Here, I’ll first (Sect. 4.1) lay out Read’s functionalist approach to gravitational energy. Its logical structure will be made explicit. Subsequently, (Sect. 4.2) I’ll critically examine three of its crucial premises. I reject them for multiple reasons. Notwithstanding my sympathies to his overall functional approach, and to the Dennettian ontological framework, I conclude that Read’s realism should be dismissed.
Functional Gravitational Energy
Here, I’ll expound Read’s realism about gravitational energy–stress (Read 2018, Sects. 3.3.2, 3.3.3), and the logical structure of his argument for it. Read proposes to embrace the background relativity of gravitational stress–energy (in the sense of Sect. 3.3). As this background-relative notion is both useful, and satisfies the functional role of gravitational, according to Read, we should be realists about it.
By “background” Read (and Lam, see below) mean (asymptotic) symmetries, encapsulated in asymptotic Killing fields, and suitable fall-off conditions, both implemented via asymptotic flatness. Lam and Read suggest that one should regard local and global gravitational and total energy as quantities well-defined relative to this background.
Let’s unravel his reasoning in more detail. Read picks up an earlier intimation by Lam (2011): On the one hand, “[…] within [GR] all meaningful notions of (gravitational and nongravitational) energy–momentum […] require the introduction of some background structures” (p. 1023); on the other hand, if these structures are present, genuine gravitational and non-gravitational energy exists: “they make only sense in particular (but very useful) settings” (ibid.).
Read’s realism, (REALLOC) & (REALGLOB), can now be cashed out as positive, principled answers to the following two questions (p. 19):
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(a)
Does the pseudotensor \(\vartheta_{a}^{ b}\) in (REALLOC) and its associated integral (“charge”) in (REALGLOB) represent anything real? Are these formal terms grounded in physical (but not necessarily fundamental) quantities?
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(b)
Suppose a positive answer to (a). Are we then licenced to identify the quantities that \(\vartheta_{a}^{ b}\) and its associated charge represent as gravitational energy–stress? “(I)s it correct to call the quantity appearing in [the continuity equation of (REALLOC) and its integral form in (REALGLOB)] […] ‘gravitational stress–energy’”?
The questions in (a) require a reality criterion. Echoing Lam, Read appeals to the explanatory and predictive utility of the gravitational pseudotensor and its associated charge: “[…] (they) are only well defined in a certain subset of [dynamically possible models, DPMs] of GR”; (n)evertheless, in such instances it is extremely useful to make use of this term, within that subclass of DPMs. Hence, at a practical level, it is legitimate to call such a quantity gravitational stress–energy.”
This is an instance of the following principle for realist commitment towards a theoretical, higher-level concept Q (cf. Dennett 1991a; Ladyman and Ross 2007, esp. Ch. 4)—what Wallace (2012, Ch. 2) dubs “Dennett’s Criterion”:
(DC)
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Whenever Q is definable and explanatorily or predictively useful, it captures a real structure (“real pattern”) in the world.
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Real patterns are higher-level structures: They are formulated in non-fundamental terms. (Think of molecules and their shapes as treated in chemistry. A satisfactory fundamental account isn’t available at present (see Hettema 2012 for the chemical case). Of course, this doesn’t imply that real patterns are “strongly autonomous” (Fodor), i.e. unrelated to the most fundamental level.)
To complete his affirmative answer to (a), Read needs to assume that the quantities conventionally labelled “(formal) gravitational energy”, gravEf,Footnote 14 indeed satisfy (DC):
(DC)[gravEf]
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For certain DPMs, gravEf is definable and explanatorily/predictively useful.
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It now follows from (DC) that gravEf captures a real pattern in the world (“is real”):
$$\left( {DC} \right) \& \left( {DC} \right)\left[ {{\text{gravE}}_{f} } \right] \to {\text{gravE}}_{f} {\text{is real}}.$$
Having established the reality of formal gravitational energy, Read’s next step is to affirm (b): The real pattern gravEf captures should be identified as genuine gravitational energy–stress; it represents gravitational energy–stress also in a substantive, physical sense.
Read’s rationale encompasses three elements: a general functionalist principle for characterising quantities, a particular functional profile for genuine gravitational energy–stress, and the premise that gravEf exhibits this profile.
Read deploys what he terms a “functionalist” (p. 20) general strategy: “In our view, it is plausible to maintain that in situations such as those in which [the integral conservation law] holds, there exists a quantity in GR that fulfils the functional role of gravitational stress–energy” (pp. 19).
