No Time for ( No ) Change

The paper provides an overview of the main metaphysical theories of persistence —that is, of three and four-dimensionalism, as applied to a relativistic setting. It then addresses and undermines one of the most powerful objections against one of those views —i.e., four-dimensionalism. That objection is labeled the No-Change objection in the literature and has it that four-dimensionalism abolishes genuine change. Finally, building on the case against the No-Change objection, I contend there is some pressure to abandon the widespread assumption that change requires reference to distinct earlier and later times, let alone the passage of time. The aim of this paper is threefold. First, I provide an overview of theories of the metaphysics of persistence in a relativistic setting.1 Then, I offer a new discussion of the so-called “No-Change objection”,2 —which is considered one of the most powerful objections against a particular metaphysics of persistence, i.e., four-dimensionalism, and undermine it. Finally, building on my case against the No-Change objection, I contend there is some pressure to abandon a widespread intuition and widely held philosophical thesis according to which genuine change requires reference to two distinct times, if not its passage. It should already be clear from my plan that the discussion of the No-Change argument plays a twofold role. On the one hand, it is supposed to provide a way to resist one of the objections that has been considered, and still to this day is considered3 —for better or worse-one of the most significant objections against fourdimensionalism. On the other hand, it paves the way to address some crucial metaphysical questions about the relation between time and change4. Note: I want to thank audiences at eidos and at L’Aquila for wonderful discussions. I would also like to thank two anonymous referees and the book editors for the comments and suggestions. This work was generously supported by the Swiss National Science Foundation, Project

Ishould note from the start that some of my arguments will cruciallydepend upon relativisticconsiderations and cannot be translated in anystraightforward wayi nto am ore classicals etting.A lso, they relyonw hat Is hallc all the spacetime conception of time,and although such aconception does not directlyrequire Relativity theory,itseems to be far more palatable once the latterisaccepted. If so, my arguments can alsoberead as acontribution to the debate about whether relativisticp hysics favors some metaphysics of persistenceo vers ome others.⁵ , ⁶ The plan is as follows. In §1Istart with abrief introduction about some general assumptions at work in the paper.In §2Iprovide aformulation of different metaphysical theses about persistenceinr elativistic spacetimes.Ithen sketch in §3 the so-called "puzzle of change" and its solutions,a nd Is how how the four-dimensionalist account is allegedlyvulnerable to some variants of the No-Change objection, which Ir eject.I nt he conclusiono ft hat section Ip ut forward what I take to be the most interesting variant of the No-Changeargument.The final section, §4,i sd edicated to undermining such av ariant and to offering ag eneral suggestion about changea nd its relation to time. Once again, the rebuttal of the last variant of the No-Changea rgument is not onlyi nteresting in itself, but also for its wide-rangingc onsequences.

1S pace, Time,a nd Spacetime
In this section Ilay down some general assumptions Iwill rely upon in what follows. They are assumptions in that Iwill not arguef or them in this paper.H owever,this does not mean they cannot be argued for.I nf act,t hey have been extensively argued for in the literature.⁷ The first general assumption concerns the very talk of time in ar elativistic setting.F ollowingL ockwood (2005,C h. 3), Sattig (2006,C h. 2;Ch. 8, §1), Skow (2015;Ch. 2),and Gilmore, Costa and Calosi(2016, §1) we can distinguish two very general conceptions of time. Ishalllabel them the spacetime conception of time and the classical conception of time. To give but afirst intuitive characterization, accordingt ot he spacetime conception of time we inhabit as ingle fourdimensionalm anifold-spacetime, and time is just an aspect-for lack of ab etter word-of this more fundamental four-dimensionalentity. Contra this,the classical conception of time has it that we literallyi nhabit two different manifolds: a three-dimensional spatial manifold and ao ne-dimensionalt emporalo ne.⁸ Alittle more precisely, accordingtothe spacetime conception of time there is onlyafour-dimensionals patiotemporalm anifold; instants or intervals of time, on the one hand, and regions of space on the other,are simplyspacetime regions of different sorts thats hare some constituents (that is, some spatiotemporal points).⁹ By contrast,a ccordingt ot he classicalc onception of time,t ime is a one-dimensionalmanifold, and it is completelydistinct from athree-dimensional spatial manifold: the two fail to share anyc onstituents.
Though the spacetimeconception of time can alsobeheld in aclassicalsetting,Iwill simply assume that it is the most natural choice once some features of Relativity are taken into account.¹⁰ As Ia lreadym entioned, Iw ill not arguef or this claim; as ap iece of evidence though,Iwill simplyq uote some passages that point in this direction. Arguably the most famousr eference goes backt o Minkowski (1908): "Henceforth space by itself and time by itself are doomed to fade away into mere shadowsa nd onlyaunion of these two will preserve an independent reality" (Minkowski 1952, 75). The same spirit is vindicated in the following,m ore recent excerpts: "Spacea nd time are just different ways in which the same real thing (spacetime) appears from ag iven perspective" (Lange2002, 220); "There is no one-dimensionaltime distinct from three-dimensional spacebut rather afour-dimensional spacetime of which time is merelyan aspect" (Sattig 2006,1 ); "Ia ssume as pacetime framework accordingt ow hich the notion of at ime is not fundamental, but rather is to be defined in terms of the notion of ar egion of spacetime […]r egion of spacetime are ontologically prior to times" (McDaniel 2014,15). Allofthese passages suggest that spacetime is the fundamental entity in Relativity theory,s ot hat in order to take Relativity seriously we should refert os pacetime regions rather than to spatial regions or temporali ntervals.¹¹  Skow (2015) calls them 4D view and 3+1 view. Gilmore, Costa and Calosi (2016) calls them unitism and separatism. Iadopt the terminologyinthe main text mainlybecause Iwill use "4D" for the metaphysics of persistence.  Modulo Whiteadean worries about the existenceo fp oints.  It should be notedt hat this conception can be challengedi nc ertain formulations of Quantum Gravity.S ee, e. g., Monton (2006).  The standards et-theoretic construction of the spacetime manifold is assumedt hroughout. "According to this view,w ei nhabit af our-dimensional manifold of spacetime points,w here points arem ereologicallys imple and unextended, spacetime regions aren on-empty sets of points,a ny such set counts as ar egion, and one region is as ubregion of another iff the first is as ubset of the second" (Gilmore2 006,1 99). Spacetime regions aret hus am odel of the socalled (Atomistic) "General Extensional Mereology";s inceIwill be using mereological notions

