Cartography of the space of theories: an interpretational chart for fields that are both (dark) matter and spacetime
aa r X i v : . [ phy s i c s . h i s t - ph ] S e p Cartography of the space of theories: an interpretationalchart for fields that are both (dark) matter andspacetime
Niels C.M. Martens and Dennis Lehmkuhl DFG Research Unit “The Epistemology of the Large Hadron Collider” (grant FOR 2063) martens@ physik. rwth-aachen. de , dennis. lehmkuhl@ uni-bonn. de Lichtenberg Group for History and Philosophy of Physics, University of Bonn Institute for Theoretical Particle Physics and Cosmology, RWTH Aachen University
This is a postprint (i.e. post-peer-review but pre-copyedit) version of an article acceptedfor publication in
Studies in History and Philosophy of Modern Physics . The finalauthenticated version is/will be available online (open access) at: . September 10, 2020
Abstract
This paper pushes back against the Democritean-Newtonian tradition ofassuming a strict conceptual dichotomy between spacetime and matter. Ourapproach proceeds via the more narrow distinction between modified grav-ity/spacetime (MG) and dark matter (DM). A prequel paper argued that thenovel field Φ postulated by Berezhiani and Khoury’s ‘superfluid dark mattertheory’ is as much (dark) matter as anything could possibly be, but also—below the critical temperature for superfluidity—as much (of a modificationof) spacetime as anything could possibly be. Here we introduce and crit-ically evaluate three groups of interpretations that one should consider forsuch Janus-faced theories. The consubstantiality interpretation holds that Φis both (dark) matter and a modification of spacetime, analogously to thesense in which Jesus (according to catholicism) is both human and god. Thefundamendalist interpretations consider for each of these roles whether theyare instantiated fundamentally or emergently. The breakdown interpretationsfocus on the question of whether Φ signals the breakdown, in some sense to bespecified, of the MG-DM dichotomy and perhaps even the broader spacetime–matter distinction. More generally, it is argued that hybrid theories urge amove towards a single space of theories, rather than two separate spaces ofspacetime theories and matter theories, respectively. ontents Φ matters and spacetimes 74 Interpretations by analogy with theology 85 Fundamentalist interpretations 10 Democritus claimed that everything in our universe is ultimately reducibleto atoms (matter) or void (space). This strict conceptual and metaphysicaldichotomy between matter and space(time)—everything in our universe fitsinto exactly one of those categories—has reigned supreme ever since Newtonrevived it. Recently, several approaches to the problem of quantum gravityhave put pressure on these two categories and the strict dichotomy betweenthem. There is however reason to worry long before reaching the region wherequantum gravity becomes relevant. (Moreover, focusing on these cases has theadditional benefit of staying closer to experiment/observation.) How to cate-gorise the metric in plain old general relativity (GR) is already controversial,as it exhibits both properties that are standardly associated with spacetimeas well as properties that are standardly associated with matter [1–19] [20,p.354] [21, Ch.9] [22, § §
8] [24, p.36]. We contend that the pressurebecomes even more acute when considering certain theories featuring in thedebate between dark matter (DM) and modified gravity/spacetime (MG). n particular, the prequel [19] to this second of two companion papersargued that the novel field Φ postulated by Berezhiani and Khoury’s ‘superfluiddark matter theory’ (SFDM) [25, 26] is as much (dark) matter as anythingcould possibly be, but also —below the critical temperature for superfluidity—as much (of a modification of) spacetime as anything could possibly be. Thisfirst result leaves open certain interpretational questions. For instance, does thefact that Φ instantiates the spacetime role only below the critical temperaturefor superfluidity imply that it is ‘really’ dark matter, acting (in the sub-critical-temperature regime) ‘as if’ it were a modification of gravity and spacetime?Or is the sense in which Φ is dark matter on a par with the sense in which it isa modification of gravity and spacetime? Does the result of the prequel paperput pressure on the contemporary importance, usefulness or even coherence ofthe dark matter/ modified gravity dichotomy and perhaps even on the broadermatter–spacetime distinction?It is the aim of this paper to unpack these and other interpretational ques-tions. After introducing ‘superfluid dark matter theory’ in more detail in sec-tion 2 and reiterating the conclusion of the prequel paper in section 3, we willintroduce and critically evaluate nine potential interpretations—distributedacross three groups—that we believe one should consider when attempting tounderstand theories with objects that do not all fall neatly into one of the twocategories (i.e. spacetime and matter). The goal is not so much to advocate asingle interpretation that is definitely to be preferred—we argue that four outof nine interpretations are currently viable for SFDM—but to put on the tablea chart of viable ways of understanding theories like SFDM and advantages ofand problems with each approach. This is partially because the theory is verynew (and, in particular, known to be incomplete, as will be discussed below),and partially because the difference between some interpretations might bebest fought out at the level of the space of theories rather than within a singletheory. This case study of SFDM serves as a bottom-up approach to the car-tography of this space of theories, by inspiring a chart of interpretations thatmay help us understand and navigate that large space. Section 4 introduces theconsubstantiality interpretation, which holds that Φ is both (dark) matter anda modification of spacetime, analogously to the sense in which Jesus (accord-ing to catholic doctrine) is both human and god. The (four) fundamendalistinterpretations (section 5) consider for each of these roles whether they are in-stantiated fundamentally or emergently. The (four) breakdown interpretations(section 6) focus on the question of whether Φ signals the breakdown, in somesense to be specified, of the MG-DM dichotomy and perhaps even the broaderspacetime–matter distinction. Section 7 discusses the relations between theseinterpretations, and reflects upon the lessons learned regarding the DM/MGdistinction, the broader spacetime–matter distinction, and the cartography ofthe space of theories. Note that this cartographic project differs from the related cartographic project by Slowik,who aims to provide a map of substantivalist and relationalist positions [27]. ‘Superfluid Dark Matter theory’ (SFDM) In order to make this paper somewhat self-contained and in order to be ableto refer back to the relevant Lagrangians, we almost verbatim reproduce theintroduction to ‘superfluid dark matter theory’ from the previous paper [19],followed by comments regarding its completeness. Readers familiar with theprequel paper may wish to skip ahead to the final paragraph of this section.Theories labeled as dark matter theories have traditionally done well atthe level of cosmology and galaxy clusters, but less so at the level of galaxies.The opposite is the case for theories labeled as modifications of gravity and/orspacetime. A promising approach to breaking this stalemate is to find a singlenovel entity for which there is a natural, physical, dynamical reason why itbehaves like DM on large scales and like MG on galactic scales. In this respectit is relevant to note that to mediate a long-range force in galaxies, a mass-less messenger (force carrier) is needed. A natural candidate presents itself:the quantised soundwaves of a superfluid, i.e. phonons, which are Goldstonebosons and thus massless. In the Standard Model of Particle Physics, matter(in the broad sense used in this paper) is divided into bosonic force carriersand fermionic matter (in a narrower sense of matter). But there is no rea-son why matter, in this narrow sense, could not also be bosonic, and it is notuncommon for new dark matter theories to postulate a bosonic dark matterfield. If the associated particles self-interact (repulsively), they can form asuperfluid Bose-Einstein condensate (BEC), which carries phonons. In otherwords, in this phase one cannot associate with the field a set of individual(nearly) collisionless particles; it is best described in terms of collective exci-tations. If these phonons cohere they can mediate a long-range force. If thephonons are described by the appropriate Lagrangian, they mediate a MOND-ian force. In order for the superfluid BEC phase to obtain, the De-Brogliewavelength of the Φ-particles needs to be larger than the mean interparticleseparation [25, p.5] [28, § U (1) symmetry, via the following term: L Φ = − (cid:0) | ∂ µ Φ | + m | Φ | (cid:1) − Λ c + | Φ | ) (cid:0) | ∂ µ Φ | + m | Φ | (cid:1) (1)with Λ c and Λ two constants introducing two mass/energy scales. (The main We are excluding the Higgs boson here, since, although it is a boson, it is not associated withone of the four fundamental forces. Λ c ensures that the theory admits a Φ = 0 vacuum [25, p.15]. ustification for this choice of Lagrangian comes from reverse engineering: it isthis Lagrangian that will ultimately reproduce MOND in the Galactic regime,as discussed below.)Spontaneous symmetry breaking of the global U (1) symmetry yields, in thenon-relativistic regime, the following Lagrangian for the associated Goldstonebosons—these massless phonons being represented by the scalar field θ , thephase of Φ = ρe i ( θ + mt ) : L T =0 , ¬ rel,θ = 2Λ(2 m ) / X p | X | , (2)with X ≡ ˙ θ − m Φ − ( ~ ∇ θ ) m , (3)where Φ is now interpreted as the external gravitational potential. (This iden-tification receives its justification when one derives this non-relativistic descrip-tion from the relativistic description, i.e. Eq.(4), with ˜ g = ˜ g SF DM as given byEq.(6).) To lowest order in the derivatives, superfluid phonons are in generaldescribed by a scalar field θ governed by a Lagrangian L = P ( X ), with X givenby Eq.(3) [25, p.3] [29]. The specific choice of P given by Eq.(2) uniquely de-termines the specific type of superfluid, namely one that interacts primarilythrough three-body interactions, i.e. with an equation of state P ∝ ρ .The interaction between the phonons and the regular (i.e. luminous) mat-ter fields is then added to this Lagrangian as an “empirical term” [25, p.8](as opposed to being derived from an interaction term in the fundamental La-grangian). In the relativistic regime we may describe this via the metric ˜ g µν ,sometimes referred to as the physical metric—why such metrics deserve thisname is discussed in [19, § g µν only indirectly, via thisphysical metric): L rel, int = L (˜ g µν , ψ α , ψ α ; µ | ˜ g ) (4)with ψ α the luminous-matter fields, and ψ α ; µ | ˜ g denoting the covariant derivativewith respect to ˜ g . The physical metric of SFDM is inspired by that of TeVeS.TeVeS, a theory usually referred to as a modification of gravity, postulatestwo new dynamical fields—a real scalar field φ and a 4-vector field A µ —which,together with the Einstein metric g µν , constitute the effective metric ˜ g T V Sµν .This physical metric is disformally related to the Einstein metric [31] [32, § stretches it by a factor of e − φ in the directionsorthogonal to A µ ≡ g µν A ν while shrinking it by the same factor in the directionparallel to A µ , where the Sanders 4-vector field A µ is unit-time-like with respect In a later paper [30], photons are not coupled to the effective metric, but to the Einsteinmetric. Some conclusions reached in section 5.2 of the prequel paper [19] do not apply to thatalternative version of the SFDM theory. o the Einstein metric ( g µν A µ A ν = −
1) [31–33]:˜ g T V Sµν = e − α Λ MPl φ ( g µν + A µ A ν ) − e α Λ MPl φ A µ A ν = e − α Λ MPl φ g µν − A µ A ν sinh( α Λ M Pl φ ) ≈ g µν − α Λ M Pl φ ( g µν + 2 A µ A ν ) , (5)with M P l the Planck mass and α a dimensionless coupling constant.SFDM modifies TeVeS in two ways: one semantic and one syntactic mod-ification. The semantic revision is to not add yet another (vector) field, butto identify the four-vector A µ with the unit four-velocity u µ of the previouslyadded scalar field Φ. Syntactically, the TeVeS factor of 2 is generalised toobtain: ˜ g SF DMµν ≈ g µν − α Λ M P l θ ( γg µν + (1 + γ ) u µ u ν ) , (6)with TeVeS being recovered for γ = 1 (and a metric conformally related to g µν for γ = − L ¬ rel,int = − α Λ M P l θρ b , (7)where ρ b is the baryon density. The total effective, non-relativistic Lagrangiancan then be shown to reproduce [25, p.10-11], under suitable approxima-tions (such as θ being static and spherically symmetric), the MONDian result a MOND = q a GMb ( r ) r for a baryonic particle with mass M b , after identifica-tion of a = α Λ M Pl .To a particle physicist, the fractional power of X (Eq.2), 3/2, althoughrequired if one aims to eventually regain MONDian behaviour [25, § IV], mightseem strange—it is, for instance, less straightforward to draw correspondingFeynman diagrams. In condensed matter theory, such powers are far fromrare. As mentioned, this specific fractional power corresponds to a phononsuperfluid, with equation of state P ∝ ρ .Superfluidity only occurs at sufficiently low temperature. This naturallydistinguishes between galaxies and galaxy clusters. Due to the smaller veloci-ties in galaxies, the superfluid description is appropriate there, exactly whereMOND is successful. In the clusters the velocity/ temperature is too high, andone finds either a mixture of the superfluid and normal phase, or only thenormal phase, suggesting that the theory might exemplify the usual successesof dark matter theories at that level. Of the normal fluid component; see below. Given a mass m and density ρ [25, Eqs. 8 & 80]. Since the local phonon gradient induced by the Sun is too large to satisfy the criteria for asuperfluid Bose-Einstein condensate, the condensate loses its coherence, which allows SFDM toavoid solar system constraints [25, § V]. nfortunately it is non-trivial to piece together a complete, general (i.e. validat all temperatures) theory from the separate pieces provided above. Assumingthat we would be able to find a relativistic generalisation of the free phononLagrangian (Eq.2), we take it that Berezhiani and Khoury intend the following(relativistic) Lagrangian to correspond to a complete description of the super-fluid regime, that is below T c —or strictly speaking only at T = 0, since atfinite temperatures correction terms would need to be added [25]: L T =0 = L Einstein − Hilbert ( g µν )+ L θ ( θ, g µν )+ L luminous (˜ g SF DMµν , ψ α , ψ α ; µ | ˜ g SFDM ) . (8)It is then tempting to postulate as the general Lagrangian: L gen = L Einstein − Hilbert ( g µν )+ L Φ (Φ , g µν )+ L luminous (˜ g SF DMµν , ψ α , ψ α ; µ | ˜ g SFDM ) . (9)In order for this to reproduce the successes of dark matter theories in galaxyclusters and at the cosmological level, we would like the theory to take on or ap-proximate L Einstein − Hilbert ( g µν )+ L DM (DM-field , g µν )+ L luminous ( g µν , ψ α , ψ α ; µ | g µν )in that regime (i.e. above T c ). For that to be the case, ˜ g SF DM needs to re-duce to the Einstein metric in that regime.
