to choose in principle infinitely many times between infinitely many possible (symmetric) energy–momentum sources.

That is, the Hilbert definition is claimed to select one out of the infinitely many possibilities in equation (1); ^{Footnote 8} however, this claim is not true in general—the result of the Hilbert definition does not generally correspond to the symmetric Noether energy–momentum tensor for flat (Minkowski) spacetime theories (Baker et al. Reference Baker, Kiriushcheva and Kuzmin2021), and in the example provided, only the Noether method was able to recover the accepted energy–momentum tensor of the theory. ^{Footnote 9} But even if the methods do align as in the case of Fierz–Pauli (Fierz–Pauli is one of a small number of models where on-shell improvements can be used to reconcile the results of the Hilbert and Noether methods) understanding why the Hilbert definition should be chosen to select the right symmetric energy–momentum tensor still requires understanding the concept of Hilbert definition itself—which as such, again, comes from GR and other curved spacetime settings. Since the Hilbert definition relies on the notion of curved spacetime, using it undermines the notion that the curved spacetime of GR is being introduced purely through some iterative procedure starting from spin-2 theory. At the purely practical level, it also needs to be pointed out that if we considered only the Noether approach without a priori knowledge of the Hilbert definition, it would be impossible to determine which one of the infinitely many superpotentials recovers (on-shell) the Hilbert expression.

With the relevance of the flat spacetime Hilbert definition called into question, and without an alternative to it, we have no reason to think that there is a coherent sense of energy–momentum sourcing at play in the self-interaction program.

3.2. The second horn

If one accepts that the identification of a dynamical coupling to self-energy is blocked, one might remain unimpressed and take the mere existence of a defining recursive relation, say for the action, as sufficient to establish that the $g$ -field arises from the $h$ -field relative to flat spacetime. This leads to the second horn of the dilemma.

Consider, following Butcher (Reference Butcher2012, section 2.3), how one can expand a gravitational action $S\left[ g \right]$ such as the Einstein–Hilbert action ${S_{{\rm{EH}}}}$ for $g = \gamma + \kappa h$ in orders of $\kappa $ around some arbitrary static background metric $\gamma $ such that $S\left[ g \right] = S\left[ {\gamma + \kappa h} \right] = \mathop \sum \nolimits_{i = 0}^\infty {\kappa ^i}{S_i}\left[ {\gamma, h} \right]$ , where the $i$ th partial action is given by ${S_i}\left[ {\gamma, h} \right] = \left( {{1}\over{i!}} \partial _\kappa ^iS\left[ {\gamma + \kappa h} \right] \right) \vert_{\kappa = 0}$ . It straightforwardly follows that

(recursive relation) $${{\delta {S_i}\left[ {\gamma, h} \right]} \over {\delta {h^{\mu \nu }}}} = {{\delta {S_{i - 1}}\left[ {\gamma, h} \right]} \over {\delta {\gamma^{\mu \nu }}}}.$$

We are interested in the case where $S = {S_{{\rm{EH}}}}$ is the Einstein–Hilbert action and $\gamma = \eta $ the Minkowski background. As worked out before, it is not clear then that $\left({ \delta {S_{i - 1}}\left[{\gamma, h} \right]} \over{\delta {\gamma^{\mu \nu }}} \right){\vert_{\gamma = \eta }}$ can be associated with any sort of special relativistic concept of energy–momentum tensors (as usually done via Noether’s theorem for the special relativistic context, i.e., in relation to the Poincaré symmetries)—and thus regarded as clearly physically sensible contributions to $h$ ’s self-energy. (As long as there is no sensible notion of energy–momentum tensor defended here, there is no clear sense in which the energy-self-sourcing interaction can literally be taken as a physical mechanism.)

What we want to bring to attention now is that the mere fact that the relation between a few terms of a perturbative series of a function(al) can be used to define the whole function(al) if suitable operations are allowed is nothing special as such; in particular, this fact alone cannot render the perturbative picture physically more fundamental than the non-perturbative one. For instance, the sequence elements ${({f_i})_{i \in \mathbb{N}_0}}$ in the series expansion of ${\rm{exp}}\left( x \right) - 1$ as $\sum_{i = 0}^{\infty} f_i$ can be re-expressed recursively as $f_{i+1} = {x f_i \over ({{max} \{j | \partial^j f_i \ne 0\} + 1})}$ with $f_0(x) = x$ analogously to how the Einstein–Hilbert action is fixed by a recursive relation in equation (recursive relation). All of this suggests that the self-interaction spin-2 approach involves interpreting a perturbative expansion to look like a physical energy-sourcing relation instead of simply seeing it as part of a standard mathematical procedure.

Perhaps a perturbative spin-2 view could still be seen to offer a decompositional picture of the metric field as composed of some background metric and an $h$ field with corresponding dynamics. Given the arbitrariness in background metric (in particular, the Minkowski metric is neither the only, nor a preferred, option in such a classical decompositional picture), the question arises then why any perturbative picture of GR should be more fundamental than the non-perturbative picture. Arguably, the converse question(s) as to why the non-perturbative picture is more fundamental than the perturbative one (or, at least, all perturbative pictures taken together) is also possible. If so, we seem to be left with many possible ways to identify the fundamental dynamical degrees of freedom of the gravitational field, without sufficiently clear grounds for privileging one. The existence of multiple viable ways of understanding a theory should come as no surprise. Philosophers often turn to super-empirical virtues to provide new grounds to single out a preferred reading. Now, the standard non-perturbative picture offers an transparent path to the spin-2 equations of motion/actions; the converse, as we have seen with all the involved ambiguities, is not equally true (section 2.1). Secondly, the standard non-perturbative picture has a wider scope since it requires only very weak restrictions on the background manifold (section 2.2). If at all, all perturbative pictures taken together have the same explanatory scope as the non-perturbative picture alone. But more than that: the myriad possible perturbations are even explained by assuming a non-perturbative picture as fundamental.

But what if one takes into account the special status of flat spacetime in quantum theory? Admittedly special relativistic field theory is a powerful framework. One might object then that that spin-2 theory cannot dynamically reduce GR should really not bother us; rather, what should be stressed is that it is the (specific) perturbative picture of flat spin-2 theory (and only that) which is continuous with all of our other best other physical theories. The flat-spacetime-based perturbative picture offers prospects of a unified account of all field theories within one framework. This kind of unificationist argument is much less appealing now than in the 1950s, however: quantized perturbative gravity is generally seen as a non-renormalizable theory while it is non-perturbatively quantized GR—so exactly not the quantization of gravity around flat spacetime—that is generally expected to be UV complete (for an overview, see Crowther and Linnemann Reference Crowther and Linnemann2019). ^{Footnote 10} More precisely, the general expectation within the quantum gravity community seems by now to be that the (Minkowski-based) spin-2 view is, again, only one out of a myriad complementary effective field theory views on general relativity—including the de Sitter- and anti-de Sitter-based spin-2 views (Huggett and Wüthrich Reference Huggett and Wüthrichforthcoming, chapter 9) but more generally perturbative quantizations around arbitrary background spacetimes. ^{Footnote 11} Unification with quantum field theory seems simply to get immediately trumped by renormalizable quantization, the posit that gravity should lead to a renormalizable quantum theory—an explanatory standard with more relevance than unification for its own sake.