philosophy of physics

Substantivalist and Relationalist Approaches to Spacetime and c onc eptually elegant (and radic al) alternative to the standard spac etime perspec tive. The key issue is not an ontological one about the reality of instantaneous spatial (that is, spac etime) points (the theory is naturally understood as c ommitted to them); it c onc erns the relative priority of spatial versus spatiotemporal ideology. Despite Barbour's c laims, the loc al c onformal degrees of freedom of CMC spac elike hypersurfac es are not obviously philosophic ally superior to the standard spac etime quantities: they are not (more) direc tly observable (rec all footnote 36), nor are primitive temporal intervals, or primitive c omparisons of distant lengths, somehow inherently suspec t. (In fac t, an argument c ould even be made that observability c onsiderations favor spac etime over instantaneous quantities.) Even the parsimony argument in favor of the Mac hian theory is less c lear-c ut in GR than in Newtonian mec hanic s. In GR it is no longer the c ase that the kinematic struc tures of the Mac hian theory are simply a proper subset of those ac c epted in the spac etime theory.8 6 The true test of the Mac hian program will be its physic al fruitfulness, in partic ular whether, as its advoc ates hope, it leads to progress in the searc h for a theory that suc c essfully rec onc iles quantum mec hanic s and general relativity. 6.3 Have-it-all Relationalism The relationalist strategies examined in sec tions 6.1 and 6.2 involve a c ertain honesty. They ac c ept that restric ted dynamic al symmetries betoken spac etime struc ture with symmetries that are at least as restric ted and seek to square this with relation-alism, either by showing how such structure can be both primitive and anchored in a relationalist ontology or by seeking new dynamic s with expanded symmetries. The approac h reviewed in this sec tion is a c ase of trying to have one's c ake and eat it. It seeks a way to rec onc ile restric ted dynamic al symmetries with more permissive spac etime symmetries. On this approac h, therefore, some of the spac etime struc ture implic it in the dynamic s is judged to have only an effec tive status, ultimately grounded in a less struc tured relationalist ontology. Huggett's “regularity approac h” is an explic it proposal about how to do this for Newtonian mec hanic s. The dynamic al approac h to spec ial relativity, defended by Brown (2005) and Brown and Pooley (2006), c an be understood along similar lines. 6.3.1 The Regularity Approach to Relational Spacetime Huggett draws inspiration from some remarks of van Fraassen's on the meaning of Newton's laws. Having posed the problem of how the relationalist c an ac c ount for the privileged status of the inertial frames, van Fraassen seeks to dissolve it by asserting that inertial frames do not have a privileged status at all (van Fraassen 1970, 116). In c laiming this, he is not asserting that Newton's laws fail to differentiate between frames of referenc e. He ac knowledges, of c ourse, that they do. His c laim, rather, is that there need be nothing more to the inertial frames' being privileged than their being exac tly those frames with respec t to whic h c ertain statements about mass, motion and forc e hold true. The differenc e between this point of view and the standard substantivalist position might seem elusive, but it c onc erns whic h fac ts are to be taken as basic . For the substantivalist, the basic fac ts inc lude fac ts about suc h things as the relative temporal distanc es between pairs of events, about what c ounts as a straight spac e-time trajec tory, and so on. While the dynamic al laws are to be understood in terms of suc h fac ts, and while the suc c ess of those laws is ac knowledged as our only evidenc e for there being suc h fac ts, the fac ts are not to be c onc eived of as in any way dependent on the dynamic al laws. Ac c ording to the relationalist view now under c onsideration, they are so dependent. Beyond the fac ts that the Leibniz ian relationalist ac knowledges as primitive, the most basic spatiotemporal fac t for van Fraassen is the existenc e of privileged c oordinate systems with respec t to whic h the dynamic s of matter takes on a partic ularly simple form. Nonrelational quantities of motion are to be thought of as defined in terms of the very laws in whic h they feature. Dynamic al laws, and the equations that express them, figure prominently in the c harac teriz ation of this position. Exac tly what it amounts to, therefore, will depend on how laws themselves are to be c onc eived. Suppose, for example, that laws of nature are held to involve some kind of primitive natural nec essity. The position then bec omes the c laim that the relative distanc es between all partic les in the universe are c onstrained as a matter of nomologic al nec essity to evolve over time so that they satisfy c ertain simple equations with respec t to a privileged c lass of c oordinate systems. Suc h a view, while c onsistent, has little to rec ommend it over the substantivalist's ac c eptanc e at fac e value of the quantities featuring in the dynamic al laws. The relationalist is effec tively c laiming that relative distanc es between bodies are c onstrained to evolve as if eac h body had an independent quantity of motion that was governed by c ertain simple laws. This looks like a c ase where Earman's c harge that the Page 24 of 48

Substantivalist and Relationalist Approaches to Spacetime relationalist position is “hardly distinguishable from instrumentalism” is justified (Earman 1989, 128). Whether or not that spells trouble for the relationalist, their debate with the substantivalist has been replac ed by a more generic dispute and has lost its distinc tive c harac ter.8 7 Things look more interesting if one adopts a Humean approac h to laws. The most promising Humean view is the Mill– Ramsey– Lewis “Best Systems” ac c ount, ac c ording to whic h the laws of nature are statements that appear as theorems of those axiom systems true of the totality of Humean fac ts that best c ombine the c ompeting virtues of simplic ity and strength (see, e.g., Lewis 1973, 72– 73; Earman 1986, c h. 5). Without some c onstraints on admissible voc abulary, the simplic ity requirement is not straightforward, bec ause a theory's simplic ity appears to be language-dependent. Lewis's later preferred constraint invokes a primitive distinction among properties: the formulations of c andidate laws with respec t to whic h simplic ity is to be judged are in languages whose predic ates denote perfec tly natural properties and relations (Lewis, 1983). Huggett's idea, effec tively, is that this requirement c an be liberaliz ed without bec oming vac uous. In partic ular, it is very plausible that, (i) if one assumes the ontology and ideology of Leibniz ian relationalism and (ii) if one allows, as c andidate Humean laws, systems formulated in terms of supervenient properties, as well as perfec tly natural properties, Newton's laws will c onstitute by far and away the simplest and strongest systematiz ation of a typic al Leibniz ian relational history c ompatible with those laws (Huggett 2006, 48–50). Unbeknown to Huggett, a parallel liberaliz ation of Lewis's Best Systems presc ription had already been outlined by Sider, as a possible response to Kripke's “rotating disks” argument against perduranc e (Sider 2001, 230– 234). Sider's goal was to ground a distinc tion between rotating and nonrotating homogeneous matter in the primitive ontology and ideology of the perdurantist (that is, someone who analyz es material persistenc e in terms of numeric ally distinc t temporal parts of the persisting objec t loc ated at the different times at whic h the objec t exists). The tric k is to suppose that Best Systems laws might be formulated in terms of “physic al c ontinuants,” that is, aggregates of genidentity-interrelated material events where, crucially, the non-Humean genidentity relation is not a primitive relation but supervenes on the total history of Humean properties together with the laws in whic h it features: Consider various ways of grouping stages together into physic al c ontinuants. Relative to any suc h way, there are c andidate laws of dynamic s. The c orrec t grouping into physic al c ontinuants is that grouping that results in the best c andidate set of laws of dynamic s; the c orrec t laws are the members of this c andidate set. (Sider 2001, 230) The c omparison of Huggett's and Sider's proposals prompts the following worry. If Huggett's reduc tion of inertial struc ture relies on primitive transtempo-ral partic le identity and Sider's reduc tion of material genidentity relies on primitive inertial struc ture, one or the other of the reduc tions must be untenable. In response, the liberal Humean might embrac e both moves at onc e: if one strips fac ts about transtemporal partic le identity from a typic al Leibniz ian relational history c ompatible with Newton's laws, it remains very plausible that those laws will form part of any Best System theory of suc h a world, if one is permitted to express the theory in terms of supervenient genidentities with respec t to supervenient privileged c oordinate systems. But c ombing both proposals into a single pac kage highlights a related diffic ulty. Onc e the stric t requirement that primitive voc abulary should express primitive, perfec tly natural properties and relations is relaxed, what governs whic h quantities are part of the supervenienc e base and whic h quantities are supervenient? Why stop at a reduc tion of genidentity and inertial struc ture? Why not seek to offer a reduc tive ac c ount of mass and c harge too? Why not even seek a reduc tive ac c ount of the temporal metric and instantaneous spatial distanc es? Onc e the reduc tion via the dynamic al laws of some apparently natural properties to the others is on the table, we need some princ iples to determine whic h properties are ripe for reduc tion and whic h are to be part of the basic ideology.8 8 Huggett himself rec ogniz es the issue. He notes that he has inc luded masses and c harges but not forc es in his supervenienc e base bec ause “a quantity c an only be said to be a forc e if it plays the right kind of role in the laws and so c annot be metaphysic ally prior to the laws” (Huggett, 2006, 47). This is a surprising thing for a Humean to say. As Huggett c onc edes, one might (as many non-Humeans do) say the same about mass and c harge. Later, when worrying that his supervenient quantities proposal is “too easy,” he c ites “serious objec tions, with a long history, against the supposition of a non-material, physic al substanc e” (ibid., 70) as reason to pursue a reduc tion that at least allows one to do without spac etime. But, as we have seen, (i) what prima facie strong objec tions there are to substantivalism c an be met and (ii), in the c ontext of Newtonian theory, there are relationalist alternatives to Page 25 of 48

Substantivalist and Relationalist Approaches to Spacetime Huggett's program that do not suffer from this partic ular problem for regularity relationalism. Huggett does offer a c riterion for determining a point beyond whic h reduc tion should not be pursued: the laws should be suc h that they determine the supervenient quantities in all nomic ally possible worlds (ibid., § 4). However, this does not address the possibility that, with respec t to the same set of laws, two distinc t sets of putative subvening properties might share this property. In suc h a c ase, how does one disc over whic h set c ontains the “real” fundamental properties? In his 2006 paper, Huggett only c onsiders Newtonian worlds. We should c onsider how the program looks from the perspec tive of relativistic physic s. Spec ial relativity does not provide a very hospitable arena for the view. The relative attrac tiveness of regularity relationalism in the c ontext of Newtonian physic s is due to a c ouple of fac tors. First, the fac t that the ideology of the Leibniz ian relationalist forms a natural subset of Newtonian ideology means that it is relatively natural to seek a reduc tive ac c ount of the additional (inertial) struc ture in terms of Leibniz ian relations. Sec ond, as reviewed in sec tion 6.1.1, the full neo-Newtonian ideology, when restric ted to a (point partic le) relationalist ontology, is not suffic ient for a relationalist ac c ount of standard physic s, whic h undermines one obvious relationalist alternative. Neither fac tor remains true in the c ontext of SR. In partic ular, Leibniz ian relations are quite unmotivated as a supervenienc e base; it is far more natural to take the spac etime interval as basic and to understand the spatial distanc e relations assoc iated with any partic ular family of simultaneity surfac es in terms of it. Moreover, if the relationalist is happy to c ountenanc e spa-tiotemporal relations between material events as primitive, there is no longer a need for a reduc tion of some spatiotemporal quantities in terms of others for, as reviewed in sec tion 6.1.2, the Minkowski interval restric ted to material events looks like a viable basis for a relational interpretation of standard spec ially relativistic physic s. Things are more interesting when one moves to GR. One aspec t of Huggett's proposal that I have not so far highlighted is that Huggett sees it as a way to allow the geometry of empty spac e to supervene on the geometric al relations instantiated by material bodies.89 Consider, for example, a history of instantaneous spatial relations between bodies that are initially Euc lidean but that depart from Euc lidic ity after some moment, perhaps then to return to Euc lidic ity after some finite further time. Suppose that this history is a solution of a (generaliz ed) Newtonian theory set in a three-dimensional spac e, G, of a fixed geometry that is everywhere Euc lidean exc ept for some finite, geometric ally simple, non-Euc lidean region. The partic les start out in the Euc lidean region and eventual stray into the non-Euc lidean region. Huggett's idea is that the relationalist c an view both the geometry of the total spac e, and the partic les' partic ular embedding in it, as supervenient on the history of relations via (his liberaliz ed version of) the Best Systems approac h to laws. The idea is that the following hypotheses jointly c onstitute the simplest and strongest systematiz ation of the relational history: (i) the history of instantaneous relations is c onstrained to be embeddable at all times into G and (ii) the relational history follows, at any moment, from the instantaneous relations and the embedding into G at that moment, together with a c ertain set of Newtonian laws. In partic ular, the simplic ity requirement fixes G over other more c omplic ated geometries into whic h the partic ular relational history c an also be embedded, e.g., geometries with additional non-Euc lidean regions unsurveyed by the material partic les.9 0 Now rec all the problem that the variable geometry of empty regions of spac etime c an c ause for a relationist who would simply restric t spatiotemporal distanc e relations to material events: a partic ular partial history of pseudo- Riemannian relations instantiated within an island configuration of material events, together with the laws of GR, might not fix the future history bec ause of the possible influenc es of the geometry of empty spac etime beyond the material c onfiguration (sec tion 6.1.2). In the natural extension of Huggett's sc heme, one takes the entire history of instantiated spatiotemporal relations between material events as the supervenience base. One and only one future evolution of the material world c ompatible with the c onsidered initial segment and laws of GR is, of c ourse, inc luded in this. The interesting question for the liberaliz ing Humean is whether, if facts about the geometry of empty spacetime are allowed to supervene, together with the laws, on the material relational history, the laws of GR c onstitute the Best System laws of suc h a world. 6.3.2 The Dynamical Approach to Relativity I finish this sec tion by highlighting some of the similarities between the dynamic al approac h to spec ial relativity, defended by Brown (2005) and by Brown and Pooley (2006), and Huggett's proposal for Newtonian physic s. The dynamic al approac h seeks to offer a reduc tive ac c ount of the Minkowski spac etime interval in terms of the dynamic al symmetries of the laws governing matter. It therefore qualifies as a type of relationalism, although this is not something that Brown himself emphasiz es. Page 26 of 48

Substantivalist and Relationalist Approaches to Spacetime One of the guiding intuitions behind the dynamic al approac h c onc erns explanatory priority. Consider, for example, the relativistic phenomenon of length c ontrac tion. Do rods behave as they do in virtue of the spatiotemporal environment in whic h they are immersed, or are fac ts about the geometric al struc ture of spac etime reduc ible to (inter alia) the behavior of rods? And if one opts for the latter point of view, what explanation is to be given of why measuring rods in motion are c ontrac ted relative to similarly c onstituted rods at rest? Brown reads Bell (1976) as seeking to demonstrate that “a moving rod c ontrac ts, and a moving c loc k dilates, because of how it is made up and not because of the nature of its spatio-temporal environment” (Brown 2005, 8, emphasis in the original). And, Brown thinks, Bell was surely right. This, though, is to present a false dic hotomy. The substantivalist should c laim that a moving rod's c ontrac tion reflec ts both how it is made up and the nature of its spatiotemporal environment.9 1 Rec all the disc ussion of the explanatory role of substantival geometry in sec tion 4.3. The substantivalist should agree that a c omplex material rod does not c onform to the axioms of some geometry simply bec ause that is the substantival geometry in whic h it is immersed; the rod would not do what it does were the laws governing its mic rophysic al parts different in key respec ts. But equally, ac c ording to the substantivalist, the c oordinate-dependent equations that are appealed to in, for example, Bell's toy-model derivation of length c ontrac tion make implic it referenc e, via the c hoic e of c oordinate system, to primitive spatiotemporal geometry. What features of the laws governing the c onstituents of a rod are responsible for the rod's c harac teristic relativistic behavior suc h as its length c ontrac tion? In an important sense, the details of the dynamic s are irrelevant. If subjec t to appropriately nondestruc tive ac c elerations, rods made of steel, wood, and glass will c ontrac t by the same amount, and for the same reason, namely, the Lorentz covariance of the laws governing their constituents.92 In rec ent disc ussion of the dynamic al approac h (e.g., Janssen 2009, Frisc h 2011), this point is widely agreed upon. As Frisc h emphasiz es, what genuine disagreement there is c enters on the status of the dynamic al symmetries to whic h suc h explanations appeal. For Balashov and Janssen, these are ultimately to be explained in terms of the geometry of spac etime. To the question: “Does the Minkowskian nature of spac etime explain why the forc es holding a rod together are Lorentz invariant or the other way around?” they reply: “Our intuition is that the geometric al struc ture of spac e (-time) is the explanans here and the invarianc e of the forc es the explanandum” (Balashov and Janssen, 2003, 340) and Janssen likes to talk of the symmetries of Minkowski geometry as the common origin of the symmetries of the various laws governing matter. For geometry to play this role, its instantiation in the physic al world had better not depend on fac ts about the dynamic al laws. This is true on the substantivalist view reviewed in sec tion 4.3 but, note, that it is also true on the Minkowski relationalist view disc ussed in sec tion 6.1.2, whic h likewise takes both the ideology of the spac etime interval and its satisfying the c onstraints of Minkowski geometry as primitive. How does this alleged explanation of dynamic al symmetries in terms of spac etime symmetries go? Clearly it will not be any kind of c ausal explanation. Moreover, as the examples of Galilean (or Maxwellian) invariant Newtonian physic s set in Newtonian (or Galilean) spac etime illustrate,9 3 the explanation must be c ompatible with the logic al possibility of theories in whic h there is a mismatc h between dynamic al symmetries and the symmetries of independently postulated spac etime struc ture (Brown and Pooley 2006, 83– 84). In these c ases, the mismatc hes are all in one direc tion; the spac etime symmetries are a proper subset of the dynamic al symmetries. It might be thought that the substantivalist c an readily explain this.9 4 On their view, dynamic al laws ultimately involve c oordinate-independent c laims desc ribing how dynamic ally varying matter is c onstrained by and adapted to spac etime struc ture. If the properties of spac etime struc ture are desc ribed explic itly, these laws should be expressible by equations that hold good in any c oordinate system. But if the spac etime struc ture has symmetries that allow for a privileged set of adapted c oordinate systems, one expec ts these equations will (apparently) simplify, as some aspec ts of the spac etime struc ture will now be enc oded in the c oordinate system. Rec all that, in c oordinate terms, dynamic al symmetries are transformations between c oordinate systems in whic h the equations expressing the laws take the same form. If the equations in question are the spec ial, simplified equations, then, on the substantivalist's understanding of these equations, (i) they should hold in all c oordinate systems appropriately adapted to spac etime struc ture, and (ii) they need not hold in others. But, in terms of c oordinates, spac etime symmetries just are the transformations between adapted c oordinate systems. Henc e, the dynamic al symmetries should inc lude the spac etime symmetries. And, very c rudely, the possibility that dynamic al symmetries outstrip spac etime symmetries arises bec ause the dynamic al laws governing matter might Page 27 of 48

Substantivalist and Relationalist Approaches to Spacetime exploit only some of the spac etime struc ture, so that the c oordinate systems in whic h dynamic s simplifies need be adapted to only some of the struc ture postulated.9 5 Given the substantivalist's understanding of the coordinate-dependent forms of dynamical equations, therefore, it follows that the symmetries of these equations c annot be more restric ted than the symmetries of the full set of postulated spac etime struc tures. In at least this sense, the substantivalist c an explain dynamic al symmetries in terms of spac etime symmetries. Ac c ording to the dynamic al approac h, however, this gets things exac tly the wrong way round. Fac ts about dynamic al symmetries c ome first and are the ground of true c laims about the geometry of spac etime: “the Minkowskian metric is no more than a c odific ation of the behavior of rods and c loc ks, or equivalently, it is no more than the Kleinian geometry assoc iated with the symmetry group of the quantum physic s of the non-gravitational interac tions in the theory of matter” (Brown 2005, 9). If spac etime geometry is to be grounded in the symmetries of the dynamic al laws governing matter, it had better be the c ase that the very idea of suc h a law and its symmetries does not presuppose spac etime geometry. That it need not do so is partic ularly c lear if a Humean c onc eption of laws is adopted. This will also bring out the parallels with Huggett's proposal. Rec all that Huggett's regularity relationalist postulates primitive Leibniz ian relations but no ideology c orresponding to inertial struc ture. The latter is grounded in the existenc e of a proper subset of the c oordinate systems adapted to the Leibniz ian relations with respec t to whic h the desc ription of the entire relational history is the solution of partic ularly simple equations (Newton's laws expressed with respec t to inertial frame c oordinates). The dynamic al approac h involves a similar but muc h more radic al move: the metric al relations themselves are to be grounded in exactly the same way. The idea is best illustrated with a simple example. The advoc ate of the dynamic al approac h need not be understood as esc hewing all primitive spatiotemporal notions (pace Norton 2008). In partic ular, one might take as basic the “topologic al” extendedness of the material world in four dimensions. Imagine suc h a world whose only material dynamic al entity has pointlike parts whose degrees of freedom c an be modeled by the real numbers. One obtains a c oordinate desc ription of suc h an entity by assoc iating, in a way that respec ts its loc al topology, eac h of its pointlike parts with distinc t elements of ℝ4 , and assoc iating with eac h of these a real number representing the dynamic al state of the c orresponding part. In other words, we direc tly map the parts of the material field postulated to be the sole entity in the world into ℝ4 and c hoose a way to represent its dynamic al state so as to obtain a sc alar field on ℝ4 . Different c hoic es of c oordinate system will yield different mathematic al desc riptions. Suppose, now, that for some spec ial c hoic e of c oordinate system the desc ription one obtains is the solution of a very simple equation. Moreover, suppose that (i) the desc riptions one obtains relative to c oordinate systems related to this first c oordinate system by Lorentz transformations yield (distinc t) desc riptions that are solutions of the very same equation but that (ii) desc riptions with respec t to other c oordinate systems, if they c an be represented as solutions of equations at all, are solutions of more c omplic ated equations. If all this were the case, the simplest equation might be considered one of the Humean laws of this world.96 The Lorentz group's being their dynamic al symmetry group is c onstituted by its being the group that maps between the c oordinate systems with respec t to whic h desc riptions of the material world satisfy the simple equation. And finally, the spatiotemporal geometry of the world is defined in terms of the invariants of the symmetry group so identified. In partic ular, for the spatiotemporal interval between two parts of the material world p, q to be I just is for (tp − tq)2 − ∣x ⃗p − x ⃗q∣2   =   ±  I(p, q)2 with respect to the privileged coordinate systems. Spacetime geometry is reduc ed to a notion of dynamic al symmetry that does not presuppose it. The example c onsidered is, of c ourse, very simple, and a number of issues will arise when fleshing out an analogous story for more realistic physic s. Some of the c hoic es to be made are highlighted by Norton (2008), who denies the feasibility of exac tly this kind of projec t. Two c harges he makes are worth dwelling on. First, he c onsiders the c ase where the world c ontains several matter fields, eac h desc ribed by a distinc t theory. He grants that eac h of these might be Lorentz invariant. His c hallenge to the advoc ate of the dynamic al approac h (dubbed the “c onstruc tivist”) is to justify the assumption that the sets of c oordinate systems with respec t to whic h these c ases of Lorentz invarianc e are manifest c oinc ide. The simple answer is that the spatiotemporally coincident parts of distinct matter fields should be assigned the same element of ℝ4. The issue is how this relation of c oinc idenc e between matter fields is to be understood. For the substantivalist it involves c oloc ation at the same spac etime point. The Minkowski relationalist, who takes interval fac ts as primitive, c an analyz e it in terms of these (though not, of c ourse, straightforwardly in terms of the vanishing of the interval, for this will not exc lude Page 28 of 48