Prima facie , this seems to workout nicely: ˜ g SF DM is given by the Einstein metric plus a phonon modificationterm which vanishes when θ is zero. Above the critical temperature there areindeed no phonons. However, although θ may indeed represent the phononfield below T c , it is, in general, the phase of Φ. As such there is no reason whyit would, generally speaking, be zero or even very small at large temperatures.It is thus an open question what the full SFDM theory is. Φ matters and spacetimes In the companion paper, we introduced a family of criteria for something to bematter [19]. The strongest such criterion was ‘having mass’. It also introduceda set of jointly sufficient criteria for allowing something to be interpreted asspacetime structure: a) being an object that is faithfully representable by adifferentiable manifold with an affine connection and a Lorentzian conformalstructure (i.e. an equivalence class of Lorentzian metrics) such that b) there is awell-defined initial value problem, and such that c) if test particles and/or lightrays were around they would, for any choice of initial conditions, follow thetimelike and null geodesics, respectively, of the same family of geodesics (i.e. the(strong) geodesic criterion), and such that d) special relativity is locally valid,in the sense that no non-negligible curvature terms appear in the equations ofmotion (i.e. the chronogeometricity criterion).We argued that, according to these criteria, it is clear that the scalar fieldΦ occuring in SFDM is as much ‘matter’ as it could possibly be: it fulfils the More precisely: The object under consideration changes in a regular fashion. This change is(partially) in response to something external, and the interaction describable in terms of coupleddifferential equations, in such a way that the object can be said to carry exchangable energy (andmomentum) representable by a stress-energy tensor, T µν , part of which is due to the rest mass ofthe object. In other words: the object is massive. trongest matter citerion of having rest mass, and all the prior criteria thatthe mass criterion builds on. Denying that Φ is (dark) matter would implythat even paradigmatic instances of matter such as electrons (as they appearin the Standard Model of particle physics) are not matter either.However, the Φ-field arguably scores just as well in terms of being (anaspect of) spacetime as it does in terms of being matter. We argued that,in the superfluid regime, ˜ g µν satisfies the above criteria for something to beinterpreted as (an aspect of) spacetime. Given that ˜ g µν is defined in terms ofΦ, one could argue that Φ is thus just as much (an aspect of) spacetime as it ismatter. The rest of the paper is devoted to introducing and evaluating threegroups of viable interpretations of Φ as it features within SFDM, in light of theabove, with the aim of providing an interpretational chart that can serve asa tool for describing and ultimately understanding (theories with) fields thatare both (dark) matter and (aspects of) spacetime. As we have seen in the previous section, the Φ-field in SFDM scores just ashigh in terms of being matter as it scores in terms of being (an aspect of)spacetime. In the following sections we will see interpretations that claim thatit is fundamentally one and merely emergently the other, or even that whatwe see is a breakdown of the very distinction between spacetime and matter.Both of these (families of) interpretations are rather radical if one is used tothe classical distinction between spacetime and matter. And a conservativelyminded philosopher/physicist might well ask: are we not overreacting to theexistence of Φ with either option?By analogy, let us consider corresponding scenarios from theology where wemay ask what a new type of entity meant for the distinction between humanand god, which arguably was challenged just as much—or just as little—bythis new entity as the distinction between spacetime and matter is challengedby Φ.Consider first mythologies with demigods, like Heracles in Greek mythology.A demigod has two parents, one a god and one a human, and a demigod wasthus 50% human and 50% god. Zeus (Heracles’ father) was 100% god, andAlcmene (Heracles’ mother) was 100% human. Contrast this with the ChristianJesus, understood according to the consubstantiality thesis or Trinitarianism: Jesus is not a demi-god, nor indeed either purely human or purely god; he issupposed to be 100% god and 100% human, even while on Earth. Puzzlingabout how this could be is one of the great mysteries of catholic scholarship.But despite all this, it would not be sensible to say that this concept of Jesusbrought about the breakdown of the very distinction between god and human.Jesus is, in a conceptual sense, an anomaly, the single being that is seen asboth fully human and fully god. However, this does not mean that any human(even a faithful catholic) should have any problem with classifying themselvesand their neighbours as clearly human and non-god. See, for instance, Catechism 467. ow is this analogous to the physical fields that we are attempting tointerpret? Recall first, from the prequel paper [19], that non-trivial metricsolutions g µν in GR satisfy up to matter criterion D (action-reaction) or E(energy definable in region). This suggests that g µν is not 0% matter, but,if one requires a stronger matter criterion as a sufficient criterion for beingmatter, then g µν is not 100% matter either. Combining this with the claimthat g µν is 100% spacetime (modulo the caveats mentioned in the prequelpaper) suggests that the analogue of non-trivial metrics in GR lies somewherebetween Heracles and Jesus. When it comes to Φ, being fully (dark) matterand (an aspect of) spacetime, there is no mistaking its analogue: Jesus. Hence, we call the interpretation of Φ discussed in this section, by analogy,the consubstantiality interpretation. A similar interpretation for objects thatare only partially matter and partially spacetime would be called the demiinterpretation, by analogy with demigods such as Heracles.Advocates of the two other, revisionary groups of interpretations discussedin the following sections may or may not grant that the existence of ‘demigodobjects’ (such as Heracles and g µν ) leaves the associated distinctions (god–human or matter–spacetime) untouched, but they insist that ‘full hybrid ob-jects’ (such as Jesus or Φ) are either problematic for the associated distinctionsor at least require us to say something more. This is one of the ways in whichconsidering the issue of distinguishing matter and spacetime in the SFDMcontext goes beyond considering it within GR. On the consubstantiality inter-pretation however, the existence of a field like Φ that is just as much matter,namely maximally, as it is an aspect of spacetime structure, does not causeany big problems for the spacetime/matter distinction. The senses in which Φis matter and spacetime are on a par. Nothing further needs to be said. Sure,Φ is rather unusal and weird in this way, but as long as it is in a clear minoritywith respect to other entities, the spacetime/matter distinction stands strong:it remains coherent, applicable, objective and useful. In a strange way, onemight even say that the analysis of Φ has shown us what it really means to bematter, just as the concept of Jesus, some would say, has shown us what it isto be human.The consubstantiality interpretation is a viable interpretation of Φ. To the Other similar analogues that one might wish to consider are the two-faced gods Janus fromRoman mythology and Culsans from Etruscan mythology, although the possibility of having asecond face is not as conceptually surprising and controversial as one might think it is to beboth spacetime and matter or a human and a god. Perhaps the double personality of Voldemort-Quirrell is a somewhat better analogy in this regard. If one were to consider a field which playsonly the spacetime role when it is in one phase and only the matter role when it is another phase—as Berezhiani and Khoury consider to be the case for Φ (Section 5)—then the transformationsbetween Bruce Banner and the Hulk, and between Dr. Jekyll and Mr. Hyde may be good analogies.If instead—as we believe to be the case for Φ (Section 5)—a field is always and everywhere matterbut only in one specific phase also spacetime, the comparison with Jennifer Walters and She-Hulkseems more appropriate, as the personality of Walters is still (mostly) there even when in She-Hulk form. Finally, Hossenfelder [34] uses ‘imposter’ terminology, which may suggest a ClarkKent/Superman analogy. However, this would not suggest an interpretation equivalent to theconsubstantiality interpretation, but something more in the spirit of the elitist interpretations(Section 5) (but see fn.12). (MG+DM)-fundamentalism
DM-fundamentalism (or: dualism) emergent
MG-fundamentalism non-fundamentalism (or: neutral monism)
Table 1: Matrix of views that for each of the DM and MG aspects distinguishbetween them being fundamental or merely emergent aspects. Egalitarian viewsare in italics; elitist views are underlined. extent that there might be an objection to this interpretation it would be thatone or both of the other two groups of options on the table, discussed below,provide a better or more complete account. Both groups of alternatives insistthat there is more to be said about Φ being 100% matter and 100% spacetime.Hence, we will compare and contrast the consubstantiality interpretation withthese other interpretations in Section 7, after the latter have been introducedand analysed.
The fundamentalist interpretations go beyond the consubstantiality interpre-tation by claiming, for each of the DM and MG aspects, whether Φ exhibitsthem fundamentally or not. This results in a 2 by 2 matrix of logically possibleviews, see Table 1. On the (MG+DM)-fundamentalist interpretation, both as-pects are fundamental aspects of Φ. On the non-fundamentalist interpretationΦ is fundamentally neutral, i.e. neither DM nor MG; both of these aspects arenon-fundamental, derivative, or even emergent. Besides these two egalitarianpositions there are two elitist positions, on which only one of the DM and MGcategories is a fundamental aspect or property of Φ, with the other categorybeing merely a derivative or emergent aspect or property—as opposed to thedivinity and humanity of Jesus being, in all relevant senses, on a par. There-fore the egalitarian positions can be considered as being closer to (the spiritof) the consubstantiality interpretation. Similarly, the non-fundamentalist in-terpretation is closest (in spirit) to the breakdown interpretations (see below),as both aspects being metaphysically non-fundamental opens the door to themnot being conceptually fundamental either. (The relations between these in-terpretations will be discussed further in Section 7.)Within the category of elitist interpretations, it may seem most plausi-ble that Φ is fundamentally dark matter and emergently a modification ofgravity and spacetime (DM-fundamentalism), rather than vice versa (MG-fundamentalism). The MG-nature is important only in galaxies. Hence, Φ Not to be mistaken with the notion of ‘egalitarian interpretation’ in [11]. erely plays the role of a modification of spacetime in that regime; it is anMG-impersonator, not fundamentally a modification of spacetime.Arguably, this is what the creators of SFDM have in mind. The followingtextual evidence places them in the right column of Table 1.In contrast with theories that propose to fundamentally modifyNewtonian gravity, in this case the new long-range force mediatedby phonons is an emergent property of the DM superfluid medium.(italics in the original) [30, p.3]Unlike most attempts to modify gravity, there is no fundamen-tal additional long-range force in the model. Instead the phonon-mediated force is an emergent phenomenon which requires the co-herence of the underlying superfluid substrate. (italics in the origi-nal) [30, p.16]The name given by them (but not by Hossenfelder, who calls it a modifiedgravity theory! [34]) to their theory, i.e. superfluid dark matter, further narrowsit down to DM-fundamentalism. In the remainder of this subsection we willfocus on evaluating this interpretation—it being further removed from theconsubstantiality interpretation than the egalitarian interpretations and primafacie more plausible than MG-fundamentalism. Comments regarding the otherthree interpretations will be made along the way.Before proceeding it is important to note that Berezhiani and Khoury seemto operate under a different understanding of the dark matter aspect of Φ.Whereas we take the property of ‘having mass’ (or one of the weaker mat-ter criteria [19]) to be what makes Φ dark matter, they seem at places toidentify the DM nature of Φ with its particle nature, that is the phase inwhich Φ is appropriately described as individual collisionless particles, i.e. thephase above T c (as opposed to the MG-phase below T c where a description interms of collective excitations—phonons—is considered more appropriate). Intheir words: “DM and MOND components have a common origin, represent-ing different phases of a single underlying substance” [25, p.3]. It turns out,ironically, that not only does their understanding of the DM nature of Φ un-dermine the main argument in favour of DM-fundamentalism more obviouslythan our criterion does ( § § As briefly hinted at before, one line of argument in favour of DM-fundamentalismis the following. Φ exhibits its DM nature everywhere in the universe; this isnot the case for its MG nature (which appears predominantly in galaxies).Hence, the DM nature of Φ is more fundamental than the MG nature. Let us Hossenfelder [34] uses ‘imposter’ terminology, but we do not want to suggest that what isimpersonated/mimicked is fictive; it is real; it is just not fundamental. Compare this to a chair,which despite not being fundamental is still real rather than a fiction. all this the scope argument. It is not immediately obvious why this conclu-sion would follow. There are at least three ways of fleshing out this argument.Two flawed ways will be discussed here. A better version is touched upon in § on our preferredunderstanding of the DM-nature of Φ, i.e. its massiveness: Φ is always andeverywhere massive, but does not always and everywhere form a superfluidBEC. (It should of course be noted though that there is not yet a completetheory, exactly because there is no justification for the θ -modification to theEinstein metric vanishing outside of galaxies.)A first way of cashing out the scope argument starts from the claim thatbecause of the universal scope of the DM aspect of Φ and the non-universalscope of the MG aspect, the former is more important than the latter. Hence, Φis DM in a ‘more real’, a ‘more fundamental’ sense than that it is MG. However,not only is ‘importance’ a vague, subjective, anthropocentric term, but thisversion of the argument conflates the colloquial meaning of fundamental—important—with the more philosophical, technical meaning that is intended inthe formulation of DM-fundamentalism (see Subsection 5.3).A second way of cashing out the argument phrases the non-universalityof the MG-aspect of Φ as that aspect ‘appearing’ in galaxies and ‘disappear-ing’ outside of galaxies. On an understanding of ‘to emerge’ as this notionof ‘to appear’, DM-fundamentalism follows. But this similarly mistakes therelevant technical notion of emergence with the colloquial meaning. That aproperty does not appear always and everywhere does not imply that it isnon-fundamental. Consider electromagnetism. The magnetic field inside aninfinitely long solenoid will be non-zero, but outside the solenoid it will be zero.We do not usually take this to imply that the magnetic field is non-fundamental(within electromagnetism). Similarly, in electrostatics a Faraday cage may beused to shield off an external electric field. That the electric field disappearswithin the cage is not usually taken to imply that it is non-fundamental. Thissecond version of the argument seems to mistake essentialism—for a propertyto be essential to an object it must be instantiated always and everywhere inall possible worlds inhabitated by that object—for fundamentalism.What perhaps inspires these two versions of the argument is that paradig-matic emergent theories, such as Newtonian gravity as compared to Generalrelativity, are not universally valid but only (approximately) valid in certain Thanks to Katie Robertson and Alex Franklin for pushing us on this line of argument. However, if the final theory of SFDM is indeed such that the modification of the Einsteinmetric at T ≫ T c goes to zero—see the end of Section 2—then it seems to be the case that theMG aspect is instantiated everywhere after all, albeit (partially) trivially. imiting regimes (see the ‘formal asymmetry criterion’ in Subsection 5.3). It isindeed the case that emergent theories that are the limit of a more fundamentaltheory therefore have a limited scope of applicability. But it does not followthat the implication goes in the other direction: from a non-universal scope ofsome property it does not follow that that property can only be described byan emergent theory that is the limit of a more fundamental theory.On colloquial understandings of the notions ‘fundamental’ and ‘emergent’the scope argument is not valid. For it to stand any chance it must con-nect the non-universality of the MG nature of Φ to the technical notions of‘fundamental’ and ‘emergent’ that are intended in the formulation of DM-fundamentalism. These technical notions will be outlined in § Remember that the superfluid BEC, which carries the phonons that mediatethe MONDian force, is a phase of Φ; this effect appears at temperatures belowthe superfluid phase transition. If one follows Berezhiani and Khoury in takingthe DM-aspect to be the phase of Φ above the critical temperature (but noton our understanding of DM), DM-fundamentalism starts to sound a lot likesaying that water vapour is more fundamental than liquid water, or liquidwater more fundamental than ice. We do not tend to think of the differentphases of water in that way; they are just different but equal states of beingof a single underlying entity. (Moreover, if we would encounter one phaseonly or mainly in galaxies, and the other phase all over the universe, thiswould not change anything.) This resonates with egalitarian interpretationsof the different phases of water. On a strong reading of ‘underlying’ in In the phase diagram of water, one can even move between the liquid and vapour phaseswithout crossing any phase transition, namely by circumventing the critical point. The onlydifference between these two ‘phases’ is then their density (which would justify a fundamental–emergent distinction even less). (Thanks to Stephen Blundell for this point.) The two phases of Φare thus perhaps better compared to ice and liquid water. At least some phase transitions (such as the transition of Φ at T c ) are associated with a breakingof the symmetries of the system. From a particle physics perspective, e.g. in the context of grandunified theories, it may seem common to associate with this change in symmetry a difference infundamentality. However, this notion of fundamentality in terms of unification or scope, if indeed arelevant notion of fundamentality at all, differs from the concepts of fundamentality and emergenceas intended in this paper ( § In § et al. do in fact label that contribution as the DM contribution [30, Eq. he earlier quote by Berezhiani and Khoury—“DM and MOND componentshave a common origin, representing different phases of a single underlyingsubstance” [25, p.3]—non-fundamentalism would be the favoured egalitarianinterpretation. At the fundamental level there is just a neutral entity, water,H O (Φ), which can appear in different forms—ice, liquid, vapour (phonons,individual collisionless particles)—without any single one of these phases beingessential to water (Φ), or a fundamental aspect of it.To sum up: if one were to equate DM to the phase in which an parti-cle description is appropriate—which Berezhiani and Khoury do but we donot—and MG to the phase in which a phonon description is appropriate, DM-fundamentalism would be problematic.
