Dark Matter = Modified Gravity? Scrutinising the spacetime-matter distinction through the modified gravity/ dark matter lens
aa r X i v : . [ phy s i c s . h i s t - ph ] S e p Dark Matter = Modified Gravity?
Scrutinising the spacetime–matter distinction through themodified gravity/ dark matter lens
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 scrutinises the tenability of a strict conceptual distinction be-tween space(time) and matter via the lens of the debate between modifiedgravity and dark matter. In particular, we consider Berezhiani and Khoury’snovel ‘superfluid dark matter theory’ (SFDM) as a case study. Two familiesof criteria for being matter and being spacetime, respectively, are extractedfrom the literature. Evaluation of the new scalar field postulated by SFDMaccording to these criteria reveals that it 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. Asequel paper examines possible interpretations of SFDM in light of this result,as well as the consequences for our understanding of (the importance of) themodified gravity/ dark matter distinction and the broader spacetime–matterdistinction. ontents If one goes to almost any talk directed at a general audience concerned withcosmology and astrophysics, the speaker is likely to declare that one of thebiggest mysteries of contemporary physics is that only 4 or 5% of the matterin the universe is ‘normal’ matter, the luminous matter we see all around us,21-25% is dark matter and the rest is dark energy. Let’s bracket the questionof whether dark energy really belongs into that list, and whether it reallyis energy in the sense in which both normal and dark matter posses energy.What, one might ask, has led to the conviction that there is roughly 5 timesas much dark matter in the universe as the kind of matter we see all aroundus, especially given that we have neither directly detected nor produced a darkmatter particle on the Earth? Why do we believe in the existence of darkmatter?Well, the truth is that not everybody does. There is a much smaller com-munity of physicists that see themselves in direct opposition to the dark mattercartel. They say that it would be too hasty to conclude that dark matter ex-ists, that indeed the observational data that has led the majority of physiciststo this conviction could be accounted for equally well, if not better, by stickingto the belief that the majority of matter in the universe is luminous baryonicmatter, and by modifying instead the laws of gravity and spacetime physics.True, members of this camp are less often invited to give public lectures toa general audience. But they have produced impressive results, accountingfor the rotation curves of galaxies without the introduction of a new kind ofmatter. Of course, this does not mean that they are right, either. he fact of the matter is that the conjunction of the assumptions i) Gen-eral Relativity (GR), and Newtonian Gravity as its non-relativistic limit, is thecorrect theory of gravity (and spacetime) and ii) most of the matter in theuniverse is luminous baryonic matter (the stuff that stars and planets consistof), leads to predictions that have been falsified by observations [1, 2]. Wewill call these observations ‘dark phenomena’ or ‘dark discrepancies’. Thesediscrepancies show up at a large range of scales. Starting with the smallestscale: Galaxy rotation curves—orbital speeds of visible stars or gas vs. radialdistance from the galactic centre—are expected to decline towards the edgeof the galaxy, but remain constant. Whole galaxies move faster than is tobe expected from the gravitational pull of the galaxy cluster that they arepart of. Gravitational lensing of either galaxies or galaxy clusters indicates astronger bending of light than expected from the mass of the luminous matter.Finally, problems appear even at the cosmological scale. Density fluctuationsof luminous matter in the early universe are washed out as long as that mat-ter is in thermal equilibrium with photons. Only after freeze-out can densityfluctuations grow under the influence of gravity, suggesting that structure for-mation should not have progressed as much as we do in fact observe it to haveprogressed at current times.Essentially, the difference between the Dark Matter (DM) community andthe Modified Gravity (MG) community is that, in response to these discrep-ancies, the former gives up premise ii) and introduces dark matter as the mainkind of matter in the universe, whereas the latter gives up on premise i) andintroduces different gravitational field equations.On galactic scales, using GR over its Newtonian limit likely makes no ob-servable difference for the predictions concerning rotation curves. Thus, thefirst proposal of the MG community consisted of a modification of the New-tonian gravitational equations, in the form of Mordehai Milgrom’s ModifiedNewtonian Dynamics (MOND). One way of presenting the theory is as a mod-ification of Newton’s inverse square law of gravity, which is replaced by: F G = G M mµ ( aa ) r (cid:26) µ ≈ , if a ≫ a (1) µ ≈ a/a , if a ≪ a (2)where a is a new constant of nature, which has been empirically determined tobe 1 . × − ms [3, 4]. This law approximates Newton’s law for gravitationalaccelerations much larger than a . However, at accelerations much smallerthan a the gravitational force is enhanced in comparison to Newton’s law. Itfollows that there is no practical difference for the planets in our solar system, Except for, e.g. Standard Model neutrinos, but we know that their masses and cosmic densitiesare not sufficient to account for what we below call the ‘dark phenomena’. Many in the literature speak of the ‘missing mass problem’, but we chose not to follow thisterminology because it seems to presuppose (rather than to argue) that premise ii) above is wrong. The common terminology ‘modified gravity’ is unfortunate in our context, in that one of thethings at stake here is exactly whether the gravitational field (in relativistic theories) is to beclassified as part of the matter or part of the spacetime sector—if such a classification is possiblein the first place. Nevertheless, we will in the first instance adopt the standard terminology, ratherthan, for instance, ‘modified spacetime’. ut there is for stars in the outer regions of their galaxy. Once this value for a is obtained, MOND uniquely and adequately predicts the observed galaxyrotation curves.As noted above, MOND described thus seems a clear example of an MGtheory, which modifies premise i) while leaving premise ii) untouched.Likewise, paradigmatic DM theories can easily be introduced within theNewtonian regime. Choosing (by hand) a suitable distribution of additional,electrically neutral, massive Newtonian objects obeying the (unmodified) lawsof Newtonian Gravity can also account for the observed rotation curves ofindividual galaxies.At the very least at the cosmological scale, and in the context of gravi-tational lensing also at the level of galaxies and galaxy clusters, we have tomove beyond (modified or unmodified) Newtonian physics and use a relativis-tic theory—and sometimes also a quantum theory—that is approximated by(modified or unmodified) Newtonian gravity at low velocities and for weakgravitational fields [5]. The question that we want to address in this paperis whether a clear distinction remains between modifying the gravitational in-teraction/ spacetime structure (premise i) on the one hand and modifying theassumption of what the matter content in our universe consists of (premise ii).In other words: once we enter the relativistic (and quantum) regime, is therestill a clear distinction between gravitational fields and matter fields?This distinction between gravity/spacetime and matter has been questionedalready in the context of classical GR, nevermind dark phenomena. For in-stance, Earman and Norton argued that the metric field in GR, the successorof the Newtonian gravitational potential in the context of GR, should be seenas belonging to the same ontological category as what represents matter inthe context of GR: both are fields defined on the spacetime manifold M [19]. Earman and Norton took inspiration directly from Einstein, who did not seethe metric field as categorically different from the electromagnetic field, orindeed any other fields, and staunchly argued that the claim that GR had‘geometrized’ the gravitational field (but not yet the electromagnetic field)did not have any meaning at all [22]. Similarly, Rynasiewicz argued that theold debate between substantivalism and relationalism (the question of whetherspace or spacetime is a fundamental entity in its own right or derivative ofthe relationships between material bodies) is ill-founded in the context of GR,primarily because it is just not clear that the metric field g µν should be seenas corresponding to (aspects of) spacetime (structure) [6]. In a similar vein,one of us has argued that the property that makes a field a matter field in Skepticism about the space(time)–matter distinction may be traced back even further. Aroundthe very end of the 19th century, it was widely held that “the question whether the aether is reallya special form of matter or really just space endowed with certain physical properties is not one ofserious consequence, but only a question of what mode of expression one prefers to adopt” [6, p.290].Of course the concept of the aether, at least in any of its historical forms, has long been abandoned,but one might wonder—as Rynasiewicz does, and we will do, in the main text—to what extentthe (manifold plus) metric field in GR is not a new aether in disguise, in the sense of having bothproperties that are historically associated with matter and with spacetime. See also [7–14] [15, p.354] [16, p.36] [17, § § See also [8, 20] [21, Ch.9]. he context of relativistic field theories is that it has a mass-energy-momentumtensor T µν asssociated with it, but that the definition of this matter-makingproperty definitionally depends on g µν in a variety of ways [23]. This, too,suggests that the line between ‘matter’ and ‘gravity / spacetime structure’ isnot as sharp as has often been claimed. Finally, of course, the distinction also comes under fire in various ap-proaches to solving the problem of quantum gravity. However, in this paperwe are interested in the degree to which this distinction becomes problematiceven before reaching the regime were such theories of quantum gravity areexpected to reign—this has the additional benefit of staying closer to experi-ment/observation.In these two companion papers, we shall provide a new argument againstthe idea that in relativistic and/or quantum theories the distinction betweenmatter and spacetime, and related to that the distinction between modify-ing the matter content or modifying the gravitational laws, is clear-cut. Weshall question the tenability of the spacetime/matter distinction by scruti-nising it through the lens of the modified gravity/ dark matter distinction(MG/DM) —which is, in some senses, more clear, more straightforward thanthe GR lens. This is one of many possible ways in which to make a small contri-bution to the ‘cartography research programme’. This research programme isconcerned with navigating the space of theories via a dynamic back-and-forthbetween, on the one hand, understanding and interpreting individual theoriesand, importantly, the relations between neighbouring theories and the con-cepts they use, and, on the other hand, exploring the larger space of theories(with the help of the lessons learnt from the individual case studies) in orderto understand that space as a whole and to streamline theory generation. (More on this in the sequel paper [24].)Deflating the strict dichotomy or even the distinction between dark mattertheories and modified gravity theories might contribute to undermining the We will return to the question of whether ‘possessing’ mass-energy-momentum is the rightcriterion for defining ‘matter’ in section 2.2. A particularly interesting context would be string theory: is the dilaton best categorised asspacetime, matter, both or neither? We would like to thank an anonymous reviewer for thissuggestion. Although the case study in this paper, superfluid dark matter theory, is a theory of quantumgravity in the sense that a (macroscopic) quantum