That is, Read adopts the following “functionalism about gravitational energy–stress”:
(FUNCgravE)
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For a quantity Q to be (represent, “\(\doteq\)”) genuine gravitational energy–stress is to exhibit a certain profile \({\mathcal{F}}\left( {\text{gravE}} \right)\) of functional roles:
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\(\left( {{\mathcal{F}}\left( {\text{gravE}} \right)} \right)\left[ Q \right] \Leftrightarrow Q \doteq {\text{gravE}}\)
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How to flesh out the functional profile of gravitational energy–stress, \({\mathcal{F}}\left( {\text{gravE}} \right)?\) Read determines it to comprise two functional roles:
(\({\mathcal{F}}\left( {\text{gravE}} \right)\))
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(i)
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balancing the non-gravitational energy such that the sum is conserved
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&
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(ii)
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“(bearing) some relation to the ‘gravitational’ degrees of freedom in the theory in question” (p. 20).
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To complete his argument, a final premise is needed—viz. that gravEf plays the preceding two functional roles:
(\({\mathcal{F}}\left( {\text{gravE}} \right)\))[gravEf]
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gravEf instantiates the profile (\({\mathcal{F}}\left( {\text{gravE}} \right)\)).
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By construction, gravEf obeys a (formal) balance equation. Hence, (i) is satisfied. Likewise, (ii) looks harmless: It’s customary (e.g. Misner et al. 1973, passim) to identify the metric with the gravitational degrees of freedom (the “gravitational field”); gravEf is directly and solely built from it.
From the conjunction of (FUNCgravE) and (\({\mathcal{F}}\left( {\text{gravE}} \right)\)) now follows that gravEf earns the label “gravitational energy”. It represents genuine gravitational energy–stress:
$$\left( {FUNC_{\text{gravE}} } \right) \& \left( {{\mathcal{F}}\left( {\text{gravE}} \right)} \right) \& \left( {{\mathcal{F}}\left( {\text{gravE}} \right)} \right)\left[ {{\text{gravE}}_{f} } \right] \to {\text{gravE}}_{f} \doteq {\text{gravE}} .$$
In summary, Read has thus given a formally valid argument for (REALLOC) & (REALGLOB). Based on the alleged expedience of the gravitational pseudotensor and its associated charge, Read argued for a realist stance towards them. Furthermore, meeting his functional desiderata of gravitational energy, they indeed represent, on his proposal, gravitational energy–stress.
What to make of Read’s proposal? Is the appeal to functionalism convincing? Does gravitational energy–stress in GR really satisfy the functional roles, stipulated by Read? Does his proposal overcome the difficulties that undergird Hoefer’s eliminativism (Sect. 3.3)? To these questions we now turn.
Objections
In this subsection, I’ll evaluate Read’s realism about gravitational energy. Apart from Dennett’s Criterion (DC), and the fact that the formal notions of gravitational energy play the two functional roles stipulated by \(\left( {{\mathcal{F}}\left( {\text{gravE}} \right)} \right)\left[ {{\text{gravE}}_{f} } \right]\), I’ll question each assumption in his reasoning sketched above.
I’ll discuss each premise separately and in increasing order of generality: (\({\mathcal{F}}\left( {\text{gravE}} \right)\)), (FUNCgravE) and (DC)[gravEf].
Is Read’s functional characterisation of gravitational energy–stress adequate?
Consider first Read’s functional profile of gravitational energy–stress, i.e. (\({\mathcal{F}}\left( {\text{gravE}} \right)\)): Are the functional roles of gravitational energy–stress adequately characterised by (i) and (ii)? I dispute that: They are neither jointly sufficient nor necessary.
Two facts cast doubt upon the view that (i) and (ii) are jointly sufficient: the triviality of continuity equations, and ambiguity, respectively.
Firstly, formal continuity equations are too easily procurable (Goldberg 1958, p. 17). For any symmetric quantity \(\gamma^{\mu \nu }\), one can always construct a symmetric quantity \(\Gamma ^{\mu \nu }\) that satisfies continuity equation \(\partial_{\nu } \left( {\sqrt {\left| g \right|} T^{\mu \nu } +\Gamma ^{\mu \nu } } \right) = 0\)—viz. \(\Gamma ^{\mu \nu } : = \partial_{\varrho ,\sigma } \left( {\gamma^{\mu \nu } \gamma^{\varrho \sigma } - \gamma^{\nu \varrho } \gamma^{\mu \sigma } } \right) - \sqrt {\left| g \right|} T^{\mu \nu }\). (Recall that that the energy–stress tensor \(T^{\mu \nu }\) also depends on the metric, cf. Lehmkuhl 2011). If one now chooses for \(\gamma^{\mu \nu }\) some arbitrary function, e.g. \(\gamma^{\mu \nu } = \sin \left( R \right)R^{\mu \nu }\), one obtains a quantity that satisfies (i) and (ii). Nonetheless, one would hesitate to ascribe it physical significance as a candidate gravitational energy.