It is about time that we spell out in a bit more detail what one might meanby ‘fundamental’ and ‘emergent’ in the context of DM-fundamentalism (andthe other three fundamentalist interpretations)—other than the too colloquialmeanings of ‘important’ and ‘appearance’, respectively, as discussed in § Metaphysical asymmetry: the emergent description (i.e. MG) onto-logically depends on the fundamental description (i.e. DM) [24];2.
Formal asymmetry: the emergent description is obtained from thefundamental description via an irreversible mathematical operation [35];3.
Scale separation/asymmetry: the fundamental description is a de-scription of smaller length scales and higher energies; the emergent de-scription is a description of larger length scales and lower energies.4.
Dynamical separation/autonomy: the fundamental description is adescription in terms of fundamental concepts (i.e. DM) only, and, ideally,the emergent description is a description in terms of emergent concepts(i.e. MG) only.These criteria may be illustrated by comparing, say, an elephant as de-scribed by zoology with its description in terms of the underlying, more fun-damental Standard Model of particle physics. There could be quarks withoutthere being elephants, but not vice versa . The emergent elephant featuring inZoology is a coarse-grained version of the underlying description; from it onecannot retrieve the full state of the quarks. Zoology is a low-energy/ large dis-tance limit of the Standard Model of particle physics. Zoologists do not needto talk about quarks; particle physicists do not need to refer to elephants.We will discuss each of these components in turn. Although the metaphys-ical symmetry condition is satisfied in some sense in SFDM, there is strongreason to doubt that the formal asymmetry and dynamical separation apply.The scale asymmetry definitely does not apply. .3.1 Metaphysical & formal asymmetry Is there a metaphysical and/or formal asymmetry between the two descrip-tions in virtue of which we may call the DM description the fundamental oneand the MG description the emergent one? Start with the formal asymme-try. The literature is riddled with examples of such irreversible mathematicaloperations—idealisations, such as the thermodynamical limit, or abstractions,such as coarse-graining or summation [35,36]. Is the phonon description withinSFDM arrived at via such an irreversible operation?Franklin and Knox describe phonons as they appear in a crystal lattice ofatoms [36]. They convincingly argue that moving from the atomic displace-ment variables to a description in terms of phonon variables provides novelexplanatory power. (Similarly, in SFDM, the phonon description explains theMONDian behaviour in galaxies.) They suggest that this explanatory nov-elty constitutes a relevant notion of emergence, differing from the standardnotions associated with mathematically irreversible operations. In fact, thetwo descriptions are dual. Complete translations between the descriptions arepossible in either direction. In light of this, they report David Wallace aspointing out that their notion of emergence fails to satisfy what we have calledthe metaphysical asymmetry criterion. They concede that a full defense ofthe emergent nature of phonons would require a further asymmetric relationbetween the two descriptions.This opens up an interesting possibility. What if there is no such asym-metric relation? Would both descriptions then be equally fundamental? Or,arguably, with the phonon description providing novel explanatory power, thatdescription seems the more fundamental one. On the alternative definition ofDM as the particle nature of an underlying field, this would suggest, in thecontext of SFDM, an inverted fundamentalist interpretation, with the MG as-pect of Φ being fundamental and the DM aspect being merely emergent: theMG-fundamentalist interpretation.Is there then a further, metaphysical asymmetry that blocks the conclu-sion that phonons are fundamental? One may wish to consider several specificoptions. One specific option would be composition: phonons in an atomiccrystal are ‘composed’ of the collective behaviour of many many atoms. An-other specific option would be a part-whole (i.e. mereological) asymmetry. Ourpreferred candidate for the relevant metaphysical asymmetry is more general,namely ontological dependence, which holds if and only if the phonons cannotexist without the atomic displacements but the atomic crystal could exist with-out there being phonons. One possible negative response is that in the SFDMcontext there is no underlying atomic crystal that could compose the phonons;it is much less clear that particle excitations of Φ have the same metaphysicalrobustness as atoms. A stronger negative response points out that, since thetwo descriptions are dual, it is not only the case that the phonon variables arelinear functions of the atomic displacement variables, but the latter are alsolinear combinations of the former [36, esp. p.75]. Duality is a symmetric rela-tion; phonons are as much composed of or a part of atomic displacements as vice versa . Now, classically we can indeed conceive of an atomic crystal at rest(i.e. all atomic displacements are 0 and remain 0), which would seem to imply hat it is possible for an atomic crystal to exist without any phonons existing‘in’ it. It might be retorted that there would still be an ideal phonon, namelyone of infinite wavelength and hence zero energy. A stronger response is that,regardless of phonons being ontologically dependent or not on a classical crys-tal, it is quantum mechanically not possible for a crystal to exist without therealso being phonons. It is not dynamically possible for quantum mechanicalconstituents of a crystal to be at rest; the lowest energy state is not zero anddoes consist of a phonon (cf. the zero-point energy of a quantum harmonicoscillator). Phonons are thus not ontologically dependent on an atomic crys-tal. That being said, we will see below when discussing ‘dynamical separation’(and in particular the two-fluid model), that the SFDM case is much morecomplex than the atomic crystal that Franklin and Knox concern themselveswith. Hence, a full evaluation of the metaphysical and formal asymmetry cri-teria is outside of the scope of this paper. However, the above discussion servesto show that one should by no means assume that a phonon, being a ‘collectiveexcitation’, by default satisfies the metaphysical asymmetry criterion, let alonethe formal asymmetry criterion.Phonons are not ontologically dependent on the atomic crystal: althoughthey indeed cannot exist without the crystal, it is not the case that the crystalcould exist without there being phonons. If a crystal exists, then phononswill also exist. However, there is one obvious way to get rid of the phonons:melt the crystal. In that sense, even though a crystal ensures the existenceof phonons, it is still possible for atoms to exist (albeit not in crystal form)without phonons existing. Phonons do not ontologically depend on an atomiccrystal, but they do seem to ontologically depend on the atoms. Similarly,even if phonons are indeed not ontologically dependent on the superfluid BEC,one could ‘melt’ this BEC by increasing the temperature, which makes thephonons disappear whilst leaving individual collisionless (massive) particlesbehind. The possible existence of particle excitations of Φ without there be-ing phonons (which occurs typically in galaxy clusters and at the cosmologicalscale) then seems to suggest that phonons are ontologically dependent on theparticles. This then finally suggests the most viable form of the scope argu-ment: the universal occurrence of the DM-nature of Φ and the non-universaloccurrence of its MG-nature suggests that the latter is ontologically dependenton the former, and since ‘ontological dependence’ is (part of) the relevant un-derstanding of the fundamental-emergent dichotomy, DM-fundamentalism isfavoured.A first thing to note is that, even though in our universe it might well be thecase that there are spacetime regions with particles but without phonons, it isstill true (for finite m ) that in each world dynamically allowed by SFDM, therewill be a finite critical temperature below which Φ will be describable in termsof phonons such that the spacetime criteria are satisfied. In other words, theexistence of Φ does guarantee that in each dynamically allowed world phononscould be made to exist as long as you (can) cool down the system sufficiently.To evaluate this final version of the scope argument, we need to againdistinguish between our definition of DM as ‘massiveness’ (or a weaker mat-ter criterion [19]) and the alternative definition ‘being describable in terms ofindividual particles’. Start with the latter. One way to question the scope ar- ument is as follows. Sure, the ‘melting story’ above shows us that the particleexcitations of Φ (DM) could exist without there being phonons (MG). But toprove ontological dependence of the latter on the former, it also needs to bethe case that phonons could not exist without there being particles or atoms.In the case of an atomic crystal that is obviously the case. But in the caseof a quantum field and especially a superfluid BEC one might well push backagainst the claim that there are, in any meaningful sense, individual particles.The literature on this is vast and we will not dwell on it here [37–55]. If itis correct that there are not in any meaningful sense individual particles in asuperfluid BEC, it would be the case that phonons and particles are both onto-logically independent of each other, and that neither of them exists everywherein a SFDM-model that could describe our actual universe.Consider now the scope argument on our (massiveness) definition of DM.First we may ask whether the MG-behaviour of Φ is ontologically dependent onits mass. The phonon-mediated force is clearly not reducible to a gravitationalcontribution of Φ to the dynamics of luminous matter: it arises from a directcoupling of the phonons to the baryons (Eqs. 7 & 4). It should be noted thatthe evolution of the phonons depends on m , as is clear from L θ (Eqs. 2 & 3).But this is a form of dynamical dependence. We are here interested in whetherthe existence of the phonons (and the phonon-mediated force) depends on m ,not their behaviour. The answer is negative, in the following sense. For avery small m , the De-Broglie wavelength and the critical temperature becomevery large. Hence, for a very small m , Φ is practically always and everywherea superfluid BEC, with the associated phonons mediating the modification ofgravity.The MG-aspect of Φ does not ontologically depend on its DM-aspect. Nev-ertheless, the MG-aspect may still depend on Φ, and it does: Φ can existwithout phonons, as it does, for instance, outside of galaxies in a model thatis supposed to represent our actual world, whereas the phonons cannot existwithout Φ. In that sense, the MG description does ontologically depend onthe Φ description, and since we have chosen m to be finite, this Φ descriptionis a DM description.In conclusion: given the above mentioned lack of a complete underlyingtheory of SFDM, it is at this stage not possible to determine whether thephonon description is dual to the missing underlying description; but givenFranklin & Knox’s discussion of phonons in an atomic crystal we should refrainfrom simply assuming that the formal asymmetry condition is satisfied bySFDM. Regarding the metaphysical asymmetry condition: phonons are notontologically dependent on the superfluid BEC, nor on the mass of Φ. However,it is possible for Φ to exist without carrying phonons, but not for phonons toexist without Φ. For finite m , the phonons are thus ontologically dependenton the Φ description, which is a DM description (on our criteria for DM). Next we consider the scale separation criterion. Although galaxies are in thiscontext (i.e. compared to galaxy clusters and the cosmological scale) indeed elatively small, they correspond to relatively low energies, as superfluidityoccurs below a critical temperature. One might retort: so be it—the con-cepts ‘fundamentality’ and ‘emergence’ do not necessitate this type of particle-physics-inspired scale separation; it is the other conditions, such as dynamicalseparation, that form the essence of emergence.