Substantivalist and Relationalist Approaches to Spacetime nonc oinc ident, lightlike related events). What options are open to the c onstruc tivist? The most natural is to take spatiotemporal c oinc idenc e as primitive (as many relationalists have done; e.g., Rovelli (1997, 194)). After all, the projec t was to reduc e c hronogeometric fac ts to symmetries, not to rec over the entire spatiotemporal nature of the world from no spatiotemporal assumptions whatsoever. The c onstruc tivist's projec t might need a primitive notion of “being c ontiguous,” but Norton is wrong to think that it follows from this that c onstruc tivists are illic itly c ommitted to the independent existenc e of spac etime.9 7 The other of Norton's objections that I wish to highlight involves what the constructivist must say about the geometry of empty regions of spac etime and of regions c ontaining homogeneous matter. Suppose some way K of c oordinatiz ing the material world satisfies the type of c ondition desc ribed above. Now suppose that the world c ontains an empty region of spac etime. Translated into our terms, Norton's point is that any K′ that agrees with K on its assignment of c oordinates to material events will yield the same desc ription. K′ c an differ from K in any way one likes over the c oordinates it assigns to the empty region. Does this leave the geometry of the empty region indeterminate? Put this way, that there really is no problem here should be obvious: for the c onstruc tivist there is literally nothing in an empty region and so nothing whose geometric al properties might be indeterminate. The c onstruc tivist does not believe in the existenc e of an independently existing spac etime! The c ase of homogeneous matter is more problematic . Now one is supposing there are entities—the material pointlike parts of the homogeneous region—whose spatiotemporal relatedness one would like to be able to enquire after. Suppose that the c onstruc tivist has attributed some primitive topogic al properties to matter. Even so, we c an respec t these properties and smoothly alter K to K′ within the region to obtain exac tly the same desc ription. The c onstruc tivist is forc ed to c onc lude that for any two material events in the region there is no fac t of the matter c onc erning the interval between them. How bad is this? Note that a number of other geometric al properties will be determinate (bec ause invariant under all c oordinate transformations that leave the desc ription of matter unaltered). For example, the spacetime volume of the homogeneous region might be determinate even though the spa- tiotemporal relatedness of the points within it is not.9 8 This is surely a pec uliarity of the c onstruc tivist's position. But, like Huggett's regularity relationalist in the fac e of analogous problems (Huggett, 2006, 55– 56), they might argue that it is not suc h a painful bullet to have to bite. 7. Substantivalism in Light o f the Ho le Argument For muc h of the last 25 years, arguments about spac etime substantivalism have been dominated by disc ussion of the Hole Argument. This is not the plac e for a thorough review of the siz eable literature that the argument has spawned.99 Here I wish only to highlight one form of substantivalism that evades the Hole Argument and to emphasiz e an important disanalogy between the Hole Argument and the arguments against Newtonian and Galilean substantivalism that were c onsidered in earlier sec tions. Originally due to Einstein, who used it prior to 1915 to explain away his inability (at that point in time) to find satisfac tory generally c ovariant field equations, the Hole Argument was rehabilitated by John Stac hel (1989) before being put to work against spacetime substantivalism by Earman and Norton (1987). Let M1   =  ⟨M ,  gab ,   Tab⟩ be a model of a generally relativistic theory.10 0 It follows from the diffeomorphism invariance of GR that, for an arbitrary diffeomorphism d, M2   =  ⟨M ,  d *  gab ,  d *  Tab⟩ will also satisfy the theory's equations. The natural (though not ineluctable) conclusion is that M 1 and M 2 jointly represent spacetimes (call them W1 and W2) that are physically possible ac c ording to the theory. In M 1 , each p ∈ M is assigned certain properties encoded by gab(p), Tab(p)); in M 2 , p is assigned the (in general) distinct properties encoded by d* gab(p), d* Tab (p) . But, according to the substantivalist, M represent physic al spac etime. This means that (on one natural understanding of how the points of M represent physic al spacetime points) M 1 and M 2 represent one and the same spacetime point as having different properties. This gives us the next ingredient in the argument: the c laim that the substantivalist is c ommitted to regarding W1 and W2 as distinct possible worlds.101 The problem is that, if this interpretation of spac etime models is permitted, GR is radic ally indeterministic . Let d be a hole diffeomorphism, a map that it is only nontrivial within a restric ted region of M (the so-c alled hole). Suppose that, relative to the metric of M1, d is nontrivial only to the future of some spacelike surface, σ. M 1 and M 2 will then be identic al struc tures up to and inc luding this surfac e but differ to its future. On the proposed interpretation of Page 29 of 48

Substantivalist and Relationalist Approaches to Spacetime M 1 and M 2 , they represent spacetimes that are identical up to the spacelike surface represented by σ but that differ to its future. It follows that the equations of GR, together with a c omplete spec ific ation of the history of the world up to some spac elike surfac e, fail to fix the future. Earman and Norton do not see this as a problem for substantivalism bec ause they think indeterminism is objec tionable per se. Their c laim, rather, is that determinism should fail only for reasons of physic s and not as the result of a metaphysic al c ommitment and in a theory- independent way (Earman and Norton, 1987, 524). Note that M 1 and M 2 are isomorphic structures. The possibilities they represent, therefore, involve exactly the same patterns of qualitative features. If W1 and W2 are distinc t possibilities, they differ only over whic h spac etime points instantiate whic h of the partic ular features c ommon to both worlds. In the terminology of modal metaphysic s, the differenc e between the possibilities is merely haecceitistic (Kaplan, 1975). Many of the pro-substantivalist responses to the argument make c ruc ial use of this aspec t of the setup. For example, a substantivalist might agree that ac c epting GR involves a c ommitment to suc h haec c eitistic distinc tions and ac c ept that the theory is indeterministic . However, they might deny that this indeterminism is in any sense troublesome prec isely bec ause it is an indeterminism only about whic h objec ts instantiate whic h properties and not about whic h patterns of properties are instantiated. A c losely related response ac c epts that GR is c ommitted to haec c eitistic distinc tions but denies that it follows that GR is indeterministic bec ause the c orrec t definition of determinism, it is c laimed, is only sensitive to qualitative differenc es.10 2 The most popular response, however, has been to advoc ate some variety of sophisticated substantivalism, that is, a version of substantivalism that denies the existenc e of physic ally possible spac etimes that differ merely haec c eitistic ally. The simplest way to sec ure this is to endorse antihaecceitism, that is, the general denial of merely haec c eitistic distinc tions between possible worlds.10 3 Two arguments disc ussed earlier in the c hapter also involved the c laim that, bec ause of the dynamic al symmetries of the relevant physic al theory, the (relevant stripe of) substantivalist was c ommitted to distinc t physic ally possible worlds, the nonidentity of whic h was alleged to be problematic . The important differenc e between these c ases and those of the Hole Argument is that the former involve qualitative differenc es between the relevant worlds. In the c ase of the kinematic shift, the worlds differ over the absolute veloc ities assigned to bodies. In the c ase of Maxwellian invariant dynamic s set in Galilean spac etime, they differ over the absolute ac c elerations assigned to bodies.10 4 The fac t that these differenc es are qualitative has two important c onsequenc es. First, that the possibilities differ qualitatively c reates an epistemologic al problem (given that one c annot observationally distinguish between the relevant quantities) that is not present in the case of merely haecceitistic differenc es.10 5 Even if diffeomorphic models of GR are to be interpreted as representing distinc t possibilities, there is no substantive fac t, about whic h I c ould be ignorant despite knowing all the observable fac ts, c onc erning whic h model really represents the ac tual world. Eac h model is equally apt, and whic h model represents the ac tual world will be a matter of (arbitrary) representational c onvention. In c ontrast, models of Galilean invariant physic s set in Newtonian spac etime that differ by boosts of their material c ontent are not equally suited to represent any given possibility. Even once representational conventions are fixed, the Newtonian substantivalist does not know whether the model that attributes a veloc ity of 10ms −1 to the Eiffel Tower, the one that attributes 20ms −1, or yet some other model, c orresponds to the ac tual world. Sec ond, the antihaec c eitist way out of the Hole dilemma is of no use in the c ontext of the kinematic shift argument. The argument is evaded if any two models related by Galilean boosts c an be shown to be different representations of the same state of affairs. Sinc e the models represent qualitatively distinc t possibilities ac c ording to the Newtonian substantivalist, merely embrac ing antihaec c eitism does not c ollapse the distinc tion between them. A substantivalist position that c an view Galilean boosted models as distinc t representations of one and the same state of affairs requires substantive work, viz ., the replac ement of Newton's substantival spac e with neo-Newtonian spac etime. (A similar observation holds c onc erning the passage from neo-Newtonian to Newton-Cartan spac etime.) This is in c ontrast to the so-c alled static shift argument against Newton's absolute spac e, whic h exploits the Euc lidean symmetries of Newtonian mec hanic s and c ompares only models related by time-independent rotations or translations.10 6 In this c ase antihaec c eitism does c ollapse the number of relevant physic al possibilities to one. It is enough to note that antihaec c eitism is a live view within metaphysic s in order to see that substantivalism need not fall to the Hole Argument. More c ontroversial is how well-motivated the position is from the perspec tive of the Page 30 of 48

Substantivalist and Relationalist Approaches to Spacetime interpretation of physic s. Here a c ouple of remarks are in order. First, as Belot and Earman (1999, 2001) have stressed, several physic ists grappling with the c onc eptual and tec hnic al problems of unifying quantum mec hanic s and general relativity do c laim to draw substantive morals from the Hole Argument. What is not clear, however, is whether the genuinely substantive interpretational questions that have c ome to the fore as a result of work on the quantiz ation of GR have anything to do with the kind of diffeomorphism invarianc e that lies at the heart of the Hole Argument. One key issue c onc erns the nature of the “observables” (that is, the genuine physical magnitudes) of diffeomorphism-invariant theories. Another concerns differenc es between GR and pre-generally relativistic theories. In partic ular, are the true physic al magnitudes of GR essentially different in kind to those of pre-GR theories (when the latter are properly understood)? While Earman (2006a,b) believes that the right answers to these questions will be inc onsistent with anything like a substantivalist interpretation of GR, even of the sophistic ated variety, it is not obvious that some of the views about the nature of “observables” advoc ated by the physic ists Earman c ites, suc h as those of Rovelli (2002), are inc ompatible with sophistic ated substantivalism. Sec ond, some of the work on “struc tural realist” interpretations of spac etime, at least where these do not involve an eliminativism about spac etime points, c an be understood as varieties of antihaec c eitist substantivalism.10 7 It is possible that the development of one of these will provide an additional motivation for sophistic ated substantivalism.108 The upshot of this sec tion is that the substantivalist understanding of spac etime physic s, as set out in sec tion 4, is not undermined by the Hole Argument. What, then, should one c onc lude about the relative merits of substantivalism versus relationalism? In sec tion 5 I c onsidered and rejec ted two other strands of antisubstantivalist argument that have motivated rec ent relationalists. That leaves substantivalism as a going c onc ern. What about relationalism? Of the three general strategies outlined, the most promising is the Mac hian, 3-spac e approac h of Barbour and c ollaborators. But, rec all, this turned out not to be a form of relationalism in the traditional, ontologic al sense. It does represent an approac h that is metaphysic ally very different from spac etime orthodoxy, but the dividing issue is not the existenc e of spac etime points but the relative priority of 3-dimensional versus 4- dimensional c onc epts. The other two relationalist approac hes fare less well. Rec ogniz ing that the Maxwell group is a symmetry group of Newtonian physic s allows for an intriguing and relatively overlooked form of enric hed relationalism, but it does not generaliz e to relativistic physic s. In the c ontext of SR, the restric tion of Minkowski distanc es to a material ontology already provides for a viable, if unexc iting, form of relationalism. In the c ontext of GR, however, the same move does not work: in general, the dynamic ally signific ant c hronometric fac ts outstrip the c hronometric fac ts about matter, as is most vividly illustrated by the abundanc e of interesting vac uum solutions. The relationalist approac h reviewed in sec tion 6.3 has not been pursued in the c ontext of GR. Instead, a popular move for relationalists is to treat the metric field as just another material field (see, e.g., Rovelli, 1997, 193– 195). This, it turns out, is also the view endorsed by Brown (2005, c h. 9). So, while the “dynamic al approac h” to relativity provides a reduc tive ac c ount of the metric —that is, a form of have-it-all relationalism—in the c ontext of SR (sec tion 6.3.2) the same is not true, for Brown at least, in GR. Brown stresses that the metric field only gains its usual “c hronometric al signific anc e” (that is, only c orresponds to the prac tic al geometry manifest by the behavior of material rods and c loc ks) in virtue of the partic ular way it dynamic ally c ouples to matter, but, as I hope to have made c lear, no sensible substantivalist should demur. What, then, is at stake between the metric -reifying relationalist and the traditional substantivalist? Both parties ac c ept the existenc e of a substantival entity, whose struc tural properties are c harac teriz ed mathematic ally by a pseudo-Riemannian metric field and whose c onnec tion to the behavior of material rods and c loc ks depends on, inter alia, the truth of the strong equivalenc e princ iple. It is hard to resist the suspic ion that this c orner of the debate is bec oming merely terminologic al. At least this muc h that c an be said for the c hoic e of substantivalist language: it underlines an important continuity between the “absolute” spacetime structures that feature in pre- generally relativistic physic s and the entity that all sides of the c urrent dispute admit is a fundamental element of reality. To the extent that one should seek to understand the c ontent and suc c ess of previous theories in terms of our c urrent best theory, this arguably vindic ates the substantivalist interpretation of Newtonian and spec ially relativistic physic s.10 9 Page 31 of 48

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Substantivalist and Relationalist Approaches to Spacetime ———. (1993). A paradox in Newtonian gravitation theory. In Proceedings of the 1992 biennial meeting of the Philosophy of Science Association, ed. M. Forbes, D. Hull, and K. Okruhlik, Vol. 2, 412–420. East Lansing, MI: Philosophy of Sc ienc e Assoc iation. ———. (2008). Why c onstruc tive relativity fails. British Journal for the Philosophy of Science 59: 821– 834. ———. (2011). The hole argument. In The Stanford encyclopedia of philosophy (Fall 2011 edn), ed. E. N. Zalta. http://plato .stanfo rd.edu/archives/fall2011/entries/spacetime- ho learg/. Patriz i, Franc esc o (1943). On physic al spac e. Journal of the History of Ideas 4: 224–245. Trans. Benjamin Bric kman. Pooley, Oliver (2001). Relationism rehabilitated? II: Relativity. http://philsci-archive.pitt.edu/id/eprint/221. ———. (2004). Comments on Sklar's “relationalist metric of time”. Chronos 6: 77– 86. http://philsci- archive.pitt.edu/id/eprint/2915. ———. (2006). Points, partic les and struc tural realism. In The structural foundations of quantum gravity , ed. D. Ric kles, S. Frenc h, and J. Saatsi, 83– 120. Oxford: Oxford University Press. Preprint: http://philsci- archive.pitt.edu/2939/. ———. (unpublished). Substantivalism and haec c eitism. Unpublished manusc ript. ———. (in preparation). The reality of spacetime. Oxford: Oxford University Press. Pooley, Oliver, and Brown, Harvey R. (2002). Relationalism rehabilitated? I: Classic al mec hanic s. British Journal for the Philosophy of Science 53: 183–204. Preprint: http://philsci-archive.pitt.edu/id/eprint/220. Quine, Willard Van (1951). Ontology and ideology. Philosophical Studies 2: 11–15. Reic henbac h, Hans (1924). The theory of motion ac c ording to Newton, Leibniz , and Huygens. In Modern philosophy of science, ed. M. Reic henbac h. London: Routledge and Kegan Paul (1959), 46– 66. Reissner, H. (1914). Über die Relativität der Besc hleunigungen in der Mec hanik [on the relativity of ac c elerations in mec hanic s]. Phys. Z. 15: 371– 375. Trans. J. B. Barbour in Barbour and Pfister (1995), 134– 142. Ric kles, Dean (2008). Symmetry, Struc ture and Spac etime. Amsterdam: Elsevier. Rovelli, Carlo (1997). Halfway through the woods: Contemporary researc h on spac e and time. In The cosmos of science. ed. J. Earman and J. Norton, 180–223. Pittsburgh: University of Pittsburgh Press. ———. (2001). Quantum spac etime: What do we know? In Physics meets philosophy at the Planck scale, ed. C. Callender and N. Huggett, 101–122. Cambridge: Cambridge University Press. http://arxiv.o rg/abs/gr- qc/9903045. ———. (2002). Partial observables. Physical Review D 65: 124013. ———. (2004). Quantum gravity. Cambridge: Cambridge University Press. Rynasiewic z , Robert A. (1994). The lessons of the hole argument. British Journal for the Philosophy of Science 45: 407–436. ———. (1995). Absolute vs. relational theories of spac e and time: A review of John Earman's “World Enough and Spac etime”. Philosophy and Phenomenological Research 55: 675– 687. ———. (1996). Absolute versus relational spac etime: An outmoded debate? Journal of Philosophy 93: 279– 306. ———. (2004). Newton's views on spac e, time, and motion. In The Stanford encyclopedia of philosophy , ed. E. N. Zalta. Saunders, Simon W. (2003a). Indisc ernibles, general c ovarianc e, and other symmetries. In Revisiting the Page 37 of 48

Substantivalist and Relationalist Approaches to Spacetime foundations of relativistic physics: Festschrift in honour of John Stachel, ed. A. Ashtekar, D. Howard, J. Renn, S. Sarkar, and A. Shimony. Dordrec ht: Kluwer. http://philsci- archive.pitt.edu/id/eprint/459. ———. (2003b). Physic s and Leibniz 's princ iples. In Symmetries in physics: Philosophical reflections, ed. K. Brading and E. Castellani, 289–307. Cambridge: Cambridge University Press. http://philsci- archive.pitt.edu/2012/. ———. (Forthc oming). Rethinking Principia. Philosophy of Science. Saunders, Simon W., and Brown, Harvey R., eds. (1991). The philosophy of the vacuum. Oxford: Oxford University Press. Sc hrödinger, Erwin (1925). Die Erfüllbarkeit der Relativitätsforderung in der klassisc hen Mec hanik [The possibility of fulfillment of the relativity requirement in c lassic al mec hanic s]. Ann. Phys. 77: 325– 336. Trans. J.B. Barbour in Barbour and Pfister (1995, 147–156). Sider, Ted (2001). Four dimensionalism: An ontology of persistence and time. Oxford and New York: Oxford University Press. Sklar, Lawrenc e (1974). Space, time and spacetime. Berkeley: University of California Press. Skow, Bradford (2006). Physic al relativity: Spac etime struc ture from a dynamic al perspec tive. Notre Dame Philosophical Reviews. http://ndpr.nd.edu/news/25025- physical- relativity- space- time- structure- fro m- a-dynamical-perspective/ ———. (2007). Sklar's maneuver. British Journal for the Philosophy of Science 58: 777–786. Stac hel, John (1989). Einstein's searc h for general c ovarianc e, 1912– 1915. In Einstein and the history of general relativity, ed. D. Howard and J. Stac hel. Boston: 63– 100. Birkhäuser. ———. (1993). The meaning of general c ovarianc e. In Philosophical problems of the internal and external worlds: Essays on the philosophy of Adolph Grünbaum, ed. J. Earman, A. Janis, and G. Massey, 129–60. Pittsburgh: University of Pittsburgh Press. ———. (2002). “The relations between things” versus “the things between relations”: The deeper meaning of the hole argument. In Reading natural philosophy. Essays in the history and philosophy of science and mathematics, ed. D. B. Malament, 231– 266. Chic ago: Open Court. ———. (2006). Struc ture, individuality and quantum gravity. In Structural foundations of quantum gravity , ed. D. Ric kles, S. Frenc h, and J. Saatsi, 53– 82. Oxford: Oxford University Press. Stein, Howard (1967). Newtonian spac etime. Texas Quarterly, 10, 174– 200. Reprinted in Robert Palter, ed., 1970, The Annus Mirabilis of Sir Isaac Newton 1666–1966 (Cambridge, MA: MIT Press), 258–284. ———. (1977). Some pre-history of general relativity. In Foundations of spacetime theories, ed. J. Earman, C. Glymour, and J. Stac hel, Vol. 8 of Minnesota studies in the philosophy of science. Minneapolis: 3– 49. University of Minnesota Press. ———. (1991). One relativity theory and openness of the future. Philosophy of Science 58: 147–167. ———. (2002). Newton's metaphysic s. In The Cambridge companion to Newton, eds. I. Bernard Cohen and George E. Smith, 256–307. Cambridge: Cambridge University Press. Sus, Adán (2011). On the explanation of inertia. Unpublished. Synge, John Lighton (1960). Relativity: The general theory . Amsterdam: North-Holland Public ation Co. Teller, Paul (1987). Spac etime as a physic al quantity. In Kelvin's Baltimore lectures and modern theoretical physics, ed. P. Ac hinstein and R. Kagon, 425– 448. Cambridge, MA: MIT Press. Trautman, Andrz ej (1962). Conservation laws in general relativity. In Gravitation: An introduction to current Page 38 of 48