Read may demur at continuity equations thus constructed as they hold irrespective of any field equations (and furthermore that they also depend on the matter degrees of freedom). They are indeed mathematical identities. Read might parry by supplementing (i) with a proviso: the conservation law not be a mathematical identity (and not directly depend on the matter degrees of freedom).Footnote 15
This doesn’t alleviate the above worry, though: The previous argument can just be rehashed for \({\tilde{\varGamma }}^{\mu \nu } : = \partial_{\varrho ,\sigma } \left( {\gamma^{\mu \nu } \gamma^{\varrho \sigma } - \gamma^{\nu \sigma } \gamma^{\mu \sigma } } \right) - \frac{1}{\kappa }\sqrt {\left| g \right|} G^{\mu \nu } .\) The continuity equation continues to hold—but now in virtue of the Einstein Equations.
Another problem arises from ambiguity. Recall from Sect. 2.3: There exist infinitely many pseudotensors satisfying a local continuity equation. All are built solely from the metric. One needn’t even restrict oneself to pseudotensors. Nothing in Read’s proposal seems to prevent one from introducing e.g. additional flat background metrics, an orthonormal tetrad or a flat connection (Pitts 2011b for a survey of such options). Ditto quasi-local notions (see e.g. Szabados 2009).Footnote 16 Objects with the functional profile \({\mathcal{F}}\left( {\text{gravE}} \right)\) abound.
Unless their mutual consistency can be established, this proliferation of candidate objects that satisfy \({\mathcal{F}}\left( {\text{gravE}} \right)\) should unsettle Read. (Recall our discussion of (R2) in Sect. 3.2.) I therefore conclude: (i)&(ii) is an insufficient characterisation of the functional profile of gravitational energy.
Further scepticism about the functional roles of \({\mathcal{F}}\left( {\text{gravE}} \right)\) is in order. 1. Conserved quantities are contingent on symmetries. Hence, criterion (i) isn’t necessary. 2. Criterion (ii) is bedevilled by general fuzziness, as well as equivocation about the gravitational degrees of freedom.
I’ll first argue that (i) imparts a spurious essentiality to a contingent feature of our most familiar spacetime settings.
Underlying Read’s stipulation is the intuition that total energy should be conserved. This intuition stems from our habituation to classical theories in flat spacetime (cf. Nerlich 1991). Why expect this to carry over to GR?
The principal motivation stems from the Noether theorems. They establish a general correlation between symmetries of the action and conserved quantities (see e.g. Brading and Brown 2000). Due to its general covariance, GR’s action has infinitely many rigid symmetries (see Bergmann 1949, 1958; Brown and Brading 2002; Brading 2005). The Noether Theorems then guarantee, at least formally, infinitely many conservation laws of the pseudotensorial type. To take these formal infinitely many conservation laws seriously, i.e. to regard them as also physically meaningful, leads us back to Pitts’ proposal. Whether it deserves realism, remains controversial, as we saw.
One source of reservations about the infinitely many conservation laws may derive from GR’s general covariance. Because of the latter, they belong to so-called “improper conservation laws” (Hilbert). These arise from Noether’s theorems for all theories with local symmetry group that have a global subgroup (see e.g. Bergmann 1949; Brading and Brown 2000). Their interpretation and physical significance—as Hilbert’s label intimates—is subtle: Under certain circumstances, they seem to be (at least, individuallyFootnote 17) trivial, i.e. mathematical identities (see e.g. Brading 2005; Sus 2017), and hence devoid of physical content. What those circumstances exactly are, is a question of current dispute (closely related to the empirical significance of symmetries, see e.g. Kosso 2000; Brading and Brown 2004; Wallace and Greaves 2014; Teh 2015; Murgueitio Ramirez 2019). On a recent proposal (Barnich and Brandt 2002; Sus 2017), GR’s improper energy conservation laws can be salvaged from triviality, if the dynamically possible spacetime models considered possess (asymptotic) symmetries. Whether in our world we should take these infinitely many conservation laws seriously, thus depends on whether we should believe that our world instantiates such asymptotic background structure. And indeed, I’ll argue below that one should—however, the asymptotic structure is that of a de Sitter space. But that entails two problems. The first is that the integrals of the pseudotensor-based continuity equations diverge. Thereby, the conserved global/integral charges aren’t well-defined. But with the symmetries of de Sitter space, also the motivation for a local/differential conservation law, based on pseudotensors, becomes moot: Using the the associated so-called Killing vectors (see e.g. Read 2018, Sect. 2.4), one can define bona fide (covariant) matter energy–stress fluxes that are covariantly conserved—with no (overt) gravitational contributions (Duerr 2018a, Sect. 2).