Consider finally dynamical separation. This feature would be violated if theobservations require a dynamical mixture (or dynamical dualism ), i.e. if thetotal dynamics depends on non-negligible contributions arising both from theDM nature and from the MG nature of the basic objects of a theory, withoutthe MG contribution being reducible to the DM contribution. One logicallypossible way in which this might happen is if there are (meta)physical reasonsto distinguish two numerically and qualitatively distinct ‘entities’, one darkmatter and the other a modification of gravity. Call this ontological dualisma metaphysical mixture . If these ‘entities’ are equally fundamental, this wouldresonate with the egalitarian positions. One specific way in which a meta-physical mixture might occur is if these two entities are (at least partially)separated in space(time), like a mixture of water and wine. Call this a spatial mixture. Given that SFDM introduces only a single field Φ, it seems primafacie to be the case that if the DM and MG aspects were to form a dynam-ical mixture, this would have to be so without them forming a metaphysicallet alone a spatial mixture. However, Lagrangian 2 only describes the highlyidealized case of a pure superfluid at zero temperature. In reality, at finitesubcritical temperature, the system is better described phenomenologically byLandau’s two-fluid model [25, 30]: some sort of mixture of both a superfluid‘phase’ and a normal ‘phase’. We would do well to investigate the metaphysicsof this model, in order to determine whether it supports a metaphysical oreven a spatial mixture.Landau [57–59] developed his two-fluid model (2FM) of superfluidity inresponse to a tension between several experiments with Helium II. One suchexperiment concerns Helium II flowing through a very narrow capillary with-out any apparent resistance due to viscosity or friction (as long as the velocityof the flow stays below some critical value) [66]. However, if the viscosity ofHelium II is measured by the damping of oscillating discs immersed in Helium It should be noted that, at a yet smaller scale, namely that of individual stars, no condensationoccurs and thus no MONDian behaviour emerges (which happens to be good news in light of solarsystem constraints [25, 30]). Note that not only are the scales inverted from what is required or expected, it is not even thecase that there is a universal length scale. The second condition for the existence of a coherentBEC (section 2) requires sufficient time for thermalisation to occur, which will differ per galaxy. Entities is meant in a broad sense here. A metaphysical mixture need not be a spatial mixture. Examples are Khoury’s two scalarfields (see the conclusion of the prequel paper [19]) [56] in regions where both fields are non-zero,or the Minkowski metric field and the electric field of a charged point particle in special relativity. Building upon earlier work by Tisza [60–64]. See e.g. Dingle [65] for other such experiments. I at a temperature just below the critical temperature T c = 2 . K , one ob-tains a substantive value, albeit one that decreases with temperature below T c [67]. Landau’s 2FM deals with this “viscosity paradox” [68, p.34] [69, p.13]by considering Helium II as a ‘mixture’ (in some sense to be determined) oftwo parts/components/phases/liquids: a superfluid component, which has noviscosity and carries no entropy and heat; and a normal component, which isviscous and carries entropy and heat. There is negligible interaction betweenthe components. The normal component is responsible for the damping ofthe discs, which allows one to assign an ‘effective inertial mass density’ to it.An effective mass density for the superfluid component can then be definedby substracting the normal effective mass density from the total mass densityof Helium II. The ratio of the densities of each component is a function oftemperature only. At T c the superfluid component vanishes; at T = 0 the nor-mal component vanishes. The capillary experiment is explained by the normalcomponent remaining stationary with respect to the capillary, whilst the su-perfluid component moves without dissipation—this is called ‘superflow’. Thedecrease with temperature of the damping of the oscillating discs is explainedby the density of the normal component decreasing with temperature.What type of mixture do the two components form? Rice [70,71] has arguedthat, for empirical reasons, these components form a spatial mixture: below T c the superfluid component has a fibroid structure; just above T c it startsto appear in small globules. This suggestion has not gained traction in theliterature. The received view is that the two components are “superimposed”[65], or “interpenetrating” [72–75], i.e. not separated nor separable in positionspace. Rather than a spatial mixture the components are, in some sense tobe specified, two different modes of motion of a single underlying substance.Landau and Lifshitz therefore use the terms ‘superfluid’ and ‘normal flow ’,rather than ‘part’ or ‘component’ [59].Is the 2FM then best interpreted as a mere metaphysical mixture of twoingredients, i.e. are there two numerically distinct entities even though theyare not spatially separated? One way to answer this is to get a grip on whatis meant by ‘two modes of motion’. Sometimes the received view is (implic-itly [70]) portrayed as the two ‘components’ being separated not in positionspace but in momentum space. One might even be tempted—Berezhiani andKhoury seem to have fallen for this temptation [25, p.4]—to identify the su-perfluid component of the Bose-Einstein condensate (i.e. ‘condensate’ in thebroad sense) with the particles in the zero momentum ground state (i.e. thecondensate’ in the narrow sense) and the normal component with the particleswith non-zero momentum (i.e. the “thermal cloud” [73]).An initial response might be that this would be to commit to a spatialmixture after all: the superfluid component is comprised of the particles atrest, with the normal component being comprised of moving particles at otherlocations than the particles at rest. This would be the case for a classicalsystem, but not for a quantum liquid. The indistinguishable ground state‘particles’ of a free BEC are maximally smeared out; they are everywhere. In the subsequently developed microscopic model underlying the phenomenological 2FM onemay derive the effective mass density of the superfluid component independently [69]. hat may still seem to leave open the possibility that the thermal particles arenot everywhere, suggesting that there are regions with ground state particlemush, but no thermal particles. However, in order for a liquid to form a(superfluid) BEC in the first place, the thermal De-Broglie wavelength needsto be at least comparable with the interparticle distance (determined from theinterparticle potential). There is thus no region where the wavefunction of thethermal particles is negligible. The ground state and the thermal cloud are notspatially separated.A stronger response is that it is impossible in the first place to identifythe two ingredients of the 2FM with the ground state and thermal cloud,respectively. For interacting BECs, at low temperatures including T = 0, onlya small fraction of the particles—roughly 10% for Helium II—is actually inthe ground state, due to the zero point kinetic energy [28, § T = 0 all the particles participate in the superflow [28, § Then, if the 2FM is a metaphysical mixture, it is highly unclear what typeof beasts these two ingredients are. Nevertheless, the viscosity experimentsstrongly suggest that there are two distinct ‘things’ with metaphysically dis-tinct properties. One thing moves through the capillary at rest while the otherstays behind; the latter thing does interact with oscillating disks with the for-mer thing now staying behind. The normal component carries entropy andheat; the superfluid component does not. With each component we can asso-ciate a different effective inertial mass density. We might want to conclude thatthere is a metaphysical mixture of two ‘things’ simply in virtue of their distinctproperties, even if we are in the dark as to the carriers of these properties (butwe will see below that there are reasons against this).Another argument that suggests this same conclusion is the following. Inthe literature a distinction is being made between first and second sound;further empirical support for a metaphysical mixture seems to come from theobservation of this second type of sound in the 2FM. First sound, i.e. phonons,consists of pressure waves. This means that the total density at each pointvaries over time. However, if there is a metaphysical mixture of two componentseach with their own effective mass density, these densities can also wave out ofphase, in such a way that the total density stays constant. Since the ratio ofthe densities of each component is a function of temperature, this correspondsto a temperature wave. This second sound has been observed in Helium II.These two reasons for the existence of a metaphysical mixture can be crit-icised. Firstly, although such a mixture indeed implies second sound, secondsound does not imply a metaphysical mixture. Second sound can be describedpurely in terms of temperature, without needing to split the constant density Alonso et al. [75] claim that even if it would be possible to assign each Helium atom to a com-ponent at a specific time, this assignment would change over time, even in dynamical equilibrium.This claim is superseded by Landau and Lifshitz’s claim that one cannot assign Helium atoms toa specific component even at a single instant of time. nto two components that wave out of phase.Secondly, how relevant and metaphysically distinct are the differences inentropy, heat and mass between the two components? One can always andtrivially make the non-naturalistic metaphysical stipulation that instead ofone entity with a value for the entropy S of s ∈ R + , there is one entity withtwo partial entropies, namely 0 and s ; or even that there are two co-locatedentities, one with entropy 0 and the other with entropy s . But that one canmake that trivial stipulation does not mean that we should; Ockham’s razoreven speaks against it. That being said, we do observe something coming outof the capillary, and it does not carry entropy, even though the superfluid BECas a whole does carry entropy.Let us turn to the effective inertial mass. If this is indeed a physical massin the relative sense, a weak equivalence principle—associating with each com-ponent a gravitational mass numerically identical to its inertial mass—wouldensure that the components have gravitational masses that are (in general) dis-tinct. Whether the weak equivalence principle holds in this novel context wouldof course need to be tested. But it is not obvious that the effective inertialmass is indeed a physical mass in the first place. That we are able to mathe-matically define a function of temperature does not mean that this representsa ratio of physical mass densities out there in the real world. For one thing,these supposed mass densities are not conserved (but vary with temperature).Moreover, we know that phonons are in fact massless, in the sense of their dis-persion relation being gapless. To be fair, we do know that the Helium atomshave rest masses, which are simultaneously inertial and gravitational masses.But, since we have seen that we cannot say which Helium atoms are involved ineach component of the 2FM, we cannot simply add their gravitational massesto obtain the gravitational mass of each component. For the same reason theeffective inertial mass of the normal component is not and could not be definedby summing over the inertial masses of the constituent atoms; instead it arisesfrom massless phonons (and higher excitations, called rotons) being excitedby the oscillating disks and thereby influencing its inertia. It is not directlyclear why gravity would interact with a quantity so defined. Dingle refers tothis phonon contribution to the effective mass as non-material [65, § Feynman states in 1954 that the effective mass (of the normal component) “isnot the average value of any quantity that can reasonably be ascribed to anindividual excitation. It appears to have meaning only for the entire group ofexcitations in, or near, thermal equilibrium” [72, p.271]; and more strongly in1972: it “is a derived concept and not the density of anything” [76, p.318]. Insummary, it is far from obvious that one can defend a metaphysical mixtureby claiming that there are two distinct components in virtue of them havingdistinct entropies (in a non-trivial sense) or them having distinct gravitationalmasses (in a strong physical sense). However, perhaps it is possible to determine how many particles are (and thus how much massis) involved in total per component. Dingle takes the effective mass to also include the material mass, i.e. presumably the gravita-tional mass of the Helium atoms, but this does not seem to follow from the way the effective massis calculated. oreover, even if a metaphysical mixture were defensible, it does not seemto be a mixture of one pure DM component and one pure MG component.The normal component—as the name suggests—acts like a conventional vis-cous fluid [28, p.29], i.e. like the phase of Φ (or Helium) above T c which isinterpreted as DM. The superfluid component exhibits the novel features, suchas superflow, that sparked the interest in superfluids as opposed to normalfluids. One might thus be forgiven for thinking that the normal componentis the DM component and the superfluid component the MG component ofthe mixture. However, the MG behaviour has nothing to do with superflow.And it is the normal component that carries the phonons. After all, it is invirtue of these and higher excitations that the normal but not the superfluidcomponent increases the effective inertia of the oscillating disk, and that thesuperfluid but not the normal component flows through the capillary withoutdissipation. The phonon-mediated MOND force is thus associated with thenormal component. Can we then interpret the superfluid component (only)as dark matter? On our criterion for dark matter this is equivalent to askingwhether we can associate gravitational mass with the superfluid component(only). In the previous paragraph we have seen why one may be skeptical thatthe effective inertial mass of a component ensures that it has a gravitationalmass. If this skepticism is unwarranted, both components would (in general)have a gravitational mass, not just the superfluid component. If the skepticis right, we can only associate a gravitational mass with the whole system,not with each component separately. Either way, it is not the case that thesuperfluid component is purely DM-like and the normal component not at allso. In summary: we have investigated what kind of ontology is being de-scribed by Landau’s two-fluid model of superfluidity. A spatial mixture is notvery plausible, nor is it obviously a metaphysical mixture—although the phe-nomenology still very strongly suggests that there must be such a mixture—anddefinitely not of one purely DM-like component and another purely MG-likecomponent. It is only “formally” [65, p.114] [69, p.15] or “artificial[ly]” [72,p.272] that a distinction between two components may be made; “it is no morethan a means of expression convenient for describing the phenomena” [58,p.357] [59, p.515]. If the condition of dynamical separation is to be violated,it must be because of a mere dynamical mixture.Is there then a dynamical mixture of dark matter and modified gravity ingalaxies? That is, does the total galactic dynamics depend on simultaneous,non-negligible contributions arising both from the DM nature and from theMG nature of Φ, without the MG contribution being reducible to the DMcontribution? Yes, it does. The MONDian features arise from the phononscoupling directly to the baryons (Eq.7), not from the gravitational mass of Φ.