effect, namely a superfluid Bose-Einstein con-densate, leads to a modification of gravity (at the scale of galaxies), it is not a theory of quantumgravity in the sense of solving the problem of quantum gravity (which concerns a regime that isnot standardly associated with the galactic scale). Other interesting pre-quantum-gravity contexts in which one can question the tenability ofthe spacetime/matter distinction—thereby further contributing to the cartography research pro-gramme (see main text and [24])—include spin-2 gravity, Newton-Cartan theory, f(R) gravity andJordan-Brans-Dicke theories [25], the cosmological constant, black holes, unified field theories andsupersymmetric theories. These companion papers thereby cohere with an argument by one of us that (even) if one (only)wants to understand and interpret a (single) theory, in particular our ‘best’ spacetime theory, GR,and figure out what makes it special, one (still) needs to look at the neighbourhood of that theoryin the space of spacetime theories. See [26] for details, and for a strategy in a similar spirit see [27]. utual hostility that is prevalent between the two camps that have formed inresponse to one of modern physics’ most pressing problems: the dark discrep-ancies. Deflating the broader distinction between spacetime and matter could,quite literally, make the oldest foundations of natural philosophy crumble, in-cluding the debate that has dominated the philosophy of space(time) over thepast three centuries: the substantivalism/relationalism debate. We will go about this by investigating a particularly promising theory dueto Lasha Berezhiani and Justin Khoury [29,30]. They call it “superfluid darkmatter theory”; we shall call it SFDM in the following. Their theory comes as aresponse to the stalemate between the DM and the MG research programmes—DM theories have traditionally done well at the level of cosmology and galaxyclusters, but less so at the level of galaxies, with the opposite being thecase for MG theories. Superfluid dark matter theory has gained quite someattention in the physics community, particularly because it seems that givenSFDM you can have your cake and eat it: in their seminal paper of 2015, theauthors announce that SFDM “matches the successes of the Λ cold dark matter(ΛCDM) model on cosmological scales while simultaneously reproducing themodified Newtonian dynamics (MOND) phenomenology on galactic scales” [30,p.1]. The way in which this is achieved is by introduction of a new self-interacting, massive and complex scalar field Φ which the authors classify asa dark matter field. However, Φ has two phases corresponding to the twodomains in which typical DM and MG theories are successful; galaxy clustersand galaxies, respectively. On the length scale of galaxy clusters, Φ is in the A third option in addition to substantivalism and relationalism that is often swept under thecarpet is that of super-substantivalism. Where relationalism argues that space(time) is derivablefrom matter, super-substantivalism argues that matter is derivable from space(time). However,most versions of super-substantivalism do not thereby identify the two notions, ontologically, letalone conceptually. Rynasiewicz argues that not even the arch-super-substantivalist Descartes’definintion of matter as res extensa was intended to destroy the conceptual distinction [6, p.281-2].Skow [28] and Lehmkuhl [16] distinguish modest and radical supersubstantivalism. The former isthe metaphysical view that, regardless of the specific physical theory (albeit most naturally in afield theory [6, p.298-9]), one can simply stipulate that paradigmatic matter properties such as massand colour are all directly attributed to spacetime. The latter is the physics research programmeof determining to what extent specific physical theories, such as Kaluza-Klein theory or Wheeler’sgeometrodynamics, manage to reduce these apparent non-geometrical properties to (or have thememerge from) geometrical (or topological) properties. In a sense this paper concerns somethingeven more radical: the research programme of determining to what extent the neighborhood of ourbest theories in the space of theories suggests a conceptual identification, or at least a blurring, ofthe orthodox distinction between spacetime and matter. For potential further interesting DM/MG case studies, see [31–43]. This may seem to contradict our earlier claim that one can account for the observed rotationcurves of individual galaxies by a suitable distribution of dark Newtonian objects obeying the(unmodified) laws of Newtonian gravity. It is indeed true that if one models the dark matterdistribution with sufficiently many parameters and adjusts them by hand for each individual galaxy,that one can fit the observed rotation curve of that galaxy. However, if one does not put in eachgalactic distribution by hand, but uses the output of computer simulations of the evolution ofthe whole universe, one has trouble reproducing the correct locations, numbers and shapes ofdark matter halos, as well as several empirical correlations across galaxies, such as the baryonicTully-Fisher relation [44]. ormal fluid phase, giving rise to axion-like particles with masses in the range ofeVs and strong self-interactions. This allows for reproduction of the successesof typical dark matter theories. On the length scale of single galaxies, Φ entersa superfluid phase, giving rise to superfluid phonons that are governed by aMOND-like action (in the non-relativistic regime). It is thus that they canreproduce MOND phenomenology in galaxies. Borrowing terminology from anapproach of kindred spirit due to Sabine Hossenfelder [45], one might call Φ inthe superfluid phase an “imposter field”: a field that is ‘really’ dark matter,but acts as if it was a modification of gravity, a new gravitational degree offreedom, in a certain domain. (Interestingly, Hossenfelder refers to SFDM as avariant of modified gravity. SFDM is formally very similar to her own covariantversion of Verlinde’s emergent/entropic gravity. She interprets the new field inher own theory as a modification of gravity, with plays the role of dark matterwithout ‘being’ dark matter. It is hence considered an imposter field (but inthe sense opposite to that above).)The aim of this pair of companion papers will be to unpack and analysesuch interpretational claims. Is Φ ‘really’ a dark matter field and only actslike a modification of gravity in certain regimes? Or should one say that itis fundamentally a new gravitational field that acts like a dark matter fieldin certain regimes? If not, why not? Or is the sense in which Φ is darkmatter on a par with the sense in which it is a modification of gravity andspacetime? And indeed, what are the criteria that justify calling a scalar field(and its excitations) a dark matter field in the first place, or a new gravitationalfield / gravitational degree of freedom? Or does SFDM put pressure on thecontemporary usefulness or even the coherence of these old categories?We will proceed in the following way. In this first paper, we will discuss thecriteria that might be put forward to categorise a newly introduced tensor (orspinor) field as a matter field (section 2) or an aspect of spacetime (section 3),respectively. In the latter case, we will focus on the question of what counts asmodifying the representation of gravitational fields as compared to GR, whichis typically the starting point of Modified Gravity approaches. Since in GRthe representation of gravity is based on pseudo-Riemannian geometry, manymodifications of GR are modifications of that underlying geometrical structure.However, note that we do not thereby commit to the claim that every theory ofgravity will conceive of gravity as something necessarily connected to spacetimegeometry; indeed, this opinion is controversial even in the context of GR. Inshort, some but not all theories of gravity are spacetime theories, just like somebut not all spacetime theories are theories of gravity. Still, within the realm ofmodified gravity theories aiming to give an account of the dark discrepancies,the decisive change as compared to GR is typically a geometrical one; as weshall see, SFDM, by following a prominent relativistic extension of MONDcalled TeVeS, involves the introduction of a second, ‘physical’ metric tensor,and a coming apart of the different roles the metric tensor plays in GR.Following the discussion of the criteria that might lead one to considera new field a matter field or a modification of spacetime and gravity (or anew gravitational and/or spatiotemporal degree of freedom), we shall—afterintroducing SFDM in more detail (section 4)—apply these criteria to the caseof SFDM (section 5). We shall argue that the often unquestioned assump- ion that every field is either a matter field or a gravitational/spacetime fieldis put under severe pressure by SFDM. In particular, we will argue that thenewly introduced scalar field in SFDM is both a matter field and a gravita-tional/spacetime field. This result will then be the starting point of the secondcompanion paper [24], where we will unpack the interpretational questionsraised above, both in the context of SFDM as well as in the broader contextof charting (the neighbourhood of SFDM in) the space of theories. The first dominant family of criteria in the literature are often considered thelitmus test for calling something matter. , Explicit discussions of a (natu-ralistic) definition of matter (as opposed to spacetime) are surprisingly rare.Interestingly, one of the clearest discussions of these criteria occurred whenEarman and Norton’s hole argument [19] lead to a discussion on the conceptof spacetime . The ensuing debate targeted a conception of substantival space-time that excluded the metric field:Norton and Earman provide several considerations designed to pro-mote the bare manifold as the most plausible candidate for sub-stantival space-time. The leading arguments are: a) In the GTR[General Theory of Relativity] the metric field is just like any otherfield, both mathematically and physically. It is represented by atensor similar to, e.g., the electromagnetic field; it is governed bylocal differential equations; it carries energy and momentum. Henceit should not be essential to space-time (Earman & Norton 1987,p. 519). b) In the GTR the metric becomes a dynamical object.Hence it should not be considered essential to space-time (Earmanforthcoming, chapter 10). [52, p.97]This line of reasoning could be interpreted as the metric field being just likeany other matter field—which would make the mentioned properties aspectsof the definition of matter. Rovelli uses the term ‘matter’ explicitly:[I]n the general relativistic world picture [the spacetime-versus-matterdistinction] collapses. In general relativity, the metric/gravitational Sometimes some of these criteria are used as criteria for being physical—for being real [46,Chs.5,8] [47, § a priori rule out thepossibility of our world (or other worlds) containing pure, physical spacetime. Baker takes the negation of some of these criteria as counting toward an object satisfying thespacetime concept [51]. Another possible interpretation however, closer to Einstein (and arguably intended by Earmanand Norton), is that the metric field and paradigmatic ‘matter’ fields such as the electromagneticfields are all just fields, with the further distinction between matter fields and other fields beingunimportant, or even misleading. Einstein’s position on this will be spelt out and discussed indetail in [53]. eld has acquired most, if not all, the attributes that have character-ized matter (as opposed to spacetime) from Descartes to Feynman:it satisfies differential equations, it carries energy and momentum,and, in Leibnizian terms, it can act and also be acted upon , and soon. [8, p.193] Let us pick these matter criteria, often put forward as necessary and/orsufficient criteria for being matter, apart systematically, and order them ac-cording to logical strength. Initially, their differences will be illustrated byapplying them, as in the above quotes, to the metric field in GR. In some casesthis application is controversial. We will see below that in the context of MGand DM theories their application is more straightforward.