The connection with Killing vectors can be developed further along a different direction. Unless the spacetime possesses symmetries (to which special coordinates could be adapted (see e.g. Pooley 2017, sect.), coordinates that would be able to single out pseudotensor-based continuity equations) the pseudotensorial conservation laws thus seem to lack intrinsic meaning. But such spacetime symmetries are contingent: Generic spacetimes lack them; even most do. Why, therefore, cling to energy conservation as a default? It seems more natural to reverse the familiar explanatory asymmetry: Energy conservation, not its failure, needs explanation—in terms of a spacetime’s special symmetries (see Carroll 2010 for a slightly brutal way of putting it; cf. Duerr 2018a, Sect. 2).Footnote 18
Of course, one might resist this whole reasoning by pointing to the mathematical fact that, due to general covariance, GR’s action has symmetries. But as mentioned before, it’s unclear that this, by itself, warrants wider-reaching physical conclusions. (Also bear in mind that that most will hesitate to regard an action as more than a merely auxiliary construct—not a physical quantity. Hence, inferences from its properties to properties of physical systems must be handled with care.)
Let’s move on to Read’s second functional characteristic of gravitational energy, (ii). It can be opposed for two reasons. One is its vagueness: What exactly is the relation that should hold between a candidate for gravitational energy–stress and the gravitational field?
A second worry is more subtle: What are the gravitational degrees of freedom—the “gravitational field”?Footnote 19 Which quantity represents them, e.g. the metric \(g_{\mu \nu }\), the connection coefficients \(\Gamma _{\mu \nu }^{\lambda }\) (Einstein’s choice, see Lehmkuhl 2014), the Riemann tensor (Synge’s choice, Synge 1960), or the deviation from flatness \(g_{\mu \nu } - \eta_{\mu \nu }\) (Pooley’s choice, Pooley 2013, fn. 20)?Footnote 20 Each choice has some merits in its favour (Lehmkuhl 2008). Read rightly cautions against any premature a priori preference for one.
Yet, it’s not obvious that his second functional role for gravitational energy, (ii), can avoid an a priori choice. The pseudotensors in (REALLOC) are the canonical energy-momenta associated with the metric as the gravitational field.
Suppose, however, that we identify the connection coefficients \({{\varGamma }}_{\beta \gamma }^{\alpha }\) as the gravitational field. Then, the associated canonical energy–stress is the Palatini-pseudotensor
$$\vartheta_{\mu }^{ \nu } \left[\Gamma \right] = \frac{{\partial {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\mathfrak{L}} }}}{{\partial \left( {\partial_{\nu }\Gamma _{\beta \gamma }^{\alpha } } \right)}}\partial_{\mu }\Gamma _{\beta \gamma }^{\alpha } - \delta_{\mu }^{\nu } {\hat{\mathfrak{L}}},$$
where \({\hat{\mathfrak{L}}} = {\hat{\mathfrak{L}}}\left( {g_{\mu \nu } ,\Gamma _{\mu \nu }^{\lambda } ,\partial_{\kappa }\Gamma _{\mu \nu }^{\lambda } } \right)\) is the (full) Einstein–Hilbert Lagrangian as a functional of the metric, the connection coefficients, and their first derivatives.Footnote 21 (Note that it satisfies the continuity equation: \(\partial_{\nu } \left( {\sqrt {\left| g \right|} \left( {\vartheta_{\mu }^{ \nu } \left[\Gamma \right] + T_{\mu }^{\nu } } \right)} \right) = 0\) with the standard energy–stress tensor \(T_{\mu }^{\nu }\). Again, \(\vartheta_{\mu }^{ \nu } \left[\Gamma \right]\) is determined only up to a superpotential term.)
Being 2nd order in the metric compatible with the connection, \(\vartheta_{\mu }^{ \nu } \left[ {{\varGamma }} \right]\) manifestly differs from the Einstein or Landau–Lifshitz pseudotensor, which Hoefer, Lam and Read are considering. What is more, its properties are physically implausible: For instance, it yields divergent integrals for radiating systems (see Murphy 1990 for details).