Substantivalist and Relationalist Approaches to Spacetime research, ed. L. Witten, 169–198. New York: Wiley. van Fraassen, Bas C. (1970). An introduction to the philosophy of space and time. New York: Columbia University Press. Weatherall, James Owen (2011a). The motion of a body in Newtonian theories. Journal of Mathematical Physics 032502/52: ———. (2011b). On the status of the geodesic princ iple in Newtonian and relativistic physic s. http://philsci- archive.pitt.edu/8662/. Weinstein, Steven (2001). Absolute quantum mechanics. British Journal for the Philosophy of Science 52(1): 67– 73. Westman, Hans, and Sonego, Sebastiano (2009). Coordinates, observables and symmetry in relativity. Annals of Physics, 1585–1611. Weyl, Hermann (1922). Space–time–matter. 4th ed. Trans. H. L. Brose. London: Methuen and Co. Ltd. Wheeler, John Arc hibald, and Feynman, Ric hard Phillips (1949). Classic al elec trodynamic s in terms of direc t interpartic le ac tion. Reviews of Modern Physics 21: 425– 433. Notes: (1) Stric tly speaking, the c ontroversy has c onc erned two c andidate entities. Prior to Minkowski's reformulation of Einstein's spec ial theory of relativity in four-dimensional form, the debate was about the existenc e of spac e. Sinc e then, the debate has been about the existenc e of spac etime. For the sake of brevity, I will often only mention spac etime, leaving the “and/or spac e” implic it. (2) For a varied sample of c ompeting interpretations, see Laymon (1978), Rynasiewic z (1995, 2004), and DiSalle (2002, 2006), who goes so far as to c laim that Newton was not a substantivalist. (3) In 1633, on hearing of the Churc h's c ondemnation of Galileo for c laiming that the Earth moved, Desc artes suppressed an early statement of his physic s, whic h did not c ontain his later relational c laims about the nature of motion. It is frequently (and plausibly) c onjec tured that Desc artes's offic ial views on motion were devised to avoid Churc h c ensure. However, the prec ise manner in whic h Desc artes's definitions sec ure the Earth's lac k of true motion suggest that he was genuinely c ommitted to a relational c onc eption of motion. What does the work in sec uring the Earth's rest is not that, in Desc artes's c osmology, there is no relative motion with respec t to immediately c ontiguous bodies (Desc artes explic itly says there is suc h motion; ibid., III: 28); it is that Cartesian true motion is motion with respec t to those c ontiguous bodies that are regarded as at rest. (4) The paragraph desc ribing the buc ket experiment c ompletes Newton's arguments for his ac c ount of true motion in terms of absolute spac e but it is not the end the Sc holium. After a brief paragraph that explic itly c onc ludes: “Henc e relative quantities are not the quantities themselves, whose names they bear, but are only sensible measures of them,” there follows a long, final paragraph desc ribing a thought experiment involving two globes attac hed by a c ord in a universe in whic h no other observable objec ts exist. The purpose of this thought experiment is not to further argue for absolute spac e by, e.g., desc ribing a situation in whic h there is absolute motion (revealed by a tension in the c ord) and yet no relative motion whatsoever. Instead, Newton's purpose is to demonstrate how true motion c an (partially) be empiric ally determined, despite the imperc eptibility of the spac e with respec t to whic h it is defined: the tension in the c ord is a measure of the rate of rotation and, by measuring how this tension c hanges as different forc es are applied to opposite fac es of the globes, one c an also determine the axis and sense of the rotation. (5) As has been emphasiz ed by Stein (1967, 269–271); the argument is also singled out by Barbour (1989, 616– 617). (6) Sinc e Newton held that everything that exists exists somewhere, the existenc e of any other being entails the existenc e of spac e (see Stein 2002, 300, n. 32, for further disc ussion). Page 39 of 48

Substantivalist and Relationalist Approaches to Spacetime (7) Rec all, also, that for the Newtonian substantivalist the basic entity is still spac e. Until spac etime substantivalism is explic itly introduc ed in sec tion 4, referenc e to spac etime points should be understood as referenc e to ordered pairs of pointlike substantival plac es with instants of time, and referenc e to point-like material events as referenc e to instantaneous states of persisting point partic les. (8) This terminology is established (see. e.g., Maudlin, 1993, 187). No suggestion that the historic al Leibniz was a Leibniz ian relationalist is intended. (9) There is no established label for these transformations. I follow Bain (2004, 350, fn6); see also Earman (1989, 30– 31). Ehlers (1973a, 74) c alls this group of transformations the kinematical group. (10) Subjec t to the qualific ations in footnote 7. (11) Cf Earman (1989, 34). Ehlers (1973a, 74) follows Weyl in referring to this group of transformations as the elementary group. (12) In spac etime terms, the notion of a frame of referenc e implic it in this stipulation c orresponds to the following: a fibration of spac etime that spec ifies a standard of rest; a foliation of spac etime that spec ifies a standard of distant simultaneity; a temporal metric on the quotient of spac etime by the foliation; and a spatial metric on the quotient of spac etime by the fibration. (13) Supposing, for the sake of argument, that the ac tual universe is Newtonian. (14) In his c orrespondenc e with Clarke (Alexander 1956), Leibniz is sometimes read as offering kinematic -shift arguments somewhat different to the one just sketc hed. The idea is that kinematic ally shifted possible worlds would violate the Princ iple of Suffic ient Reason (PSR) and the Princ iple of the Identity of Indisc ernibles (PII). Sinc e these princ iples are a priori true, ac c ording to Leibniz , there c an be no suc h plurality of possibilities. A “Leibniz ian” argument from the PSR would ask us to c onsider what reasons God c ould have had for c reating the ac tual universe rather than one of its kinematic ally shifted c ousins. An argument from the PII would c laim that, sinc e kinematic ally shifted worlds are observationally indistinguishable, they direc tly violate the PII. Neither argument is c onvinc ing (nor is either faithful to Leibniz ; see Pooley, unpublished). The sense of indisc ernibility relevant to kinematic shifts is not that whic h has been the foc us of c ontemporary disc ussion of the PII. This takes two entities to be indisc ernible just if they share all their (qualitative) properties. In general, two kinematic ally shifted worlds do differ qualitatively; given how the qualitative/nonqualitative distinc tion is standardly understood, a body's absolute speed is a qualitative property, and differenc es in absolute veloc ity are (typic ally) qualitative differenc es. Suc h qualitative differenc es are empiric ally inac c essible but, theoretic ally, they c ould ground a reason for an all-seeing God's preferenc e for one possibility over another. A PSR dilemma for God is c reated if we c onsider kinematic ally shifted worlds that differ, not in terms of the magnitude of their objec ts' absolute veloc ities, but only over their direc tions. These are worlds that are qualitatively indistinguishable. Disc ussion of how the substantivalist should treat these is postponed until section 7. (15) In fac t, it is too strong; see sec tion 6.1 below. (16) These notions are standard, although terminology varies; see, e.g., Anderson (1967, 74) and Friedman (1983, 48). (17) A locus classicus in the philosophic al literature for a disc ussion of Newtonian theory formulated in this style is Friedman (1983, c h. III). I adopt the following widespread notational c onventions: Roman indic es from the start of the alphabet do not denote components—they are “abstract indices” merely indicating the type of geometric- objec t field; Greek indic es denote that the c omponents of the objec ts relative to some spac etime c oordinate system are being considered; repeated indices indicate a sum over those indices (the Einstein summation c onvention). (18) That is, the trajec tory is parametriz ed so that ta ξa = 1, where ta is a one-form related to the temporal metric via tab = ta tb. (19) In general, an action of group G on a space K is a function ϕ : (g,  k)  ∈   G  ×  K   ↦  g  ⋅  k  ∈  K such that, for all g,h ∈ G and k  ∈  K ,  g  ⋅  (h  ⋅  k)  =  (gh)  ⋅  k and e · k = k, where e is the identity element of G. To avoid Page 40 of 48

Substantivalist and Relationalist Approaches to Spacetime triviality, we should also require that the ac tion is faithful, that is, that for any g, if g · k = k for all k, g = e· G is a symmetry group if and only if g · s ∈ S for all g ∈ G and s ∈ S. The symmetry group of a theory c harac teriz ed in this way is referred to as the theory's covariance group by Anderson (1967, 75). (20) See Gordon Belot's c hapter in this volume for further disc ussion. (21) The diffeomorphism group Diff(M) is the group of all differentiable one-one mappings from M onto itself. The definition of the map d*, whic h ac ts on geometric al objec ts on M and is induc ed by the manifold mapping d: p ∈ M M ↦ dp ∈ M, will depend on the type of field. For a sc alar field, ϕ , d* ϕ (dp) : = ϕ (p). The ac tion of d* on sc alar fields c an then be used to define its ac tion on tensor fields. For example, for the vec tor field V, we require that d* V(d* ϕ )| dp = V(ϕ )| p for all points p and sc alar fields ϕ . (22) More c arefully, the requirement that d* Ai = Ai for eac h Ai pic ks out a spec ific subgroup of Diff(M) relative to a particular choice of Ai. Suppose that (M,Ai,Pj) and ⟨M ,  A′i,  P ′j ⟩ are both models of our theory and that there is a diffeomorphism ϕ: M − M such that A′i  =  ϕ * for each Ai,  A′i (all models of a theory set in Galilean spacetime will have this property). Although it will not be true, in general, that {d  ∈  Diff (M ) :  d *  Ai  =  Ai}  =  {d′   ∈  Diff (M ) :  d′ * A′i  = A′i}, the groups will be isomorphic to the same (abstrac t) group; cf. Earman (1989, 45). (23) As noted in footnote 22, different c hoic es of Ai will, stric tly speaking, yield distinc t subgroups of Diff (M) but (for well-behaved theories) these will simply correspond to different representations of the same abstract group. (24) The dynamic al symmetry group of Newtonian theory set in neo-Newtonian spac etime in fac t turns out to be a larger group if the theory inc orporates gravitation in a field-theoretic way. See sec tion 6.1.1 below. (25) This might seem like a banal observation but I take it to be signific ant bec ause it c onflic ts with prevalent c laims about the meaning of preferred c oordinates in non-generally c ovariant theories made by, e.g., Rovelli (2004, 87– 88) and Westman and Sonego (2009, 1952– 1953). Their c onc eption of the signific anc e of suc h c oordinates implies that there is a differenc e in kind between the observables of nonc ovariant and generally c ovariant theories. On the view outlined above, there is no such difference. (26) Nerlic h (1979, 2010) is staunc h advoc ate of the explanatory role of the geometry of spac etime, realistic ally c onstrued. He c lassifies the role of physic al geometry in suc h explanations as nonc ausal, but, on c ertain plausible understandings of c ausation (e.g., Lewis, 2000), it does c ount as c ausal (see also Mellor, 1980). (27) The genesis of Einstein's general theory has been subject to extensive historical and philosophical scrutiny. For an exc ellent introduc tion to the topic , see Janssen (2008). (28) Gab   :   =  Rab   −   1  Rgab , where Rab (the Ricci tensor) and R (the Riemann curvature scalar) are both 2 measures of c urvature. gab is the metric tensor and enc odes all fac ts about the spatiotemporal distanc es between spac etime points. Rab and R are offic ially defined in terms of the Riemann tensor, itself defined in terms of the c onnec tion ∇a. However, sinc e we are c onsidering the unique torsion-free, metric -c ompatible c onnec tion, we c an view these quantities as defined in terms of the metric and, indeed, they c an be given natural geometric interpretations direc tly in terms of spac etime distanc es. Tab enc odes the net energy, stress, and momentum densities assoc iated with the material fields in spac etime. (29) For a c ritic al disc ussion of Einstein's various formulations of the princ iple, see Norton (1985). (30) I should note that some still hold out against this orthodoxy (e.g., Dieks, 2006). (31) In this case, the four-force on a particle with charge q and four-velocity ξa is given by qF  a b ξb and the equation is simply the c oordinate-free version of the Lorentz forc e law. (32) For example, the c omponents of Fab relative to an inertial c oordinate system are F0 i = −Ei, Fij = ϵijk Bk, where Ei and Bi are the c omponents of the elec tric and magnetic three-vec tor fields in that frame. (33) Ehlers (1973b, 18); see Brown (2005, 169–172) for a recent discussion. (34) Some authors favor talk of “tidal forc es” or state that there is a real “gravitational field” just where the Page 41 of 48

Substantivalist and Relationalist Approaches to Spacetime Riemann tensor is nonz ero (e.g., Synge, 1960, ix). As far as I c an see, this is simply a misleading way of talking about spac etime c urvature and (typic ally) nothing of c onc eptual substanc e is intended by it. For a disc ussion of some of the pros and c ons of identifying various geometric al struc tures with the “gravitational field,” see Lehmkuhl (2008, 91– 98). Lehmkuhl regards the metric gab as the best c andidate. My own view is that c onsideration of the Newtonian limit (e.g., Misner et al., 1973, 445–446) favors a c andidate not on his list, viz ., deviation of the metric from flatness: hab, where gab = ηab + hab. That this split is not prec isely defined and does not c orrespond to anything fundamental in c lassic al GR undersc ores the point that, in GR, talk of the “gravitational field” is at best unhelpful and at worst confused. The distinction between background geometry and the graviton modes of the quantum field propagating against that geometry is fundamental to perturbative string theory, but this is a feature that one might hope will not survive in a more fundamental “background-independent” formulation. (35) Instantaneous relative distanc es and their first derivatives are the natural Leibniz ian relational data. As reviewed in sec tion 6.2, Barbour's preferred framework for understanding c lassic al mec hanic s also dispenses with a primitive temporal metric and an absolute length sc ale. With respec t to these more frugal initial data, five, not three, additional numbers are needed. See Barbour (2011, §2.2). (36) Nor are distanc e ratios, Barbour's preferred relational quantities. For an illuminating disc ussion of how instantaneous quantities are detected only indirectly, in measurements that necessarily take finite time, see Stein (1991, 157). (37) De Sitter first pointed out to Einstein that, in addition to spec ific ation of Tab, one needs to spec ify boundary c onditions at infinity in order to determine gab. This prompted Einstein to searc h for spatially c ompac t solutions to the EFEs and to introduc e the c osmologic al c onstant to allow for a static , spatially c losed universe. This in turn led de Sitter to the disc overy of the de Sitter universe: a spatially c ompac t vac uum solution to the modified EFEs. See Janssen (2008, § 5) for a summary of this episode and for further referenc es. (38) It is also worth stressing that the stress-energy properties of matter, as enc oded in Tab, c annot even be defined independently of gab; see Lehmkuhl (2011). (39) The idea that something should be c apable of ac ting if and only if it c an also be affec ted by those things that it c an influenc e is known as the action– reaction principle (see Anandan and Brown, 1995, for a disc ussion). (40) That is, there are c oordinate systems with respec t to whic h the partic les' spatial c oordinates are linear func tions of their time c oordinates. In Brown's view,“anyone who is not amaz ed by this c onspirac y has not understood it” (Brown, 2005, 15). (41) The idea that spacetime geodesics are defined as the trajectories of force-free bodies is defended by DiSalle (1995, 327), whom Brown quotes approvingly. Elsewhere Brown, ostensibly to make a point against the substantivalist explanation of inertia, stresses that the princ iple that the trajec tories of forc e-free bodies are geodesic s in fac t has limited validity in GR (Brown 2005, 141, see also 161– 168). What this observation in fac t undermines is a relationalist approac h to spac etime geometry that tries to define geodesic s in terms of “basic physic al laws” (DiSalle 1995, 325). More rec ently, DiSalle makes c lear that he differs from the logic al positivists in not regarding the c oordination of geodesic s with free-fall trajec tories as a matter of arbitrary stipulation. Instead it is said to be “a kind of disc overy, at onc e physic al and mathematic al, that … the only objec tively distinguishable state of motion c orresponds to the only geometric ally distinc tive path in a generally c ovariant geometry” (DiSalle, 2006, 131– 132). Nothing in the substantivalist's metaphysic s is inc onsistent with this position; it is less c lear what other metaphysic al views are c ompatible with it. DiSalle does not share the substantivalist's and relationalist's preoc c upation with ontologic al questions but nor does he offer reasons to see suc h questions as illegitimate. (42) Note that Einstein had in mind desc riptions of interac ting systems in different states of ac c eleration, and not simple inertial motion, when he c laimed that “something real has to be c onc eived as the c ause for the preferenc e of an inertial system over a noninertial system” (Einstein, 1924, 16). (43) See Malament (2010) for a c ritic al disc ussion of this result. (44) A c losely parallel derivation is also possible in the geometriz ed form of Newtonian gravity; see Weatherall (2011a,b). This might be taken to further undermine the c laim that only in GR is inertia explained. Page 42 of 48

Substantivalist and Relationalist Approaches to Spacetime (45) See, e.g., Trautman (1962, 180–181). (46) Brown's thesis that inertia rec eives a dynamic al explanation only in GR has rec ently been defended by Sus (2011). Sus emphasiz es that in GR the metric is a genuinely dynamic al entity and that one c an derive ∇a Tab = 0 from the very equations that govern the metric 's behavior. In c ontrast, SR, as standardly c onc eived, involves fixed inertial struc ture whose properties are postulated by fiat. However, this differenc e between the theoriesis c ompatible with the theories agreeing on the fundamental reasons why forc e-free bodies are related to inertial struc ture in just the way they are. (47) The terminology is Quine's, who c harac teriz es a theory's ontology as “the objec ts over whic h the bound variables of the theory should be c onstrued as ranging in order that the statements affirmed in the theory be true” (Quine 1951, 11). (48) The need for this sec ond step is emphasiz ed by Earman (1989, 128), though not in prec isely these terms. (49) Related versions of relationalism, ac c ording to whic h absolute veloc ity (or even absolute position) is inter preted as a primitive, monadic property of partic les, have been disc ussed by Horwic h (1978, 403) and Friedman (1983, 235) (see also Teller, 1987). In addition to being less natural than the form of Newtonian relationalism identified by Maudlin, they are vulnerable (like Newtonian relationalism) to the kinematic shift argument. The absolute position version is also vulnerable to the static shift argument mentioned in sec tion 7. (50) The obvious constraints are that the embedding respects the temporal separation between material events and that there is a single c ongruenc e of inertial geodesic s suc h that, for any two material events e1, e2 loc ated on geodesic s from the c ongruenc e v1,v2 , the Newtonian relational distanc e between e1 and e2 equals the (c onstant) spatial distanc e between simultaneous points of v1 and v2 . (51) This is the princ ipal inadequac y of Newtonian relationalism that Maudlin identifies (1993, 193). Friedman (1983, 235) makes the same c ritic ism of the postulation of a primitive property of “absolute veloc ity” (52) Maudlin restric ts the extension of col to nonsimultaneous events, but there is no reason why mutually simultaneous events should not be inc luded, with col(e1, e2 , e3 ) holding just if the sum of the distanc es between two of the pairs of events equals the distanc e between the third pair. (53) A similar example involving Minkowski spac etime is disc ussed by Mundy (1986), Catton and Solomon (1988), and Earman (1989, 168– 169). The relations of spac elike separation, lightlike separation and timelike separation determine the struc ture of Minkowski spac etime up to an overall sc ale fac tor. However, these relations instantiated between material events need not fix their embedding into Minkowski spacetime up to Poincaré transformations. The examples disc ussed by Mundy et al. involve a small finite number of events, but the problem generaliz es to c ertain c onfigurations of c ontinuum many For example, c onsider two partic les whic h move so that any two events from distinc t trajec tories are always spac elike (the events on eac h trajec tory are all mutually timelike). We know that, as t − ±oo, the partic les must be ac c elerating in opposite direc tions, but not muc h more. (54) This objec tion to neo-Newtonian relationalism, reported by Huggett (1999, 26), is again due to Maudlin. (55) Sinc e these are func tions of the rijs, just as Sklarations are func tions of the rijs, they would not ac tually be of any help either. The situation c hanges if higher derivatives are allowed. (56) Friedman's expression of the law is F i/m  =  ¨xi  +  ai  +  2˙aji ˙xj   +  ¨akixk (Friedman, 1983, 226, eqn 8; I have slightly altered the notation). Fi is the ith c omponent of Newtonian (three-)forc e on the partic le we are c onsidering and xi (t) is the ith c omponent of its position vec tor with respec t to some rigid Euc lidean c oordinate system. ai is the ith c omponent of the absolute ac c eleration of the origin of the coordinate system (that is, the Sklaration that a hypothetic al partic le would have were it c omoving with the c oordinate origin). aji is the rotation of the c oordinate system about its origin with respec t to an inertial frame. Thus only the first of the three additional terms on the right- hand side of the equation is direc tly interpretable in terms of a Sklaration, and then only if we pic k a c oordinate system that happens to have a partic le c omoving with its origin. Cruc ially, we need to be told how to interpret the rotation pseudo-vec tor aji in terms of Sklarations. (57) Mundy (1983, 224) even interprets the Euc lidean c onstraints on instantaneous distanc es similarly, so that his Page 43 of 48