No Time for( No) Change
The second general assumption concerns the contents of the spacetime manifold and spacetime regions.Iwill assume that the physical content of relativistic physics does not mention (nor does it need to mention) frames of reference. Rather,itissimplygiven by the contents (of regions) of spacetime, and the latter can be described independentlyo fa ny reference frames. The most vivid formulation of such av iew is givenb yG ibson and Pooley (2006): What is wrongw ith the first¹² […]( is that) it seeks to attach metaphysical weight to frames of reference that they simplydonot carry.Inertial frames of reference[ … ]a re no more thant he spatiotemporala nalogues of Cartesian coordinate systems […]B ut just as no one would attach ontological weight to features of an object that are relative to ac hoice of Cartesian coordinates,s on oo ne should attach significancet op roperties of objects that are essentiallyd efined in terms of canonicalf rames of reference. From the physicist'sp erspective,t he content of spacetime is as it is. One can choose to describe this content from the perspective of aparticular inertial frame […]But one can equallywell choose to describet he content of spacetime with respect to some frame that is not optimallyadapted to the geometric structure of spacetime, or indeed, choose to describe it in some entirelyframe independent manner" (Gibson and Pooley 2006, 161-162). ¹³ Gibson and Pooley alsod eclare: "It should be clear from our discussion of perdurance thatthe wrong waytorelativisticallyre-construe all this is to replace reference to moments andtimes with reference to times of inertial frames" (Gibson and Pooley 2006,164). Finally, Iwill assume that Relativity favors some sort of Btheoretic eternalist metaphysics of time,a ccordingt ow hich all spatiotemporal regions of the entire spacetime manifold are ontologicallyo napar and there is nothing objectivelya nd metaphysicallyp rivileged about one such region, called "the present".¹⁴ It aket hese assumptions to suggest-to sayt he least-that anym etaphysical thesis, insofar as it is meant to seriously engage with the hints comingfrom Relativity,should replace talk about spatial distances,temporalintervals and frames of referencewith talk about spacetime regions of different sorts that build up the entire relativistic spacetime.
to define persistingo bjects already, Iw ill use mereological notions as opposed to set-theoretic ones,t om inimize ideology.  I.e., with the option of relativizing times to reference frames.  This assumption might be morecontroversial in the case of Special Relativity.However,itis possible to give aformulation of Special Relativity that is frame-independent as well. See, e. g., Malament (2009).  The literatureonthis last issue is simplyt oo wide to even be brieflym entioned. Forrecent, opposingv iews,s ee Zimmerman (2011) and Wüthrich (2013) and references therein.
Therefore, in what follows, whenever reference to e. g., temporalf acts is explicitlym entioned, an attempt should be made to identify the sorto fm oref undamental spatiotemporal facts behindthem, so to speak.¹⁵ Itake this to be afirst step towards engagingw ith ac hallenget hat was raised in Gibson and Pooley (2006,157). The goal in engagingt hat challengew ould be to start with the relativistic world-picture, and then ask how thingsp ersist and changea ccordingt o such ap icture.