One may object: By comparing the Einstein pseudotensor and the Palatini pseudotensor, aren’t we comparing apples and oranges? The Palatini pseudotensor \(\vartheta_{\mu }^{ \nu } \left[\Gamma \right]\) is based on the full Einstein–Hilbert Lagrangian—not (as is the Einstein pseudotensor) on the truncated, “\({\varGamma \varGamma }^{\prime \prime }\) Lagrangian \({\bar{\mathcal{L}}}_{\left( g \right)}\). If one determines the corresponding Einstein pseudotensor for the full Einstein–Hilbert Lagrangian, both expressions coincide (Novotný 1993).
Prima facie, this is satisfying (and a remarkable property of the Einstein–Hilbert Lagrangian!). Nonetheless, it spells a dilemma for Read. One horn is that Read’s criterion seems incomplete: It can’t decide between the Palatini pseudotensor and the metric-based pseudotensors. If then, in light of the above considerations, one rules out the former, one thereby has to identify the metric as the gravitational field. (Prima facie this isn’t implausible: It surely plays a privileged role. For instance, one cannot write down matter coupling to gravity locally using only a connection. One also needs the metric or something equivalent.Footnote 22) But even so, the metric’s special status doesn’t by itself justify its elevation to the gravitational field—as Read himself admits.Footnote 23
The worry about the right identification of the gravitational field is even more general: Why assume that in GR there exists an ambiguously identifiable gravitational field to begin with? It’s not implausible that no choice for the gravitational field is ultimately unique across different contexts (Rey 2013).
In short, Read’s second functional role, (ii), on pain of incompleteness, cannot remain neutral on the identification of a gravitational field—against his express intentions.
Is a functionalist strategy appropriate for gravitational energy–stress?
Now turn to (FUNCgravE): Why appeal to functionalism in the specific context of gravitational energy in GR? I’ll launch two lines of attack against it: First, I’ll rebut Read’s explicit argument for it; secondly, I’ll rehearse the reasons that motivate functionalism in the philosophy of mind, and try to ascertain their analogues.
First, let’s examine Read’s own argument for a functionalist stance towards gravitational energy. I reject it as unfounded.
What primarily bolsters (FUNCgravE) for Read is the sterility of its negation: “[…] the alternative to functionalism is to say that ‘the structure of certain DPMs of GR is such that it appears that there exists gravitational stress–energy in those models, but really there is no such stress–energy there’; the payoff to be gained from making such a claim is unclear” (p. 20). In particular, he cautions that without (FUNCgravE), one may be barred from potentially more perspicuous avenues for explaining some gravitational phenomena, e.g. binary systems.
I concur with Read on the infertility of dogmatically boycotting higher-level explanations from the outset. Sundry examples from non-gravitational physics attest to that (e.g. Falkenburg 2015; Knox 2016; 2017; Knox and Franklin 2018). Yet, the use of higher-level concepts doesn’t per se imply functionalism.Footnote 24 The latter is a specific thesis about the meaning and/or the ontological nature of certain quantities (depending on the strain of functionalism, see below). The purported explanatory pay-off of recourse to gravitational energy–stress as a non-fundamental explanans doesn’t per se warrant functionalism about gravitational energy–stress.
Moreover, not even the explanatory pay-off of gravitational energy as a higher-level explanans is obvious. Read concedes that appeal to gravitational energy–stress isn’t necessary: “one could indeed explain all general relativistic phenomena, in any model of the theory, simply using the apparatus used to pick out the DPMs of the theory” (p. 20).Footnote 25 The existence of two alternative explanations prompts the question: Which of the two achieves the pay-off that Read extolls? (Contrast this with the case of quasi-particles. Fundamentally, they are collective excitations in a solid. In some regards, they behave like particles. A bottom-up, statistical mechanical treatment would require utopian computational power: We’d have to solve typically ~ 1023 coupled differential equations. The pay-off of the higher-level description is manifest.)
What about binary stars, which Read adduces as an example? The case isn’t as clear-cut as Read suggests. GR predicts that two stars revolving each other emit gravitational radiation, and increase their orbital frequency. With marvellous accuracy, this has been confirmed (e.g. Will 2014). In line with Read’s claim, the standard account indeed involves gravitational (wave) energy as an explanans (cf. e.g. Hobson et al. 2006, Ch. 18): The gravitational wave is supposed to carry away the binary system’s total (kinetic plus gravitational) energy; as a result, the stars’ orbital frequency increases, with the stars spiralling in towards each other.