Substantivalist and Relationalist Approaches to Spacetime relationalist does not need a primitive notion of geometric possibility over and above that of physic al possibility. (58) I explore some of the options in Pooley (in preparation). As with Sklarations, the required “kinematic al” c onstraints on the instantiation of suc h relations suggest that the proposals are really substantivalism in disguise. (59) My terminology again follows Earman (1989, 31) and Bain (2004, 351). Ehlers (1973a, 78–79) discusses the group but leaves it unnamed. (60) There is no c anonic al differential-geometric way of c apturing this struc ture. Earman (1989, 32) resorts to an equivalenc e c lass of c onnec tions whose c ongruenc es of geodesic s are nonrotating with respec t to one another. Saunders (Forthc oming, §7) offers an elegant c harac teriz ation of a similar but stric tly weaker struc ture. (61) This problem for Newtonian gravitation set in neo-Newtonian spac etime is well-known. For a related disc ussion in the philosophic al literature, see, e.g., Friedman (1983, 95– 97). (62) This theory is presented as the solution to the problem fac ed by the Galilean substantivalist by Friedman (1983, 97– 104; 120– 124). See also Malament (1995), who presents it as a solution to a c losely related problem raised by Norton (1993). (63) Malament (2012, c h. 4) reviews these results and Newton– Cartan theory more generally; see also Bain (2004). (64) As far as I am aware, Simon Saunders was the first to stress that transtemporal c omparison of direc tions are obviously c ompatible with relationalist ontology. Saunders (Forthc oming) is a rec ent disc ussion of related topic s. I am grateful to him for disc ussion. Earman (1989, 78– 81) c omes c lose to attributing the basic idea to James Clerk Maxwell, who, when disc ussing absolute rotation in Maxwell (1877, § 104), wrote: “in c omparing one c onfiguration of the system with another, we are able to draw a line in the final c onfiguration parallel to a line in the original c onfiguration.” Earman's assessment is that “Maxwell's set of parallel direc tions is, of c ourse, inertial struc ture, and in modern terms what he seems to be proposing is that neo-Newtonian spac etime is the appropriate arena for the sc ientific desc ription of motion” (Earman, 1989, 80). However, it is c lear that Maxwell here only assumes a standard of parallelism for spacelike lines whic h, as we have seen, does not require the full struc ture of neo- Newtonian spac etime. Perhaps Earman did not realiz e how apt his label Maxwellian spacetime is. (65) One “best matc hes” instantaneous c onfigurations only with respec t to rigid translations and not, as Barbour does, by translations, rotations, and dilations. Barbour-type partic le theories that do not implement rotations as gauge symmetries have been disc ussed rec ently by Anderson (2012, sec tion 2.4). (66) I am grateful to David Wallac e for highlighting this possibility. (67) The main discussions of a position of this sort are Earman (1989, 128–130) and Maudlin (1993, 196–199). (68) A treatment of ac c eleration along these lines c an be found in Minkowski's original presentation (Minkowski, 1909, 85–86). (69) Relativistic theories in whic h the four-forc e on a partic le at a point is determined direc tly by the properties of other partic les at other spac etime loc ations are not impossible; Feynman and Wheeler's version of elec tromagnetism is suc h a “pure partic le theory” (Wheeler and Feynman, 1949). These theories, however, have various unwelc ome features, and their empiric al adequac y remains an open question; see Earman (1989, 155– 158) for discussion. (70) See Malament (1982, 532, fn 11), who is responding to Field's argument. Other c lear expressions of this view c an be found in Belot (1999, 45) and Rovelli (2001, 104). (71) As the rest of this sec tion illustrates, still less does the move trivializ e the substantivalist–relationalist debate (pace Field 1985, 41), although it does exc use the relationalist from replac ing field theories with ac tion-at-a- distanc e theories. (72) Note that this c onstitutes an answer to Earman's c hallenge that the relationalist must provide a “direc t c harac teriz ation” of the reality underlying the substantivalist's desc ription of fields (Earman, 1989, 171). Page 44 of 48

Substantivalist and Relationalist Approaches to Spacetime (73) See, for example, Malament (2004, § 3), where the tensorial properties of the elec tromagnetic field Fab are derived from assumptions about its ac tion on c harges. (74) The relationalist c an also question whether one should regard regions of z ero field strength as regions where the material field literally does not exist. This might be the natural interpretation of fields that represent “dust” in models of GR, but it is at least c ontroversial for, e.g., the elec tromagnetic field. The stipulation is yet more problematic when one moves to quantum field theory. I am grateful to David Wallac e for pressing this point. (75) For related reasons, Earman defines Machian spacetime to be spac etime with simultaneity struc ture and Euc lidean metric al struc ture on its simultaneity surfac es but with no temporal metric (Earman 1989, 27– 30). (76) It should be stressed that Barbour initially postulated a Jac obi-like ac tion on purely Mac hian grounds and only learned of the c onnec tions with standard dynamic s several years afterward. (77) Note that c hoosing a simplifying parameter for Equation (4) is quite unlike c hoosing a time c oordinate that is adapted to the spac etime substantivalist's temporal metric . The latter also simplifies the (generally c ovariant) equations, but these equations explic itly refer to an independent standard of duration. Ac c ording to the Mac hian interpretation of Jacobi's Principle, fundamental dynamics is formulated without reference to such an external time. See Pooley (2004, 78–79). (78) Noteworthy examples are Hofmann (1904), Reissner (1914), Schrödinger (1925), Barbour (1974a), Barbour and Bertotti (1977), and Assis (1989). For further disc ussion, see Earman (1989, 92– 96) and Barbour and Pfister (1995, 107– 178). It turns out that Barbour's later partic le theories (disc ussed immediately below) c an themselves be formulated in a natural way direc tly in terms of the right c hoic e of relative c oordinates. For details, see Anderson (2012, Chapters 2 and 3), where a wider c lass of suc h theories is c onsidered. (79) For an informal disc ussion of the c entral idea, see Barbour (1999, c h. 7). For the extension to dilations, see Barbour (2003). For more formal and general treatments, see Anderson (2006) and Gryb (2009). (80) Similarly, one c an argue that the Mac hian relationalist is able to explain formal features of the potential, suc h as its dependenc e only on the ṙijs, that are again nonessential aspec ts of standard Newtonian theory (Barbour 2011). (81) W can depend on hab and its spatial derivatives up to some finite order; the presence of h  :   =  √−d−e−t− h−a−b is simply to ensure that the integration is invariantly defined. d(8if2fe) oIfm ξoar ipsh aisnm in gfienniteersaimteadl b3y-v ξeac t⋅o rL fiξe hlda,b (, hthaeb −Lie L dξe hriavba)t iisv eth oef rheasbu wlt itohf raecstpinegc ot nto h ξaab, bisy g aivne inn fbinyit esimal Lξ hab  =  ∇a ξb   +  ∇b ξa , where Δa is the derivative operator associated with the unique torsion-free connection c ompatible with hab. For the reason why TBM is defined in terms of the Lie derivatives with respec t the velocity of a 3-vec tor field, see Barbour et al. (2002, 3219) and Barbour (2003, § 4). (83) Gryb (2010, 16– 18) c ontains a brief disc ussion of these theories. (84) The importanc e of the distinc tion between (5) and (6), and the fac t that GR c ould be c ast in the form of (6), was first pointed out to Barbour by Karel Kuc hař (Barbour and Bertotti 1982, 305). (85) This is one of the main results of Barbour et al. (2002), who also c laim to rec over the equivalenc e princ iple and Maxwellian elec tromagnetism from the c onstraints that c onsistenc y alone plac es on how matter fields c an be added to the theory. The results are extended to Yang Mills theory in Anderson and Barbour (2002). It should be stressed that these results do not amount to a derivation of GR and the equivalenc e princ iple from Mac hian first princ iples alone. In addition to the c hoic e of (6) over (5), the form of (6) embodies a number of simplic ity assumptions, the relaxing of whic h permits a range of other Mac hian theories; see Anderson (2007). (86) There may be good reason to see c onformal geometrodynamic s as superior to alternative 3 +1 approac hes to GR: its basic quantities are dimensionless and the true degrees of freedom are transparent (Barbour 2011, 24, 39). This, though, does not speak direc tly to the preferability of a 3 + 1 over a spac etime perspec tive. (87) One might also worry that if the laws are about c oordinate systems it will be hard for the relationalist to avoid Page 45 of 48

Substantivalist and Relationalist Approaches to Spacetime what Field (1985) c alls heavy duty platonism. (Thanks to Jeremy Goodman for highlighting this.) I take the role of c oordinate systems in the spec ific ation of the Humean alternative disc ussed next to be less problematic . (88) If one pursues the program too far, the supervenienc e base will eventually bec ome too impoverished to subvene Newtonian laws. Suppose, for example, that the only spatiotemporal information one retains is that whic h is c ommon to all c oordinatiz ations of the partic le trajec tories obtainable from an initial inertial c oordinate system by smooth but otherwise arbitrary c oordinate transformations that preserve the timelike direc tedness of the trajec tories (that is, that they are nowhere tangent to surfac es of c onstant time c oordinate). Suc h topologic al data inc ludes information about whether any two trajec tories ever intersec t, and information about how the trajec tories are “knotted”, but little else. Many Newtonian worlds involving c omplex histories of relative distanc es and interac tions will be topologic ally equivalent to histories where all partic les maintain c onstant distanc e from one another. If one inc ludes only suc h topologic al information in the supervenienc e base, worlds like this will not be worlds where Newton's laws are laws of nature. One might also wonder whether any degree of topologic al c omplexity (that is, any degree of c omplex entwining of the trajec tories) will promote Newton's laws to Best System status. Might not simpler, equally strong alternatives always be available? (89) This aspec t of the proposal is sc rutiniz ed by Belot (2011, c h. 3). (90) Simplic ity will not determine a unique geometry for G in all c ases, but Huggett makes a persuasive c ase that the underdetermination is benign and that the regularity relationalist should be content to live with the possibility that there may be no determinate fac t of the matter about the geometry of physic al spac e (Huggett 2006, 55– 56). (91) This is not to say that every explanatory question one might ask about the phenomenon of length c ontrac tion requires an appeal to dynamic al laws; in some c ontexts it is enough to c ite the relevant geometric al fac ts in order to provide an explanation. This is a point explic itly emphasiz ed in Brown and Pooley (2006, 78–79, 82), where paradigm explanatory uses of Minkowski diagrams (e.g., to highlight that observers in relative motion c onsider different c ross-sec tions of a rod's world tube when judging its length) are said to c onstitute “perfec tly ac c eptable explanations (perhaps the only ac c eptable explanations) of the explananda in question.” Our emphasis of this fac t seems to have been overlooked by some authors (Skow 2006, Frisc h 2011). (92) As it was put in Brown and Pooley (2006, 82): “it is suffic ient for these bodies to undergo Lorentz c ontrac tion that the laws (whatever they are) that govern the behavior of their mic rophysic al c onstituents are Lorentz c ovariant. It is the fact that the laws are Lorentz covariant …that explains why the bodies Lorentz c ontrac t. To appeal to any further details of the laws that govern the c ohesion of these bodies would be a mistake.” Janssen's (2009) c arefully argued c ase that phenomena rec ogniz ed to be kinematic al (in his sense) should not be explained in terms of the details of their dynamic s is therefore one that we had antec edently c onc eded. The explanation of the phenomena in terms of symmetries nonetheless deserves the label “dynamical” (though not, as acknowledged in Brown and Pooley (2006, 83), “c onstruc tive”) bec ause the explanantia are (in the first instanc e) the dynamical symmetries of the laws governing the material systems manifesting the phenomena. (93) Other examples are provided by Lorentz invariant dynamic s set in Newtonian spac etime; see, e.g., Earman (1989, 50–55). (94) I am grateful to Hilary Greaves for disc ussion of this point. The story given here c an also be told, mutatis mutandis, by a relationalist who posits primitive spatiotemporal relations held to satisfy primitive geometric al c onstraints. (95) See Earman (1989, 45– 47) for a related disc ussion of the c onnec tion between spac etime symmetries and dynamic al symmetries. (96) This law c ould be expressed in a c oordinate-independent manner if one introduc es an auxiliary devic e, the Minkowski metric , whic h would then be “no more than a c odific ation of the Kleinian geometry assoc iated with the symmetry group” of the laws. (97) More radic al options c ould also be pursued. Starting with the idea that there are no primitive fac ts about the contiguity or otherwise of distinct material events, one might nonetheless map them into a single copy of ℝ n. The coincidence of events (which events are to be mapped to the same element in ℝ n) is then to be thought of as determined in the same manner as the spac etime interval, that is, determined by those c oordinatiz ations that yield Page 46 of 48

Substantivalist and Relationalist Approaches to Spacetime total desc riptions of all events that satisfy some simple set of equations. Perhaps one c ould even view the value of n (that is, the dimensionality of spac etime itself) as determined in this way too. As with generaliz ations of Huggett's proposal (see footnote 88), the more one views as grounded via some kind of Best System presc ription, the more unc onstrained the problem bec omes; it c eases to be plausible that the c omplexity of the postulated supervenienc e base will be suffic ient to underwrite the target quantities and the laws they obey. (98) Compare how, on some treatments of vagueness, disjunc tions c an be determinately true (Fred is either bald or not bald) even though neither disjunct is determinately true. (99) A good introduc tion is provided by Norton (2011). (100) Rec all (sec tion 4) that the pseudo-Riemannian metric tensor gab enc odes all of the geometric al properties of spac etime, itself represented by the four-dimensional manifold M. Stric tly speaking, the stress– energy tensor Tab does not direc tly represent the fundamental matter c ontent of the model. This will be represented by other fields, in terms of whic h Tab is defined. (101) This amounts to a denial of Leibniz Equivalence. Earman and Norton take suc h a denial to be the ac id test of substantivalism (Earman and Norton, 1987, 521). (102) For further disc ussion of the definition of determinism appropriate to GR, and of the merits of these options, see Butterfield (1989b), Rynasiewic z (1994), Belot (1995), Leeds (1995), Brighouse (1997), and Melia (1999). (103) This is my preferred option (see Pooley 2006, 99– 103). Despite the important differenc es between them, I take Maudlin (1989), Butterfield (1989a), Maidens (1992), Stachel (1993, 2002), Brighouse (1994), Rynasiewicz (1994), Hoefer (1996), and Saunders (2003a) all to deny that the relevant haec c eitistic differenc es c orrespond to distinc t physic al possibilities. For several of these authors (though notably not for Maudlin), the c ommitment follows from a c ommitment to some kind of antihaec c eitism, at least c onc erning spac etime points, whether on general philosophic al grounds (as in Hoefer's c ase), or as a perc eived lesson of the diffeomorphism invarianc e of the physics (as in Stachel's case). (104) Note one parallel between the Hole Argument and the argument against Galilean spac etime that exploits the Maxwell group. The fac t that the Maxwell group involves a parameter that is an arbitrary func tion of time means that the Galilean substantivalist interpretation of the models of a Maxwellian invariant theory involves regarding the theory as indeterministic (c f Stein 1977, Saunders, 2003a). The fac t that the indeterminism involves qualitative differenc es (ac c ording to the Galilean substantivalist) arguably makes the argument more effec tive against Galilean substantivalism than the Hole Argument is against GR. (105) This point is disc ussed by Horwic h (1978), Field (1985), and Maudlin (1993). (106) An argument like this was made by Leibniz in his c orrespondenc e with Clarke (Alexander, 1956). That Leibniz makes a prec isely parallel argument, exploiting permutation invarianc e, against the existenc e of atoms, should give those sympathetic to the static shift argument pause for thought. Consistenc y should lead one either to embrac e or rejec t both c onc lusions. (107) Self-dec lared struc turalist approac hes to spac etime that are best desc ribed as varieties of substantivalism (in the sense that they include spacetime points among the ground-floor ontology) include those of Stachel (2002, 2006), Saunders (2003a), Esfeld and Lam (2008), and Muller (2011). For an overview of a wider range of struc turalist approac hes, see Greaves (2011), who gives reasons to be skeptic al that a c oherent position that does not c ollapse into sophistic ated substantivalism (or relationalism) has yet to be c learly identified. Bain (2006) and Ric kles (2008) are two more advoc ates of spac etime struc turalism, not c ited by Greaves. (108) I am attrac ted to the view that sees individualistic fac ts as grounded in general fac ts (Pooley, unpublished). However, as Dasgupta (whose terminology I adopt) has rec ently stressed (Dasgupta, 2011, 131– 134), this requires that one's understanding of general fac ts does not presuppose individualistic fac ts. Sinc e the stan dard understanding of general fac ts arguably does take individualistic fac ts for granted, the spac etime struc turalist/sophistic ated substantivalist must show that they are not illic itly making the same presupposi tion. (Dasgupta's own view is that something quite radic al is needed (2011, 147– 152).) The rec ent literature on “weak disc ernibility” (see, e.g., Saunders, 2003b) has made muc h of the fac t that numeric al diversity fac ts c an Page 47 of 48

Substantivalist and Relationalist Approaches to Spacetime supervene on fac ts statable without the identity predic ate even when traditional forms of the Princ iple of the Identity of Indisc ernibles are violated. Note, however, that merely showing that one set of fac ts supervene on another set of fac ts is not suffic ient to show that the former are grounded in the latter (or even that it is possible to think of them as so grounded). (109) The skepticism concerning the substantiveness of the debate expressed in this paragraph is therefore not that of Rynasiewic z (1996). For a c onvinc ing response to many Rynasiewic z 's c laims, see Hoefer (1998). O liver Pooley Oliver Pooley is University Lecturer in the Faculty of Philosophy at the University of Oxford and a Fellow and Tutor at Oriel College, Oxford. He works in the philosophy of physics and in m etaphysics. Much of his research focuses on the nature of space, tim e, and spacetim e.

Global Space Time Structure John Byron Manchak The Oxford Handbook of Philosophy of Physics Edited by Robert Batterman Abstract and Keywords This c hapter, whic h examines global spac etime struc ture and the qualitative, primarily topologic al and c ausal, aspec ts of general relativity, proposes an abstrac t c lassific ation of loc al and global spac etime properties and identifies a set of c ausal c onditions that form a stric t hierarc hy of possible c asual properties of spac etime. It also addresses the philosophical questions concerning the physical reasonableness of these various spacetime properties and c onsiders the notion of geodesic inc ompleteness. K ey words: spaceti me stru ctu re, gen eral rel ati v i ty , cau sal con di ti on , ph i l osoph i cal qu esti on s, ph y si cal reason abl en ess, geodesi c i n compl eten ess 1. Intro ductio n The study of global spac etime struc tureis a study of the more foundational aspec ts of general relativity. One steps away from the details of the theory and instead examines the qualitative features of spac etime (e.g., its topology and c ausal struc ture). We divide the following into three main sec tions. In the first, we outline the basic struc ture of relativistic spac etime and rec ord a number of fac ts. In the sec ond, we c onsider a distinc tion between loc al and global spac etime properties and provide important examples of eac h. In the third, we examine two c lusters of global properties and question whic h of them should be regarded as physic ally reasonable. The properties c onc ern “singularities” and “time travel” and are therefore of some philosophical interest. 2. Relativistic Spacetime We take a (relativistic ) spacetime to be a pair (M, gab). Here M is a smooth, c onnec ted, n-dimensional (n ≥ 2) manifold without boundary. The metric gab is a smooth, nondegenerate, pseudo-Riemannian metric of Lorentz signature (+,−,…,−) on M.1 2.1 Manifold and Metric Let (M,gab) be a spac etime. The manifold M c aptures the topology of the universe. Eac h point in the n-dimensional manifold M represents a possible event in spac e-time. Our experienc e tells us that any event c an be c harac teriz ed by n numbers (one temporal and n − 1 spatial c oordinates). Naturally, then, the loc al struc ture of M is identic al to ℝn. But globally, M need not have the same struc ture. Indeed, M c an have a variety of possible topologies. In addition to ℝn, the sphere Sn is c ertainly familiar to us. We c an c onstruc t a number of other manifolds by taking Cartesian produc ts of ℝn and Sn. For example, the 2-c ylinder is just ℝ1 × S1 while the 2-torus is S1 × S1 (see figure 16.1). Any manifold with a c losed proper subset of points removed also c ounts as a manifold. For example, Sn − n Page 1 of 15

Global Space Time Structure {p} is a manifold where p is any point in Sn. We say a manifold M is Hausdorff if, given any distinc t points p,pʹ ∈ M, one c an find open sets O and Oʹ suc h that p ∈ O, pʹ ∈ Oʹ, and O ∩ Oʹ = ∅. Physic ally, Hausdorff manifolds ensure that spac etime events are distinc t. In what follows, we assume that manifolds are Hausdorff.2 We say a manifold is compact if every sequenc e of its points has an ac c umulation point. So, for example, Sn and Sn × Sm are c ompac t while ℝn and ℝn × Sm are not. It c an be shown that every nonc ompac t manifold admits a Lorentz ian metric . But there are some c ompac t manifolds that do not. One example is the manifold S4 . Thus, assuming spac etime is four dimensional, we may deduc e that the shape of our universe is not a sphere. One c an also show that, in four dimensions, if a c ompac t manifold does admit a Lorentz ian metric (e.g., S1 × S3 ), it is not simply c onnec ted. (A manifold is simply connected if any c losed c urve through any point c an be c ontinuously deformed into any other c losed c urve at the same point.) We say two manifolds M and Mʹ are diffeomorphic if there is a bijec tion φ : M → Mʹ suc h that φ and φ are smooth. Diffeomorphic manifolds have identic al manifold struc ture and c an differ only in their underlying elements. Figure 16.1 The cylinder ℝ1 × S1 and torus S1 × S1. Figure 16.2 Timelike, null, and spacelike vectors fall (respectively) inside, on, and outside the double cone struc ture. The Lorentz ian metric gab c aptures the geometry of the universe. Eac h point p ∈ M has an assoc iated tangent spac e Mp. The metric gab assigns a length to eac h vec tor in Mp. We say a vec tor ξa is timelike if gabξaξb 〉 0, null if gabξaξb = 0, and spacelike if gabξaξb 〈 0. Clearly, the null vec tors c reate a double c one struc ture; timelike vec tors are inside the c one while spac elike vec tors are outside (see figure 16.1). In general, the metric struc ture c an vary over M as long as it does so smoothly. But it c ertainly need not vary and indeed most of the examples c onsidered below will have a metric struc ture that remains c onstant (i.e., a flat metric ). For some interval I ⊆ ℝ,a smooth c urve γ: I → M is timelike if its tangent vec tor ξa at eac h point in γ[I] is timelike. Similarly, a c urve is null (respec tively, spacelike) if its tangent vec tor at eac h point is null (respec tively, spac elike). A c urve is causal if its tangent vec tor at eac h point is either null or timelike. Physic ally, the worldlines of massive partic les are images of timelike c urves while the worldlines of photons are images of null c urves. We say a c urve γ: I → M is not maximal if there is another c urve γʹ: Iʹ → M suc h that I is a proper subset of Iʹ and γ(s) = γʹ(s) for all s ∈ I. We say a spac etime (M, gab) is temporally orientable if there exists a c ontinuous timelike vec tor field on M. In a temporally orientable spac etime, a future direc tion c an be c hosen for eac h double c one struc ture in way that involves no disc ontinuities. A spac etime that is not temporally orientable c an be easily c onstruc ted by taking the underlying manifold to be the Möbius strip. In what follows, we will assume that spac etimes are temporally orientable and that a future direc tion has been c hosen.3 Page 2 of 15