The four weakest matter criteria might be dubbed the kinetic criteria . In orderof logical strength:
Matter criterion strength A:
The object under consideration is not con-stant/ static, but varies/ changes. Matter criterion strength B:
The object under consideration changes in aregular fashion.
Matter criterion strength C:
The object under consideration changes in aregular fashion that is describable by non-trivial differential equations.
Matter criterion strength D:
The object under consideration changes ina regular fashion. This change is (partially) in response to somethingexternal, and (thus) describable in terms of coupled differential equationsthat describe the coupling of the object to the external factor.Newton’s absolute space has none of these properties; Newtonian matter—such as planets and billiard balls—has all of them. Although the first criterionis thus in principle sufficient to separate space(time) and matter in the New-tonian framework (and also in special relativity), we will see below that New-ton nevertheless had a much richer conception of matter. A Humean mosaicwith mass values randomly sprinkled on top would be of exactly strength A.Non-trivial sourceless solutions to GR, such as those containing gravitationalwaves, satisfy up to criterion C. Generic metrics with sources that are solutionsto GR satisfy all of the above criteria.Criterion D is a strong version of what is often called the action-reactionprinciple. Obeying a general action-reaction principle is logically speaking notstronger than criterion B, if it is logically possible to regularly change due to See also [13]. Within a model/possible world that is; we are not considering an inter-model/inter-worldcomparison. Note that we do not insist on the variation being across time. These are often referred to as vacuum solutions, but the notion of vacuum becomes ambiguousif the spacetime–matter distinction is ambiguous. What we are referring to is solutions with noother non-zero fields besides the metric. omething external without that implying that the change becomes describablein terms of differential equations. However, such a possibility is not relevantin the context of this paper.In the context of criterion D, we could make the further distinction betweenobjects whose dynamics is totally determined by external factors, and thosewhich have their dynamics only partially determined externally, that is thosewith partially internal dynamics. A special case of demanding total determina-tion is Einstein’s 1918 version of Mach’s principle: the demand that the metricfield g µν be uniquely determined by the the mass-energy-momentum distribu-tion of matter, T µν [54]. This criterion was put under pressure in the debatebetween Einstein and De Sitter following the discovery of the De Sitter solu-tion to the Einstein field equations with cosmological constant in 1917 [55–57].We now know that the metric field in general relativity is merely constrainedrather than determined by matter. Hence, the set of solutions to the Einsteinequations satisfying exactly up to criterion C is the set of sourceless solutions;the same can be said for other field equations in which the gravitational field(in the case of GR the metric g µν ) has independent degrees of freedom. Hav-ing independent degrees of freedom is, however, not a necessary condition forbeing matter in any of these four senses. The next three stronger criteria may be called the energetic criteria . Theyrequire the object under consideration to carry energy , [48], a conceptthat can be cashed out in three distinct ways. In order of logical strength: Matter criterion strength E–G:
The object under consideration changesin a regular fashion. This change is (partially) in response to somethingexternal, and (thus) describable in terms of coupled differential equations,in such a way that the object can be said to carry exchangeable ‘energy’(and momentum)( E ) ascribable to a particular spacetime volume;( F ) ascribable to each point in spacetime;( G ) representable by a stress-energy tensor, T µν . This becomes particularly clear from the decomposition of the Riemann curvature tensor intoterms featuring only the Ricci tensor and terms featuring only the Weyl tensor. Only Ricci cur-vature is determined by the Einstein equations; the Weyl tensor encodes the free gravitationaldegrees of freedom—they are constrained but not fixed by the Einstein equations. We thus strongly disagree with Bunge, who equates energy to changeability and thereby con-flates matter criteria A through G [48]. Why is energy more relevant than, say, entropy? At least within thermodynamics it doesnot seem to be more special. It would be interesting to consider whether carrying entropy isin any way an indicator of being matter. If carrying entropy is considered necessary for beingmatter, one consequence would be that (the superfluid component of the two-fluid model of)perfect superfluids, such as those appearing in SFDM, would not count as matter. (The two-fluidmodel will be discussed in the second companion paper [24].) riterion E insists on the energy being exchangeable, that is it ‘interacts’with some measurement apparatus whenever the required matter for such ap-paratus is present. After all, it might seem conceivable that an object has(non-exchangeable) energy, without satisfying criterion D—being influencableby external objects—and even its converse—being able to influence externalobjects. To make it crystal clear that criterion E is strictly stronger than Dwe insist on the energy being exchangeable. Being kinetical (in any of the senses of criteria A–D) means that some-thing is changing. Changing often means changing with respect to spacetime,in which case kinetic energy is indeed definable (relative to a frame), and ifthere is an action-reaction principle with (other) matter then this energy isexchangeable. Doesn’t criterion D imply E and F and perhaps even G? Arethey truly distinct—extensionally, or at the very least intensionally? Well, inGR the metric field, often taken to represent spacetime, is itself dynamical,which makes it less clear if and how energy is involved. What we can definitelysay, in the restricted context of GR, is that criterion G is not satisfied by themetric, since gravitational energy is represented by a pseudo-tensor, ratherthan a tensor (and thus, a fortiori , not by a stress-energy tensor T µν ). ForHoefer [58] and D¨urr [59, 60] this implies that criteria E and F are also notsatisified by the metric. Hoefer believes that the concepts of carrying energyand energy being representable by a stress-energy tensor do not come apart (hence the quotation marks around ‘energy’ in criterion E and F):I am inclined to believe ... that gravitational waves do not in factcarry substantival energy content. ... [T]he fact that gravitationalenergy must be represented by a pseudo-tensor (and that no onepseudo-tensor has a privileged status for this representation) are thereasons for denying that gravitational energy is truly substantial.[58, p.13; italics in original] This implication goes against a long tradition of taking gravitational waves(GWs) to carry energy (despite this energy not being represented by a tensor),whether at each point or only in each finite region. This tradition started withEinstein’s first derivation of gravitational waves in the linearized limit [63, 64],and was continued with Feynman and Bondi’s sticky bead thought experiment Here we assume that measurement apparatus must be made of matter. This assumption is ofcourse questionable in light of the general thesis of this paper, and also, more specifically, in thecontext of attempts to reduce supposed matter to (or unify it with) spacetime. One might counter that this situation is not coherently conceivable, because such non-exchangeable energy would have no operational meaning and thereby not really any meaningwhatsoever. Adding the qualifier ‘exchangeable’ would thus be a pleonasm; no information isadded. Such reasoning would open the door to finding the first three criteria meaningless as well.We will thus assume for now that the notions of change and energy still have some technical, if notoperational, meaning in these situations. After all, as noted in fn. 24, the notion of ‘operational’may have to be rethought anyway if the spacetime–matter distinction becomes ill-defined. D¨urr is more liberal, in that he allows all so-called ‘geometric objects’ to potentially representenergy, even non-tensorial geometric objects. Pseudo-tensors are however not geometric objects inhis sense [60]. See also Hoefer’s 2000 paper [61] and [62]. esigned to show that GWs can heat up matter [65]. Subsequent advocatesoften invoke binary systems of which the orbital decay is claimed to be bestexplained by the system’s energy and momentum being carried away by GWs.The tradition is perhaps most dramatically proclaimed by Rovelli, who doesnot only suggest that this carrying of energy puts the metric field on a parwith the electromagnetic field, but that this shatters the spacetime–matterdistinction altogether:Let me put it pictorially. A strong burst of gravitational wavescould come from the sky and knock down the rock of Gibraltar,precisely as a strong burst of electromagnetic radiation could. Whyis the first “matter” and the second “space”? Why should we regardthe second burst as ontologically different from the second? Clearlythe distinction can now be seen as ill-founded. [8, p.193]Read argues, against Hoefer and D¨urr, that under certain assumptions and ona functional understanding of energy, there exist solutions to GR in which themetric can be attributed gravitational stress-energy (represented by a pseudo-tensor), albeit only in finite regions, not at a point (i.e., criterion E but not Fis satisfied) [66]. We will not attempt to settle this debate here. That the issues of GWscarrying energy and there being any substantial notion of gravitational energyin GR are controversial is sufficient reason to, at least prima facie , separatecriterion E and F from G, and D from E and F, and entertain the possibilitythat GR may contain solutions that satisfy up to criterion D only as well assolutions that satisfy up to criterion E or F.The final criterion perhaps does not deserve to be called a matter criterion,but it follows neatly in the hierarchy, and will become important when applyingthe matter criteria to SFDM:
Matter criterion strength H:
The object under consideration changes ina regular fashion. This change is (partially) in response to somethingexternal, and (thus) describable in terms of coupled differential equations,in such a way that the object can be said to carry exchangeable energy(and momentum) representable by a stress-energy tensor, T µν , part ofwhich is due to the rest mass of the object. In other words: the objectis massive.The tradition of considering ‘having mass’ to be the litmus test for mat-ter started with Newton in his
Principia , where he takes mass (arrived at via The modern debate could profit from a more detailed study of the original debate on thenature of gravitational energy and the role of tensorial vs. pseudo-tensorial objects; see sectionII of Volume 7 and section VIII of Volume 8 of the Collected Papers of Albert Einstein for anoverview, and [53] for analysis of and connections between the positions advocated by Einstein,Lorentz, Klein, Schr¨oedinger and Levi-Civita on these questions, as well as a discussion of howtheir insights could be used in the context of the modern debate. Note that this criterion is fulfilled by the relativistic fluids but not by electromagnetic fields.For the trace of the energy-momentum tensor of the former does not vanish (and thus allows fora rest frame and for the possession of rest mass), while the energy-momentum tensor of the lattervanishes: the continuum counterpart of the principle of the constancy of the speed of light. ass density times volume) to be the quantity of matter [67, p.1] [68]. Thistradition was embraced by Maxwell, Kelvin, Tait and Clifford [69], and wasstill influential in the 1920s when the aether was judged not to be matter ex-actly because it lacked rest mass. In Newtonian times such a criterion madeperfect sense, as all well-understood (particle) matter had mass. In light of thecurrent standard model of particle physics, and especially the existence of themassless photon, one may of course ask why mass would be so special. Whynot consider spin, or isospin, or any other quantum number to be the essenceof matter? Or, more generally, a disjunction of all of the above? Alterna-tively one may simply consider matter criterion G, at most, to be sufficient forsomething to be matter; any stronger condition is asking too much.