In a recent detailed analysis, however, Duerr (2018b) compares this standard interpretation of the binary stars to the alternative without gravitational (wave) energy which Read adumbrates. The latter is found to trump the former on the four explanatory virtues of parsimony, scope, depth, and unificatory power. At least pro tempore, this diminishes the force of Read’s argument, or even shifts the burden of proof upon Read’s shoulders.Footnote 26
Two caveats are in order. First, examples might eventually be found in favour of Read’s claim (e.g. in a similar analysis of instabilities in rotating neutron stars, induced by gravitational radiation, see e.g. Schutz and Ricci 2010, Sect. 6.2).Footnote 27 But for the dialectic of the debate to progress, detailed case-studies of such examples are needed. At the moment, they aren’t available. Secondly, some of the persuasiveness of Duerr’s (2018a, b) arguments depends on whether one shares his GR-exceptionalist creed (see Sect. 1). But Read gives no explicit reasons for or against it.
My second line of attack against (FUNCgravE) adverts to the motivation for functionalism in the philosophy of mind. I submit, it doesn’t carry over to the case at hand.
The functionalism, which Read (via Wallace) imports into the philosophy of physics, stems from the philosophy of mind (see e.g. Van Gulick 2009; Levin 2013; Braddon-Mitchell and Jackson 2007, Part I, II, IV). It’s mainly motivated by two difficulties: the non-intersubjectivity of mental states, and the identity theory’s failure to account for multi-realisability, respectively.
The first is a general and epistemological point: We can’t directly know other people’s mental states. They defy inter-subjectivity: A tooth-ache is inherently “private”. At best, we can infer mental states indirectly from external indicators (screams, tears, etc.). If thus we want to attribute mental states to other people, prima facie we have to postulate them as entities whose intrinsic nature is elusive. (Mental states might—at best—be accessible introspectively.Footnote 28) It’s sound philosophical advice to strive to minimise the gap between our speculations about the world and our knowledge. How then to accommodate for mental states?
A second motivation for functionalism arises from a shortcoming of the preceding identity theory. According to the latter, mental states (or properties) are identical with physical states (or properties). Mental states are multiply realisable: It seems unduly chauvinistic to decree apriori that organisms can’t be ascribed the same (or sufficiently similar) mental states, despite neuroanatomical and neurophysiological differences. Why shouldn’t, say, Read and an octopus both be able—at least in principle—to experience pain and pleasure? But on the identity theory it remains mysterious, how two intrinsically sufficiently different brain states can be identical with the same mental state.
Both difficulties can be eschewed by characterising mental states not via intrinsic properties of brain states, but via their function: They are individuated by the structural roles they play in a (neuronal) network.
Do these two motivations have counterparts for the case of gravitational energy–stress in GR? Three disanalogies speak against it: its absence in the manifest image, its non-privacy and absence of multi-realisability.
First, on the one hand, gravitational energy–stress—unlike mental states—isn’t an empirical phenomenon that needs to be accounted for. On the other hand, unlike (say) belief states, even as a theoretical concept, gravitational energy scarcely counts as a robust folk-theoretic notion in our manifest image that an adequate scientific theory in one way or the other must save.Footnote 29 Read himself acknowledges that it’s—at least conceivably—dispensable.
Secondly, being a physical quantity, gravitational energy doesn’t suffer from the privacy of mental states: Nobody is endowed with a privileged introspective access to gravitational energy–stress, opaque to lesser mortals.
A less quirky sense of “privacy” in this context takes its cue from Dennett (1991a, b, cf. Ladyman and Ross 2007, pp. 161).Footnote 30 For him, it’s typical of real patterns to become visible only on higher-levels of description. On the fundamental level, one may lose their salience out of sight: One doesn’t see the wood for the trees. (This is the sense in which the higher-level explanations, discussed by Knox (2016, 2017) and Knox and Franklin (2018) reveal the salient features, otherwise opaque on the microphysical level.)
Is gravitational energy “private” in this sense? Can it only be properly understood on the coarse-grained, higher-level which Read’s functionalist perspective envisions? That, too, I impugn. Formal notions of gravitational energy aren’t higher-level concepts in the relevant sense: They are non-fundamental in that they are only definable in certain subclass of models. Again, the motivation from “privacy” founders.
Thirdly, multi-realisability has no obvious counterpart. Recall that it’s an inter-level relationship: It links higher-level and lower-level (more fundamental) entities. (FUNCgravE) presupposes that the functional profile of gravitational energy is supplied from gravitational theories other than GR.