Global Space Time Structure Naturally, a timelike c urve is future-directed (respec tively, past-directed) if all its tangent vec tors point in the future (respec tively, past) direc tion. A c ausal c urve is future-directed (respec tively, past-directed) if all its tangent vec tors either point in the future (respec tively, past) direc tion or vanish. Two spac etimes (M,gab) and (Mʹ, gʹab) are isometric if there is a diffeomorphism φ : M → Mʹ suc h that φ* (gab) = gʹab. Here, φ* is a map that uses φ to “move” arbitrary tensors from M to Mʹ. Physic ally, isometric spac etimes have identic al properties. We say a spac etime (Mʹ,gʹab) is a (proper) extension of (M,gab) if there is a proper subset N of Mʹ suc h that (M,gab) and (N,gʹab|N) are isometric . We say a spac etime is maximal if it has no proper extension. One c an show that every spac etime that is not maximal has a maximal extension. Finally, two spac etimes (M,gab) and (Mʹ,gʹab) are locally isometric if, for eac h point p ∈ M, there is an open neighborhood O of p and an open subset Oʹ of Mʹ suc h that (O,gab|O) and (Oʹ,gʹab|O,) are isometric , and, c orrespondingly, with the roles of (M,gab) and (Mʹ,gʹab) interc hanged. Although loc ally isometric spac etimes c an have different global properties, their loc al properties are identic al. Consider, for example, the spac etimes (M,gab) and (Mʹ,gʹab) where M = S1 × S1, p ∈ M, Mʹ = M − {p}, and gab and gʹab are flat. The two are not isometric but are loc ally isometric . Therefore, they share the same loc al properties but have differing global struc tures (e.g., the first is c ompac t while the sec ond is not). One c an show that for every spac etime (M,gab), there is a spac etime (Mʹ,gʹab) suc h that the two are not isometric but are loc ally isometric . 2.2 Influence and Dependence Here, we lay the foundation for the more detailed disc ussion of c ausal struc ture in later sec tions. Consider the spac etime (M,gab). We define the two-plac e relations ≪ and 〈 on the points in M as follows: we write p ≪ q (respec tively, p 〈 q) if there exists a future-direc ted timelike (respec tively, c ausal) c urve from p to q. For any point p ∈ M, we define the timelike future (domain of influenc e) of p, as the set I+(p) ≡ {q: p ≪ q}. Similarly, the causal future (domain of influenc e) of p is the set J+(p) ≡ {q: p ≪ q}. The c ausal (respec tively, timelike) future of p represents the region of spac etime that c an be possibly influenc ed by partic les (respec tively, massive partic les) at p. The timelike and c ausal pasts of p, denoted I−(p) and J−(p), are defined analogously. Finally, given any set S ⊂ M, we define I+[S] to be the set ∪{I+(p): p ∈ S}. The sets I−[S] and J +[S], and J −[S] are defined analogously. We shall now list a number of properties of timelike and c ausal pasts and futures. For all p ∈ M, the sets I+(p) and I−(p) are open. Therefore, so are I+[S] and I−[S] for all S ⊆ M. However, the sets J +(p), J−(p), J +[S] and J −[S] are not, in general, either open or c losed. Consider Minkowski spac etime4 and remove one point from the manifold. Clearly, some c ausal pasts and futures will be neither open nor c losed. Figure 16.3 Cylindrical Minkowski spacetime containing a closed causal curve (e.g., the dotted line) but no c losed timelike c urves. By definition, I+ (p) ⊆J+(p) and I−(p) ⊆ J− (p). And it is c lear that if p ∈ I+(q), then q ∈ I−(p) and also that if p ∈ I−(q), then q ∈ I+ (p). Analogous results hold for c ausal pasts and futures. We c an also show that if either (i) p ∈ I+ (q) and q ∈ J+(r) or (ii) p ∈ J+(q) and q ∈ I+(r), then p ∈ I+(r). Analogous results hold for the timelike and c ausal pasts. From this it follows that ¯I¯¯+¯¯¯(¯p¯¯)  =  ¯J¯¯+¯¯¯(¯p¯¯)¯ ,  ¯I¯¯−¯¯¯(¯p¯¯)  =  ¯J¯¯¯−¯¯¯(¯p¯¯),  I˙+  (p)  =  J˙+ (p), and I˙− (p)  =  J˙−  (p).5 Page 3 of 15

Global Space Time Structure Bec ause future-direc ted c asual c urves c an have vanishing tangent vec tors, it follows that for all p, we have p ∈ J+(p) and p ∈ J−(p). Of c ourse, a similar result does not hold generally for timelike futures and pasts. But there do exist some spac etimes suc h that, for some p ∈ M, p ∈ I+(p) (and therefore p ∈ I−(p)). Gödel spac etime is one famous example (Gödel 1949). We say the chronology violating region of a spac etime (M,gab) is the (nec essarily open) set {p ∈ M: p ∈ I+ (p)}. We say a timelike c urve γ: I → M is closed if there are distinc t points s,sʹ ∈ I suc h that γ(s) = γ(sʹ). Clearly, a spac etime c ontains a c losed timelike c urve if and only if it has a nonempty c hronology violating region. One c an show that, for all spac etimes (M,gab), if M is c ompac t, the c hronology violating region is not empty (Geroc h 1967). The c onverse is false. Take any c ompac t spac etime and remove one point from the underlying manifold. The resulting spac etime will c ontain c losed timelike c urves and also fail to be c ompac t. We define a c ausal c urve γ: I → M to be closed if there are distinc t points s,sʹ ∈ I suc h that γ(s) = γ(sʹ) and γ has no vanishing tangent vec tors. It is immediate that c losed timelike c urves are nec essarily c losed c ausal c urves. But one c an find spac etimes that c ontain the latter but not the former. Consider, for example, Minkowski spac etime (M,gab), whic h has been “rolled up” along one axis in suc h a way that some null c urves but no timelike c urves are permitted to loop around M (see figure 16.3). Other c onditions relating to “almost” c losed c ausal c urves will be c onsidered in the next sec tion. Finally, we say the spac etimes (M,gab) and (M,gʹ ab) are conformally related if there is a smooth, stric tly positive func tion Ω: M → ℝ suc h that gʹ ab = Ω2 ,gab (the func tion Ω is c alled a conformal factor). Clearly, if (M,gab) and (M,gʹ ab) are c onformally related, then for all points p,q ∈ M, p ∈ I+(q) in (M,gab) if and only if p ∈ I+(q) in (M,gʹ ab). Analogous results hold for timelike pasts and c ausal futures and pasts. Thus, the c ausal struc tures of c onformally related spac etimes are identic al. A point p ∈ M is a future endpoint of a future-direc ted c ausal c urve γ :I → M if, for every neighborhood O of p, there exists a point sʹ ∈ I suc h that γ(s) ∈ O for all s 〉 sʹ. A past endpoint is defined analogously. For any set S ⊆ M, we define the future domain of dependence of S, denoted D+(s), to be the set of points p ∈ M suc h that every c ausal c urve with future endpoint p and no past endpoint intersec ts S. The past domain of dependence of S, denoted D−(s), is defined analogously. The entire domain of dependence of S, denoted D(s), is just the set D−(s) ∪ D+(s). If “nothing c an travel faster than light,” there is a sense in whic h the physic al situation at every point in D(s) depends entirely upon the physical situation on S. Clearly, we have S ⊆ D+(s) ⊆ J+[S] and S ⊆ D−(s) ⊆ J−[S]. Given any point p ∈ D+(S), and any point q ∈ I+[S] ∩ I−(p), we know that q ∈ D+ (S). An analogous result holds for D−(s). One c an verify that, in general, D(s) is neither open nor c losed. Consider Minkowski spac etime (M,gab). If S = {p} for any point p ∈ M, we have D(s) = S, whic h is not open. If S = I+(p) ∩ I−(q) for any points p & M and q ∈ I+(p), we have D(s) = S, whic h is not c losed. A set S ⊂ M is a spacelike surface if S is a submanifold of dimension n − 1 suc h that every c urve in S is spac elike. We say a set S ⊂ M is achronal if I+[S] ∩ S = ∅. One c an show that for an arbitrary set S, I+ [S] is ac hronal. In what follows, let S be a c losed, ac hronal set. We have D+(s) ∩ I−[S] = D−(S) ∩ I+[S] = ∅. We also have int(D+(S)) = I−[D+(S)] ∩ I+[S] and the analogous result for D−(s). Finally, we have int(D(s)) = I−[D+(S)] ∩ I+[D−(S)] = I+[D−(S)] ∩ I−[D+(S)]. We say a c losed, ac hronal set S is a Cauchy surface if D(s) = M. Physic ally, c onditions on a Cauc hy surfac e S (nec essarily a submanifold of M of dimension n − 1) determine c onditions throughout spac etime (Choquet-Bruhat and Geroc h 1969). Clearly, if S is a Cauc hy surfac e, any c ausal c urve without past or future end-point must intersec t S, I +[S], and I −[S]. One c an verify that Minkowski spac etime admits a Cauc hy surfac e. We define the future Cauchy horizon of S, denoted H+(s), as the set ¯D¯¯¯+¯¯¯(¯¯S¯¯)  −  I −  [D+ (S)]. The past Cauchy horizon of S is defined analogously. One c an verify that H + (s) and H − (s) are c losed and ac hronal. The Cauchy horizon of S, denoted H(s), is the set H +(S) ∪ H −(s). We have H(s) = D(s) and therefore H(s) is c losed. Also, a nonempty, c losed, ac hronal set S is a Cauc hy surfac e if and only if H(S) = ∅. Page 4 of 15

Global Space Time Structure Figure 16.4 Minkowski spacetime with one point removed contains a slice S but no Cauchy surface. The region above the dotted line is not part of D(s). The edge of a c losed, ac hronal set S ⊂ M is the set of points p & S suc h that every open neighborhood O of p c ontains a point q ∈ I+(p), a point r ∈ I−(p), and a timelike c urve from r to q whic h does not intersec t S. A c losed, ac hronal set S ⊂ M is a slice if it is without edge. It follows that every Cauc hy surfac e is a slic e. The c onverse is false. Consider Minkowski spac etime with one point removed from the manifold. It c ertainly admits a slic e but no Cauc hy surfac e (see figure 16.4). Of c ourse, not every spac etime admits a slic e. For a c ounterexample, c onsider any spac etime that has a c hronology violating region identic al to its manifold. 3. Spacetime Pro perties We say a property P on a spac etime is local if, given any two loc ally isometric spac etimes (M,gab) and (Mʹ,gʹ ab), (M,gab) has P if and only if (Mʹ,gʹ ab) has P. A property is global if it is not loc al. Below, we will introduc e and c lassify a number of spacetime properties of interest. 3.1 Local Properties The most important loc al spac etime property is that of being a “solution” to Einstein's equation. There are a number of ways one c an understand this property and we shall investigate eac h of them in what follows. Figure 16.5 The vector ηa is parallelly transported along a closed curve γ. Note that the vector returns to the point p orientated differently. Let (M, gab) be a spac etime. Assoc iated with the metric gab is a unique (torsion-free) derivative operator ∇a suc h that ∇agbc = 0. Given a smooth c urve y: I → M with tangent field ξa, we say a vec tor)]a, defined at every point in the range of γ, is parallelly transported along γ if ξb ∇b ηa = 0 (see figure 16.1). We say a smooth c urve γ: I → ℝ is a geodesic (i.e., nonac c elerating) if its tangent field ξa is suc h that ξb ∇bξa = 0. Given any point p ∈ M, there is some neighborhood O of p suc h that any two points q,r ∈ O c an be c onnec ted by a unique geodesic c ontained entirely in O. Suc h a neighborhood is said to be convex normal . The derivative operator ∇a c a2n∇ b[ce∇ u ds]e ξda t.o Ade mfineetr itch eg R ioenm Man ins fclaurt vifa atunrde otnelnys iof ri.t sIt aiss sthoec iautneiqdu Rei etemnasnonr cRubarcvda stuucreh that for all ξa ,  Rabcd ξb   =   − Page 5 of 15

Global Space Time Structure Rtenba(scod)r =Rba c0d,  vRaa[nbicsdh]  e=s e 0v,e ∇ry[wn h Re∣arbe∣c do]n  = M. 0 T,h  Re ateb(ncds)o r=s R 0ba,c d R aan[bdc dR] a=bcd  0h,a Rve(a ba) cndu  m=b0e,r aonf du sReafbucld s=y mRmcdeabtr.ies: dWeefi ndeedfi naes tRhea bR  −icc  i12 tRengsabo.r ItR pabla tyos b ae cReacnbtcr aal nrdo leth ien swchaalat rf oclulorvwast.u Oren eR ctoa nb ev eRria fay. tThhaet ∇Eian Gstaeb in= t0e.nsor Gab is then We suppose that the entire matter c ontent of the universe c an be c harac teriz ed by smooth tensor fields on M. For example, a sourc e-free elec tromagnetic field is c harac teriz ed by an anti-symmetric tensor Fab on M, whic h satisfies Maxwell's equations: ∇[aFbc] = 0,∇ a Fab = 0. Other forms of matter, suc h as perfec t fluids and Klein-Gordon fields, are c harac teriz ed by other smooth tensor fields on M. Assoc iated with eac h matter field is a smooth, symmetric energy-momentum tensor Tab on M. For example, the energy-momentum tensor Tab associated with an electromagnetic field Fab is Fan  Fbn  gab (F nm F nm   +   1 ). Note that 4 Tab is a func tion not only of the matter field itself but also of the metric . Other matter fields, suc h as those mentioned above, will have their own energy-momentum tensors Tab. Fix a point p ∈ M. The quantity Tabξaξb at p represents the energy density of matter as given by an observer with tangent ξa at p. The quantity Tba ξb −  Tnb ξn  ξb  ξa at p represents the spatial momentum density as given by the same observer at p. We require that any energy-momentum tensor satisfy the conservation condition: ∇a Tab = 0. Physic ally, this ensures that energy-momentum is loc ally c onserved. Finally, we c ome to Einstein's equation: Gab = 8π Tab.7 It relates the c urvature of spac etime with the matter content of the universe. In four dimensions, Einstein's equation can be expressed as Rab   =  8π (Tab − 1  Tgab ) where T = Taa. 2 Of c ourse, any spac etime (M,gab) c an be thought of as a trivial solution to Einstein's equation if Tab is simply defined to be 1  G ab . Note that Tab automatic ally satisfies the c onservation c ondition, sinc e ∇a Gab = 0. But, in 8π general, the energy-momentum tensor defined in this way will not be assoc iated with any known matter field. However, if the Tab so defined is also the energy-momentum tensor assoc iated with a known matter field (or the sum of two or more energy momentum tensors assoc iated with known matter fields) the spac etime is an exact solution. We say an exac t solution is also a vacuum solution if Tab = 0. And, in four dimensions, one c an use the alternate version of Einstein's equation to show that Tab = 0 if and only if Rab = 0. Between trivial and exact solutions, there are the constraint solutions. These are spacetimes whose associated energy-momentum tensors (defined via Einstein's equation) satisfy one or more c onditions of interest. Here, we outline three. We say Tab satisfies the weak energy condition if, for any future-direc ted unit timelike vec tor ξa at any point in M, the energy density Tab ξa ξb is not negative. We say Tab satisfies the strong energy condition if, for any future-direc ted unit timelike vec tor ξa at any point in M, ) ξa  ξb the quantity (Tab   −   1  Tgab is not negative. The strong energy c ondition c an be interpreted as the 2 requirement that a c ertain effec tive energy density is not negative. Note that, in four dimensions, the strong energy c ondition is satisfied if and only if the (timelike) convergence condition, Rab ξa ξb ≥ 0, is also satisfied. This latter c ondition c an be understood to assert that gravitation is attrac tive in nature. Finally, we say Tab satisfies the dominant energy condition if, for any future-direc ted unit timelike vec tor ξa at any point in M, the vector Tba ξb is causal and future-directed. This last condition can be interpreted as the requirement that matter c annot travel faster than light. Indeed, if Tab vanishes on some c losed, ac hronal set S ⊂ M and satisfies the dominant energy and c onservation c onditions, then Tab vanishes on all of D(s) (Hawking and Ellis 1973). Clearly, the dominant energy c ondition implies (but is not implied by) the weak energy c ondition. One c an show that being a trivial, exac t, or vac uum solution of Einstein's equation is a loc al spac etime property. In addition, being a c onstraint solution is also a loc al spac etime property if the c onstraint under c onsideration is one of the three energy conditions considered here. 3.2 Global Properties A large number of important global properties c onc ern either “c ausal struc ture” or “singularities.” Here we Page 6 of 15

Global Space Time Structure investigate them. There is a hierarc hy of c onditions relating to the c ausal struc ture of spac e-time.8 Eac h c ondition c orresponds to a global spac etime property (the property of satisfying the c ondition). We say a spac etime satisfies the chronology c ondition if it c ontains no c losed timelike c urves (equivalently, p ∉ I+(p) for all p ∈ M). A spac etime satisfies the causality c ondition if there are no c losed c ausal c urves (equivalently, J+ (p) ∩ J−(p) = {p} for all p ∈ M). As mentioned previously, causality implies chronology but the implication does not run in the other direction (see figure 16.2). The next few c onditions serve to rule out “almost” c losed c ausal c urves. We say a spac etime (M,gab) satisfies the future distinguishability c ondition if there do not exist distinc t points p,q ∈ M suc h that I+(p) = I+(q). The past distinguishability c ondition is defined analogously. One c an show that a spac etime (M,gab) satisfies the future (respec tively, past) distinguishability c ondition if and only if, for all points p ∈ M and every open set O c ontaining p, there is an open set V ⊂ O also c ontaining p suc h that no future (respec tively, past) direc ted c ausal c urve that starts at p and leaves V ever returns to V. We say a spac etime satisfies the distinguishability c ondition if it satisfies both the past and future distinguishability c onditions. Future or past distinguishability implies c ausality. But the c onverse is not true. Of c ourse, distinguishability implies past (or future) distinguishability. But one c an c ertainly find spac etimes that satisfy future (respec tively, past) distinguishability but not past (respec tively, future) distinguishability (Hawking and Ellis 1973). Consider two distinguishing spac etimes (M,gab) and (Mʹ,gʹab) and a bijec tion φ : M → Mʹ suc h that for all p,q ∈ M, p ∈ I+(q) if and only if φ (p) ∈ I+(φ (q)). One c an show (Malament 1977) that φ is a diffeomorphism and φ*(gab) = ω2 gʹab for some c onformal fac tor ω: Mʹ → ℝ. Thus, if the c ausal struc ture of spac etime is suffic iently well-behaved, that struc ture alone determines the shape of the universe, as well as the metric struc ture up to a c onformal fac tor. We say a spac etime satisfies the strong causality c ondition if, for all points p ∈ M and every open set O c ontaining p, there is an open set V ⊂ O also c ontaining p suc h that no c ausal c urve intersec ts V more than onc e. If a spac etime (M,gab) satisfies strong c ausality, then, for every c ompac t set K ⊂ M, a c ausal c urve γ: I → K must have future and past endpoints in K. Thus, in a strongly c ausal spac etime, an inextendible c ausal c urve c annot be “imprisoned” in a c ompac t set. Clearly, strong c ausality implies distinguishability. One c an show that the implic ation does not run in the other direc tion (Hawking and Ellis 1973). Figure 16.6 Cylindrical Minkowski spacetime with three horizontal lines removed as shown. The spacetime is strongly c ausal but not stably c ausal. A spac etime (M,gab) satisfies the stable causality c ondition if there is a timelike vec tor field ξa on M suc h that the spac etime (M, gab + ξa ξb) satisfies the c hronology c ondition. Physic ally, even if the light c ones are “opened” by a small amount at eac h point, the spac etime remains free of c losed timelike c urves. We say a spac etime (M,gab) admits a global time function if there is a smooth func tion t: M → ℝ suc h that, for any distinc t points p,q ∈ M, if p ∈ J+(q), then t(p) 〉 t(q). The func tion assigns a “time” to every point in M suc h that it inc reases along every (nontrivial) future-direc ted c ausal c urve. An important result is that a spac etime admits a global time func tion if and only if it satisfies stable c ausality (Hawking 1969). One c an also show that stable c ausality implies strong c ausality but the c onverse is false (see figure 16.6). The remaining c ausality c onditions not only require that there be no almost c losed c ausal c urves but, in addition, that there be limitations on the kinds of “gaps” in spac etime (Hawking and Sac hs 1974). We say a spac etime (M,gab) satisfies the causal continuity c ondition if it satisfies distinguishability and, for all p, q ∈ M, I+ (p) ⊆ I+(q) if and only if I−(q) ⊆ I−(p). Physic ally, c ausal c ontinuity ensures that points that are c lose to one another do not have wildly different timelike futures and pasts. One c an show that c ausal c ontinuity implies stable Page 7 of 15

Global Space Time Structure c ausality. The c onverse is not true. A c ounterexample c an be c onstruc ted by taking Minkowski spac etime and exc ising from the manifold a c ompac t set with nonempty interior. The resulting spac etime satisfies stable c ausality but not c ausally c ontinuity. A spac etime (M,gab) satisfies the causal simplicity c ondition if it satisfies distinguishability and, in addition, for all p ∈ M, the sets J+ (p) and J−(p) are c losed. One c an show that c ausal simplic ity implies c ausal c ontinuity. The c onverse is false, sinc e Minkowski spac etime with a point removed from the manifold satisfies c ausal c ontinuity but not c ausal simplic ity. Finally, we say a spac etime (M,gab) satisfies global hyperbolicity if it satisfies strong c ausality and, in addition, for all p,q ∈ M, the set J+ (p) ⊂ J−(q) is c ompac t. A fundamental result is that a spac etime satisfies global hyperbolic ity if and only if it admits a Cauc hy surfac e (Geroc h 1970b). In addition, one c an show that the manifold of any spac etime that satisfies global hyperbolic ity must have the topology of ℝ × ∑ for any Cauc hy surfac e ∑. Global hyperbolic ity implies c ausal simplic ity but the c onverse is not true. Anti-de Sitter spac etime is one c ounterexample (Hawking and Ellis 1973). In sum, we have the following implic ations (none of whic h run in the other direc tion): global hyperbolic ity → c ausal simplic ity → c ausal c ontinuity → stable c ausality → strong c ausality → distinguishability → future (or past) distinguishability → c ausality → c hronology. There are a number of senses in whic h a spac etime may be said to c ontain a “singularity.”9 Here, we restric t attention to the most important one: geodesic inc ompleteness. We say a geodesic γ: I → M is incomplete if it is maximal and suc h that I ≠ ℝ. We say a future-direc ted maximal timelike or null geodesic γ: I → M is future incomplete (respec tively, past incomplete) if there is a r ∈ ℝ suc h that r 〉 S for all s ∈ I. A past incomplete geodesic is defined analogously. Naturally, a spac etime is timelike geodesically incomplete if it c ontains a timelike inc omplete geodesic . In a timelike geodesic ally inc omplete spac etime, it is possible for a nonac c elerating massive partic le to experienc e only a finite amount of time. We c an define spacelike and null geodesic incompleteness analogously. Finally, we say that a spac etime is geodesically incomplete if it is either timelike, spac elike, or null geodesic ally inc omplete. If a spac etime has an extension, it is geodesic ally inc omplete. The c onverse is false. Consider Minkowski spac etime (M,gab) and let Mʹ be the manifold M − {p} for any p ∈ M. Let ω: Mʹ → ℝ be a c onformal fac tor that approac hes z ero as the missing point p is approac hed. The resulting spac etime (Mʹ, ωgab|Mʹ) is maximal but c ontains timelike, spac elike, and null inc omplete geodesic s. Other maximal spac etimes exist whic h are geodesic ally inc omplete and have a flat metric .10 In other words, one c an have singularities without any spac etime c urvature at all. Sinc e there are c ertainly flat spac etimes that are geodesic ally c omplete (e.g., Minkowski spac etime), it follows that geodesic inc ompleteness is a global property. We mention in passing that the property of being maximal is also global. Finally, one c an show that timelike, spac elike, and null inc ompleteness are independent c onditions in the sense that there are spac etimes that are inc omplete in any one of the three types and c omplete in the other two (Geroc h 1968). Additionally, one c an show that c ompac t spac etimes are not nec essarily geodesic ally c omplete (Misner 1963). These two results suggest that geodesic inc ompleteness fails to mesh c ompletely with our notion of a “hole” in spac etime. 4. Which Pro perties are Reaso nable? So far, we have provided examples of a number of spac etime properties. In this sec tion, we ask: Whic h properties are “physic ally reasonable”? It is usually taken for granted that “the normal physic al laws we determine in our spac etime vic inity are applic able at all other spac etime points” (Ellis 1975, 246). This assumption allows us to stipulate that the loc al property of being a solution to Einstein's equation is a physic ally reasonable one. And often this means that we take the energy c onditions as nec essarily satisfied. However, some have argued that even the energy c onditions c an be violated in some physic ally reasonable spac etimes (Vollic k 1997). Page 8 of 15