2P ersistencei nR elativistic Spacetimes
In this section, Iw ill provide ar igorous formulation of the main metaphysical theories of persistence¹⁶ in relativistic spacetimes.¹⁷ This formulation takes the lead from the pioneering works of Gilmore (2006;2 008;2 018) and Balashov (2008;2 010),¹⁸ though in certain respects Iw ill depart considerablyf rom them.¹⁹ Iw ill use onlyt wo primitive notions-plus first-order plural logic with identity:²⁰ the mereologicaln otiono fparthood and the locational notion of exact location. Gilmore (2018) offers the following intuitiveg loss on the latter: "an entity x is exactlyl ocated at ar egion y if and onlyi fxhas (or has-at-y)e xactlyt he same shape and size as y and stands (or stands-at-y)i na ll the same spatial or spatiotemporal relations to other entities as does y".
 This characterization, though deliberately vague, is sufficient for my aims:i ti sm eant to point out that facts about spacetime arei ns ome sense morefundamental than facts about (respectively)spaceand time considered separately, without anycommitment as to the specific nature of the relation between facts of the threecategories.S attig( 2006) mentions supervenience, while Skow (2015) seems to suggest some stronger notion of dependence-possiblys omething likethe grounding relation. Foranintroductiontothis last relation and cognatenotions,see Correiaa nd Schnieder (2012); Bliss and Trogdon (2014).  Iw ill focus on "three-dimensionalism" (or "endurantism")o nt he one hand, and "four-dimensionalism" (or "perdurantism")o nt he other.Iwill not discuss the so-called "Stage-View" (or "exdurantism"). See Sider (1996) for ad efense.  This section is not intended to be exhaustive.I nterested readers arer eferredt oG ilmore (2006); Balashov (2010); Calosi and Fano (2015).  To be fair,B alashov (2010) mentions frames of reference. His formulation can however be slightlym odified to avoid such ar eference.  My claims will considerably depart from Balashov'sf ormulation. It does not coincide with Gilmore'sf ormulation either,e speciallyi nt he definitiono fatemporal part.Iwill require temporal parts to be proper parts of four-dimensional objects.This seemingly small differenceh as significant consequences. AlthoughIcannot enteri ntod etails here,s ee footnotes 27 and 29.  From now on double signss uch as "xx" stand for pluralt erms (both constants and variables).