The most straightforward such “reference theory” is Newtonian Gravity. GR reduces to it in the weak gravity limit.Footnote 31 Hence, the functional role would be fixed by GR itself in a particular regime. (One may already ponder: Isn’t it ad-hoc to accord an ontological privilege to this particular regime? The more modest goal of identifying rough-and-ready functional counterparts of quantities in antecedent theories is, of course, harmless. See below.) Suppose now that in another regime, GR exhibits some structural similarity to the weak-field regime. This similarity doesn’t constitute an inter-level relationship of the kind required for multi-realisability. It doesn’t link a fundamental and a less fundamental level of description. Rather it’s an intra-level relationship. The same applies to different reference theories, say massive graviton gravity.Footnote 32 Both GR and it vie for providing the best description of the same domain. They operate on the same ontological level. Again, we aren’t dealing with multi-realisability.
It’s terminological confusion to say that GR “instantiates” or “realises” some quantity, defined in massive graviton gravity. Of course, one could meaningfully ask: “What structures of a GR spacetime are the (rough) analogues or counterparts of some quantity in massive graviton gravity?”Footnote 33 But gleaning structural similarities is ontologically much less ambitious than Read’s realism.Footnote 34
In short: The two main motivations for functionalism in the philosophy of mind—non-intersubjectivity of mental states and multi-realisability—lapse for gravitational energy–stress. This corrodes any tangible motivation for (FUNCgravE).
This conclusion calls for qualification. Plausibly, the above motivations are (individually) sufficient conditions for the application of functionalism. I don’t claim that they are necessary. But to my knowledge, there aren’t any other motivations for functionalism in the literature. Hence, it seems not unfair to request of Read a justification of his functionalist strategy, should it be motivated “non-standardly”.
Is Dennett’s Criterion really satisfied?
Let’s eventually revert to Read’s reality criterion, (DC). To decide whether the formal concepts of gravitational energy, \({\text{gravE}}_{f}\), capture real structures, Read employs Dennett’s reality criterion (DC): If a higher-level quantity is well-defined and explanatorily/predictively useful, it merits realist commitment. Are the antecedent conditions really satisfied?
Let’s hark back to the main finding of our more careful exegesis of Hoefer’s first objection, (H1), in Sect. 3: Realism about local and global pseudotensorial gravitational energy–stress, (REALLOC) and (REALGLOB), is obstructed both by the pseudotensor’s ambiguity/non-uniqueness and vicious coordinate-dependence (unless Read’s position collapses onto Pitts’—a position for which Read gives no independent arguments). Read’s responses to Hoefer were seen to be either ineffective or incomplete. His functional approach discussed added nothing relevant as regards these problems: The first antecedent condition of (DC) isn’t satisfied: Gravitational energy–stress isn’t well-defined (except for Pitts’ object).
What about the other condition—explanatory utility? Read still owes us an argument, or full-fledged example, for why gravitational energy–stress is a powerful explanans.Footnote 35 To my mind, this can only be satisfactorily gauged through detailed case studies (e.g. of energy extraction processes in Black Holes, see e.g. Geroch 1973). Nonetheless, a strong argument for eliminativism can already be made, turning on a wide range of astrophysical and cosmological phenomena.
Recall that Read’s (REALGLOB) hinges on realism about asymptotic flatness. In Sect. 3.3, I suggested that the disagreement between Hoefer and Read over the acceptability of asymptotic flatness best be understood as a disagreement between different classifications: Whereas for Read asymptotic flatness is a good approximation, for Hoefer it’s an idle posit in an idealisation. The bone of contention is therefore: Do the salient features of empirically confirmed asymptotically flat models successfully refer? Primarily in light of contemporary cosmology, I contend, they don’t.
On the one hand, many spacetimes utilised for modelling the exterior of stationary astrophysical objects are indeed asymptotically flat. Apart from the Schwarzschild metric, the the Kerr-Newmann solution for the exterior of a rotating, charged black hole is a case in point (cf. Reiris 2014 for a proof of a large class of spacetimes). But unfortunately, no interior solution for the (uncharged) Kerr metric is known whose source is a perfect fluid—the simplest model for a star.
This may merely be regrettable (perhaps even a temporary issue). But more generally, Christodoulou and Klainerman (1993, p. 10) warn: “[…] (I)t remains questionable whether there exists any nontrivial (non-stationary) solution of the field equations that satisfies the Penrose requirements [i.e. the geometric conditions encoding asymptotic flatness]. Indeed, his regularity assumptions translate into fall-off conditions of the curvature that may be too stringent and thus may fail to be satisfied by any solution that would allow gravitational waves. Moreover, the picture given by the conformal compactification fails to address the crucial issue of the relationship between the conditions in the past and in the behaviour in the future.”