The philosophical literature contains a second major family of criteria associ-ated with the spacetime–matter distinction. We will refer to this family as theset of spacetime criteria. They are usually proferred as necessary and/or suffi-cient criteria for something to be spacetime or an aspect of spacetime structurein the subtantivalism–relationalism debate.Both substantivalism and relationalism are realist positions about space(time),they merely differ on the relative fundamentality of space(time) and spati(otempor)alrelations between material objects. Within the context of the modern substantivalism–relationalism debate one needs to consider two issues:i) the core issue: whether spacetime and matter are separate fundamentalentities (substances) or whether spacetime is derivable from matter insome sense (for example from the relationships between material bodies).This part of the debate is the direct continuation of the debate betweenLeibniz and Clarke in the context of pre-relativistic mechanics.ii) the prior issue of fixing the referents of the terms in the core issue, in par-ticular what the referent of the term ‘spacetime’ is in GR. The main candi-dates in the philosophical literature on the substantivalism–relationalismdebate are a) the manifold M by itself, b) the manifold M and the metricfield g µν together, c) the metric field g µν by itself [19, 58, 70].Only the second issue matters to us in the context of this paper, or ratherthe connected question of what is a lower bound, a minimal set of propertiesfor something to count as spacetime or as an aspect of spacetime structure. Inthe following we will consider some candidate criteria for calling something aspacetime. It will be important to keep two things in mind. In fact, many particles within the standard model are not always massive; they only becomemassive via the Higgs mechanism once the electroweak symmetry spontaneously breaks in the earlyuniverse. We would like to thank Radin Dardashti for this point. Both of these caveats cohere with Baker’s understanding of the notion of spacetime as a clusterconcept [51], with which we sympathise to some extent. A cluster concept is a way of makingprecise how certain properties can count toward the application of the concept being appropriate. irstly, the list of three structures discussed in the philosophical literature(issue ii) is too limited for the space of theories we envisage, although this isperhaps not surprising given that the philosophical debate presupposed GR.The physics literature considers a much larger variety of possibilities, see forinstance Sharpe [71], Vizgin [72], Goenner [73] and Blagojevi´c & Hehl [74]. Inthe following, we will go slightly beyond these three structures, but will stillfall short of considering the full range. Instead, we will stay close to generalrelativity and consider one set of (many possible sets of) criteria that are jointlysufficient for something to be spacetime, namely the set that is relevant to thecase study at hand. Note that we are thus not claiming that these criteria arenecessary for being spacetime —as in fact we do not think they are, althoughwe do believe that many of these criteria are typically fulfilled by spacetimes.Defining spacetime exhaustively in the context of the full space of theories willrequire going beyond this set of criteria.Secondly, although some of these criteria are logically stronger or weakerthan others, others are logically independent of each other. Hence, we will notlabel them alphabetically as we did to indicate the total logical ordering of thematter criteria. A first subset of the spacetime criteria might be called the structural [76] or abstract or mathematical criteria . A first candidate is as follows: Spacetime criterion M:
The object under consideration is (faithfully rep-resentable by) a differentiable manifold, i.e. a (topological) manifold plusdifferentiable structure.Leibnizian spacetime [77, p.30–31] satisfies this criterion. However, New-ton’s bucket experiment is designed to show that we need a way of distinguish-ing straight motion from curved motion —it is exactly for this reason thatabsolute space has empirical consequences, according to Newton. As any affineconnection grounds a distinction between geodesics and non-geodesics, thissuggests adding at least one affine connection to obtain the following strongercriterion: Although these properties are jointly sufficient for the concept, say spacetime, to apply, they arenot jointly nor even individually necessary—but, in order to avoid triviality, they are disjunctivelynecessary in the sense that at least some of the properties must obtain in order for the conceptto apply. (Note that understanding spacetime as such a cluster concept, rather than insisting ona set of criteria that is both jointly necessary and sufficient, immediately suggests that we shouldnot expect every (logically possible) object to fall into exactly one of two categories, spacetime ormatter [24, § For instance, neither superspace [75] nor discrete ‘spacetimes’ satisfy these criteria, but we donot believe that this suffices to rule them out as spacetimes. For simplicity’s sake, we are ignoring that strictly speaking one only needs a standard ofrotation, not a full standard of curved motion [78–80]. pacetime criterion M ∇ : The object under consideration is (faithfully rep-resentable by) a (differentiable) manifold endowed with an affine connec-tion.But what about physical theories where a distinction between geodesicand non-geodesic paths is not empirically significant, say theories for whichLeibnizian space(time) would be a sufficiently rich space(time)?Moreover, objects that satisfy either of these two criteria (and feature in adiffeomorphism invariant theory) seem to run up against the hole argument. Of course, one might attempt to avoid the hole argument using one of severalresponses in the literature. For instance, it may be pointed out that Earmanand Norton assume primitive/fundamental transworld identities of the space-time points, and that the hole argument dissolves if one strips these identitiesaway [81–84]. The objects that remain would still satisfy criterion M or M ∇ ,and therefore deserve the name ‘spacetime’. Moreover, even if the hole argu-ment would go through, that would not imply that these two criteria are badcriteria for denoting something as spacetime. For instance, in the context ofGR, it would merely imply that GR is best interpreted relationally, and there-fore that spacetime is not fundamental—albeit still real. The hole argumentis thus not a reason against taking either of these two conditions as criteria forbeing spacetime.Some considerations that do pose a real problem turn out to arise evenprior to the introduction of matter dynamics, as was the case for the bucketand hole arguments. Firstly, configuration spaces or phase spaces could alsosatisfy the above two criteria. How to distinguish them from the type of spacethat we are after? Secondly, these criteria do not distinguish n+1 dimensionalspace from n+1 dimensional spacetime. Let us consider the following strongercriterion: Spacetime criterion M ∇ g: The object under consideration is (faithfullyrepresentable by) a (differentiable) manifold with an affine connection,plus a Lorentzian metric field , on that manifold (which may or maynot be compatible with the connection).This criterion is too strong, or rather something weaker will do the job ofdistinguishing between two types of dimensions—and thus between space andspacetime: The manifold substantivalism that Earman and Norton target uses criterion M explicitly, buttheir argument would work equally well against a form of substantivalism that adds a connection. Of course, such a non-fundamental spacetime would not feature fundamental transworld iden-tities either—but see Maunu [85]. cf. [76, § For a more nuanced position, see Dewar & Eisenthal [86], who argue for a middle way betweenbare-manifold and manifold-plus-metric accounts of spacetime. Of course the topological and differentiable structure and a fortiori the connection mentionedin the first two spacetime criteria also require a metric in order to be defined, namely a metricon the space of coordinates, but that metric is in general not the same metric as the metric fieldliving on the manifold. For instance, in GR the topology is induced from a Euclidean metric onthe space of coordinates, but has a Lorentzian metric living on the manifold. We are grateful toTushar Menon for pointing this out. pacetime criterion M ∇ C: The object under consideration is (faithfullyrepresentable by) a (differentiable) manifold with an affine connection,plus a Lorentzian conformal structure, i.e. an equivalence class of Lorentzianmetrics, on that manifold.Conformal structure suffices to distinguish between space and spacetime—itdistinguishes two types of dimensions, one usually referred to as temporal, theother as spatial—while affine structure suffices to distinguish between geodesicand non-geodesic paths through spacetime.One problem that remains is that we do not know how to distinguish ann+1 spacetime from a 1+n spacetime. For instance, why do we take fourdi-mensional spacetimes to have one temporal dimension and three spatial di-mensions, rather than the other way round? Is it simply essential to time thatit is a single dimension, and essential to space that it may have multiple di-mensions? But where would this leave 1+1 spacetimes? Or is the distinctionbetween space and time less relevant in such a context—after all, GR in twodimensions exhibits a cornucopia of pecularities [27]?A further problem arises. Reference to the metric (or conformal structure)in some of the above criteria is ambiguous, as there may be more than one. Forone thing, one can always just define extra metrics, such as Finsler metrics.One might respond that the above criteria implicitly refer to the metrics thatare fundamental within the theory under consideration, that is metrics thatare postulated as part of the fundamental ontology of the theory. This wouldhowever rule out, by fiat, mainstream positions such as relationalism—whichis a realist but non-fundamentalist position about spacetime—as well as thedynamical approach to special relativity [21]. It would also be very muchagainst the spirit of spacetime functionalism. Even if we were to take this route,we would expect the problem of multiple metrics to resurface in some theoriesof modified gravity/spacetime—where we might expect the modification togenerate a second effective metric—and explicitly in bimetric theories, whereboth metrics are part of the fundamental ontology. Application of spacetimecriterion M ∇ g (and in some cases also criterion M ∇ C) would then suggest thatthis theory exhibits two spacetimes, which both share the same differentiablemanifold, but consist of a different metric. It seems a sensible restriction on theconcept of spacetime that there can be at most one spacetime in each model ofa theory at each level of description. Worries similar to those about multiplemetrics apply to theories with multiple affine connections, and even in theorieswith just one metric and one affine connection if those are not compatible. Inthe latter case, it is ambiguous whether a trajectory (say of a test particle) isa geodesic, as it may be an affine geodesic but not a metric geodesic, or viceversa.Furthermore, we still have not managed to distinguish configuration andphase spaces from the space(time) that we are after. (Imposing that the metric(or conformal structure) be dynamic would certainly rule out configurationspace and phase space, but at the cost of denying Special Relativity the statusof a spacetime theory.) The reason is that the criteria so far talk only ofabstract, mathematical objects but have not made any connection with physics, hat is with the (observable) behaviour of matter [58, p.12] [12, esp. p.89]. Perhaps by moving from mathematics to physics we can thereby also resolvethe impasse arising from spacetime and configuration space being representableby the same type of mathematical object.