Global Space Time Structure One global property that is usually taken to be physic ally reasonable is that spac etime be maximal. Metaphysic al c onsiderations seem to drive the assumption. One asks (Geroc h 1970a, 262), “Why, after all, would Nature stop building our universe … when She c ould just as well have c arried on?” Of c ourse, suc h reasoning c an be questioned (Earman 1995, Norton (2011)). What about the global properties concerning singularities and causal structure? Which of them are to be c onsidered physic ally reasonable? 4.1 Singularities Muc h of the work in global struc ture has c onc erned singularities. The task has been to show, using fairly c onservative assumptions, that all physic ally reasonable spac etimes must be (null or timelike) geodesic ally inc omplete. The projec t has produc ed a number of theorems of this type. Here, we examine an influential one due to Hawking and Penrose (1970). Three preliminary c onditions are c ruc ial and eac h has been taken to be satisfied by all (or almost all) physic ally reasonable spac etimes. We shall temporarily adopt these bac kground assumptions in what follows. The first is c hronology (no c losed timelike c urves). The sec ond is the c onvergenc e c ondition (Rab ξa ξa ≥ 0 for all unit timelike vec tors ξa). Rec all that the c onvergenc e c ondition is satisfied in four dimensions if and only if the strong energy c ondition is. In this sec tion, we will restric t attention to four-dimensional spac etimes. The third is the generic condition—that eac h c ausal geodesic with tangent ξ a c ontains a point at whic h ξ[aRb]c d[e ξf ] ξc ξd ≠ 0. Physic ally, the generic c ondition requires that somewhere along eac h c ausal c urve a c ertain effec tive c urvature is enc ountered. Although highly symmetric spac e-times may not satisfy the generic c ondition (e.g., Minkowski spac etime), it is thought to be satisfied by all suffic iently “generic ” ones. Now, c onsider the following statement. (S) Any spac etime that satisfies c hronology, the c onvergenc e c ondition, the generic c ondition, and, must be timelike or null geodesic ally inc omplete. We seek to fill in the blank with physic ally reasonable “boundary” c onditions that make (S) true. Hawking and Penrose (1970) considered three of them (see also Earman 1999). First, if the boundary c ondition is the requirement that there exist a c ompac t slic e, (S) is true. So, a “spatially c losed” universe is singular if it is physic ally reasonable. One c an show that the existenc e of a c ompac t slic e is a nec essary c ondition for predic ting future spac etime events (Manc hak 2008). Thus, we have the somewhat c ounterintuitive result that predic tion is possible in a physic ally reasonable spac etime only if singularities are present.11 Sec ond, (S) is true if the boundary c ondition is the requirement that there exist a trapped surfac e. A trapped surface is a two-dimensional c ompac t spac elike surfac e T suc h that both sets of “ingoing” and “outgoing” future- direc ted null geodesic s orthogonal to T have negative expansion at T.12 Physic ally, whenever a suffic iently large amount of matter is c ontained in a small enough region of spac etime, a trapped surfac e forms (Sc hoen and Yau 1983). Third, (S) is true if the boundary c ondition is the requirement that there is a point p ∈ M suc h that the expansion along every future (or past) direc ted null geodesic through p is somewhere negative. Physic ally, a spac etime that satisfies this c ondition c ontains a c ontrac ting region in the c ausal future (or past) of a point. It is thought that the observable portion of our own universe c ontains suc h a region (Ellis 2007). Additional examples of boundary c onditions that make (S) true c ould be multiplied (Senovilla 1998). And instead of boundary c onditions, one c an also find c ausal c onditions that make (S) true. We mention one here. It turns out that (S) is true if the c ausal c ondition is the requirement that stable c ausality is not satisfied (Minguz z i 2009). Thus, physic ally reasonable spac etimes (whic h are assumed to be c ausally well behaved in the sense that they satisfy c hronology) are singular if they are not too c ausally well behaved. One naturally wonders if it is possible for physic ally reasonable spac etimes to avoid singularities if the c hronology c ondition is dropped. But this seems unlikely (Tipler 1977; Kriele 1990). A large number of physic ally reasonable spac etimes (inc luding our own) seem to satisfy at least one of the above Page 9 of 15

Global Space Time Structure mentioned boundary conditions and hence contain singularities. And the worry has been that these singularities c an be observed direc tly—that they are “naked” in some sense. So, one would like to show that all (or almost all) physic ally reasonable spac etimes do not c ontain naked singularities. This is the “c osmic c ensorship” hypothesis. There are a number of ways to formulate the hypothesis (Joshi 1993; Penrose 1999). Here, we c onsider one. Figure 16.7 Minkowski spacetime with one point removed is nakedly singular. The future incomplete geodesic γ , c ontained in the timelike past of p, approac hes the missing point. We do not wish to c ount a “big bang” singularity as naked and therefore restric t attention to future (rather than past) inc omplete timelike or null geodesic s. We say a spac etime (M, gab) is nakedly singular if there is a point p ∈ M and a future inc omplete timelike or null geodesic γ: I → M suc h that the range of γ is c ontained in I−(p) (see figure 16.7). One c an show that a nakedly singular spac etime does not admit a Cauc hy surfac e (Geroc h and Horowitz 1979). Thus, if all physic ally reasonable spac etimes are globally hyperbolic , then the c osmic c ensorship hypothesis is true. And Penrose (1969, 1979) has suggested that one might be able to show the antec edent of this c onditional. The idea would be to show that spac etimes that fail to be globally hyperbolic are unstable under c ertain types of perturbations. However, suc h a c laim is diffic ult to express prec isely (Geroc h 1971). And although some evidenc e does seem to indic ate that instabilities are present in nonglobally hyperbolic spac etimes (Chandrasekhar and Hartle 1982), still other evidenc e suggests otherwise (Morris, Thorne, and Yurtsever 1988). There is also an epistemological predicament at issue. An observer never can have the evidential resources to rule out the possibility that his or her spac etime is not globally hyperbolic —even under any assumptions c onc erning loc al spac etime struc ture (Manc hak 2011b). And how c ould we ever know that all physic ally reasonable spac etimes are globally hyperbolic if we c annot even be c onfident that our own spac etime is? 4.2 Time Travel If the c osmic c ensorship hypothesis is false, there are physic ally reasonable spac e-times that do not satisfy global hyperbolic ity Might there be some physic ally reasonable spac etimes that do not even satisfy c hronology? We investigate the question here. One way to rule out a number of chronology-violating spacetimes concerns self-consistency constraints on matter fields of various types. Here, we examine sourc e free Klein-Gordon fields. Let (M, gab) be a spac etime. We say an open set U ⊂ M is causally regular if, for every func tion φ : U → ℝ whic h satisfies ∇a ∇a φ = 0, there is a func tion φ ʹ: M → ℝ suc h that ∇a ∇a φ ʹ = 0 and φ ʹ.| U = φ . We say (M,gab) is causally benign if, for every p ∈ M and every open set U c ontaining p, there is an open set Uʹ ⊂ U c ontaining p whic h is c ausally regular. It has been argued that a spac etime that is not c ausally benign is not physic ally reasonable. We c ertainly know that every globally hyperbolic spac etime is c ausally benign. But although some c hronology violating spac etimes are not c ausally benign, a number of others are (Yurtsever 1990; Friedman 2004). Given the existence of causally benign yet chronology violating spacetimes, another area of research seems fruitful to pursue. One wonders if c hronology violating region c an, in some sense, be “c reated” by rearranging the distribution and flow of matter (Stein 1970). In other words, c an a physic ally reasonable spac etime c ontain a “time mac hine” of sorts? Here, we examine one way of formaliz ing the question given by Earman, Smeenk, and Wüthric h 13 Page 10 of 15

Global Space Time Structure (2009).13 First, in order to c ount as a time mac hine, a spac etime (M, gab) must c ontain a spac elike slic e S ⊂ M representing a “time” before the time mac hine is switc hed on. Sec ond, the spac etime must also have a c hronology violating region V after the mac hine is turned on. So we require V ⊂ J+[S]. Finally, in order to c apture the idea that a time mac hine must “c reate” a c hronology violating region, every physic ally reasonable maximal extension of int(D(s)) must c ontain a c hronology violating region Vʹ .14 Consider the following statement. (T) There is a spac etime (M, gab) with a spac elike slic e S ⊂ M and a c hronology violating region V ⊂ J+[S] suc h that every maximal extension of int(D(s)) whic h satisfies c ontains some c hronology violating region Vʹ. We seek to fill in the blank with physic ally reasonable “potenc y” c onditions that make (T) true. And we know from c ounterexamples c onstruc ted by Krasnikov (2002) that (T) will be false unless there is a potenc y c ondition and this c ondition limits spac etime “holes” in some sense. But Hawking (1992) has suggested that limiting holes may not be enough. Indeed, he c onjec tured that all physic ally reasonable spac etimes are “protec ted” from c hronology violations and provided some evidenc e for the c laim. We say H+(s) is compactly generated if all past direc ted null geodesic s through H+(s) enter and remain in some c ompac t set. Any spac etime with a slic e S suc h that H+(s) is nonempty and c ompac tly generated does not satisfy strong c ausality. And Hawking showed there is no spac etime that satisfies the weak energy c ondition whic h has a nonc ompac t slic e S suc h that H+(s) is nonempty and c ompac tly generated. Figure 16.8 Misner spacetime. Every maximal, hole-free extension of int(D(s)) (the region below the dotted line) c ontains some c hronology violating region. But some have argued that insisting on a c ompac tly generated Cauc hy horiz on rules out some physic ally reasonable spac etimes (Ori 1993; Krasnikov 1999). And, of c ourse, a slic e S need not be nonc ompac t to be physic ally reasonable. Thus, Hawking's c hronology protec tion c onjec ture remains an open question. Are there any potenc y c onditions that make (T) true? We say a spac etime (M,gab) is hole-free if, for any spac elike surfac e S in M there is no isometric embedding θ : D(S) → Mʹ into another spac etime (Mʹ, gʹ) suc h that θ(D(s)) ≠ D(θ(S)). Physic ally, hole-freeness ensures that, for any spac elike surfac e S, the domain of dependenc e D(s) is “as large as it c an be.” And one c an show that any spac etime with one point removed from the underlying manifold fails to be hole-free. It has been argued that all physic ally reasonable spac etimes are hole-free (Clarke 1976; Geroc h 1977). And it turns out that (T) is true if the potenc y c ondition is hole-freeness (Manc hak 2009b). The two- dimensional spac etime of Misner (1967) c an be used to prove the result (see figure 16.8). However, hole-freeness may not be a physic ally reasonable potenc y c ondition after all. Indeed, some maximal, globally hyperbolic models, inc luding Minkowski spac etime, are not hole-free (Manc hak 2009a; Krasnikov 2009). But, another more reasonable “no holes” potenc y c ondition c an be used to make (T) true: the demand that, for all p ∈ M, J+(p) and J−(p) are c losed (Manc hak 2011a). Call this c ondition causal closedness and rec all that c ausal c losedness is used, along with distinguishability, to define c ausal simplic ity. Not only is c ausal c losedness satisfied by all globally hyperbolic models, inc luding Minkowski spac etime, but it is also satisfied by many c hronology violating spac etimes as well (e.g., Gödel spac etime, Misner spac etime). In this sense, then, it is a more appropriate c ondition than hole-freeness. But is c ausal c losedness satisfied by all Page 11 of 15

Global Space Time Structure physic ally reasonable spac etimes? The question is open. So too is the question of whic h other potenc y c onditions make (T) true. 5. Co nclusio n Here, we have outlined the basic struc ture of relativistic spac etime. As we have seen, general relativity allows for a wide variety of global spac etime properties—some of them quite unusual. And one wonders whic h of these properties are physically reasonable. Early work foc used on singularities. Initially, a number of results established that all physic ally reasonable spac etimes are geodesic ally inc omplete. Next, the relationship between these singularities and determinism was investigated: Can a physic ally reasonable (and therefore geodesic ally inc omplete) spac etime fail to be globally hyperbolic ? The question remains open. Rec ently, foc us has shifted somewhat toward ac ausality: Can physic ally reasonable spac etimes c ontain c losed timelike c urves? If so, c an these c losed timelike c urves be “c reated” in some sense by rearranging the distribution and flow of matter? Again, these questions remain open. References Carter, B. (1971). Causal struc ture in spac etime. General Relativity and Gravitation 1: 349– 391. Chandrasekhar, S., and J. Hartle (1982). On c rossing the Cauc hy horiz on of a Reissner–Nordström blac k–hole. Proceedings of the Royal Society (London) A. 348: 301–315. Choquet– Bruhat, Y., and R. Geroc h (1969). Global aspec ts of the Cauc hy problem in general relativity. Communications in Mathematical Physics 14: 329–335. Clarke, C. (1976). Spac etime singularities. Communications in Mathematical Physics. 49: 17– 23. ——— (1993). The analysis of spacetime singularities. Cambridge: Cambridge University Press. Curiel, E. (1999). The analysis of singular spac etimes. Philosophy of Science. 66: S119– S145. Earman, J. (1995). Bangs, crunches, whimpers, and shrieks: Singularities and acausalities in relativistic spacetimes. Oxford: Oxford University Press. ——— (1999). The Penrose– Hawking singularity theorems: History and implic ations. In The expanding worlds of general relativity, Einstein Studies, ed: H. Goerher, J. Renn, and T. Sauer, Vol. 7, Boston: Birkhäuser. 235–267. ——— (2001). Lambda: The c onstant that refuses to die, Archives for History of Exact Sciences, 51: 189– 220. ——— (2002). What time reversal invarianc e is and why it matters, International Journal for the Philosophy of Science, 16: 245–264. ——— (2008). Pruning some branc hes from ‘Branc hing Spac etimes’, in The ontology of spacetime II, ed. D. Dieks, 187–205, Amsterdam: Elsevier. Earman, J., and C. Wüthric h (2010). Time Mac hines. In Stanford Encyclopedia of Philosophy , ed. E. Zalta. http://plato .stanfo rd.edu/entries/time- machine/. Earman, J., C. Smeenk, and C. Wüthric h, (2009). Do the laws of physic s forbid the operation of time mac hines? Synthese 169: 91–124. Ellis, G. (1975). Cosmology and verifiability. Quarterly Journal of the Royal Astronomical Society, 16: 245–264. ——— (2007). Issues in the philosophy of c osmology. In Handbook of the philosophy of physics , ed. J. Butterfield and J. Earman, 1183–1286. Oxford: Elsevier. Page 12 of 15

Global Space Time Structure Ellis, G., and B. Sc hmidt (1977). Singular spac etimes. General Relativity and Gravitation 8: 915– 953. Friedman, J. (2004). The Cauc hy problem on spac etimes that are not globally hyperbolic . In The Einstein equations and the large scale behavior of gravitational fields, ed. P. Chrusc iel and H. Friedric h, 331– 346. Boston: Birkhäuser, Geroc h, R. (1967). Topology in general relativity. Journal of Mathematical Physics 8: 782– 786. ——— (1968). What is a singularity in general relativity? Annals of Physics 48: 526–540. ——— (1970a). Singularities. In Relativity, ed. M. Carmeli, S. Fic kler, and L. Witten, 259– 291. New York: Plenum Press. ——— (1970b). Domain of dependenc e., Journal of Mathematical Physics 11: 437– 449. ——— (1971). General relativity in the large. General Relativity and Gravitation. 2: 61–74. ——— (1977). Predic tion in general relativity. In Foundations of spacetime theories, Minnesota Studies in the Philosophy of Sc ienc e Vol. 8, ed. J. Earman, C. Glymour, and J. Stac hel, 81– 93. Minneapolis: University of Minnesota Press. Geroc h, R., and G. Horowitz (1979). Global struc ture of spac etimes. In General Relativity: An Einstein Centenary Survey, ed. S. W. Hawking and W. Israel, 212–293. Cambridge: Cambridge University Press. Geroc h, R., C. Liang, and R. Wald (1982). Singular boundaries of spac etime. Journal of Mathematical Physics . 23: 432–435. Gödel, K. (1949). An example of a new type of c osmologic al solutions of Einstein's field equations of gravitation. Reviews of Modern Physics 21: 447–450. Hawking, S. (1969). The existenc e of c osmic time func tions. Proceedings of the Royal Society A 308: 433– 435. ——— (1992). The c hronology protec tion c onjec ture. Physical Review D 46: 603– 611. Hawking, S., and G. Ellis (1973). The large scale structure of spacetime. Cambridge: Cambridge University Press. Hawking, S., and R. Penrose, (1970). The singularities of gravitational c ollapse and c osmology. Proceedings of the Royal Society of London A 314: 529–548. Hawking, S., and B. Sac hs (1974). Causally c ontinuous spac etimes. Communications in Mathematical Physics 35: 287–296. Hogarth, M. (1997). A remark c onc erning predic tion and spac etime singularities. Studies in History and Philosophy of Modern Physics 28: 63–71. Joshi, P. (1993). Global aspects in gravitation and cosmology. Oxford: Oxford University Press. Krasnikov, S. (1999). Time mac hines with non-c ompac tly generated c auc hy horiz ons and “Handy Singularities”. In Proceedings of the Eighth Marcel Grossmann Meeting on General Relativity, ed. T. Piran and R. Ruffini, 593–595. Singapore: World Scientific. ——— (2002). No time mac hines in c lassic al general relativity. Classical and Quantum Gravity 19: 4109– 4129. ——— (2009). Even the minkowski spac e is holed. Physical Review D 79: 124041. Kriele, M. (1990). Causality violations and singularities. General Relativity and Gravitation, 22: 619–623. Malament, D. (1977). The c lass of c ontinuous timelike c urves determines the topology of spac etime. Journal of Mathematical Physics 18: 1399–1404. ——— (2012). Topics in the foundations of general relativity and Newtonian gravitation theory . Chic ago: University of Chic ago Press. Page 13 of 15

Global Space Time Structure Manc hak, J. (2008). Is predic tion possible in general relativity? Foundations of Physics, 38: 317– 321. ——— (2009a). Is spac etime hole-free? General Relativity and Gravitation 41: 1639– 1643. ——— (2009b). On the existenc e of “time mac hines” in general relativity. Philosophy of Science 76: 1020– 1026. ——— (2011a). No no-go: A remark on time mac hines. Studies in History and Philosophy of Modern Physics . 42: 74–76. ——— (2011b). What is a physic ally reasonable spac etime? Philosophy of Science 78: 410– 420. Minguz z i, E. (2009). Chronologic al spac etimes without lightlike lines are stably c ausal. Communications in Mathematical Physics 288: 801–819. Misner, C. (1963). The flatter regions of Newman, Unti, and Tamburino's generaliz ed Sc hwarz sc hild spac e. Journal of Mathematical Physics 4: 924–937. ——— (1967). Taub– NUT spac e as a c ounterexample to almost anything. In Relativity theory and astrophysics I: relativity and cosmology, ed. J. Ehlers, 160– 169. Providenc e: Americ an Mathematic al Soc iety. Morris, M., K. Thorne, and U. Yurtsever, (1988). Wormholes, time mac hines, and the weak energy c ondition. Physical Review Letters 61: 1446–1449. Norton, J. (2011), “Observationally indistinguishable Spac etimes: A c hallenge for any induc tivist,” in G. Morgan, Philosophy of Sc ienc e Matters. Oxford: Oxford University Press, p. 164– 176. Ori, A. (1993). Must time-mac hine c onstruc tion violate the weak energy c ondition? Physical Review Letters 71: 2517–2520. Penrose, R. (1969). Gravitational c ollapse: The role of general relativity. Revisita del Nuovo Cimento, Serie I, 1: 252–276. ——— (1979). Singularities and time-asymmetry. In General Relativity: An Einstein Centenary Survey, ed. S. Hawking and W. Israel, 581–638. Cambridge: Cambridge University Press. ——— (1999). The question of c osmic c ensorship. Journal of Astrophysics and Astronomy 20: 233– 248. Senovilla, J. (1998). Singularity theorems and their c onsequenc es. General Relativity and Gravitation 30: 701– 848. Sc hoen, R., and S. Yau (1983). The existenc e of a blac k hole due to c ondensation of matter. Communications in Mathematical Physics 90: 575–579. Smeenk, C., and C. Wüthric h (2011). Time travel and time mac hines. In The Oxford Handbook of Time, ed. C. Callender, 577–630. Oxford: Oxford University Press. Stein, H. (1970). On the paradoxic al time-struc tures of Gödel. Philosophy of Science 37: 589– 601. Tipler, F. (1977). Singularities and c ausality violations. Annals of Physics 108: 1– 36. Vollic k, D. (1997). How to produc e exotic matter using c lassic al fields. Physical Review D 56: 4720– 4723. Wald, R. (1984). General relativity. Chic ago: University of Chic ago Press. Yurtsever, U. (1990). Test fields on c ompac t spac etimes. Journal of Mathematical Physics 31: 3064– 3078. Notes: (1) In what follows, the reader is enc ouraged to c onsult Hawking and Ellis (1973), Geroc h and Horowitz (1979), Wald (1984), Joshi (1993), and Malament (2012). Page 14 of 15

Global Space Time Structure (2) See Earman (2008) for a discussion of this condition. (3) See Earman (2002) for a discussion of this condition. (4) Minkowski spac etime (M,gab) is suc h that M = ℝn , gab is flat, and there exist no inc omplete geodesic s (defined below). See Hawking and Ellis (1973). (5) In what follows, for any set S, the sets S¯¯, S˙, and int(S) denote the closure, boundary, and interior of S, respec tively (6) In what follows, square brac kets denote anti-symmetriz ation. Parentheses denote symmetriz ation. See Malament (2012). (7) Here, we drop the c osmologic al c onstant term −λgab sometimes added to the left side of the equation for some λ ∈ ℝ. For more on this term, see Earman (2001). (8) Although we only consider a small handful here, there are an infinite number of conditions in the causal hierarc hy (Carter 1971). (9) See Ellis and Schmidt (1977), Geroch, Liang, and Wald (1982), Clarke (1993), and Curiel (1999) for details. (10) Here is one example. Remove a point from ℝ2 and take the universal c overing spac e. Let the resulting spac etime manifold have a flat metric . (11) For a related discussion, see Hogarth (1997). (12) The (sc alar) expansion of a c ongruenc e of null geodesic s is a bit c omplic ated to define (see Wald 1984). But one c an get some idea of the quantity by noting that the expansion of a c ongruenc e of timelike geodesic s with unit tangent field ξa is ∇aξa. (13) See also Earman and Wüthric h (2010) and Smeenk and Wüthric h (2011). (14) Here we abuse the notation somewhat. Properly, we require that every physically reasonable maximal extension of (int(D(s)),gab|int(D(S))) must c ontain a c hronology violating region Vʹ. John Byron Manchak J ohn Manchak is an Assistant Professor of Philosophy at the University of Washington. His prim ary research interestes are in Philosophy of Physics and Philosophy of Science. His research has focused on foundational issues in general relativity.