Φ of course still has a mass, which means that its gravitational contributionneeds to be added to the phonon contribution. Berezhiani et al. refer to this The relative importance of each contribution depends on the values of the parameters. Inref. [30, p.3-7,11-12] the DM contribution is more important than in ref. [25, p.13]. In fact,in the latter paper the DM contribution is considered negligible for radii much smaller than acertain transition radius. However, this does not change the fact that the remaining phonon s the hybrid method of calculating the total dynamics [30, p.4,11-12]. Thisdynamical mixture violates the criterion of dynamical separation/autonomy(i.e. the fundamental description is a description in terms of fundamental con-cepts (i.e. DM) only, and, ideally, the emergent description is a description interms of emergent concepts (i.e. MG) only).However, a friend of the DM-fundamentalist approach might push backagainst this: it would indeed have been pleasant if the emergent descriptionwere in terms of emergent concepts only, but the only thing that is crucialis that the fundamental description is a description in terms of fundamentalconcepts only. Perhaps the hybrid method belongs to the emergent descriptiononly, with the fundamental description being able to provide a non-MG storyunderlying the phonon contribution (even though we have seen that that storywould also have to be something other than the gravitational contribution ofthe dark matter mass). If we take L Φ (Eq. 1) to be the relevant fundamentaldescription, there is indeed no MG-aspect to be identified whatsoever, as thereis no direct coupling to luminous matter which might modify the spacetimethat luminous matter would ‘experience’ (i.e. not the Einstein metric). But thisleaves only the option of influencing luminous matter indirectly by contribut-ing via the mass of Φ to the stress-energy tensor that determines the Einsteinmetric, as would be expected from dark matter. And it is for exactly this samereason that this can hardly be the complete fundamental Lagrangian, since wedo require a direct coupling between phonons (arising from Φ) and luminousmatter at the effective level, L eff, ¬ rel,int (Eq. 7), to obtain the desired resultsfor Galaxy rotation curves. It is difficult to see how an emergent Lagrangiandescribing the direct interaction between phonons and luminous matter couldarise from a fundamental Lagrangian that contains no non-gravitational inter-action between Φ and luminous matter whatsoever. Only the non-interactingeffective phonon Lagrangian 2 emerges. The creators of SFDM admit that L eff, ¬ rel,int is an “empirical term” [25, p.8] that is added to the phonon de-scription to obtain the desired results for Galaxy rotation curves. They suggestthat such a coupling may arise if baryonic matter couples to the vortex sectorof the superfluid [25, p.8], or it may be a non-perturbative effect [30, p.5]. Or,it “ could be a soft fundamental coupling between DM and baryons” [30, p.5](italics added). In any case, some interaction term needs to be included inthe fundamental Lagrangian; they explicitly acknowledge that “a fundamentaldescription of [their] DM superfluid is still lacking” [30, p.6]; L Φ is incomplete.Thus, arguably, the only reason that L Φ (Eq. 1) does not tell an MG-story un-derlying the phonon contribution to the total dynamics is because it currentlytells no such story at all. Given that a completion would have to ultimatelyaccount for the hybrid method, we may well expect this completion to exhibitMG features.To sum up, although the lack of a complete theory of SFDM makes it diffi-cult to reach a definite conclusion regarding the satisfaction of the dynamicalseparation criterion—that is, the fundamental description is a description interms of fundamental concepts (i.e. DM) only, and, ideally, the emergent de- contribution is not reducible to the gravitational mass of Φ. Moreover, near the transition radiusboth contributions do become important. cription is a description in terms of emergent concepts (i.e. MG) only—wehave good reasons to doubt that this criterion is satisfied.This then brings us to a summary of our discussion of the fundamentalistinterpretations. For each aspect of Φ, DM and MG, is this aspect fundamen-tal or emergent? On our understanding of these aspects they are not bothdistinct phases of Φ but two different types of beast. The DM aspect is themassiveness of Φ, the MG aspect is mediated by phonons carried by the normalcomponent of Φ when Φ is in the condensed phase below the critical superfluidtemperature.Within the theory as described by Lagrangian 9 the mass is a fundamentalparameter. Since the free Φ terms ( L Φ , Eq.1) are not the reason that thisLagrangian is not the complete Lagrangian, there is good reason to believethat these terms will survive in the unknown complete theory, which wouldpreserve the conclusion that the DM aspect of Φ is fundamental.The MG aspect of Φ is associated with the superfluid BEC phase (i.e. themulti-component phase below the critical temperature, as opposed to thesingle-component phase above that temperature). ‘Phase’ reminds us of ther-modynamics, a paradigm example of an emergent theory. If we want to con-sider all interesting aspects of this phase, we may well have to go to the ther-modynamic limit. But the Thermodynamics of SFDM is a different theoryfrom the theory described by Lagrangian 8. In our evaluation of Φ accordingto the spacetime criteria, no thermodynamics was invoked. Lagrangian 8 suf-ficed. Within that single Lagrangian, θ and m are on a par, suggesting that θ is as fundamental as m . This conclusion receives further confirmation fromthe scale separation and dynamical separation criteria being problematic whenapplied to SFDM. The similar case of phonons in an atomic crystal suggeststhat the formal asymmetry criterion may also be violated. The metaphysicalasymmetry criterion is however satisfied in some sense.Thus, the lower half of Table 1, MG-fundamentalism and non-fundamentalism,is ruled out, but both of the views in the top half, (MG+DM)-fundamentalismand DM-fundamentalism, are still contenders for the correct interpretation ofΦ within SFDM. Perhaps a complete theory of SFDM will break the tie, orperhaps the MG aspect of Φ is simply emergent in some senses but not inothers—we will briefly return to this in Subsection 7.1. We turn to breakdown interpretations, a weak version of which we sympathisewith in the context of SFDM. The four interpretations in this group have incommon that they take the conceptual distinction between (dark) matter and(modified) spacetime to break down, albeit for different reasons and in differ-ent ways. Rynasiewicz has argued that the history from Newtonian physics toGR, via the concept of the aether and the development to field theory, alreadybreaks down the distinction between matter and space(time) as it was clas-sically conceived [6], but he does not distinguish between four different waysin which a breakdown can happen. We will pry them apart, since only theweakest version seems to apply to SFDM—and it may be expected that other heories satisfy some of the other interpretations but again not all of them. First consider three strong versions of a breakdown interpretation. On thestrongest of these three versions, the DM–MG distinction and by extension thematter–spacetime distinction is considered incoherent . Rynasiewicz seems tohint at such an interpretation when he summarises his argument as hoping “tohave established that the substantivalist-relationist controversy is not necessar-ily well formulated in every theoretical context” [6, p.293], and when he claimsthat “[i]nsofar as [the substantivalist-relationist controversy] is intended to haveanything to do with physics, ... it is no longer ... meaningful ” [6, p.279, ital-ics added]. On this incoherentist strong breakdown interpretation, the SFDMscalar Φ seeming to be both maximally matter-like and maximally spatiotem-poral is taken to reveal that it is in fact incoherent to distinguish these conceptsin the first place. This follows, for instance, if one takes the well-known con-tainer metaphor seriously. One might phrase this metaphor as “spacetime isthe container in which matter is contained” or, on a functional understanding:“to be spacetime is to contain and to be matter is to be contained” [78, p.161-167] [6, p.306]. It is indeed incoherent to say that Φ, playing both the spacetimeand matter role, is ‘contained in itself’, since containment is an irreflexive re-lation. The main argument of the prequel paper would become a reductio adabsurdum.Another motivation for this version of the strong breakdown interpreta-tion could be the assumption that it is essential to a conceptual spacetime–matter distinction that it is a strict dichotomy: every object is either (anaspect of) spacetime or (an aspect of) matter but never both. It seems to bethe case that this is what Rovelli has in mind when he claims, in the context ofGR, that the “very distinction between spacetime and matter is likely to be ill-founded [7, p.181]”. Rynasiewicz similarly considers whether the breakdown ofthe spacetime–matter distinction in the context of the electromagnetic aetherarises from the aether seeming to fall into both categories [6, e.g. p.286,290]. Inany case, if this strict dichotomy were required, Φ would be a straightforwardcounterexample.However, none of the criteria in our two families is necessarily motivatedby the container metaphor nor requires these families to constitute a jointlyexhaustive and mutually exclusive dichotomy. Nothing in the prequel paperdepended on these motivations. That being said, it is important to notethat the two families of criteria are not totally unrelated. Within the fam-ily of spacetime criteria, the (strong) geodesic and chronogeometricity criteria(Section 3; [19, § The consubstantiality thesis/ Trinitarianism (Section 4) has similarly been accused of beingincoherent [77]. If spacetime and matter are the only two available categories, the container metaphor becomesa special case of the strict dichotomy assumption. y identifying the matter fields, and then uses those as input for the referencesto matter in the criteria for spacetime. However, in the prequel paper onlytest particles, rods and clocks consisting of luminous matter were considered;in other words, consisting only of pure matter, i.e. fields that are only matter and not also spacetime . It may thus seem that in order to evaluate the crite-ria for being spacetime in the context of SFDM, one already needs to knowbeforehand which matter fields are not also spacetime ; inferential circularitylooms.In practice there is however no such inferential problem. In the con-text of SFDM, one starts by identifying all the matter fields without yetcaring whether they are pure or hybrid matter. These are the field Φ andthe luminous-matter fields ψ α . One then inspects the Lagrangian to deter-mine how these fields couple to the metric fields in the theory. In the su-perfluid regime, the ψ α couple to the physical metric ˜ g SF DMµν , which is con-structed from the Einstein metric and Φ (Eq.4). There is no analogous term, L matter ( g SF DMµν , Φ , Φ ; µ | ˜ g SFDM ), for Φ qua matter field; the free field Lagrangianfor Φ (see eqs. 1 & 9) refers (implicitly) to the Einstein metric instead. Thus,˜ g SF DMµν satisfies the spacetime criteria with respect to the matter fields ψ α andnot with respect to the matter field Φ. Moreover, in SFDM no further metricappears that is constructed from the ψ α fields. All this suffices to simulta-neously make the following two claims: 1) the most spacelike object in thetheory (in the superfluid regime) is ˜ g SF DMµν (which, remember, is constructedfrom Φ), and 2) the ψ α are pure matter. This leaves open two options, neitherof which imply an incoherent spacetime–matter distinction. Either (1) is de-cided to be sufficient to call ˜ g SF DMµν spacetime, making Φ a hybrid object, andthe ψ α the only pure matter fields (with respect to all of which ˜ g SF DMµν sat-isfies the spacetime criteria). Or, (1) is considered insufficient to call ˜ g SF DMµν spacetime, from which it follows that there is no spacetime at all in the theory;Φ and ψ α are all pure matter fields. Hence, in the latter scenario the space-time category would not be ill-defined, but just inapplicable to SFDM. Thiswould make SFDM amenable to the next interpretation to be discussed, notthe incoherence interpretation.In the prequel paper the first option was chosen. This may seem ad hoc.Isn’t the point of spacetime that it is ‘experienced’ universally by all matter?That intuition stems from times when all theorised matter was pure matter.It is silent on the question of how to extend the concept of the universality ofspacetime to theories with (tentatively) hybrid matter. If there is an optionthat ensures (coherence and) applicability and an option that does not, isn’t thecharitable thing to do to go with the former? (That is, assuming applicationof the spacetime category is at all useful—this will be questioned below, asuselessness is much weaker a charge than incoherence or even inapplicability.) The second-strongest breakdown interpretation grants the coherence of the(dark) matter and (modified) spacetime categories, but claims that they areinapplicable to the theory or field under consideration—for reasons beyond theprevious uncharitable claim of the inapplicability of the spacetime category. On his interpretation, there is nothing inherently wrong with these concepts, butit would be a category mistake to apply them to Φ. It is like finding out thatone should not use Aristotelian elements—earth, air, fire, water, aether—butNewtonian massive point particles, or, different still, quantum fields. Perhapsthis inapplicability of the traditional concepts of matter and spacetime is thecore of Rynasiewicz’s interpretation of the history from Newtonian physics toGR: [O]ne could just as well say that [the aether] is neither [space normatter], but instead belongs to a category of its own. [6, p.290]In the course of its development, physical theory simply lost touchwith the categories necessary for the original formulation of the[substantivalist-relationist controversy, i.e. space(time) and matter].[6, p.279]Present-day physicists do not employ a language that conforms withthe original contrast [between matter and space(time)] [6, p.301,italics in original] This raises an obvious question: what, if any, are the correct categories? Ry-nasiewicz and Earman do raise this question, affirming that there must beother such categories, but they do so only in the last paragraph of their paperand book, respectively, leaving it open what exactly the alternative option issupposed to be:Some have suggested, as has Earman [see e.g. [79, p.208]], thatwhat is needed now is a tertium quid to the traditional positions,something that will require an act of scientific creativity. I submitthat this act, or rather, series of acts, has already occurred, but,for a multitude of reasons, we have simply failed to recognize it. [6,p.306]It is important to point out though that, while the old categories may well be replaced by new categories, there is no guarantee that nor any urgent reasonwhy there would be new categories: the inapplicability of the old categoriesmay simply indicate that they are eliminated rather than replaced.Does this inapplicability interpretation apply to SFDM? Are the conceptsof matter and spacetime simply the wrong conceptual tools to understand thistheory? If so, what glasses, if any, should we use? What is this tertium quid,tentatively labeled “dark stuff”? Should we modus tollens the modus ponens ofthe prequel paper—‘if the families of criteria for spacetime and matter are ap-plicable, then Φ satisfies both’ ? We do not think that the inapplicability thesisis a viable interpretation of SFDM. For one thing, the category of (‘pure’) mat-ter still applies—to the luminous matter fields, Lagrangian 4, which are onlymatter and not also an aspect of spacetime. Similarly, the category of (‘pure’)spacetime is still expected to apply in the final SFDM theory to the Einsteinmetric in the regime T ≫ T c where we expect any modifications to be negligible(see the end of Section 2). Moreover, it is not the case that the proposed new See also [12, p.80]. ategory, “dark stuff”, is unrelated to the traditional categories. It is fairlydirectly related, by being the conjunction of both traditional categories. It isa hybrid category, rather than a novel, irreducible category. Hence, this caseis not at all analogous to, say, the move from using only Aristotelian elementsto using only strings. The inapplicability breakdown interpretation does notapply to Φ within SFDM, or at least not beyond the extent one may considerit to hold already in the case of plain GR. The inapplicability interpretationwould be more appropriate for objects that belong to neither category thanfor objects that belong to both categories. The final strong breakdown interpretation does grant that there is a coherentconceptual distinction between (dark) matter and (a modification of) space-time that applies to the field under consideration, but it is a conventional,rather than an objective fact, whether that field instantiates one or the otheraspect. This interpretation is of particular interest in light of the claimed dual-ity between f(R) gravity and particular Brans-Dicke-Theories and the furtherclaimed equivalence between the Einstein and Jordan frames/representations ofBrans-Dicke Theories [80]. Rynasiewicz [6] hints that the distinction betweenspace(time) and matter is conventional in a much larger context—and with-out even requiring the existence of a duality or equivalent representations—referring to it as a “verbal [dispute] occasioned by arbitrary preference” (p.279), a “ fa¸con de parler ” (p.287; italics in original), a mere “mode of ex-pression” (p.290), a matter of “taste” (p.290), “a matter of whim” (p.299),not a matter of fact (p.301). However, it is an objective fact in which phasethe field Φ in SFDM is, and hence an objective fact when it instantiates whichrole. The conventionalist strong breakdown approach is not appropriate for Φas it features within SFDM, or at least not more so than it may already be inplain GR.