We have come to the second and final subset of the spacetime criteria, the physical criteria. ‘Physical’ is meant to oppose ‘merely mathematical’, notto be synonymous to ‘material’, a mistaken equivocation that often occurs inthe context of the matter criteria (cf. fn.15). These criteria take into accountthe connection between spacetime and matter. The first advantage that thisconnection provides is that it allows us to distinguish between n+1 and 1+nspacetimes:
Cauchy (spacetime) criterion:
A Lorentzian signature separates the di-mensions into two types. The dimension(s) with respect to which a well-defined initial value problem can be formulated—for the matter ‘livingin’ that spacetime and the dynamics governing that matter—form(s) thetemporal type; the other dimensions are of the spatial type. [88]In practice, this ensures that there is typically one temporal and n spatialdimensions, for multiple time dimensions would make a well-defined initialvalue problem all but impossible ( pace Weinstein [89]).Two further criteria concern the connection between spacetime and matter.In response to the question of what reasons we have to believe that GR canbe interpreted as geometrising the gravitational field—which we take to beequivalent to gravity becoming part of spacetime in GR—one might give atwo-part answer. First, in GR we can describe gravity solely in terms ofthe structure of spacetime. Secondly, in GR gravitationally charged (i.e.massive) test particles move on the timelike geodesics of the connection (whichis compatible with the metric) even in the presence of gravity, and lights raysmove on the null geodesics of the same connection. Our second physicalcriterion is thus: One may thus wish to call the spacetime candidates in this subsection ‘theoretical spacetimes’and such spacetimes that furthermore satisfy the physical criteria in the next subsection the ‘phys-ical spacetimes’ or ‘operational spacetimes’. It is however important to distinguish this usagefrom the usage by Read and Menon, for whom both the concepts of theoretical and operationalspacetime refer to the dynamics of matter [87]. See [76, Chapter 9] for details, and the inspiration for the weak geodesic criterion below,applied to the question of whether the gravitational field in GR, as well as both the gravitationaland electromagnetic field in Weyl’s theory and in Kaluza-Klein theories, can be interpreted as anaspect of spacetime structure. One might not describe geometrised forces in terms of curvature (only), but in terms of otherstructures, such as metric and affine properties, and those deriving from them, like torsion andcon-torsion, or even topological ones [16, 90]. Note that Einstein opposed the idea that any of this means that gravity is reducible to thegeometry of spacetime; see Lehmkuhl [22] for details. trong geodesic (spacetime) criterion: For any choice of initial condi-tions, if test particles (and massive bodies that can be idealised as testparticles) were around, then they would all follow the time-like geodesicsof the same affine connection or metric whose null geodesics light rayswould follow if they were around.It is often simply assumed that in GR the Levi-Civita connection fulfilsthis criterion. And because this connection also gives rise to the curvaturetensor that features on the left-hand side of the gravitational field equations,and is compatible to the metric tensor that solves these field equations, it ispossible to interpret GR as showing that the gravitational potential g µν ispart of spacetime structure. If one does not want to assume that that testparticles move on timelike geodesics, then different geodesic theorems (one ofwhich we discuss below) allow one to derive that they do. When it comes toderiving the assertion that light rays move on null geodesics, then the choicebetween possible ways of deriving this is much more limited; the most commonargument procceds via the geometrical-optical limit of GR, which is far fromhaving the status of a theorem. This might lead one to demand a weakened version of the geodesic crite-rion, one that remains agnostic on whether light rays do indeed move on nullgeodesics:
Weak geodesic (spacetime) criterion:
For any choice of initial conditions,if test particles (and massive bodies that can be idealised as test particles)were around, then they would all follow the geodesics of the same affineconnection or metric.For this criterion the distinction between timelike and null is not relevantanymore.But note that though this criterion is weaker, we are here searching notfor necessary conditions for something to be a spacetime or part of spacetimestructure; the array of generalizations of pseudo-Riemannian geometry thatarose after GR is far too rich for that, and each such generalization, with itsmultiple curvature tensors, torsion, contorsion and non-metricity tensors, isin principle a candidate for a way spacetime could be. For our purposes,designed to tell whether a newly introduced field in superfluid dark mattertheory counts as part of spacetime structure, a sufficient condition is enough.And here the stronger condition, the strong geodesic criterion, seems to providefor the more cautious path.One tentative possibility for a jointly sufficient (albeit not necessary) setof criteria for something to be interpretable as spacetime would then be the See e.g. [91], p.120-121. Note, however, that Geroch and Weatherall’s approach [92] towards deriving geodesic motion,which relies on a new mathematical concept they call “tracking”, allows to derive the assertionthat bodies constructed from wave packets of Maxwell fields ‘track’ null geodesics. They argue(p.626) that this result ‘reflects’ the result normally gained via the geometric-optical limit, namelythat light rays move on null geodesics, in a precise way. See [73], section 2.1, for an overview of such structures, many of which were tried out ascandidate spacetimes in the context of unified field theories. ollowing: it must adhere to spacetime criterion M ∇ C, such that the Cauchycriterion is satisfied and test particles and light rays follow the geodesics ofits metric or connection. This solves our problems of multiple metrics, multi-ple connections and incompatible connections and metrics: the unique objectwhose geodesics are followed by test particles and light rays is the object thatis part of spacetime. If a theory modifies spacetime/gravity in such a way thattest particles and light rays would follow the geodesics of this novel, modified,effective metric, one may say that particles moving ‘under the influence’ of thismodification are forcefree. Thus, when there is a candidate modified/effectivemetric, we can determine whether this is the physically relevant metric bychecking whether test particles and light rays would follow the geodesics ofthis modified object (or rather the geodesics of the connection compatiblewith that new metric) rather than those of the standard metric posited by thetheory.However, the above proposal is incomplete, as a final important physicalspacetime criterion remains, which we might call the chronogeometricity cri-terion or the Strong Equivalence Principle [93]. We need not only ensure thattest particles and light rays survey (the geodesics of) the metric or connection,but also that the (luminous) matter from which one may construct rods andclocks couples to the (same) metric or connection in the right way. That is,there should be no (non-negligible) curvature terms appearing in the dynamicalequations which would prevent those rods and clocks from functioning prop-erly, i.e. from behaving, locally, as they would in special relativity: exhibitingtime dilation, length contraction, etc.
Chronogeometricity (spacetime) criterion:
The Strong Equivalence Prin-ciple is satisfied, i.e. Special Relativity is locally valid, i.e. there are nonon-negligible curvature terms in the local equations of motion. A more demanding set of jointly sufficient criteria for an object to be inter-pretable as part of spacetime structure, and the one we will work with in thefollowing, is then that the object is defined on a manifold fulfilling spacetimecriterion M ∇ C, such that the ‘constraints’ on matter ‘living in’ that spacetime,as described by the Cauchy, strong geodesic and chronogeometricity criteria,are satisfied. Note that if all of these conditions are imposed together, thenthe metric and curvature tensor referred to in the chronometricity criterionneed to be those compatible with the metric referred to in the geodesic crite-ria. If there were no further geometric structures defined on the manifold, thenthis would imply that the conformal and affine structure referred to in crite-rion M ∇ C would be compatible with one another. This in turn means that thepart of spacetime structure that is surveyed by rods, clocks, test particles andlight rays needs to be either (effectively) pseudo-Riemannian or Weylian. Of For more precise definitions, see [93, 94]. Arguably there is one additional relevant criterion: one needs independent reasons to believethat the dimensionality of the manifold matches the dimensionality of physical spacetime. Withoutthis additional constraint, the 5-dimensional metric of a Kaluza-Klein theory would satisfy all thecriteria in the main text, even though it is arguably the 4-dimensional metric that one obtains fromprojecting the 5-dimensioal metric into four dimensions that corresponds to physical space [76]. See [95] for a proof to that effect. ourse, that seems to restrict the set of allowed spacetime structures at firstsight. But note again that we are not saying that all these conditions have tobe fullfilled for something to be a spacetime or an aspect of spacetime struc-ture; we only say that if they are fullfilled, then the object in question can be interpreted as part of spacetime structure.With the matter and spacetime criteria in hand, it is time to turn to ourcase study, superfluid dark matter theory (SFDM), and see how the newlyintroduced field of the theory fares in light of our criteria—and how the criteriafare in light of the physics. Theories labeled as dark matter theories have traditionally done well at thelevel 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 [30, p.5] [96, § We are excluding the Higgs boson here, since, although it is a boson, it is not associated withone of the four fundamental forces. ith global 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) (3)with Λ c and Λ two constants introducing two mass/energy scales. (The mainjustification 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 | , (4)with X ≡ ˙ θ − m Φ − ( ~ ∇ θ ) m , (5)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.(6), with ˜ g = ˜ g SF DM as given byEq.(8).) 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.(5) [30, p.3] [97]. The specific choice of P given by Eq.(4) 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” [30, 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—whether such metrics deservethis name will be discussed in § g µν only indirectly, viathis physical metric): L rel, int = L (˜ g µν , ψ α , ψ α ; µ | ˜ g ) (6)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] [99, Λ c ensures that the theory admits a Φ = 0 vacuum [30, p.15]. In a later paper [98], photons are not coupled to the effective metric, but to the Einsteinmetric. Some conclusions reached in § 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 respectto the Einstein metric ( g µν A µ A ν = −
1) [31, 99, 100]:˜ 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 ) , (7)with M P l the Planck mass and α a dimensionless coupling constant.SFDM modifies TeVeS in two ways: one semantic and one syntactic modi-fication. The semantic revision is to not add yet another (vector) field, but toidentify the four-vector A µ with the unit four-velocity u µ of the (normal fluidcomponent [24, § g SF DMµν ≈ g µν − α Λ M P l θ ( γg µν + (1 + γ ) u µ u ν ) , (8)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 , (9)where ρ b is the baryon density. The total effective, non-relativistic Lagrangiancan then be shown to reproduce [30, 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.4), 3/2, althoughrequired if one aims to eventually regain MONDian behaviour [30, § 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 veloc-ities in galaxies, the superfluid description is appropriate there, exactly Given a mass m and density ρ [30, 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 [30, § V]. bject Theory Matter strength Newtonian spacetime Newtonian Gravity – (static)Minkowski spacetime Special Relativity – (static)Non-trivial g µν General Relativity D (action-reaction) orE (E in region)Photon Standard Model G ( T µν )Electron Standard Model H (mass)Scalar field Φ Superfluid Dark Matter H (mass) (or G ( T µν ))Table 1: Application of the matter criteria to the case study and to other familiarobjects that provide a contrast. where MOND is successful. In the clusters the velocity/ temperature is toohigh, and one finds either a mixture of the superfluid and normal phase, oronly the normal phase, suggesting that the theory might exemplify the usualsuccesses of dark matter theories at that level. Having introduced the two families of matter and spacetime criteria, as well asSFDM with its novel complex scalar field Φ, we are finally in the position to askthe main question: which label(s)—dark matter or modified gravity/spacetime,or both, or neither—does this new field Φ within SFDM deserve? In thissection we first evaluate Φ with respect to the matter criteria (Subsection 5.1)and subsequently with respect to the spacetime criteria (Subsection 5.2).