Philosophy of Cosmology Chris Smeenk The Oxford Handbook of Philosophy of Physics Edited by Robert Batterman Abstract and Keywords This c hapter addresses philosophic al questions raised in c ontemporary work on c osmology. It provides an overview of the Standard Model for c osmology and argues that its defic ienc y in addressing theories regarding the very early universe c an be resolved by introduc ing a dynamic al phase of evolution that eliminates the need for a spec ial initial state. The c hapter also disc usses rec ent hypotheses about dark matter and energy, issues that it relates to philosophical debates about underdetermination. K ey words: cosmol ogy , S tan dard M odel , v ery earl y u n i v erse, dy n ami cal ph ase, i n i ti al state, dark matter, dark en ergy , u n derdetermi n ati on 1. Intro ductio n Cosmology has made enormous progress in the last several dec ades. It is no longer a neglec ted subfield of physic s, as it was as rec ently as 1960; it is instead an ac tive area of fundamental researc h that c an boast of a Standard Model well-supported by observations. Prior to 1965 researc h in c osmology had a strikingly philosophic al tone, with debates foc using explic itly on sc ientific method and the aims and sc ope of c osmology (see, e.g., Munitz 1962; North 1965; Kragh 1996). One might suspec t that with the maturation of the field these questions have been settled, leaving little room for philosophers to c ontribute. Although the nature of the field has c hanged dramatic ally with an inc rease of observational knowledge and theoretic al sophistic ation, there are still ongoing foundational debates regarding c osmology's proper aims and methods. Cosmology c onfronts a number of questions dear to the hearts of philosophers of sc ienc e: the limits of sc ientific explanation, the nature of physic al laws, and different types of underdetermination, for example. There is an opportunity for philosophers to make fruitful c ontributions to debates in c osmology and to c onsider the ramific ations of new ideas in c osmology for other areas of philosophy and foundations of physics. Due to the uniqueness of the universe and its inac c essibility, c osmology has often been c harac teriz ed as “unsc ientific ” or inherently more spec ulative than other parts of physic s. How c an one formulate a sc ientific theory of the “universe as a whole”? Even those who rejec t skeptic ism regarding c osmology often assert instead that c osmology c an only make progress by employing a distinc tive methodology. These disc ussions, in my view, have by and large failed to identify the sourc e and the extent of the evidential c hallenges fac ed by c osmologists. There are no c onvinc ing, general no-go arguments showing the impossibility of sec ure knowledge in c osmology; there are instead spec ific problems that arise in attempting to gain observational and theoretic al ac c ess to the universe. In some c ases, c osmologists have ac hieved knowledge as sec ure as that in other areas of physic s—arguably, for example, in the account of big bang nucleosynthesis. Cosmologists do, however, fac e a number of distinc tive c hallenges. The finitude of the speed of light, a basic feature of relativistic c osmology, insures that global properties of the universe c annot be established direc tly by observations (sec tion 5). This is a straightforward limit on observational ac c ess to the universe, but there are other Page 1 of 34

Philosophy of Cosmology obstac les of a different kind. Cosmology relies on extrapolating loc al physic al laws to hold universally. These extrapolations make it possible to infer, from observations of standard c andles suc h as Type Ia supernovae,1 the startling c onc lusion that the universe inc ludes a vast amount of dark matter and dark energy. Yet the inferenc e relies on extrapolating general relativity (GR), and the observations may reveal the need for a new gravitational theory rather than new types of matter. It is diffic ult to adjudic ate this debate due to the lac k of independent ac c ess to the phenomena (sec tion 3). The early universe (sec tion 6) is interesting bec ause it is one of the few testing grounds for quantum gravity. Without a c lear understanding of the initial state derived from suc h a theory, however, it is diffic ult to use observations to infer the dynamic s governing the earliest stages of the universe's evolution. Finally, it is not c lear how to take the selec tion effec t of our presenc e as observers into ac c ount in assessing evidence for cosmological theories (section 7). These c hallenges derive from distinc tive features of c osmology. One suc h feature is the interplay between global aspec ts of the universe and loc al dynamic al laws. The Standard Model of c osmology is based on extrapolating loc al laws to the universe as a whole. Yet, there may be global-to-loc al c onstraints. The uniqueness of the universe implies that the normal ways of thinking about laws of physic s and the c ontrast between laws and initial c onditions do not apply straightforwardly (sec tion 4). In other areas of physic s, the initial or boundary c onditions themselves are typic ally used to explain other things rather than being the target of explanation. Many lines of researc h in c ontemporary c osmology aim to explain why the initial state of the Standard Model obtained, but the nature of this explanatory projec t is not entirely c lear. And due to the uniqueness of the universe and the possibility of anthropic selec tion effec ts it is not c lear what underwrites the assignment of probabilities. What follows is not a survey of a thoroughly explored field in philosophy of physic s. There are a variety of topic s in this area that philosophers c ould fruitfully study, but as of yet the potential for philosophic al work has not been fully realiz ed.2 The leading c ontributions have c ome primarily from c osmologists who have turned to philosophic al c onsiderations arising from their work. The literature has a number of detailed disc ussions of spec ific issues, but there are few attempts at a more systematic approac h. As a result, this essay is an idiosync ratic tour of various topic s and arguments rather than a survey of a well-c harted intellec tual landsc ape. It is also a limited tour and leaves out a variety of important issues—most signific antly, the impac t of quantum mec hanic s on issues ranging from the origin of density perturbations in the early universe to the possible c onnec tions between Everettian and c osmologic al multiverses. But I hope that despite these limitations, this survey may nonetheless enc ourage other philosophers to ac tualiz e the potential for c ontributions to foundational debates within c osmology. 2. Overview o f the Standard Mo del Sinc e the early 1970s c osmology has been based on what Weinberg (1972) dubbed the “Standard Model.” This model desc ribes the universe's spac etime geometry, material c onstituents, and their dynamic al evolution. The Standard Model is based on extending loc al physic s—inc luding general relativity, quantum physic s, and statistic al physic s—to c osmologic al sc ales and to the universe as a whole. A satisfac tory c osmologic al model should be suffic iently ric h to allow one to fix basic observational relations, and to ac c ount for various striking features of the universe, suc h as the existenc e of struc tures like stars and galaxies, as c onsequenc es of the underlying dynamic s. The Standard Model describes the universe as starting from an extremely high-temperature early state (the “big bang”) and then expanding, c ooling, and developing struc tures suc h as stars and galaxies. At the largest sc ales the universe's spacetime geometry is represented by the expanding universe models of general relativity. The early universe is assumed to begin with matter and radiation in loc al thermal equilibrium, with the stress-energy dominated by photons. As the universe expands, different types of partic les “freez e out” of equilibrium, leaving an observable signature of earlier stages of evolution. Large-sc ale struc tures in the universe, suc h as galaxies and c lusters of galaxies, arise later via gravitational c lumping from initial “seeds.” Here I will give a brief sketc h of the Standard Model to provide the nec essary bac kground for the ensuing disc ussion.3 2.1 Expanding Universe Models Einstein (1917) introduc ed a strikingly new c onc eption of c osmology, as the study of exac t solutions of general relativity that desc ribe the spac etime geometry of the universe. One would expec t gravity to be the dominant forc e Page 2 of 34

Philosophy of Cosmology in shaping the universe's struc ture at large sc ales, and it is natural to look for solutions of Einstein's field equations (EFE) c ompatible with astronomic al observations. Einstein's own motivation for taking the first step in relativistic c osmology was to vindic ate Mac h's princ iple.4 He also sought a solution that desc ribes a static universe, that is, one whose spatial geometry is unc hanging. He forc ed his theory to ac c ommodate a static model by modifying his original field equations, with the addition of the infamous c osmologic al c onstant ʌ. As a result Einstein missed one of the most profound implic ations of his new theory: general relativity quite naturally implies that the universe evolves dynamic ally with time. Four of Einstein's c ontemporaries disc overed a c lass of simple evolving models, the Friedman-Lemaître-Robertson-Walker (FLRW) models, that have proven remarkably useful in representing the spac etime geometry of our universe. These models follow from symmetry assumptions that dramatic ally simplify the task of solving EFE. They require that the spac etime geometry is both homogeneous and isotropic; this is also c alled imposing the “c osmologic al princ iple.” Roughly speaking, homogeneity requires that at a given moment of c osmic time every spatial point “looks the same,” and isotropy holds if there are no geometric ally preferred spatial direc tions. These requirements imply that the models are topologic ally R, visualiz able as a “stac k” of three-dimensional spatial surfac es Σ(t) labeled by values of the c osmic time t. The worldlines of “fundamental observers,” taken to be at rest with respec t to matter, are orthogonal to these surfac es, and the c osmic time c orresponds to the proper time measured by the fundamental observers. The spatial geometry of Σ is suc h that there is an isometry c arrying any point p ∈ Σ to any other point lying in the same surfac e (homogeneity), and at any point p the three spatial direc tions are isometric (isotropy).5 The c osmologic al princ iple tightly c onstrains the properties of the surfac es Σ(t). These are three-dimensional spac es (Riemannian manifolds) of c onstant c urvature, and all of the surfac es in a given solution have the same topology. If the surfac es are simply c onnec ted, there are only three possibilities for Σ: (1) spheric al spac e, for the c ase of positive c urvature; (2) Euc lidean spac e, for z ero c urvature; and (3) hyperbolic spac e, for negative c urvature.6 Textbook treatments often neglec t to mention, however, that replac ing global isotropy and homogeneity with local analogs opens the door to a number of other possibilities. For example, there are models in whic h the surfac es Σ have finite volume but are multiply c onnec ted, c onsisting of, roughly speaking, c ells pasted together.7 Although isotropy and homogeneity hold loc ally at eac h point, above some length sc ale there would be geometric ally preferred direc tions reflec ting how the c ells are c onnec ted. In these models it is in princ iple possible to see “around the universe” and observe multiple images of a single objec t, but there is at present no strong observational evidence of such effects. Imposing global isotropy and homogeneity reduc es EFE—a set of 10 nonlinear, c oupled partial differential equations—to a pair of differential equations governing the sc ale fac tor R(t) and ρ(t), the energy density of matter. The sc ale fac tor measures the c hanging spatial distanc e between fundamental observers. The dynamic s are then c aptured by the Friedmann equation: 8 (1) and (a spec ial c ase of) the Rayc haudhuri equation: (2) Ṙ means differentiation with respec t to the c osmic time t, G is Newton's gravitational c onstant, and ʌ is the cosmological constant. The curvature of surfaces Σ(t) of constant cosmic time is given by k , where k = R2(t) −1,0,1 for negative, flat, and positive curvature (respectively). The assumed symmetries force the matter to be desc ribed as a perfec t fluid with energy density ρ and pressure p.9 The energy density and pressure are given by the equation of state for different kinds of perfec t fluids; for example, for “pressureless dust” p = 0, whereas for radiation p = ρ/3. Given a spec ific ation of the matter c ontent, there exist unique solutions for the sc ale fac tor R(t) and the energy density ρ(t) for eac h type of matter inc luded in the model. Several features of the dynamic s of these models are c lear from inspec tion of these equations. Suppose we take “ordinary” matter to always have positive total stress-energy density, in the sense of requiring that ρ + 3p 〉 0. Then, from (2), it is c lear that the effec t of suc h ordinary matter is to dec elerate c osmic expansion, R¨  ⟨ 0— Page 3 of 34

Philosophy of Cosmology reflec ting the familiar fac t that gravity is a forc e of attrac tion. But this is only so for ordinary matter. A positive c osmologic al c onstant (or matter with negative stress-energy) leads, c onversely, to ac c elerating expansion, R¨  ⟩ 0. Einstein satisfied his preferenc e for a static model by c hoosing a value of ʌ that prec isely balanc es the effect of ordinary matter, such that R¨  =  0. But his solution is unstable, in that a slight concentration (deficit) of ordinary matter triggers run-away c ontrac tion (expansion). It is diffic ult to avoid dynamic ally evolving c osmologic al models in general relativity. Restric ting c onsideration to ordinary matter and setting ʌ = 0, the solutions fall into three types depending on the relative magnitude of two terms on the right-hand side of Eq. (1), representing the effec ts of energy density and c urvature. For the c ase of flat spatial geometry k = 0, the energy density takes exac tly the value needed to c ounterac t the initial veloc ity of expansion suc h that Ṙ → 0 as t → ∞. This solution separates the two other c lasses: if the energy density is greater than c ritic al, there is suffic ient gravitational attrac tion to reverse the initial expansion, and the spatial slic es Σ have spheric al geometry (k = +1); if the energy density is less than c ritic al, the sign of Ṙ never c hanges, expansion never stops, and the spatial slic es have hyperbolic geometry (k = −1).10 This simple pic ture does not hold if ʌ ≠ 0, as the behavior then depends on the relative magnitude of the c osmologic al c onstant term and ordinary matter. The equations above lead to simple solutions for R(t) for models inc luding a single type of matter: for elec tromagnetic radiation, R(t) ∝ t1/2 ; for pressureless dust R(t) ∝ t2 /3 ; and for a c osmologic al c onstant, R(t) ∝ et.11 Obviously, more realistic models inc lude several types of matter. The energy density for different types of matter dilutes with expansion at different rates: pressureless dust—ρ(t) ∝ R−3 ; radiation—ρ(t) ∝ R−4 ; and a c osmologic al c onstant remains c onstant. As a result of these different dilution rates, a c omplic ated model c an be treated in terms of a sequenc e of simple models desc ribing the effec ts of the dominant type of matter on c osmic evolution. At t ≈ 1 sec ond, the Standard Model desc ribes the universe as filled with matter and radiation, where the latter initially has muc h higher energy density. Bec ause the energy density of radiation dilutes more rapidly than that of matter, the initial radiation-dominated phase is followed by a matter-dominated phase that extends until the present. Current observations indicate the presence of “dark energy” (discussed in more detail below) with properties like a ʌ term. Supposing these are c orrec t, in the future the universe will eventually transition to a dark- energy-dominated phase of exponential expansion, given that the energy density of a ʌ term does not dilute at all with expansion. FLRW models with ordinary matter have a singularity at a finite time in the past. Extrapolating bac k in time, given that the universe is c urrently expanding, Eq. (2) implies that the expansion began at some finite time in the past. The c urrent rate of expansion is given by the Hubble parameter, H  =   R˙ . Simply extrapolating this expansion rate R bac kward, R(t) → 0 at the Hubble time H−1 ; from Eq. (2) the expansion rate must inc rease at earlier times, so R(t) → 0 at a time less than the Hubble time before now. As this “big bang” is approac hed, the energy density and c urvature inc rease without bound. This reflec ts the instability of evolution governed by EFE: as R(t) dec reases, the energy density and pressure both inc rease, and they both appear with the same sign on the right-hand side of Eq. (2). It was initially hoped that the singularity c ould be avoided in more realistic models that are not perfec tly homogeneous and isotropic , but Penrose, Hawking, and Geroc h showed in the 1960s that singularities hold quite generic ally in models suitable for c osmology. It is essential for this line of argument that the model inc ludes ordinary matter and no c osmologic al c onstant; sinc e the ʌ term appears in Eq. (2) with the opposite sign, one c an avoid the initial singularity by inc luding a c osmologic al c onstant (or matter with a negative stress-energy). One of the most remarkable disc overies in twentieth-c entury astronomy was Hubble's (1929) observation that the red-shifts of spec tral lines in galaxies inc rease linearly with their distanc e.12 Hubble took this to show that the universe is expanding uniformly, and this effec t c an be given a straightforward qualitative explanation in the FLRW models. The FLRW models predic t a c hange in frequenc y of light from distant objec ts that depends direc tly on R(t).13 There is an approximately linear relationship between red-shift and distanc e at small sc ales for all the FLRW models, and departures from linearity at larger sc ales c an be used to measure spatial c urvature. At the length sc ales of galaxies and c lusters of galaxies, the universe is anything but homogeneous and isotropic , and the use of the FLRW models involves a (usually implic it) c laim that above some length sc ale the average matter distribution is suffic iently uniform. By hypothesis the models do not desc ribe the formation and evolution of inhomogeneities that give rise to galaxies and other struc tures. Prior to 1965, the use of the models was typic ally Page 4 of 34

Philosophy of Cosmology justified on the grounds of mathematic al utility or an argument in favor of the c osmologic al princ iple, with no expec tation that the models were in more than qualitative agreement with observations— espec ially when extrapolated to early times. The situation c hanged dramatic ally with the disc overy that the FLRW models provide an extremely ac c urate desc ription of the early universe, as revealed by the uniformity of the c osmic bac kground radiation (CBR, desc ribed below). The need to explain why the universe is so strikingly symmetric was a driving forc e for researc h in early universe c osmology (see sec tion 6 below). 2.2 Thermal History Alvy Singer's mother in Annie Hall is right: Brooklyn is not expanding. But this is not bec ause the c osmic expansion is not real or has no physic al effec ts. Rather, in the c ase of gravitationally bound systems suc h as the Earth or the solar system the effec ts of c osmic expansion are far, far too small to detec t.14 In many domains the c osmologic al expansion c an be ignored. The dynamic al effec ts of expansion are, however, the c entral theme in the Standard Model's ac c ount of the thermal history of the early universe. Consider a given volume of the universe at an early time, filled with matter and radiation assumed to be initially in loc al thermal equilibrium.15 The dynamic al effec ts of the evolution of R(t) are loc ally the same as slowly varying the volume of this region, imagining that the matter and radiation are enc losed in a box that expands (or c ontrac ts) adiabatic ally. For some stages of evolution the c ontents of the box interac t on a suffic iently short timesc ale that equilibrium is maintained through the change of volume, which then approximates a quasi-static process. When the interac tion timesc ale bec omes greater than the expansion timesc ale, however, the volume c hanges too fast for the interac tion to maintain equilibrium. This leads to a departure from equilibrium; partic le spec ies “freez e out” and dec ouple, and entropy inc reases. Without a series of departures from equilibrium, c osmology would be a boring subjec t—the system would remain in equilibrium with a state determined solely by the temperature, without a trac e of things past. Departures from equilibrium are of central importance in understanding the universe's thermal history.16 Two partic ularly important c ases are big bang nuc leosynthesis and the dec oupling of radiation from matter. The Standard Model desc ribes the synthesis of light elements as oc c urring during a burst of nuc lear interac tions that transpire as the universe falls from a temperature of roughly 109 K, at t ≍ 3 minutes, to 108 K, at about 20 minutes.17 Prior to this interval, any deuterium formed by c ombining protons and neutrons is photodissoc iated before heavier nuc lei c an build up, whereas after this interval, the temperature is too low to overc ome the Coulumb barriers between the colliding nuclei. But during this interval the deuterium nuclei exist long enough to serve as seeds for formation of heavier nuc lei bec ause they c an c apture other nuc leons. Calc ulating the primordial abundanc es of light elements starts from an initial “soup” at t ≍ 1 sec ond, inc luding neutrons, protons, elec trons, and photons in loc al thermal equilibrium.18 Given experimentally measured values of the relevant reac tion rates, one c an c alc ulate the c hange in relative abundanc es of these c onstituents and the appearanc e of nuc lei of the light elements. The result of these c alc ulations is a predic tion of light-element abundanc es that depends on physic al features of the universe at this time, suc h as the total density of baryonic matter and the baryon to photon ratio. Observations of primordial element abundanc es c an then be taken as c onstraining the c osmologic al model's parameters. Although there are still disc repanc ies (notably regarding Lithium 7) whose signific anc e is unc lear, the values of the parameters inferred from primordial abundances in conjunction with nucleosynthesis calculations are in rough agreement with values determined from other types of observations. As the temperature drops below ≍ 4,000K, “re-c ombination” oc c urs as the elec trons bec ome bound in stable atoms.19 As a result, the rate of one of the reac tions keeping the photons and matter in equilibrium (Compton sc attering of photons off elec trons) drops below the expansion rate. The photons dec ouple from the matter with a blac k-body spec trum. After dec oupling, the photons c ool adiabatic ally with the expansion, and the temperature drops as T ∝ 1/R, but the blac k-body spec trum is unaffec ted. This “c osmic bac kground radiation” (CBR) c arries an enormous amount of information regarding the universe at the time of dec oupling. It is diffic ult to provide a natural, alternative explanation for the blac k-body spec trum of this radiation.2 0 The initial detec tion of the CBR and subsequent measurements of its properties played a c ruc ial role in c onvinc ing physic ists to trust the extrapolations of physic s to these early times, and ever sinc e its disc overy, the CBR has been a target for inc reasingly sophistic ated observational programs. These observations established that the CBR has a uniform temperature to within 1 part in 105, and the minute fluc tuations in temperature provide empiric al Page 5 of 34