This leads us to a weak version of the breakdown interpretation. The matterand spacetime categories may coherently and objectively be applied to objectsin physical theories, such as SFDM. Up to this point this interpretation agreeswith the fundamentalist interpretations and the consubstantiality interpreta-tion. The difference is that whilst on those interpretations this distinction isconsidered to be useful (and hence must be retained), on the weak breakdown Similar questions of conventionalism seem to arise in string theory in the context of the claimedequivalence between the string frame and the Einstein frame [81] (which are related by a conformaltransformation involving the dilaton). We would like to thank James Read for suggesting thisinteresting comparison. This is how North interprets Rynasiewicz [82]. Vassallo refers to the distinction as “a mere choice of vocabulary”, since it is “naive andperfectible” [20, p.354], and Rovelli states that it is merely a matter of “semantics”, “only a matterof choice of words, and thus, ultimately, personal taste” [12, p.77], but it is not clear whether theymean that it is conventional, or that it is dispensable (see the weak breakdown approach). nterpretation it is considered to be dispensable. Indeed, there does not seemto be a clear benefit provided by these categories, at least within SFDM. Theyare no longer useful, or, as Rynasiewicz puts it, no longer “fruitful” [6, p.291].It is then only a small further step to abandon such surplus vocabulary, on thegrounds of conceptual parsimony.What could be a reason to retain these concepts? We have already seentwo motivations—the container metaphor and viewing the spacetime–matterdistinction as the most basic metaphysical dichotomy such that everythingfalls into exactly one of these categories—which, if they ever were a motivationfor introducing the spacetime–matter distinction, definitely can not serve asreasons anymore in light of SFDM. One might insist that we still need thisdistinction to make sense of the (modified) Newtonian gravity and specialrelativistic regimes of theories such as SFDM and GR. These concepts may wellbe useful in those limits, just as the concepts of chairs, tigers and temperatureare, but that does not make these concepts fundamental, i.e. relevant for thefundamental description of reality, just as chairs, tables and temperature arenot.Mentioning Newtonian Gravity just now does remind us of the ‘work’ thatthe concept(s) of spacetime (and matter) did in that theory: space(time) pro-vided a distinction between inertial and non-inertial motion (of all matter).Hence, when the (strong) geodesic criterion ( §
3) is included in the set of crite-ria that allow interpreting something as spacetime, it may seem that the weakbreakdown interpretation becomes analytically false: if an object is spacetimewith respect to test particles of matter in the sense that the trajectories ofall those test particles follow the geodesics of that object (and non-test parti-cles do not) then that concept of spacetime so instantiated automatically does‘work’ in that it distinguishes between inertial and non-inertial motion of allmatter, and hence the concept deserves to be retained. But this is only correctin a restricted sense. The qualifiers ‘all’ are crucial. As mentioned earlier inthis section, the prequel paper concluded that ˜ g SF DM satisfies the criteria forbeing spacetime with respect to test particles (and rods and clocks) made outof luminous matter, which is pure matter, i.e. fully matter but not at all space-time. But what about test particles made out of Φ? They couple to (only) theEinstein metric rather than to ˜ g SF DM and hence the geodesic criterion is notsatisfied for the combination of ˜ g SF DM and Φ-test-matter. And it’s exactlythe conceptually novel Φ field that we are trying to interpret. And it is notthe case that Φ, qua matter, experiences what we had identified as spacetime(˜ g SF DM ) nor is it true that Φ, qua aspect of spacetime (via ˜ g SF DM ), providesa distinction between inertial and non-inertial motion for all matter. Thus,this specific type of ‘work’, that we are used to being done by the concept(s)of spacetime (and matter), does not extend fully to hybrid theories such asSFDM.Alternatively, in response to the question of what reason one may haveto retain the concepts of spacetime and matter, one might consider the ideathat (modified) spacetime and (dark) matter have different explanatory roles toplay. One example would be the claim that spacetime and matter stand in an Depending on one’s favourite model of explanation, the previous suggestion—providing a dis- symmetric explanatory relation to each other. For instance, consider theorieswith a dynamical metric and other fields such that if we create test particles,light rays, rods and clocks out of these other fields, the metric satisfies the(strong) geodesic criterion (i.e. if test particles and/or light rays were aroundthey would, for any choice of initial conditions, follow the timelike and nullgeodesics, respectively, of this metric) and the chronogeometricity criterion(i.e. local validity of special relativity) with respect to those other fields (see[19]). We may then want to call the metric ‘spacetime’ and those other fields‘matter’. On the so-called ‘dynamical approach’ to this metric, it is because of properties of those matter fields and the way in which they are coupledto the metric that the metric field obtains its chronogeometric meaning (andpresumably also its inertial meaning, and thereby thus full spatiotemporality)[21]. Properties of matter fields (and the way in which they are coupled to themetric) explain why matter—rods, clocks, test particles—surveys the metricfield (which therefore justifies calling the metric ‘spacetime’). On the opposingapproach, the ‘geometrical approach’, the arrow of explanation is reversed:the metric is primitively spatiotemporal, in that it constrains the properties ofthe other fields such that those survey the metric field (in the same sense asbefore, i.e. the (strong) geodesic and chronogeometricity criteria). The metricfield thus explains those properties and thereby justifies referring to those fieldsas ‘matter’.Does this move apply to SFDM? We contend that it does not, regardlessof one’s preference for the direction of the arrow of explanation. Consider firstwhether Φ might be both spacetime and matter (as indeed suggested by our twofamilies of criteria which were silent on any (further) explanatory asymmetry).In other words, could it be that Φ is matter and g SF DMab (Φ) spacetime in thespecific sense of the previous paragraph? This would be guaranteed by a Φterm in the SFDM Lagrangians analogous to the last terms in Eqs. 8 and 9,i.e. L matter ( g SF DMµν , Φ , Φ ; µ | ˜ g SFDM ). There is no such term. There may be anarrow of explanation between g SF DMab (Φ) and luminous matter, but there isno parallel arrow between g SF DMab (Φ) and dark matter (Φ). Could we thenperhaps view Φ as being only spacetime but not matter, in the explanatorysense of the previous paragraph? Not only would this be to ignore that Φ scoredmaximally on our family of matter criteria, but also, once again, that as faras we know the MG-aspects only appear in the superfluid regime, and we donot yet have a complete and fully satisfactory fundamental SFDM Lagrangianavailable to determine whether Φ is in any sense a modification of spacetime atthe fundamental level. This specific attempt to justify a (modified) spacetimevs. (dark) matter distintion via differing explanatory roles is of no avail.Until now we have considered theory-internal senses of the ‘work’ that thecategories (dark) matter and (modified) spacetime might be doing. Perhapstheir usefulness arises at a more pragmatic, theory-external level: a heuris-tic role in theory-development. We will discuss this issue—the usefulness ofthe conceptual distinction between (dark) matter and (modified) spacetime atthe level of the space(s) of theories—in Subsection 7.2. However, this move tinction between inertial and non-inertial motion—may also be an instance of this ‘alternative’suggestion of considering the explanatory roles of spacetime (and matter). rom a theory-internal to a theory-external usefulness of the distinction is todeclare it dispensable within SFDM considered in itself. As it stands we thussympathise, within the group of breakdown interpretations, with the weakbreakdown interpretation of the DM/MG distinction in the context of SFDM.Nevertheless, one may argue that one specific feeling of unease remains on thisinterpretation. Historically, theories that have been labeled as modificationsof gravity/spacetime have done very well at explaining galactic data but notso well at explaining or even accounting for data at larger scales, and viceversa for theories that have been labeled as dark matter. This may give theimpression that one necessarily needs something spatiotemporal to explain thegalactic data and something (dark) matter-like to explain data at larger scales.Could it be these distinct explanatory roles that could be used to re-establish afundamental conceptual distinction between (modified) spacetime and (dark)matter? This is the central question of our next project. On the other hand,perhaps the lesson to be learnt from SFDM is that it is a crucial counterex-ample to this historical trend, highlighting its contingency. In the prequel paper it was argued that the novel field Φ postulated by Berezhi-ani and Khoury’s ‘superfluid dark matter theory’ not only maximally fits theputative conceptual category of (dark) matter, but it also maximally fits—atleast in the superfluid regime—the putative conceptual category of (a mod-ification of) spacetime. This paper has proposed a chart of nine—or ten, ifthe demi interpretation is included for the sake of completeness— prima facie possible interpretations of such Janus-faced objects and their theories, struc-tured into three groups (Table 2). A first evaluation of those interpretationsfor the specific case of the scalar field Φ featuring in SFDM has left four ofthose options on the table as viable interpretations of Φ: the consubstantialityinterpretation, the two fundamentalist interpretations that take at least theDM aspect to be fundamental, and the weak breakdown interpretation.
If such cartography of the space(s) of theories is to present us with a descriptionand ultimately an understanding of the space(s) of theories, we need not amere list of interpretations, but a full chart that includes how they are ‘located’relative to each other. While all these interpretations are mutually inconsistent,some pairs are closer in spirit to each other than other pairs. This subsectionexplores the subtle relations between the various interpretations.One might think that the consubstantiality interpretation is nothing morethan a repetition of the result of the prequel paper, and that it is therefore thebasis of all the other eight interpretations (besides the demi interpretation).The latter statement does not follow, since the incoherence and inapplicabilityversions of the strong breakdown interpretation disagree with the results ofthat paper. Moreover, the consubstantiality interpretation has more to it, roup (Sub)Name Description T h e o l og i c a l a n a l og i e s Consubstantiality The field Φ is both fully (dark) matter and fully (amodification of) spacetime—both roles are on a par.This leaves the conceptual categories of (dark) matterand (modified) spacetime and their distinctionunaffected (i.e. coherent, applicable, objective and useful).Demi The field Φ is partially (dark) matter and partially (amodification of) spacetime. This leaves the conceptual X categories of (dark) matter and (modified) spacetimeand their distinction unaffected (i.e. coherent, applicable,objective and useful). F und a m e n t a li s m (MG+DM)-fundamentalism The field Φ is fundamentally both (dark) matter(or: Dualism) and (a modification of) spacetime.DM-fundamentalism The field Φ is fundamentally (dark) matter, but onlyemergently (a modification of) spacetime.MG-fundamentalism The field Φ is fundamentally (a modification of) space- X time, but only emergently (dark) matter.Non-fundamentalism The field Φ is merely emergently (dark) matter as well(or: Neutral monism) as (a modification of) spacetime; fundamentally it is X something else altogether. B r e a k d o w n Strong Incoherence The field Φ reveals that there is no coherent conceptual X distinction between (dark) matter and (modified) spacetime.Inapplicability The field Φ reveals that the (dark) matter and (modified) X spacetime categories, despite being coherent, are thewrong conceptual categories for SFDM.Conventionalism There is a coherent conceptual distinction beween (dark) X matter and (modified) spacetime, but it is conventional(i.e. not objective) which role applies to the field Φ.Weak There is a coherent, objective conceptual distinctionbetween (dark) matter and (modified) spacetime thatapplies to the field Φ, but it is dispensable/useless. It isthen dispensed with, on grounds of conceptual parsimony. Table 2: Chart of possible interpretations of (theories with) fields that do notfall into exactly one of two putative categories, i.e. (dark) matter and (modified)spacetime. A red X indicates that the interpretation is not a viable interpretationof the scalar field Φ in SFDM. 32 aking it inconsistent with all the other interpretations. For one thing, itinsists that the DM/MG distinction is objective and useful, contra the wholeclass of breakdown interpretations. How does it relate to the fundamentalistinterpretations? On the consubstantiality interpretation, the senses in which Φinstantiates the DM and MG roles are on a par. It is thus in some sense closer tothe spirit of the egalitarian interpretations than the elitist interpretations. Thiscould easily be mistaken as agnosticism about the egalitarian interpretations—as long as one of those is viable, the consubstantiality interpretation is viable.Similarly, an agnostic version of the demi interpretation is consistent with anyof the four fundamentalist interpretations. In order for all ten interpretations tobe mutually exclusive, we focus on versions of the consubstantiality and demiinterpretations that embrace quietism about the fundamental vs. emergentnature of the DM and MG aspects of Φ. There is then simply no fact of thematter about the fundamentality of the DM and MG roles, just as one mighthold that it is not a meaningful question to ask whether Jesus is more, equallyor less fundamentally divine than he is human.The decision between the consubstantiality, DM-fundamentalist and dual-ist interpretations of SFDM thus depends on whether there is a matter of factabout the fundamentality of each of the two roles that Φ instantiates. Thatthe latter two interpretations have not yet been ruled out may be seen as rul-ing out the consubstantiality interpretation. Especiallly their agreement thatthe DM role is a fundamental aspect of Φ may seem funest for the consub-stantiality interpretation. However, it is not so much the case that both theDM-fundamentalist and dualist interpretations are unproblematic, but ratherthat the truth, if any, seems to lie in between. The MG aspect of Φ fits someof the aspects of “emergence”, but not others. Perhaps the complete theory ofSFDM will solve this. If not, it may be that a different understanding of theconcept of emergence is needed, or that a new hybrid fundamentalist view isrequired, or maybe we should take this problem to vindicate the consubstan-tiality interpretation, although that seems in tension with the unproblematicclaim that the DM aspect is fundamental.All this being said, there is one sense in which the consubstantiality and fun-damentalist interpretations are closely related to each other when contrastedwith the breakdown interpretations. The former but not the latter consider theDM/MG distinction to be coherent, applicable, objective and useful. Withinthe group of breakdown interpretations, the one that is closest to the other fiveis nevertheless the only one that is still viable: the weak breakdown interpre-tation. In contrast to the other breakdown interpretations, this interpretationdoes allow one to use the DM/MG distinction (non-conventionally). On thisinterpretation it is allowed to ponder the question of whether each role is in-stantiated fundamentally or emergently, but this would not be more than afun academic exercise. The issue is moot, it is pointless, if the distinction isnot useful.The decision between the weak breakdown interpretation on the one hand,and the other three remaining viable approaches on the other hand, depends onthe usefulness of the DM/MG distinction. In the discussion of the weak break-down approach, several proposals for the use of the DM/MG and spacetime–matter distinctions within SFDM have been argued against. The onus is thus n the other interpretations to provide a proposal for the usefulness of thesedistinctions within SFDM. We have hinted that distinct explanatory powers,for instance with respect to galactic data vs data from clusters and cosmology,may provide such a role. Without such a concrete theory-internal proposalhowever, the contest between the weak breakdown interpretation on the onehand, and the other three remaining viable approaches on the other hand,is perhaps most interesting at the level of the space(s) of theory, as will bediscussed in the next subsection.Note finally the connection between the inapplicability version of the break-down interpretations and neutral monism. Both agree that the categories areinapplicable at the fundamental level. Their difference though is that this iswhere it stops for neutral monism, which still acknowledges their applicabilityat the emergent level, whereas the inapplicability applies universally accordingto the inapplicability breakdown interpretation. It is this difference that makesthe former compatible with the conclusion of the prequel paper, but not thelatter. What are the upshots of all of this? More precisely, what are the upshots ofthis general chart of interpretations and what is the relevance of applying it toSFDM specifically?Perhaps the most important upshot is the realisation that there is a singlespace of theories, rather than a space of spacetime theories and a space ofmatter theories. This single space is not reducible to (just) those two sepa-rate spaces. This is something that could have been realised directly from thetwo families of criteria for being matter and being spacetime, respectively, butwhich had not been sufficiently noticed due to the widespread focus on fieldsthat are (thought to be) purely matter or purely spacetime. (Which raisesthe question: which other conceptual mistakes have we made by focusing onlyon our favourite theories?) Interpreting SFDM has helped highlight this real-isation, which was already noted in Section 6: the distinction between (dark)matter and (modified) spacetime is not a strict dichotomy, i.e. the distinctiondoes not necessitate that all objects fall into exactly one of these categories.Why did we introduce two independent families of criteria for spacetime