The matter criteria have been put forward, in their various strengths, as nec-essary and/or sufficient conditions for something to be matter. Before usingthem to evaluate Φ, let us briefly rank some familiar objects, in order to eventu-ally provide contrast with Φ. Newtonian spacetime and Minkowski spacetimesatisfy none of the matter criteria, and would thus, as expected, be consid-ered not to be matter (Table 1). A paradigmatic matter field such as thatof the electron in Quantum Electrodynamics satisfies all matter criteria up tostrength H. A photon satisfies up to exactly criterion G. Remember that we arein the business of distinguishing matter from spacetime, not stable (fermionic)matter from (bosonic) force carriers between that matter—we are grouping thelatter together with matter in this paper. As hinted at before, the strongestcriterion (H) thus seems to be overkill; at most criterion G should be sufficientfor labeling something matter. Depending on how much one lowers the bar forsufficiency the metric field in GR might also be considered matter according tothese criteria. Regardless of where the bar lies exactly, the scalar field added n Berezhiani and Khoury’s Superfluid Dark Matter Theory (SFDM), satisfy-ing even the strongest criterion, would definitely count as matter, making it aDark Matter theory as their choice of name suggests! One issue though is thatit may be unclear, especially in the context of our case study, what is meantexactly by having mass. Particle physicists may equate mass to the pole of thepropagator. In gravitational physics, the total mass of a gravitating systemlike a star or a black hole is defined as its Komar, ADM or Bondi mass, whichinclude contributions both from the gravitational binding energy holding thestar (say) together, and from the matter fields that make it shine. However,these three conceptions of mass don’t necessarily coincide, and can only be de-fined in special spacetimes. Nevertheless, this has not prevented condensedmatter physicists, whose discipline inspired core elements of SFDM, from usingthe concept of mass.That being said, it is also true that there are regions within galaxies wherethe mass-energy-momentum associated with Φ plays a subdominant or evennegligible role in accounting for the (MONDian) behaviour of luminous matterin galaxies. The explanatory story can then be told (almost) fully in terms ofeffective metrics. Moreover, even Rovelli admits that, although the metric fieldin GR has adopted most of the classic properties of matter, this “is not to saythat the gravitational field is exactly the same object as any other field. Thevery fact that it admits an interpretation in geometrical terms witnesses to itspeculiarity” [8, p.194]. Before passing judgement it therefore seems advisableto turn to the second family of criteria popular in the literature, the spacetimecriteria.
In this subsection we evaluate the SFDM scalar field Φ according to the space-time criteria. Crucial to this discussion will be the fact that in SFDM, withinthe superfluid regime, normal matter (i.e. luminous, non-dark matter) ‘feels’the effective metric ˜ g SF DMµν —built up out of the Einstein metric and the SFDMscalar field—instead of just the Einstein metric g µν . For, in that regime, L lum − mat is a function of ˜ g SF DMµν and of covariant derivatives with respectto that effective metric (Eq.6), rather than a function of g µν and of covariantderivatives with respect to that Einstein metric: , L lum − mat = L (˜ g SF DMµν , ψ α , ψ α ; µ | ˜ g SFDM ) . (10)Let us start with the ‘strong geodesic criterion’. We first turn to testparticles, followed by a discussion of the behaviour of light rays. According to Both the ADM mass and the Bondi mass rely on approximating the spacetime containing thestar or black hole as asymptotically flat, and rely on the resulting Killing symmetries to definethe total mass of the body in question. The Bondi mass is defined via asymptotic symmetriesat spatial infinity, the ADM mass via those at null infinity. The Komar mass does not demandasymptotic flatness but stationarity, and so here too Killing symmetries do much of the work ofdefining a concept of mass. See fn.52. We assume the simplest possible case, i.e. minimal coupling and only first order derivatives ofthe matter fields, which is typical for a matter Lagrangian. FDM, do massive test particles (of luminous matter) follow geodesics, andif so, the geodesics of which metric? The Einstein metric? The effective SFDMmetric? A different metric altogether? Let us recall the geodesic theorem byGeroch and Jang [101]. Suppose that given any open subset O of manifold M with metric g ′ µν containing a curve γ , there exists a smooth, symmetric field T µν with the following properties:1. T µν satisfies the strengthened dominant energy condition , i.e. given anytimelike vector ξ µ at any point in M , T µν ξ µ ξ ν ≥
0, and either T µν = 0or T µν ξ ν is timelike; T µν satisfies the conservation condition , i.e. ∇ ν T µν = , where the co-variant derivatives are with respect to g ′ µν ;3. supp( T µν ) ⊂ O ; and4. there is at least one point in O at which T µν = .Then γ is a timelike curve that may be reparametrized as a geodesic.The paths of test particles are supposed to correspond to γ in virtue ofcondition 4, whereby the non-vanishing of T µν is taken as corresponding tothe presence of a massive test particle. However, note that there is no furtherrequirement that T µν be the stress-energy-momentum tensor, though it needsto be a second-rank tensor that can be interpreted as an indicator of thepresence of matter in order to use the theorem for making predictions aboutthe presence of matter. The theorem was further generalized by Geroch andEhlers [102] and by Geroch and Weatherall [92], but for our purposes theoriginal Geroch-Jang theorem suffices. Our question now becomes: for which metric g ′ µν within SFDM does thegeodesic theorem hold? In particular, with respect to (the covariant derivativesassociated with) which metric is T µν conserved? For that is condition 2 of theGeroch-Jang theorem, and drives much of the proof of the theorem, telling uson the geodesics of which metric test particles move. To that end, recall thefollowing further result. Consider any matter field, describable by an actionthat is a function (only) of that field, some metric g ′ µν and covariant derivativesof that field with respect to that metric. If that action is required to bediffeomorphism invariant, it can be shown that there exists a rank-2 tensor T µν associated with that matter field which is symmetric, vanishes in openregions only if the field configuration (satisfying the field equations) vanishesthere, and which is covariantly conserved with respect to g ′ µν (in virtue ofthe matter field equations holding) [105, p.64–67] [106, p.456]. We are here not considering test matter made out of Φ, but see the section on breakdowninterpretations in the sequel paper [24]. Though the metric does not explicitly appear in the definition of the strengthenend dominantenergy condition, the latter nevertheless can only be defined with reference to at least conformal,if not metric, structure. See [23, § See also [103] for further discussion of the Geroch-Jang theorem and its conditions and appli-cations, and [104] for arguments why Einstein preferred a different type of geodesic theorem. As long as the Lagrangian density vanishes only where the field (satisfying the field equations)does and has non-trivial dependence on the metric (configuration satisfying the field equations forthat metric). he luminous-matter part of the SFDM action (applicable to the superfluidregime) has exactly the required form, with g ′ µν = ˜ g SF DMµν (Eq.10). Fromthe requirement of diffemorphism invariance of this action, one thus obtainsa symmetric rank-2 tensor T µν which, under the additional assumption of itsatisfying the strengthened dominant energy condition, satisfies the geodesictheorem with respect to g SF DMµν . Thus, within the superfluid regime, testparticles (of luminous matter) in SFDM follow the timelike geodesics of theeffective metric g SF DMµν ! The SFDM scalar field modifies the behaviour of thosetest particles, in the sense that they do not follow the timelike geodesics of theEinstein metric, as they do in GR, but instead those of an effective metric builtup out of the Einstein metric and that SFDM scalar field.To determine the behaviour of light rays in SFDM, the most standard waywould be to consider the geometrical-optical limit of electromagnetic fields de-fined on an SFDM spacetime. As of yet, no-one has worked this out in anydetail. However, what drives the argument that light rays move on null geode-scis in the context of GR is that one assumes minimal coupling between themetric and the electromagnetic field, i.e. replaces the partial derivatives in theMaxwell equations by covariant derivatives, where the covariant derivative iscompatible with the metric that solves the gravitaional field equations, andwith the help of which the curvature tensor featuring in the field equations isdefined. One then derives the wave equation for the electromagnetic vectorpotential on a curved background, which features a contraction of said curva-ture tensor. Then one solves this wave equation approximately by introducinga parameter ǫ that tracks how rapidly various terms approach zero or infin-ity in said approximation, and then observes that looking at terms of order O ( ǫ ) and O ( ǫ ) one can derive that in that limit, where electromagnetic wavescan be approximated as (light) rays, the light rays move on null geodesics ofthe connection. Now, if one assumes that in the SFDM context the partialderivatives occuring in the Maxwell equations on flat spacetime should be re-placed by covariant derivatives with respect to the effective metric ˜ g µν fromEq.8 (cf. Eq.10), then it seems plausible that one could make an analogousargument in the context of SFDM. If this is true, then light rays in SFDMwould follow geodesics of the effective metric ˜ g µν .The effective metric in SFDM (plus Maxwell) would then be as spatiotem-poral as the Einstein metric in GR (plus Maxwell) as far as the behaviour oflight rays is concerned.Finally, we move to the chronogeometricity