Philosophy of Cosmology guidanc e for the development of early universe theories. In c losing, two aspec ts of the ac c ounts of the thermal history deserve emphasis. First, the physic s used in developing these ideas has independent empirical credentials. Although the very idea of early universe cosmology was regarded as spec ulative when c alc ulations of this sort were first performed (Alpher, Bethe, and Gamow 1948), the basic nuc lear physic s was not. Sec ond, treating the c onstituents of the early universe as being in loc al thermal equilibrium before things get interesting is justified provided that the reaction rates are higher than the expansion rate at earlier times. This is an appealing feature, sinc e equilibrium has the effec t of washing away dependenc e on earlier states of the universe. As a result proc esses suc h as nuc leosynthesis are relatively insensitive to the state of the very early universe. The dynamic al evolution through nuc leosynthesis is based on well-understood nuc lear physic s, and equilibrium effac es the unknown physic s at higher energies. 2.3 Structure Formation By c ontrast with these suc c esses, the Standard Model lac ks a c ompelling ac c ount of how struc tures like galaxies formed. This reflec ts the diffic ulty of the subjec t, whic h requires integrating a broader array of physic al ideas than those required for the study of nuc leosynthesis or the FLRW models. It also requires more sophistic ated mathematic s and c omputer simulations to study dynamic al evolution beyond simple linear perturbation theory. Newtonian gravity enhanc es c lumping of a nearly uniform distribution of matter, as matter is attrac ted more strongly to regions with above average density. Jeans (1902) studied the growth of fluc tuations in Newtonian gravity and found that fluc tuations with a mode greater than a c ritic al length exhibit instability and their amplitude grows exponentially. The first study of a similar situation in general relativity (Lifshitz 1946) showed, by c ontrast, that expansion in the FLRW models c ounterac ts this instability, leading to muc h slower growth of initial perturbations. Lifshitz (1946) c onc luded that the gravitational enhanc ement pic ture c ould not produc e galaxies from plausible “seed” perturbations and rejec ted it. Two dec ades later the argument was reversed: given the gravitational enhancement account of structure formation (no viable alternative accounts had been discovered), the seed perturbations had to be muc h larger than Lifshitz expec ted. Many c osmologists adopted a more phenomenologic al approac h, using observational data to c onstrain the initial perturbation spec trum and other parameters of the model. Contemporary ac c ounts of struc ture formation treat observed large-sc ale struc tures as evolving by gravitational enhanc ement from initial seed perturbations. The goal is to ac c ount for observed properties of struc tures at a variety of sc ales— from features of galaxies to statistic al properties of the large-sc ale distribution of galaxies—by appeal to the dynamic al evolution of the seed perturbations through different physic al regimes. Harrison, Peebles, and Zel'dovic h independently argued that the initial perturbations should be sc ale invariant, that is, lac king any δρ c harac teristic length sc ale. 2 1 Assuming that these initial fluc tuations are small (with a density c ontrast ρ  ⟨ ⟨ 1), they c an be treated as linear perturbations to a bac kground c osmologic al model where the dynamic aδρlρ e  v≈ol u1t,ion of individual modes is spec ified by general relativity As the perturbations grow in amplitude and reac h perturbation theory no longer applies and the perturbation mode “separates” from c osmologic al expansion and begins to c ollapse. In c urrent models, struc ture grows hierarc hic ally with smaller length sc ales going nonlinear first. Models of evolution of struc tures at smaller length sc ales (e.g., the length sc ales of galaxies) as the perturbations go nonlinear inc orporate physic s in addition to general relativity, suc h as gas dynamic s, to desc ribe the c ollapsing c lump of baryonic matter. The c urrent c onsensus regarding struc ture formation is c alled the ʌCDM model. The name indic ates that the model inc ludes a nonz ero c osmologic al c onstant (ʌ) and “c old” dark matter (CDM). (Cold dark matter is disc ussed in the next sec tion.) The model has several free parameters that c an be c onstrained by measurements of a wide variety of phenomena. The ric hness of these evidential c onstraints and their mutual c ompatibility provide some c onfidenc e that the ACDM model is at least partially c orrec t. There are, however, ongoing debates regarding the status of the model. For example, arguably it does not c apture various aspec ts of galaxy phenomenology. Although I do not have the spac e to review these debates here, it is c lear that c urrent ac c ounts of struc ture formation fac e more unresolved c hallenges and problems than other aspec ts of the Standard Model. Page 6 of 34

Philosophy of Cosmology 3. Dark Matter and Dark Energy The main support for the Standard Model c omes from its suc c essful ac c ounts of big bang nuc leosynthesis, the redshift-distanc e relation, and the CBR. But pushing these lines of evidenc e further reveals that, if the Standard Model is basic ally c orrec t, the vast majority of the matter and energy filling the universe c annot be ordinary matter. Ac c ording to the “c onc ordanc e model,” normal matter c ontributes ≍ 4% of the total energy density, with ≍ 22% in the form of non-baryonic dark matter and another ≍ 74% in the form of dark energy.2 2 Dark matter was first proposed based on observations of galaxy c lusters and galaxies.2 3 Their dynamic al behavior c annot be ac c ounted for solely by luminous matter in c onjunc tion with Newtonian gravity. More rec ently, it was disc overed that the deuterium abundanc e, c alc ulated from big bang nuc leosynthesis, puts a tight bound on the total amount of baryonic matter. Combining this c onstraint from big bang nuc leosynthesis with other estimates of c osmologic al parameters leads to the c onc lusion that there must be a substantial amount of non-baryonic dark matter. Ac c ounts of struc ture formation via gravitational enhanc ement also seem to require non-baryonic c old dark matter. Adding “c old” dark matter to models of struc ture formation helps to rec onc ile the uniformity of the CBR with the subsequent formation of struc ture. The CBR indic ates that any type of matter c oupled to the radiation must have been very smooth, muc h too smooth to provide seeds for struc ture formation. Cold dark matter dec ouples from the baryonic matter and radiation early, leaving a minimal imprint on the CBR.2 4 After rec ombination, however, the c old dark matter perturbations generate perturbations in the baryonic matter suffic iently large to seed struc ture formation. The first hint of what is now c alled “dark energy” also c ame in studies of struc ture formation, whic h seemed to require a nonz ero c osmologic al c onstant to fit observational c onstraints (the ACDM models). Subsequent observations of the redshift-distanc e relation, with supernovae (type Ia) used as a powerful new standard c andle, led to the disc overy in 1998 that the expansion of the universe is ac c elerating.2 5 This further indic ates the need for dark energy, namely a type of matter that c ontributes to Eq. (2) like a ʌ term, suc h that R¨  ⟩ 0 .2 6 Most c osmologists treat these developments as akin to Le Verrier's disc overy of Neptune. In both c ases, unexpec ted results regarding the distribution of matter are inferred from observational disc repanc ies using the theory of gravity. Unlike the c ase of Le Verrier, however, this c ase involves the introduc tion of new types of matter rather than merely an additional planet. The two types of matter play very different roles in c osmology, despite the shared adjec tive. Dark energy affec ts c osmologic al expansion but is irrelevant on smaller sc ales, whereas dark matter dominates the dynamic s of bound gravitational systems suc h as galaxies. There are important c ontrasts in the evidential c ases in their favor and in their c urrent statuses. Some c osmologists have c alled the c onc ordanc e model “absurd” and “preposterous” bec ause of the oddity of these new types of matter and their huge abundanc e relative to that of ordinary matter. There is also not yet an analog of Le Verrier's suc c essful follow-up telesc opic observations. Perhaps the appropriate historic al analogy is instead the “z odiac al masses” introduc ed to ac c ount for Merc ury's perihelion motion before GR. Why not modify the underlying gravitational theory rather than introduc e one or both of these entirely new types of matter? The ongoing debate between ac c epting dark matter and dark energy vs. pursuing alternative theories of gravity and c osmology turns on a number of issues familiar to philosophers of sc ienc e. Does the evidenc e underdetermine the appropriate gravitational theory? At what stage should the need to introduc e distinc t types of matter with exotic properties c ast doubt on the gravitational theory and qualify as anomalies in Kuhn's sense? How suc c essful are alternative theories c ompared to GR and the Standard Model, relative to different ac c ounts of what c onstitutes empiric al suc c ess? What follows is meant to be a primer identifying the issues that seem most relevant to a more systematic treatment of these questions.2 7 Confidenc e that GR adequately c aptures the relevant physic s supports the mainstream position, ac c epting dark matter and dark energy. The applic ation of GR at c osmologic al sc ales involves a tremendous extrapolation, but this kind of extrapolation of presumed laws has been inc redibly effec tive throughout the history of physic s. This partic ular extrapolation, furthermore, does not extend beyond the expec ted domain of applic ability of GR. No one trusts GR at suffic iently high energies, extreme c urvatures, and short length sc ales. Presumably it will be superseded by a theory of quantum gravity. Disc overing that GR fails at low energies, low c urvature, and large length sc ales—the regime relevant to this issue—would, however, be extremely surprising. In fac t, avoiding dark matter entirely would require the even more remarkable c onc ession that Newtonian gravity fails at low Page 7 of 34

Philosophy of Cosmology ac c elerations. In addition to the c onfidenc e in our understanding of gravity in this regime, GR has proven to be an extremely rigid theory that c annot be easily c hanged or adjusted.2 8 At present, there is no c ompelling way to modify GR so as to avoid the need for dark matter and dark energy, while at the same time preserving GR's other empiric al suc c esses and basic theoretic al princ iples. (Admittedly this may reflec t little more than a failure of imagination; it was also not obvious how to c hange Newtonian gravity to avoid the need for z odiac al masses.) The independence of the different lines of evidence indicating the need for dark matter and dark energy provides a sec ond powerful argument in favor of the mainstream position. The sourc es of systematic error in estimates of dark matter from big bang nuc leosynthesis and galaxy rotation c urves (disc ussed below), for example, are quite different. Evidenc e for dark energy also c omes from observations with very different systematic s, although they all measure properties of dark energy through its impac t on spac etime geometry and struc ture formation. Several apparently independent parts of the Standard Model would need to be mistaken in order for all these different lines of reasoning to fail. The c ase for dark energy depends essentially on the Standard Model, but there is a line of evidenc e in favor of dark matter based on galac tic dynamic s rather than c osmology. Estimates of the total mass for galaxies (and c lusters of galaxies), inferred from observed motions in c onjunc tion with gravitational theory, differ dramatic ally from mass estimates based on observed luminous matter.2 9 To take the most famous example, the orbital veloc ities of stars and gas in spiral galaxies would be expec ted to drop with the radius as r−1/2 outside the bright c entral region; observations indic ate instead that the veloc ities asymptotic ally approac h a c onstant value as the radius inc reases.3 0 There are several other properties of galaxies and c lusters of galaxies that lead to similar c onc lusions. The mere existenc e of spiral galaxies seems to c all for a dark matter halo, given that the luminous matter alone is not a stable c onfiguration under Newtonian gravity.3 1 The c ase for dark matter based on these features of galaxies and c lusters draws on Newtonian gravity rather than GR. Relativistic effec ts are typic ally ignored in studying galac tic dynamic s, given the prac tic al impossibility of modeling a full galac tic mass distribution in GR. But it seems plausible to assume that the results of Newtonian gravity for this regime c an be rec overed as limiting c ases of a more exac t relativistic treatment.3 2 There is another way of determining the mass distribution in galaxies and c lusters that does depend on GR, but not the full Standard Model. Even before he had reac hed the final version of GR, Einstein realiz ed that light-bending in a gravitational field would lead to the magnific ation and distortion of images of distant objec ts. This lensing effec t c an be used to estimate the total mass distribution of a foreground objec t based on the distorted images of a bac kground objec t, whic h c an then be c ontrasted with the visible matter in the foreground objec t.3 3 Estimates of dark matter based on gravitational lensing are in rough agreement with those based on orbital veloc ities in spiral galaxies, yet they draw on different regimes of the underlying gravitational theory. Critic s of the mainstream position argue that introduc ing dark matter and dark energy with properties c hosen prec isely to resolve the mass disc repanc y is ad hoc . Whatever the strength of this c ritic ism, the mainstream position does c onvert an observational disc repanc y in c osmology into a problem in fundamental physic s, namely that of providing a believable physic s for dark matter and dark energy. In this regard the prospec ts for dark matter seem more promising. Theorists have turned to extensions of the Standard Model of partic le physic s in the searc h for dark matter c andidates, in the form of weakly interac ting massive partic les. Although the resulting proposals for new types of partic les are spec ulative, there is no shortage of c andidates that are theoretic ally natural (ac c ording to the c onventional wisdom) and as yet c ompatible with observations. There also do not appear to be any fundamental princ iples that rule out the possibility of appropriate dark matter c andidates. With respec t to dark energy, by c ontrast, the disc overy of ac c elerating expansion has exac erbated what many regard as a c risis in fundamental physic s.3 4 Dark energy c an either take the form of a true ʌ term or some field whose stress-energy tensor effec tively mimic s ʌ. As suc h it violates an energy c ondition assoc iated with “ordinary” matter, although few theorists now take this c ondition as inviolable.3 5 A more fundamental problem arises in c omparing the observed value of dark energy with a c alc ulation of the vac uum energy density in quantum field theory (QFT). The vac uum energy of a quantum field diverges. It is given by integrating the z ero-point c ontributions to the total energy, ½ ℏω(k) per osc illation mode, familiar from the quantum harmonic osc illator, over momentum (k). Evaluating this quartic ally divergent quantity by introduc ing a physic al c utoff at the Planc k sc ale, Page 8 of 34

Philosophy of Cosmology the result is 120 orders of magnitude larger than the observed value of the c osmologic al c onstant.3 6 This is sometimes c alled the “old” c osmologic al c onstant problem: Why isn't there a c anc ellation mec hanism that leads to ʌ = 0? Post-1998, the “new” problem c onc erns understanding why the c osmologic al c onstant is quite small (relative to the vac uum energy density c alc ulated in QFT) but not exac tly z ero, as indic ated by the ac c elerating expansion. Both problems rest on the c ruc ial assumption that the vac uum energy density in QFT c ouples to gravity as an effec tive c osmologic al c onstant. Granting this assumption, the c alc ulation of vac uum energy density qualifies as one of the worst theoretic al predic tions ever made. What turns this dramatic failure into a c risis is the diffic ulty of c ontrolling the vac uum energy density, by, say, introduc ing a new symmetry. Rec ently, however, an anthropic response to the problem has drawn inc reasing support. On this approac h, the value of ʌ is assumed to vary ac ross different regions of the universe, and the observed value is “explained” as an anthropic selec tion effec t (we will return to this approac h in sec tion 7 below). Whether abandoning the assumption that the vac uum energy is “real” and gravitates is a viable response to the c risis depends on two issues. First, what does the empiric al suc c ess of QFT imply regarding the reality of vac uum energy? The treatment of the scaling behavior of the vacuum energy density above indicates that vacuum energy in QFT is not fully understood given current theoretical ideas. This is not particularly threatening in calculations that do not involve gravity, since one can typically ignore the vacuum fluctuations and calculate quantities that depend only on relative rather than absolute values of the total energy. This c onvenient feature also suggests, however, that the vac uum energy may be an artifac t of the formalism that c an be stripped away while preserving QFT's empiric al c ontent. Sec ond, how should the standard treatment of the vac uum energy from flat-spac e QFT be extended to the c ontext of the c urved spac etimes of GR? The symmetries of flat spac etime so c ruc ial to the tec hnic al framework of QFT no longer obtain, and there is not even a c lear way of identifying a unique vac uum state in a generic c urved spac etime. Reformulating the treatment of the sc aling behavior of the vac uum energy density is thus a diffic ult problem. It is c losely tied to the c hallenge of c ombining QFT and GR in a theory of quantum gravity. In QFT on c urved spac etimes (one attempt at c ombining QFT and GR) different renormaliz ation tec hniques are used that eliminate the vac uum energy. The question is whether this approac h simply ignores the problem by fiat or reflec ts an appropriate generaliz ation of renormaliz ation tec hniques to c urved spac etimes. These two issues are instanc es of familiar questions for philosophers—what parts of a theory are ac tually supported by its empiric al suc c ess, and what parts should be preserved or abandoned in c ombining it with another theory? Philosophers have offered c ritic al evaluations of the c onventional wisdom in physic s regarding the c osmologic al c onstant problem, and there are opportunities for further work.37 Returning to the main line of argument, the prospec ts for an analog of Le Verrier's telesc opic observations differ for dark matter and dark energy. There are several experimental groups c urrently searc hing for dark matter c andidates, using a wide range of different detec tor designs and searc hing through different parts of the parameter spac e (see, e.g., Sumner 2002). Suc c essful detec tion by one of these experiments would provide evidenc e for dark matter that does not depend direc tly on gravitational theory. The properties of dark energy, by way of c ontrast, insure that any attempt at a nonc osmologic al detec tion would be futile. The energy density introduc ed to ac c ount for ac c elerated expansion is so low, and uniform, that any loc al experimental study of its properties is prac tic ally impossible given c urrent tec hnology. There are different routes open for those hoping to avoid dark energy and dark matter. Dark energy is detec ted by the observed departures from the spacetime geometry that one would expect in a matter-dominated FLRW model. Taking this departure to indicate the presence of an unexpected contribution to the universe's overall matter and energy c ontent thus depends on assuming that the FLRW models hold. There are then two paths open to those exploring alternatives to dark energy. The first is to c hange the underlying gravitational theory and to base c osmology on an alternative to GR that does not support this inferenc e. A sec ond would be to retain GR but rejec t the FLRW models. For example, models that desc ribe the observable universe as having a lower density than surrounding regions can account for the accelerated expansion without dark energy. Cosmologists have often assumed that we are not in a “spec ial” loc ation in the universe. This c laim is often c alled the “Copernic an princ iple,” to whic h we will return in sec tion 5 below. This princ iple obviously fails in these models, as our observable patc h would be loc ated in an unusual region—a large void.3 8 It has also been proposed that the ac c elerated expansion may be ac c ounted for by GR effec ts that c ome into view in the study of inhomogeneous models without dark energy. Buc hert (2008) reviews the idea that the bac k-reac tion of inhomogeneities on the Page 9 of 34

Philosophy of Cosmology bac kground spac etime leads to an effec tive ac c eleration. These proposals both fac e the c hallenge of ac c ounting for the various observations that are regarded, in the c onc ordanc e model, as manifestations of dark energy. On the other hand, dark matter c an only be avoided by modifying gravity— including Newtonian gravity—as applied to galaxies. Milgrom (1983) argued that a modific ation of Newtonian dynamic s (c alled MOND) suc c essfully c aptures several aspec ts of galaxy phenomenology. Ac c ording to Milgrom's proposal, below an ac c eleration threshold (a0 ≍ 10 m/s2 ) Newton's sec ond law should be modified to F   =  m a2 . This modific ation ac c ounts for a0 observed galaxy rotation c urves without dark matter. But it also ac c ounts for a wide variety of other properties of galaxies, many of whic h Milgrom suc c essfully predic ted based on MOND (see, e.g., Sanders and Mc Gaugh 2002, Bekenstein 2010 for reviews). Despite these suc c esses, MOND has not won widespread support. Even advoc ates of MOND admit that at first blush it looks like an extremely odd modific ation of Newtonian gravity. Yet it fares remarkably well in ac c ounting for various features of galaxies—too well, ac c ording to its advoc ates, to be dismissed as a simple c urve fit. MOND does not fare as well for c lusters of galaxies and may have problems in ac c ounting for struc ture formation. In addition to these potential empiric al problems, it is quite diffic ult to embed MOND within a c ompelling alternative to GR. In sum, it is reasonable to hope that the situation with regard to dark matter and dark energy will be c larified in the c oming years by various lines of empiric al investigation that are c urrently underway. The apparent underdetermination of different alternatives may prove transient, with empiric al work eventually forc ing a c onsensus. Whether or not this oc c urs, there is also a possibility for c ontributions to the debate from philosophers c onc erned with underdetermination and evidential reasoning. The c onsiderations above indic ate that even in a c ase where c ompeting theories are (arguably) c ompatible with all the evidenc e that is c urrently available, sc ientists c ertainly do not assign equal c redenc e to the truth of the c ompetitors. Philosophers c ould c ontribute to this debate by helping to artic ulate a ric her notion of empiric al support that sheds light on these judgments (c f. the c losing c hapter of Harper 2012). 4. Uniqueness o f the Universe The uniqueness of the universe is the main c ontrast between c osmology and other areas of physic s. The alleged methodologic al c hallenge posed by uniqueness was one of the main motivations for the steady-state theory. The c laim that a generaliz ation of the c osmologic al princ iple, the “perfec t c osmologic al princ iple,” is a prec ondition for sc ientific c osmology, is no longer ac c epted.3 9 It is, however, often asserted that c osmology c annot disc over new laws of physics as a direct consequence of the uniqueness of its object of study.40 Munitz (1962) gives a concise formulation of this common argument: With respec t to these familiar laws [of physic s] … we should also mark it as a prerequisite of the very meaning and use of suc h laws that we be able to refer to an ac tual or at least possible plurality of instanc es to whic h the law applies. For unless there were a plurality of instanc es there would be neither interest nor sense in speaking of a law at all. If we knew that there were only one ac tual or possible instanc e of some phenomenon it would hardly make sense to speak of finding a law for this unique oc c urrenc e qua unique. This last situation however is prec isely what we enc ounter in c osmology. For the fac t that there is at least but not more than one universe to be investigated makes the searc h for laws in c osmology inappropriate. (Munitz 1962, 37) Ellis (2007) reac hes a similar c onc lusion: The c onc ept of “Laws of Physic s” that apply to only one objec t is questionable. We cannot scientifically establish “laws of the universe” that might apply to the class of all such objects, for we cannot test any such proposed law except in terms of being consistent with one object (the observed universe). (Ellis 2007, 1217, emphasis in the original) His argument for this c laim emphasiz es that we c annot perform experiments on the universe by c reating partic ular initial c onditions. In many observational sc ienc es (suc h as astronomy) the systems under study also c annot be manipulated, but it is still possible to do without experiments by studying an ensemble of instanc es of a given type of system. However, this is also impossible in c osmology. Page 10 of 34