andmatter respectively? If spacetime and matter are supposed to form a strictdichotomy, would we then not expect a single criterion such that if the crite-rion is satisfied by an object, then that object is a spacetime, and otherwiseit is matter, or vice versa? There is no reason to expect that an objectsatisfying any of the spacetime criteria does not (or does) satisfy any of thematter criteria, and vice versa. There is no conceptual link between the two Even if, for independent reasons, there would be multiple spaces of theories after all, it is stillthe case that none of those is a pure space of spacetime theories or a pure space of matter theories,nor do they reduce to one of these pure spaces. The case for a strict dichotomy gets even worse if we move away from necessary and sufficientcriteria for interpreting something as matter or spacetime, but instead consider these categories tobe cluster concepts [83]. amilies of criteria, beyond the reference of the spacetime criteria to matter(e.g. test particles, rods and clocks). In the space of theories, theories withobjects that fall under exactly one family of criteria may well be the exception;the prima facie expectation should be to also encounter objects that resembleboth matter and spacetime, or neither. History has been such that two earlytheories, Newtonian physics and special relativity, were exceptions in this re-gard, rather than the norm. We should not be mislead into believing in a strictspacetime–matter dichotomy merely because of our familiarity with these earlytheories.In other words, the urge to project pure spacetime and pure matter cate-gories onto theories, such as for instance SFDM, and insisting that everythingfalls into exactly one of these categories, is an artefact of our familiarity withNewtonian theories and Special relativity. Had SFDM been the only theorywe had known, insisting on applying these pure categories would have seemedas strange as someone insisting that we use Goodman’s bleen and grue [84] todescribe trees and the sky rather than blue and green—there is nothing inco-herent about using bleen and grue, but they just are not particularly usefulconcepts. If the first few theories or observations we came across had beensuch that all red things were square or triangular and all blue things circular,we might have developed terms such as ‘POLYGON’ and ‘BLUE’ for theserespective groups of objects. If there were no observations of further objectsnot falling into these categories, we might have surmised that the most basicconceptual distinction is the strict POLYGON-BLUE dichotomy. However,once we would make further observations and/or further explore the space oftheories, we might realise that there is no reason why red things could not becircular, and blue things squared or triangular, and that the POLYGON-BLUEpattern in our first theory/data was an exception rather than the norm. Wewould immediately realise, if we hadn’t already (and of course we could have,simply by analysing the concepts of polygon and blue), that this distinction,however coherent, is not particularly useful for fundamental metaphysics—itdoes not track anything deep about the space of theories. In particular, wewould realise that there is a single space of theories with theories/objects thatare/contain blue squares, blue circles, red squares, red circles, partially blueand partially red triangles, etc —rather than all theories falling either into aspace of POLYGON theories or a space of BLUE theories.These two companion papers contribute to the ‘cartography research programme’—a generalisation of the research programme previously advocated by one of usfor the space of spacetime theories [85]—which aims to help us navigate thesingle space of theories. Analysing individual theories and the similarities anddifferences of neighbouring theories helps us to better understand the space oftheories as a whole, which in turn helps us better understand further individualtheories and pairs of theories, and also to re-evaluate familiar theories, suchas GR. The cartography research programme advocates a dynamic back-and-forth between both levels of analysis. Here we developed a new dimension to One might retort that this leaves open the possibility that there is a different distinction thatis the most basic conceptual distinction, for instance one based on size vs. hue. However, it is hardto see why that wouldn’t face the same problems as the POLYGON-BLUE dichotomy. he cartography of the space of theories, distinct from the focus of previouscartographers [85, § It would then turn out, a posteriori , that there is inpractice, i.e. extensionally, an almost strict dichotomy between pure spacetimeand pure matter. Keeping this conceptual distinction would then be useful asa basic categorisation principle. If there were many Jesusses and few or nomere humans and mere gods, it would not be as helpful to stick to those twocategories as the most basic conceptual categories. Similarly, if the regionsof pure spacetime theories (such as, perhaps, certain Palatini approaches tomodified gravity [86]) and pure matter theories (such as, perhaps, certain ex-amples of WIMP dark matter) in the space of theories turn out to be small, thespacetime–matter distinction does not play as useful a role in the cartographyof that space.All this relates to the issue of whether the spacetime–matter distinctionearns its spurs as a theory-external, heuristic tool. Given the lessons learntfrom SFDM, it seems that focusing on two separate spaces of theories has ac-tually hindered theory development, as it has blinded us to potentially vast re-gions in the space of theories. The case may thus be worse than the spacetime–matter distinction having been neutral when it comes to its fruitfulness for the-ory development: letting go of the distinction may actively help us in exploringthe space of theories. And even if there is a unique theory that describes ourworld and that theory contains only elements that are pure matter or purespacetime, we can learn about that theory by studying how it differs fromhybrid theories.In a similar vein, a strict adherence to a DM/MG dichotomy has not beenfruitful with regards to the collaboration between the DM community andthe MG community. The second upshot of these companion papers is thatit undermines the somewhat infamous hostility between these communities. Or, if, say, DM-fundamentalism is applicable to the majority of theories, one may be temptedto conclude that, in practice, only the (dark) matter category is conceptually fundamental andthat the single space of theories is, basically, a space of matter theories. Note, however, thatit is not obvious that a difference in relative fundamentality between two roles means that themore fundamental concept or the distinction between the two concepts is doing any ‘work’ in anyrelevant sense. pon realisation that a field being dark matter does not exclude it being alsoan aspect of spacetime, and vice versa, it becomes clear that allowing thatthe other community may be on the right track does not necessitate one’sown community being misguided. The lack of collaboration between the com-munities of course also depends on many other historical, social and economicfactors—e.g. differences in training, differences in funding—and (resulting) dif-ferences in preferred guiding principles and theoretical virtues (e.g. accounts ofscientific explanation). Hence, it is implausible that removing the conceptualincompatibility will by itself build a full bridge between the communities, buthopefully it opens the door, if just a little bit. A stronger focus on hybridtheories may help develop a trading zone [87, Ch.9] between the DM and MGcommunities [88].The third upshot of these companion papers—besides the move to a sin-gle space of theories and the effects on the sociology of the DM and MGcommunities—concerns their implications for one of the largest debates in thephilosophy of physics, that between substantivalism and relationalism aboutspacetime. As this debate presupposes a conceptual distinction between space-time and matter, it becomes moot for theories to which any of the breakdowninterpretations applies. On the (quietist—albeit not the agnosticist—version ofthe) consubstantiality interpretation it is claimed that nothing further can besaid beyond the claim that the field under consideration is both (dark) matterand (modified) spacetime and that these roles are on a par. This includesquietism on the issue of relative fundamentality that the substantivalism–relationalism debate is concerned with. Could the fundamentalist interpre-tations save the debate? In fact, if the debate is being glossed over as beingabout the relative fundamentality of spacetime and matter, it may even seemthat the fundamentalist interpretations just are different positions within thatdebate: DM-fundamentalism is relationalism, MG-fundamentalism is monis-tic substantivalism (also known as supersubstantivalism, dualism is (dualistic)substantivalism, and neutral monism is either so-called ‘emergent spacetime &matter’ or so-called ‘super-emergent spacetime’ [89].However, if a single object is both spacetime and matter, it cannot bethat, qua substance(s), this object is relatively fundamental or emergent to it-self, as relative fundamentality is an irreflexive relation. Moreover, the debateconcerns the relata of spatiotemporal relations: are they spacetime points ormatter? DM-fundamentalism, for instance, is perfectly compatible with theview that a field such as Φ is a matter field that is both ‘living on’ the funda-mental spacetime points of an underlying manifold and instantiating, in someemergent regime, the role of a modification of the structure of that manifold.This does not seem to resemble relationalism. Finally, an important differencein consequence that traditionally distinguishes substantivalism from relation-alism is that the former position considers vacuum solutions to be physicallypossible whereas the latter typically (but not always) judges them to be im-possible. However, it is not even clear whether one can ask whether vacuumsolutions are possible in theories with hybrid objects, since it is not clear in thefirst place what vacuum solutions are supposed to be in such theories. If oneobtains a vacuum solution by removing all matter and including all spacetimestructure, then Φ would have to be both included and excluded to obtain a acuum solution of SFDM. A detailed evaluation of (the feasibillity of) thesubstantivalism–relationalism debate in the context of hybrid theories such asSFDM goes beyond the scope of this paper. Suffice it to say that in the worstcase scenario the debate becomes ill-defined and in the best case scenario it be-comes even more complex than it had already become in the context of modernphysics.Finally one may ask why these companion papers focused on SFDM and noton, say, GR? Does GR not stand a better chance at being empirically adequate?Should we not focus all our effort on interpreting GR? Several responses areappropriate here. First of all, suggesting that there are no empirical resultsspeaking against GR is to beg the question against pure MG approaches, whichclaim that the empirical discrepancies that GR-advocates try to account forwith DM actually show the empirical inadequacy of GR. That being said,SFDM is not a pure MG approach and we expect it to coincide with GR at T ≫ T c (see the end of Section 2). Hence, analysing SFDM does not mean thatone is not also analysing aspects of GR. More generally, one can often learnmore about theories, even familiar ones such as GR, by asking interpretativequestions about other, neighbouring theories and comparing and contrastingthe answers with the answers known from the familiar theories; at the veryleast it teaches us what is special about our mainstream theories and what isnot. Finally, having a single theory, such as SFDM, that doesn’t fit a strictDM/MG dichotomy, suffices to claim that there is a single space of theories thatis irreducible to a space of spacetime theories and a space of matter theories—whether that theory is ultimately the correct description of our world doesnot change that. We have simply chosen SFDM because Φ, being massive, ismore clearly and less controversially a hybrid object than the metric in GR is(see e.g. Section 4 and [19, § § the cosmological constant, black holes, unified field theories, su-persymmetric theories, and theories of quantum gravity . Within the car-tography research programme, progress requires a continuous back and forthbetween analysis of individual theories and analysis of the space of theories.Another interesting follow-up would be to determine which modifications ofSFDM would make Φ more or less of a dark matter field, and more or less of Read et al. seem to suggest that TeVeS itself may be a hybrid theory [90]. For potentialfurther, interesting case studies, see [91–102]. A particularly interesting example could be the dilaton in string theory. We would like tothank an anonymous reviewer for this suggestion. modification of spacetime. Extending the current context and analysingother contexts are however work for another day. We hope to have providedsome useful tools—possible interpretations and arguments in favour of andagainst various versions of these interpretations—in exploring this map.
There is a single space of theories, which is not reducible to (just) a spaceof spacetime theories and another space of matter theories. This conclusionwas reached via Superfluid Dark Matter Theory, which has helped highlightan important feature of the families of criteria for being matter and for beingspacetime (which were reported in the prequel paper): they do not not forma strict dichotomy, but allow objects to satisfy criteria of either one family,of both, or of neither. The ‘cartography research programme’ aims to anal-yse individual theories or pairs of theories in the space of theories in orderto better understand the space of theories as a whole, which in turn helpsus understand further individual theories and pairs of theories. This paperhas provided a (non-exhaustive) chart of nine interpretations—structured intothree groups—that may help navigate the space of theories. The chart wasillustrated by applying it to the novel scalar field Φ within SFDM, which wasconcluded in the prequel paper to be as much (dark) matter as it could possi-bly be, and—at least in the superfluid regime—as much (of a modification of)spacetime as it could possibly be. Five of the nine interpretations have beenruled out for Φ; at least one interpretation within each of the three groups re-mains on the table: the consubstantiality interpretation, DM-fundamentalism,(MG+DM)-fundamentalism (also known as dualism), and the weak break-down interpretation. Although it is expected that finding a complete theoryof SFDM may further narrow down the interpretational options, the arenawhere the ultimate fate of the DM/MG distinction and the broader matter–spacetime distinction, i.e. of our Democritean-Newtonian goggles, will be de-cided is likely the space of theories. We hope to have provided some usefulcartographic tools—possible interpretations of Janus-faced theories and argu-ments in favour of and against these interpretations—towards this endeavour.Everyone is cordially invited to contribute to the cartography research pro-gramme. Physicists from the DM and MG communities are encouraged toconsider focusing on a single space of theories as an overarching framework,and to use hybrid theories as a bridge across the gap between the communities. For instance, a very recent modification of SFDM by Berezhiani, Famaey & Khoury [30] couplesphotons not to ˜ g µν but to g µν . Prima facie , this undermines our claim that Φ is a modificationof spacetime [19, section 5.2], and may seem to release the pressure on the matter–spacetimedistinction by rendering Φ a pure matter field. On second thought, it may indicate either a)that there are several spacetimes—one experienced by photons, the other by the other luminousmatter—or b) that both g µν and ˜ g µν are structural aspects of a single underlying spacetime; or c)that there is no spacetime at all in SFDM (which prompts the question of how to categorise g µν and ˜ g µν ); or d) that the strong geodesic criterion should be weakened to exclude light rays, i.e. itshould be replaced by what we call the weak geodesic criterion [19, section 3.2]. Neither optionpaints a simple picture of the spacetime–matter distinction. See also [101, 103]. hilosophers of physics are encouraged to consider the interpretational ques-tions voiced in these companion papers in different contexts, i.e. for differentpoints and pairs of (neighbouring) points in the space of theories, to considerthe feasibility of the substantivalism–relationalism debate when it comes tohybrid theories, and to enrich the interpretational chart; in other words, tocolour in the rough sketch of a map provided here. Acknowledgements
We would like to acknowledge support from the DFG Research Unit “TheEpistemology of the Large Hadron Collider” (grant FOR 2063). Within thisresearch unit we are particularly indebted to the other members of our ‘LHC,Dark Matter & Modified Gravity’ project team—Miguel ´Angel Carretero Sahuquillo,Michael Kr¨amer and Erhard Scholz—for invaluable and extensive discussionsand comments on many iterations of this paper. We would furthermore like tothank Florian Boge, Jamee Elder, Alex Franklin, Tushar Menon, James Read,Katie Robertson, Kian Salimkhani, Michael St¨oltzner and Adrian W¨uthrich forvaluable discussions and comments, as well as the audiences of the Dark Matter& Modified Gravity Conference (Aachen, Germany, 2019), the German Societyfor the Philosophy of Science Conference (Cologne, Germany, 2019) and theFirst Oxford-Notre Dame-Bonn Workshop on the Foundations of SpacetimeTheories (Oxford, UK, 2019).
References [1] James L. Anderson.
Principles of Relativity Physics . New York andLondon: Academic Press, 1967.[2] J. Earman and J.D. Norton. What price spacetime substantivalism? thehole story.
British Journal for the Philosophy of Science , 38:515–525,1987.[3] Tim Maudlin. The essence of spacetime. In
PSA: Proceedings of theBiennial Meeting of the Philosophy of Science Association, Volume Two:Symposia and Invited Papers , 1988.[4] Richard P. Feynman.
Feynman Lectures on Gravitation . Addison-WesleyPublishing Company, 1995.[5] Carl Hoefer. The metaphysics of space-time substantivalism.
The Jour-nal of Philosophy , 93(1):5–27, 1996.[6] Robert Rynasiewicz. Absolute versus relational space-time: An out-moded debate?
Journal of Philosophy , 93(6):279–306, 1996.[7] Carlo Rovelli. Halfway through the woods: Contemporary research onspace and time. In John Earman and John Norton, editors,
The Cosmosof Science , pages 180–223. University of Pittsburgh Press, 1997.[8] Mauro Dorato. Substantivalism, relationism, and structural spacetimerealism.
Foundations of Physics , 30(10):1605–1628, 2000.
9] Edward Slowik. On the cartesian ontology of general relativity: Or, con-ventionalism in the history of the substantival-relational debate.
Philos-ophy of Science , 72:1312–1323, 2005.[10] Mauro Dorato.