criterion. To determine the See e.g. [107], p.570-577 for details. However, note that the curvature tensor derived from ˜ g µν would contain the phase of the newlyintroduced superfluid field, θ , as well as the velocity vector field u µ of its normal fluid component.Since part of the assumption that goes into the above approximation scheme is that the ‘typicalreduced wavelength’ of the electromagnetic waves is small compared to the ‘typical component ofthe Riemann tensor in a typical Lorentz frame’, it could be that the more complicated curvaturetensor gets in the way of imposing these conditions. Berezhiani, Khoury and Famaey [98] assert that MOND, and by extension SFDM, violatesthe ‘strong equivalence principle’. It is clear though that what they refer to as (an implicationof) the ‘strong equivalence principle’, namely that “a homogeneous acceleration has no physicalconsequence” [98, p.14] (which would indeed be contradicted by MOND’s acceleration scale a ), is ocal validity of Special Relativity, we need to determine whether any curvatureterms pop up in the equations of motions of the luminous matter (that makesup rods and clocks). To this end, compare the way luminous matter is coupleduniversally to the Einstein metric g µν within General relativity, L lum − mat − in − GR = L lum − mat ( g µν , ψ α , ψ α ; µ | g µν ) (11)with the way in which the direct analogue of that Lagrangian term would beincluded in (the superfluid regime of) SFDM, that is by a universal coupling tothe SFDM metric, Eq.10. As the matter equations are determined by varying S lum − mat , which has the same form for both theories except for the role of themetric being played by different tensors, these equations of motions also havethe same form except for any potential curvature terms being with respect dodifferent metric tensors (i.e. g µν and ˜ g SF DMµν respectively). Strict satisfactionof the chronogeometricity criterion would require there to be no curvatureterms at all in the local equations of motions of the luminous matter fields ψ α .It turns out however that that not only non-minimal coupling can generatecurvature terms [21, § Let us sum up where that leaves us so far. The scalar field Φ introduced bySFDM satisfies the matter criterion of (almost) the highest strength. Regard-less of where one draws the line, that is which matter strength one considerssufficient for something being matter, the scalar field Φ will clearly count as amatter field. We have also seen that, at least within the superfluid regime, theSFDM effective metric, constructed out of the Einstein metric and Φ, satisfiesthe spacetime criteria to the same extent as the Einstein metric does withinGR. Φ is thus—at least for temperatures below the critical temperature for not what we have defined as the strong equivalence principle. We would call this a version of theEinstein equivalence principle [94]. uperfluidity—also as much of an aspect of spacetime as one can expect of adynamical field.The significance of this result becomes clear when we contrast it with a pre-vious attempt by Khoury to account for dark phenomena [108]. That theoryadds not one but two scalar fields. According to Khoury, the first scalar field“behaves as a dark matter fluid on large scales” [108, p.1]. The second “me-diates a fifth force that modifies gravity on nonlinear scales” [108, p.1]. Evenif these quotes hold up under the interpretations offered by our families of cri-teria, it would be one field being responsible for satisfying one or more of thematter criteria, and the other field being responsible for ticking off the space-time criteria. This is neither novel nor especially interesting with respect tothe interpretational questions posed in our two companion papers: both within(field-theoretic versions of) Newtonian physics and Special Relativity do themetric fields satisfy the spacetime criteria (and none of the matter criteria)and, say, electric charge density fields satisfy the matter criteria (and none ofthe spacetime criteria). What is interesting about SFDM is that, rather thanrequiring a second field, the modification of the metric is associated with a four-velocity of the only new (scalar) field Φ which also plays the dark matter role;the “DM and MOND [or Modified Gravity] components have a common origin,representing different phases of a single underlying substance” [30, p.1,3].The result of this paper still leaves open the interpretational questions men-tioned at the beginning. What follows from this for the distinction betweendark matter and modified gravity, as well as the broader distinction betweenmatter and spacetime, both within SFDM and in general? Does the fact thatΦ seems to satisfy the spacetime criteria only below the critical temperaturefor superfluidity imply that it is ‘more’ of a dark matter field than it is a mod-ification of spacetime, or are both roles on a par? These and other remaininginterpretational questions will be the focus of the companion paper [24]. 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 Radin Dardashti, Tushar Menon, James Read, Joshua Rosaler, Kian Sal-imkhani, Michael St¨oltzner and Adrian W¨uthrich for valuable discussions andcomments, as well as the audiences of the Geneva Symmetry Group ResearchSeminar (Geneva, Switzerland, 2018), the Fifth International Conference onthe Nature and Ontology of Spacetime (Albena, Bulgaria, 2018), the BritishSociety for the Philosophy of Science Conference (Oxford, UK, 2018), the Phi-losophy of Science & Technology Colloquium (Aachen, Germany, 2018), thePhilosophy Colloquium (Cologne, Germany, 2018), the Dark Matter & Mod-ified Gravity Conference (Aachen, Germany, 2019), the German Society forthe Philosophy of Science Conference (Cologne, Germany, 2019) and the First xford-Notre Dame-Bonn Workshop on the Foundations of Spacetime Theo-ries (Oxford, UK, 2019). References [1] Robert H. Sanders.
The Dark Matter Problem: A Historical Perspective .New York: Cambridge University Press, 2010.[2] Gianfranco Bertone and Dan Hooper. History of dark matter.
Rev. Mod.Phys. , 90:045002, Oct 2018.[3] Pengfei Li, Federico Lelli, Stacy McGaugh, and James Schombert. Fit-ting the radial acceleration relation to individual sparc galaxies.
A&A ,615:A3, 2018.[4] Stacy S. McGaugh, Pengfei Li, Federico Lelli, and James M. Schombert.Presence of a fundamental acceleration scale in galaxies.
Nature Astron-omy , 2:924, 2018.[5] Benoˆıt Famaey and Stacy S. McGaugh. Modified newtonian dynamics(mond): Observational phenomenology and relativistic extensions.
Liv-ing Reviews in Relativity , 15(1):10, Sep 2012.[6] Robert Rynasiewicz. Absolute versus relational space-time: An out-moded debate?
Journal of Philosophy , 93(6):279–306, 1996.[7] Richard P. Feynman.
Feynman Lectures on Gravitation . Addison-WesleyPublishing Company, 1995.[8] 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.[9] Mauro Dorato. Substantivalism, relationism, and structural spacetimerealism.
Foundations of Physics , 30(10):1605–1628, 2000.[10] 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.[11] 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.[12] 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.[13] Carlo Rovelli.
Quantum Gravity . Cambridge University Press, 2010.[14] David Rey. Similarity assessments, spacetime, and the gravitational field:What does the metric tensor represent in general relativity? 2013.[15] 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.
16] Dennis Lehmkuhl. The metaphysics of super-substantivalism.
Noˆus ,52(1):24–46, 2018.[17] Gordon Belot. Background-independence.
General Relativity and Grav-itation , 43(10):2865–2884, Oct 2011.[18] Hilary Greaves. In search of (spacetime) structuralism.
PhilosophicalPerspectives , 25:189–204, 2011.[19] J. Earman and J.D. Norton. What price spacetime substantivalism? thehole story.
British Journal for the Philosophy of Science , 38:515–525,1987.[20] James L. Anderson.
Principles of Relativity Physics . New York andLondon: Academic Press, 1967.[21] Harvey R. Brown.
Physical Relativity: Space-time structure from a dy-namical perspective . Oxford University Press, 2005.[22] Dennis Lehmkuhl. Why einstein did not believe that general relativitygeometrizes gravity.
Studies in History and Philosophy of Science PartB: Studies in History and Philosophy of Modern Physics , 46:316 – 326,2014.[23] Dennis Lehmkuhl. Mass-Energy-Momentum: Only there Because ofSpacetime?
The British Journal for the Philosophy of Science ,62(3):453–488, 06 2011.[24] [authors omitted]. Cartography of the space of theories: an interpreta-tional chart for fields that are both (dark) matter and spacetime.[25] Patrick M. D¨urr. Theory (in-)equivalence and conventionalism in f(r)gravity. ms.[26] 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.[27] Samuel C. Fletcher, J.B. Manchak, Mike D. Schneider, and James OwenWeatherall. Would two dimensions be world enough for spacetime?
Stud-ies in History and Philosophy of Science Part B: Studies in History andPhilosophy of Modern Physics , 63:100 – 113, 2018.[28] B. Skow.
Once Upon a Spacetime . PhD thesis, New York University,2005.[29] Lasha Berezhiani and Justin Khoury. Dark matter superfluidity andgalactic dynamics.
Physics Letters B , 753:639–643, 2016.[30] Lasha Berezhiani and Justin Khoury. Theory of dark matter superfluid-ity.
Phys. Rev. D , 92:103510, Nov 2015.[31] Jacob D. Bekenstein. Relativistic gravitation theory for the MONDparadigm.
Phys. Rev. , D70:083509, 2004. [Erratum: Phys.Rev.D71,069901(2005)].[32] Luc Blanchet and Alexandre Le Tiec. Model of dark matter and darkenergy based on gravitational polarization.
Phys. Rev. D , 78:024031, 72008.
33] HongSheng Zhao. Reinterpreting mond: coupling of einsteinian gravityand spin of cosmic neutrinos? 2008.[34] 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.[35] Baojiu Li and Hongsheng Zhao. Environment-dependent dark sector.
Phys. Rev. D , 80:064007, Sep 2009.[36] Chiu Man Ho, Djordje Minic, and Y. Jack Ng. Cold dark matter withmond scaling.
Physics Letters B , 693(5):567 – 570, 2010.[37] C.M. Ho, D. Minic, and Y.J. Ng. Quantum gravity and dark matter.
Gen Relativ Gravit , pages 2567–2573, 2011.[38] Chiu Man Ho, Djordje Minic, and Y. Jack Ng. Dark matter, infinitestatistics, and quantum gravity.