The only known non-stationary, asymptotically flat solutions (e.g. within the Robinson-Trautman class of metrics describing expanding gravitational waves) are marred by singularities. This threatens their physicality.
There are two responses to this. One is that singularities may not be as calamitous as orthodoxy (e.g. Earman 1995, p. 12) has it (Curiel and Bokulich 2009, Sect. 2; Lehmkuhl 2017). Another reaction points to approximate solutions based on perturbative methods. Via them one can determine the spacetime of, say, an in-spiralling compact binary system, yielding a spacetime that is non-stationary and asymptotically flat.
This leads us to the major objection to asymptotic flatness as an approximation—cosmology. Prior to that, though, let’s briefly dwell on the perturbative approximation schemes featuring in the treatment of binary systems. In a nutshell (see e.g. Maggiore 2007, Ch. 5; Poisson and Will 2014 for details), in the astrophysical system’s neighbourhood, one employs the so-called Post-Newtonian approximation scheme—an expansion in powers of a small parameter (\(1/c^{2}\))—to determine the system’s near field. But this expansion in the near-zone expansion is a singular perturbation theory: For distances tending to infinity, higher-order terms blow up; the Post-Newtonian scheme isn’t uniformly valid for all distances. In particular, it cannot incorporate the no-incoming radiation boundary conditions, apt for gravitationally radiating objects. One therefore adopts a different approximation scheme for the so-called “far-field zone”. In the intermediate region, both expansions are then smoothly glued together (“matched asymptotic expansion”). Which boundary conditions to impose for the far-field zone? A standard choice is asymptotic flatness.
Here lies the principal reason for classifying asymptotic flatness as an idealisation: According to today’s best cosmological model, we live in an FLRW universe with a positive cosmological constant \({{\varLambda }}\). It leads to infinite (albeit ever slower) expansion in our universe’s long-term future: Our universe is asymptotically deSitter; it’s not asymptotically flat (see e.g. Carroll 2003; Rubin and Hayden 2016 for details).
Already for the exterior of the simplest, i.e. spherically symmetric star model, immersed in a deSitter spacetime, asymptotic flatness breaks down. Does this vitiate all—well-confirmed!—calculations based on an asymptotically flat far-field? Luckily—no: Far from the source, but still much closer than cosmological scales, spacetime is approximately flat—for all practical purposes. So, the usual techniques apply—as long as one doesn’t venture too “far out” in space and time (Ashtekhar et al. 2016; Bonga and Hazboun 2017).
Asymptotic flatness is therefore an idealised extrapolation of the ambient spacetime at a particular phase of a star’s life: One ignores its future beyond a certain point, prescinding from the star’s cosmic embedding. The referents of asymptotically flat spacetimes are therefore ahistorical fictional objects. The practising physicist uses them as convenient surrogates for the real target objects, e.g. a pulsar, a galaxy, etc., because they share with the latter the relevant structural features up to cosmological scales. (It’s this omission of actual history that physicists mean, when taking asymptotic flatness to characterise isolated systems. An object in an asymptotically flat spacetime is dynamically isolated in the sense that it quiesces into a stationary state.Footnote 36) Asymptotically flat spacetimes thus are idealisations. Even when successful, they describe surrogate systems, distinct (with respect to their past or future evolution) from remotely physical ones.
More importantly, the working posits of successful asymptotically flat models aren’t their fall-off behaviour at infinity. Rather, they are the right fall-off behaviour up to cosmological scales: All empirical content is garnered from the properties of a finite patch of an asymptotically flat spacetime. But it’s, of course, the behaviour at infinity that is salient of asymptotic flatness. Asymptotic flatness is therefore an idle posit. Recourse to (DC) is thus blocked.
In short: Gravitational energy in Read’s proposal contravenes both conditions of Dennett’s Criterion. Owing to its coordinate-dependence and ambiguity, local and global gravitational energy is ill-defined (unless Read’s position collapses onto Pitts’, for which then he should argue explicitly). Moreover, asymptotic flatness is an idle posit. Hence, it doesn’t yield the explanatory mileage that a realist would urge.
I conclude that Read’s argument for a realism about pseudotensor-based global and local gravitational energy fails. In consequence, vis-à-vis Read’s proposal, Hoefer’s alternative seems preferable. It cuts the Gordian knot: We should indeed be eliminativists about gravitational energy, and recognise that in GR, energy just ceases to be conserved as a default (see Schroedinger 1950, p. 105 for a “singularly striking example”, cf. Misner et al. 1973, Sect. 19.4).