Is Structural Spacetime Realism Relationism in Dis-guise? The supererogatory nature of the substantivalism/relationism de-bate , pages 17–37. Elsevier Science, 2008.[11] Dennis Lehmkuhl. Is spacetime a gravitational field? In Dennis Dieks,editor,
The Ontology of Spacetime II , volume 4 of
Philosophy and Foun-dations of Physics , chapter 5, pages 83 – 110. Elsevier, 2008.[12] Carlo Rovelli.
Quantum Gravity . Cambridge University Press, 2010.[13] Dennis Lehmkuhl. Mass-Energy-Momentum: Only there Because ofSpacetime?
The British Journal for the Philosophy of Science ,62(3):453–488, 06 2011.[14] David Rey. Similarity assessments, spacetime, and the gravitational field:What does the metric tensor represent in general relativity? 2013.[15] James Read. Functional gravitational energy.
The British Journal forthe Philosophy of Science , 71:205–232, 2020.[16] Patrick M. D¨urr. It ain’t necessarily so: Gravitational waves and energytransport.
Studies in History and Philosophy of Science Part B: Studiesin History and Philosophy of Modern Physics , 65:25 – 40, 2019.[17] Patrick M. D¨urr. Fantastic beasts and where (not) to find them: Localgravitational energy and energy conservation in general relativity.
Studiesin History and Philosophy of Science Part B: Studies in History andPhilosophy of Modern Physics , 65:1 – 14, 2019.[18] Patrick D¨urr. Against ‘functional gravitational energy’. Forthcoming inSynthese.[19] [authors omitted]. Dark matter = modified gravity? scrutinising thespacetime–matter distinction through the modified gravity/ dark matterlens.[20] Antonio Vassallo. A metaphysical reflection on the notion of backgroundin modern spacetime physics. In Laura Felline, Antonio Ledda, FrancescoPaoli, and Emanuele Rossanese, editors,
New Directions in Logic and thePhilosophy of Science , pages 349–365. College Publications, LightningSource, Milton Keynes, UK, 2016.[21] Harvey R. Brown.
Physical Relativity: Space-time structure from a dy-namical perspective . Oxford University Press, 2005.[22] Gordon Belot. Background-independence.
General Relativity and Grav-itation , 43(10):2865–2884, Oct 2011.[23] Hilary Greaves. In search of (spacetime) structuralism.
PhilosophicalPerspectives , 25:189–204, 2011.[24] Dennis Lehmkuhl. The metaphysics of super-substantivalism.
Noˆus ,52(1):24–46, 2018.
25] Lasha Berezhiani and Justin Khoury. Theory of dark matter superfluid-ity.
Phys. Rev. D , 92:103510, Nov 2015.[26] Lasha Berezhiani and Justin Khoury. Dark matter superfluidity andgalactic dynamics.
Physics Letters B , 753:639–643, 2016.[27] Edward Slowik.
Substantivalism and Relationism as Bad Cartography:Why Spatial Ontology Needs a Better Map , pages 185–198. SpringerInternational Publishing, Cham, 2018.[28] James F. Annett.
Superconductivity, Superfluids and Condensates . Ox-ford University Press, 2004.[29] D.T. Son. Low-energy quantum effective action for relativistic superrflu-ids. 2002.[30] Lasha Berezhiani, Benoit Famaey, and Justin Khoury. Phenomenologicalconsequences of superfluid dark matter with baryon-phonon coupling. arXiv preprint arXiv:1711.05748 , 2017.[31] Jacob D. Bekenstein. Relativistic gravitation theory for the MONDparadigm.
Phys. Rev. , D70:083509, 2004. [Erratum: Phys.Rev.D71,069901(2005)].[32] Timothy Clifton, Pedro G. Ferreira, Antonio Padilla, and ConstantinosSkordis. Modified gravity and cosmology.
Physics Reports , 513(1):1 –189, 2012. Modified Gravity and Cosmology.[33] Jacob D. Bekenstein. Relation between physical and gravitational geom-etry.
Phys. Rev. D , 48:3641–3647, Oct 1993.[34] Sabine Hossenfelder. Covariant version of verlinde’s emergent gravity.
Physical Review D , 95(12):124018, 2017.[35] Eleanor Knox. Abstraction and its limits: Finding space for novel ex-planation.
Noˆus , 50:41–60, 2016.[36] Alexander Franklin and Eleanor Knox. Emergence without limits: Thecase of phonons.
Studies in History and Philosophy of Modern Physics ,64:68–78, 2018.[37] Henry Margenau. The exclusion principle and its philosophical impor-tance.
Philosophy of Science , 11(4):187–208, 1944.[38] Steven French and Michael Redhead. Quantum physics and the identityof indiscernibles.
The British Journal for the Philosophy of Science ,39(2):233–246, 1988.[39] David Malament. In defense of dogma: Why there cannot be a relativis-tic quantum mechanical theory of (localizable) particles. In R. Clifton,editor,
Perspectives on Quantum Reality . Kluwer Academic Publishers,1996.[40] Hans Halvorson and Rob Clifton. No place for particles in relativisticquantum theories?
Philosophy of Science , 69(1):1–28, 2002.[41] S. French and D. Krause.
Identity in Physics: A Formal, Historical andPhilosophical Approach . Oxford University Press, 2006.
42] B. Falkenburg.
Particle metaphysics. A critical account of subatomicreality . Berlin/Heidelberg: Springer, 2007.[43] F. A. Muller and Simon Saunders. Discerning fermions.
The BritishJournal for the Philosophy of Science , 59(3):499–548, 2008.[44] F. A. Muller and M. P. Seevinck. Discerning elementary particles.
Phi-losophy of Science , 76(2):179–200, 2009.[45] M. Kuhlmann.
The ultimate constituents of the material world. In searchof an ontology for fundamental physics . Heusenstamm: Ontos Verlag,2010.[46] Laura Ruetsche.
Interpreting quantum theories . Oxford/New York: Ox-ford University Press, 2011.[47] Adam Caulton. Discerning “indistinguishable” quantum systems.
Phi-losophy of Science , 80(1):49–72, 2013.[48] Nick Huggett and Josh Norton. Weak Discernibility for Quanta, theRight Way.
The British Journal for the Philosophy of Science , 65(1):39–58, 03 2013.[49] F.A. Muller. Elementary particles and metaphysics. In W.J. Gonza-lez S. Hartmann T. Uebel M.C. Galavotti, D. Dieks and M. Weber,editors,
New directions in the philosophy of science , pages 417–431.Cham/Heidelberg: Springer, 2014.[50] Florian J. Boge.
Quantum Mechanics Between Ontology and Epistemol-ogy . Springer, 2018.[51] James Robert Brown. How do feynman diagrams work?
Perspectives onScience , 26(4):423–442, 2018.[52] Mauro Dorato and Emanuele Rossanese. The nature of representationin feynman diagrams.
Perspectives on Science , 26(4):443–458, 2018.[53] Letitia Meynell. Picturing feynman diagrams and the epistemology ofunderstanding.
Perspectives on Science , 26(4):459–481, 2018.[54] Michael St¨oltzner. Feynman diagrams: Modeling between physics andmathematics.
Perspectives on Science , 26(4):482–500, 2018.[55] Adrian W¨uthrich. The exigencies of war and the stink of a theoreti-cal problem: Understanding the genesis of feynmans quantum electro-dynamics as mechanistic modelling at different levels.
Perspectives onScience , 26(4):501–520, 2018.[56] Justin Khoury. Alternative to particle dark matter.
Phys. Rev. D ,91:024022, Jan 2015.[57] L. Landau. The theory of superfluidity of helium ii.
Journal of Physics ,V(1):71, 1941.[58] L. Landau. Theory of the superfluidity of helium ii.
Physical Review ,60:356, 1941.[59] L.D. Landau and E.M. Lifshitz.
Fluid Mechanics , volume 6 of Courseof Theoretical Physics. Pergamon Press, second english edition, revisededition, 1987. Translated by J.B. Sykes and W.H. Reid.
60] L. Tisza.
C.R. Acad. Sci. Paris , 207:1035, 1938.[61] L. Tisza.
C.R. Acad. Sci. Paris , 207:1186, 1938.[62] L. Tisza. Transport phenomena in helium ii.
Nature , 141:913, 1938.[63] L. Tisza.
J. Phys. Radium , 1:164, 1940.[64] L. Tisza.
J. Phys. Radium , 1:350, 1940.[65] R.B. Dingle. Theories of helium ii.
Advances in Physics: A QuarterlySupplement of the Philosophical Magazine , 1(2):111, 1952.[66] P. Kapitza. Viscosity of liquid helium below the λ -point. Nature , 141:74,1938.[67] W.H. Keesom and G.E. Macwood. The viscosity of liquid helium.
Phys-ica , 5(8):737–744, 1938.[68] Russell J. Donnelly. The two-fluid theory and second sound in liquidhelium.
Physics Today , pages 34–39, 2009.[69] Andreas Schmitt.
Introduction to Superfluidity , volume 888. Springer,2015.[70] O.K. Rice. The thermodynamics of liquid helium on the basis of thetwo-fluid theory.
Physical Review , 76(11):1701, 1949.[71] O.K. Rice. The partial molal entropy of superfluid in pure he below the λ -point. Physical Review , 78:182, 1950.[72] R.P. Feynman. Atomic theory of the two-fluid model of liquid helium.
Physical Review , 94:262, 1954.[73] Hayder Salman, Natalia G. Berloff, and Paul H. Roberts. From clas-sical fields to the two-fluid model of superfluidity: Emergent kineticsand local gauge transformations. In Matthew Davis Nick Proukakis, Si-mon Gardiner and Marzena Szyma´nska, editors,
Quantum Gases: FiniteTemperature and Non-Equilibrium Dynamics , pages 369–384. 2013.[74] M.L. Amig´o, T. Herrera, L. Ne˜ner, L. Peralta Gavensky, F. Turco, andJ. Luzuriaga. A quantitative experiment on the fountain effect in super-fluid helium.
European Journal of Physics , 38(055103), 2017.[75] J.L. Alonso, F. Ares, and J.L. Brun. Unraveling the landau’s consistencecriterion and the meaning of interpenetration in the “two-fluid” model.
The European Physical Journal B , 91(226), 2018.[76] Richard P. Feynman.
Statistical Mechanics: A Set of Lectures . W.A.Benjamin, Inc., 1972.[77] Dale Tuggy. Trinity. In Edward N. Zalta, editor,
The Stanford Ency-clopedia of Philosophy . Metaphysics Research Lab, Stanford University,winter 2016 edition, 2016.[78] Lawrence Sklar.
Space, Time and Spacetime . Berkeley: University ofCalifornia Press, 1974.[79] J. Earman.
World Enough and Space-Time: Absolute versus RelationalTheories of Space and Time . Cambridge, Massachusetts: MIT, 1989.
80] Patrick M. D¨urr. Theory (in-)equivalence and conventionalism in f(r)gravity. ms.[81] Enrique ´Alvarez and Jorge Conde. Are the string and einstein framesequivalent?
Modern Physics Letters A , 17(7):413–420, 2002.[82] Jill North. A new approach to the relational–substantival debate. In
Ox-ford Studies in Metaphysics , volume 11. Oxford University Press, 2018.[83] David J. Baker. On spacetime functionalism.[84] Nelson Goodman.
Fact, Fiction and Forecast . Harvard University Press,1955.[85] Dennis Lehmkuhl. Introduction: Towards a theory of spacetime theories.In Dennis Lehmkuhl, Gregor Schiemann, and Erhard Scholz, editors,
To-wards a Theory of Spacetime Theories , pages 1–11. Springer/Birkh¨auser:New York, 2017.[86] Gonzalo J. Olmo. Palatini Approach to Modified Gravity: f(R) Theoriesand Beyond.
Int. J. Mod. Phys. , D20:413–462, 2011.[87] Peter Galison.
Image and Logic: A Material Culture of Microphysics .The University of Chicago Press, 1997.[88] Niels C.M. Martens, Miguel ´Angel Carretero Sahuquillo, Erhard Scholz,Dennis Lehmkuhl, and Michael Kr¨amer. Editorial: Integrating darkmatter, modified gravity and the humanities. ms.[89] Niels C.M. Martens. The metaphysics of emergent spacetime theories.
Philosophy Compass , 14(7):e12596, 2019. e12596 10.1111/phc3.12596.[90] James Read, Harvey R. Brown, and Dennis Lehmkuhl. Two miracles ofgeneral relativity.
Studies in History and Philosophy of Science Part B:Studies in History and Philosophy of Modern Physics , 2018.[91] Luc Blanchet and Alexandre Le Tiec. Model of dark matter and darkenergy based on gravitational polarization.
Phys. Rev. D , 78:024031, 72008.[92] HongSheng Zhao. Reinterpreting mond: coupling of einsteinian gravityand spin of cosmic neutrinos? 2008.[93] Jean-Philippe Bruneton, Stefano Liberati, Lorenzo Sindoni, and BenoitFamaey. Reconciling MOND and dark matter?
Journal of Cosmologyand Astroparticle Physics , 2009(03):021–021, mar 2009.[94] Baojiu Li and Hongsheng Zhao. Environment-dependent dark sector.
Phys. Rev. D , 80:064007, Sep 2009.[95] Chiu Man Ho, Djordje Minic, and Y. Jack Ng. Cold dark matter withmond scaling.
Physics Letters B , 693(5):567 – 570, 2010.[96] C.M. Ho, D. Minic, and Y.J. Ng. Quantum gravity and dark matter.
Gen Relativ Gravit , pages 2567–2573, 2011.[97] Chiu Man Ho, Djordje Minic, and Y. Jack Ng. Dark matter, infinitestatistics, and quantum gravity.
Phys. Rev. D , 85:104033, May 2012.
98] M. Cadoni, R. Casadio, A. Giusti, W. M¨uck, and M. Tuveri. Effectivefluid description of the dark universe.
Physics Letters B , 776:242–248,2018.[99] M. Cadoni and M. Tuveri. Galactic dynamics and long-range quantumgravity.
Phys. Rev. D , 100:024029, Jul 2019.[100] Erhard Scholz. A scalar field inducing a non-metrical contribution togravitational acceleration and a compatible add-on to light deflection.
General Relativity and Gravitation , (46), 2020.[101] Elisa G.M. Ferreira. Ultra-light dark matter. ms.[102] Constantinos Skordis and Tom Z lo´snik. A new relativistic theory formodified newtonian dynamics, 2020.[103] Elisa G.M. Ferreira, Guilherme Franzmann, Justin Khoury, and RobertBrandenberger. Unified superfluid dark sector.
Journal of Cosmologyand Astroparticle Physics , 2019(08):027–027, aug 2019., 2019(08):027–027, aug 2019.