Phys. Rev. D , 85:104033, May 2012.[39] 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.[40] M. Cadoni and M. Tuveri. Galactic dynamics and long-range quantumgravity.
Phys. Rev. D , 100:024029, Jul 2019.[41] 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.[42] Elisa G.M. Ferreira. Ultra-light dark matter. ms.[43] Constantinos Skordis and Tom Z lo´snik. A new relativistic theory formodified newtonian dynamics, 2020.[44] Antonino Del Popolo and Morgan Le Delliou. Small scale problems ofthe λ cdm model: A short review. Galaxies , 5(17), 2017.[45] Sabine Hossenfelder. Covariant version of verlinde’s emergent gravity.
Physical Review D , 95(12):124018, 2017.[46] Marc Lange.
An Introduction to the Philosophy of Physics: Locality,Fields, Energy, and Mass . Blackwell Publishers, 2002.[47] Dustin Lazarovici. Against fields.
European Journal for Philosophy ofScience , 8:145–170, 2018.[48] Mario Bunge. Energy: Between physics and metaphysics.
Science &Education , 9(5):459–463, 2000.[49] Charles T. Sebens. The Mass of the Gravitational Field.
The BritishJournal for the Philosophy of Science , 01 forthcoming. axz002.[50] M. Frisch.
Inconsistency, Asymmetry, and Non-locality: A PhilosophicalInvestigation of Classical Electrodynamics . Oxford: Oxford UniversityPress, 2005.[51] David J. Baker. On spacetime functionalism.[52] 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.
53] Dennis Lehmkuhl.
Einstein’s Principles. On the Interpretation of Grav-ity . Oxford University Press, forthcoming.[54] Albert Einstein. Prinzipielles zur allgemeinen Relativit¨atstheorie.
An-nalen der Physik , 55:241–244, 1918. Reprinted as Vol. 7, Doc. 4 CPAE.[55] The collected papers of albert einstein, volume 8: The berlin years: Cor-respondence, 1914-1918, 1998.[56] Carl Hoefer. Einstein’s struggle for a Machian gravitation theory.
Studiesin History and Philosophy of Science Part A , 25(3):287–335, 1994.[57] Carl Hoefer. Einstein’s formulations of Mach’s principle. In Julian Bar-bour and Herbert Pfister, editors,
Mach’s Principle: From Newton’sBucket to Quantum Gravity , volume 6 of
Einstein Studies , pages 67–90. Birkh¨auser, 1995.[58] Carl Hoefer. The metaphysics of space-time substantivalism.
The Jour-nal of Philosophy , 93(1):5–27, 1996.[59] 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.[60] Patrick D¨urr. Against ‘functional gravitational energy’. Forthcoming inSynthese.[61] C. Hoefer. Kant’s hands and earman’s pions: chirality arguments forsubstantival space.
International Studies in the Philosophy of Science ,14(3):237–256, 2000.[62] 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.[63] Albert Einstein. N¨aherunsweise Integration der Feldgleichungen derGravitation.
Sitzungsberichte der K¨oniglich Preussischen Akademie derWissenschaften , (688-696), 1916. Reprinted as Document 32, Volume 6CPAE.[64] Albert Einstein. ¨Uber Gravitationswellen.
Sitzungsberichte der K¨oniglichPreussischen Akademie der Wissenschaften , pages 154–167, 1918.Reprinted as Document 1 of Volume 7 CPAE.[65] C´ecile M. DeWitt and Dean Rickles.
The role of gravitation in physics:Report from the 1957 Chapel Hill Conference , volume 5. epubli, 2011.[66] James Read. Functional gravitational energy.
The British Journal forthe Philosophy of Science , 71:205–232, 2020.[67] Isaac Newton.
Philosophiae Naturalis Principia Mathematica . London:Joseph Streater, 1686/7. The Scholium to the Definitions has beenreprinted in Alexander [109].[68] Max Jammer.
Concepts of Mass in Contemporary Physics and Philoso-phy . Princeton University Press, 2000.
69] L. M. Hoskins. Mass as quantity of matter.
Science , 42(1080):340–341,1915.[70] Tim Maudlin. Buckets of water and waves of space: Why spacetime isprobably a substance.
Philosophy of Science , 60:183–203, 1993.[71] R.W. Sharpe.
Differential geometry: Cartan’s generalization of Klein’sErlangen program . Berlin: Springer, 1997.[72] V. Vizgin.
Unified Field Theories in the First Third of the 20th Century .Basel: Birkh¨auser, 1994.[73] Hubert F.M. Goenner. On the history of unified field theories.
LivingReviews in Relativity , 7(2), 2004.[74] M. Blagojevi´c and F.W. Hehl.
Gauge theories of gravitation. A readerwith commentaries . London: Imperial College Press, 2013.[75] Tushar Menon. Taking up superspace: The spacetime structure of su-persymmetric field theory. In
Philosophy beyond Spacetime . Oxford Uni-versity Press, forthcoming.[76] D. Lehmkuhl.
Spacetime Matters: On Super-Substantivalism, GeneralRelativity, and Unified Field Theories . PhD thesis, Oriel College, Uni-versity of Oxford, 2009.[77] J. Earman.
World Enough and Space-Time: Absolute versus RelationalTheories of Space and Time . Cambridge, Massachusetts: MIT, 1989.[78] Simon Saunders. Rethinking newton’s principia.
Philosophy of Science ,80(1):22–48, 2013.[79] Eleanor Knox. Newtonian spacetime structure in light of the equivalenceprinciple.
British Journal for the Philosophy of Science , 65(4):863–80,2014.[80] David Wallace. Fundamental and Emergent Geometry in NewtonianPhysics.
The British Journal for the Philosophy of Science , 12 2017.axx056.[81] Jeremy Butterfield. Albert einstein meets david lewis. In Arthur Fine andJarret Leplin, editors,
Proceedings of the 1988 Biennial Meeting of thePhilosophy of Science Association , volume 2, pages 65–81. Philosophy ofScience Association, 1989. Reprinted in Butterfield, Hogarth and Belot1996.[82] O. Pooley.
The Reality of Spacetime . Dphil thesis, Oriel College, Uni-versity of Oxford, 2002.[83] Oliver Pooley.
The Reality of Spacetime . Manuscript.[84] Oliver Pooley. Substantivalist and relationalist approaches to spacetime.In Robert Batterman, editor,
The Oxford Handbook of Philosophy ofPhysics . Oxford University Press, 2013.[85] Ari Maunu. Generalist transworld identitism (or, identity through pos-sible worlds without nonqualitative thisnesses).
Logique & Analyse ,48(189-192):151–158, 2005.
86] Neil Dewar and Joshua Eisenthal. A raum with a view: Hermann weyland the problem of space. ms.[87] James Read and Tushar Menon. The limitations of inertial frame space-time functionalism.
Synthese , forthcoming.[88] Craig Callender.
What Makes Time Special?
Oxford University Press,2017.[89] Steven Weinstein. Many times.
Foundational Questions Insti-tute The Nature of Time Essay Contest. fqxi.org/data/essay-contest-files/Weinstein FQXI2.pdf , 2008.[90] John Archibald Wheeler. Geometrodynamics. In
Italian Physical Society:Topics of Modern Physics, Volume I . Academic Press Inc., 1962.[91] David Malament.
Topics in the Foundations of General Relativity andNewtonian Gravitation Theory . University Of Chicago Press, 2012.[92] Robert Geroch and James Owen Weatherall. The motion of small bodiesin space-time.
Communications in Mathematical Physics , 364(2):607–634, 2018.[93] 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.[94] Dennis Lehmkuhl. The equivalence principle(s). In Eleanor Knox andAlastair Wilson, editors,
The Routledge Companion to Philosophy ofPhysics . forthcoming.[95] J. Ehlers, A. Pirani, and A.Schild. The geometry of free fall and lightpropagation. In L. O’Raifeartaigh, editor,
General Relativity . New York,Oxford, 1972.[96] James F. Annett.
Superconductivity, Superfluids and Condensates . Ox-ford University Press, 2004.[97] D.T. Son. Low-energy quantum effective action for relativistic superrflu-ids. 2002.[98] Lasha Berezhiani, Benoit Famaey, and Justin Khoury. Phenomenologicalconsequences of superfluid dark matter with baryon-phonon coupling. arXiv preprint arXiv:1711.05748 , 2017.[99] Timothy Clifton, Pedro G. Ferreira, Antonio Padilla, and ConstantinosSkordis. Modified gravity and cosmology.
Physics Reports , 513(1):1 –189, 2012. Modified Gravity and Cosmology.[100] Jacob D. Bekenstein. Relation between physical and gravitational geom-etry.
Phys. Rev. D , 48:3641–3647, Oct 1993.[101] R. Geroch and P.S. Jang. Motion of a body in general relativity.
Journalof Mathematical Physics , 16:65, 1975.[102] J. Ehlers & R. Geroch. Equation of motion of small bodies in relativity. arXiv:0309074v1 , 2008 [2003].[103] James Owen Weatherall. Conservation, inertia, and spacetime geometry.
Studies in the History and Philosophy of Modern Physics , 2017. Philosophy of Science , 84(5), 2017. Extended version at http://philsci-archive.pitt.edu/12461/.[105] S.W. Hawking and G.F.R. Ellis.
The large scale structure of space-time .New York: Cambridge University Press, 1973.[106] R.M. Wald.
General Relativity . University of Chicago Press, 1984.[107] C.W. Misner, K.S. Thorne, and J.A. Wheeler.
Gravitation . New York:W.H. Freemand and Company, 1973.[108] Justin Khoury. Alternative to particle dark matter.
Phys. Rev. D ,91:024022, Jan 2015.[109] H.G. Alexander, editor.
The Leibniz-Clarke Correspondence . Manch-ester: Manchester University Press, 1956 [1717]. Originally written byLeibniz, G.W. & Clarke, S. in 1715-16 and published by Clarke, S. in1717.. Manch-ester: Manchester University Press, 1956 [1717]. Originally written byLeibniz, G.W. & Clarke, S. in 1715-16 and published by Clarke, S. in1717.