No Time for( No) Change
Forthe sake of completeness, let me alsomention thatinwhat follows Iwill assume that parthoodi sapartial order and obeys the so-called "Weak-Supple-mentationP rinciple" (Simons 1987;Varzi 2016) and that the exactl ocation relation obeys two axioms to the effect thateverythingthat is located somewherein spacetime has an exact location and that compositeo bjectsa re located where their parts are located.²¹ Such axioms sometimes go under the names of Exactness and Expansivity in the literature.²² Before approaching the metaphysics of persistenceproperly, some mereological notions need to be defined:²³ (1) x is a proper part of y = x is part of y and x is distinct from y.
(2) x and yo verlap =t here is a z that is part of both x and y.
(3) x is the mereological fusion of the yy =e ach of the yy is part of x and each part of x overlaps one of the yy.
Iwillalsoneed to define the notion of an achronal spacetime region-which can be informallyu nderstood as the relativistic counterpart of an instantaneous, temporallyu nextended region-and the notiono fapath of an object.T ot his end, Is hall use basicg eometrical features of relativistic spacetimes: (4) R is an achronal region =for anytwo distinct points p 1 and p 2 in²⁴ R,the vector connectingt hem is space-like. (5) The spacetimeregion Path(x) is the path of an object x²⁵ = Path(x) is the mereological fusion of x'se xact locations.
It is customary to quote Lewis at this point: "Letu ss ay that something persists iff, somehow or other, it existsatvarious times; this is the neutralword" (Lewis 1986,2 02). This can be elegantlyc aptured by (6) below,f or (6) entails that the  Notethat these two axioms areinline with relativistic physics insofar as the notion of spacetime trajectory is relativisticallywell-defined. This is not so obvious in the quantum mechanical case.
 See e. g., Casati and Varzi (1999); Parsons (2007). Notet hat the latter assumesad ifferent primitive notion, that of "weakl ocation".S trictlys peaking, Exactness is a theorem in at heory of location that takes "exact location" as aprimitive.Ins uch theories,iti scustomary to define the notion of weaklocation as follows: x is weaklylocatedatR=xis exactlylocated at R* and R and R* overlap.O ncet his definition is in place, Exactness follows.S ee e. g., Gilmore( 2018).  In the following, simple signssuch as "x" will be used as singular variableswhereas double signs such as "xx" will be used as plural ones.  Point pi si nregion R =p oint p is part of region R.  The path of an object is-intuitively-its entires patiotemporal career. spatiotemporal career of an object comprises at least two points thatare causally separated, i. e., either time-likeo rl ight-likes eparated²⁶ -thereby ensuring that the entity in question existsa td ifferent times, as per Lewis' request: (6) An object x is ap ersisting object = Path(x) is not achronal One of the great merits of Gilmore (2006) and Balashov (2010) is thato fh aving clearlystatedthe differencebetween what we might call locational and what we might call mereological persistence. In the former case, conditions for persistence are giveninterms of the exactlocations of material objects; in the latter,they are giveninterms of their mereological structure. When we think in locational terms, (locational) three-dimensionalobjects are exactlylocated at achronal, temporally unextended regions,w hereas (locational) four-dimensionalo bjectsa re uniquelye xactlyl ocated at not-achronal, temporallye xtended regions.²⁷ This is supposed to captureo ne wayo fp arsing the pre-analytical intuition that three-dimensional objects are not extended in time, while four-dimensional objects are. This suggests the following definitions: (7) An object x is a locational four-dimensional object 4D L (x) = x is ap ersisting object that is uniquelye xactlyl ocated at its path. (8) An object x is a locational three-dimensional object 3D L (x) = x is apersisting object that is exactlyl ocated at achronal spacetime regions.
It follows from these definitions and the mereological and locational axiomsI assumedt hat: i. four-dimensionalo bjects have au nique exactl ocation-i. e., their paths; ii. three-dimensional objects are multi-located at different achronal regions that are proper parts of theirp aths.
The metaphysical debate about persistenceh as been traditionallyc asta sadebate about the mereological structure of persisting objects. In particular,i th as most commonlyb een approached as ad ebate as to whether persisting objects divide into temporal parts or fail to thus divide. To define at emporal part in a relativisticsetting,Ifollow Gibson and Pooley (2006) and first define an achronal maximalr egion of another region. The easiest wayt od efine such an otion is the  This ensures that photons counta sp ersistingo bjects.  Notet hat this is in line with the intuitive gloss Ig ave for the notion of "exact location".
No Time for (No) Change following: (9) Aspacetime region R is a maximalachronal region of aspacetime region R* =(i) R is part of R*;(ii) R is an achronal region, (iii) R is not aproper part of anyo ther achronal region that is part of R*.
Now we can define an achronal temporal part x of y: (10) x is an achronal temporalp art of y =( i) x is ap roper part of y;( ii) x is uniquelye xactlyl ocated at region R;( iii) R is am aximal achronal region of Path(y)²⁸ The general notion of atemporalpart can be defined as amereologicalfusion of some achronal temporalp arts: (11) x is atemporalpart of y =(i) x is aproper part of y;(ii) x is the mereological fusion of some achronal temporalp arts of y.
Note thatt his definition explicitlyr equires that temporalp arts of 4D-objectswhich will be defined below-are properp arts of such objects.²⁹ Mereological four and three-dimensional objects are defined, respectively,a sf ollows: (12) x is a mereological four-dimensional object 4D M (x) = x is ap ersisting object that is the fusion of its temporalp arts. (13) x is a mereological three-dimensional object 3D M (x) = x is apersisting object that does not divide into temporalp arts.
Form yp urposes here, it will sufficet od efine (respectively)t hreea nd four-dimensional objectsa sf ollows:  Thereare afew differences between this definitionofatemporal part and others that can be found in the literature, as Ialreadypointed out in footnote18. Sincesuch differencesdonot play anyc rucial rolef or the arguments Ia dvance in this paper,Iwill not discuss them. As far as I know,t he approach that gets the closest to the one Is uggest herec an be found in Gibson and Pooley (2006).  This immediatelyundermines VanI nwagen'si nfamous argument against four-dimensionalism in VanInwagen(1990), in that the argument cruciallydependsu pon consideringafour-dimensional object at emporal part of itself.
(14) x is at hree-dimensionalo bject 3D(x) = x is bothalocational and am ereological three-dimensional object. (15) x is af our-dimensionalo bject 4D(x) = x is both al ocational and am ereological four-dimensional object.³⁰ Usually, Three and Four-dimensionalism are construed as,r espectively,the theses thata ll persisting objects are three-dimensionala nd the thesis that all such objects are four-dimensional.

3T he Puzzle of Changea nd the No-Change Argument
In anutshell, the so-called puzzle of changearises from the question as to how it is possiblef or an object to instantiatei ncompatible properties. Ap erspicuous formulation of the puzzle was provided in Kurtz (2006,2 ).³¹ Accordingt o Kurtz,t he puzzle consists in at ension between the metaphysical theses she calls-respectively-Consistency, Change and Persistence;such theses are so deeply entrenched as to be hardlyn egotiable, and yett hey seem to be jointlyi nconsistent.The theses are the following: I. Consistency: Nothingc an instantiate incompatible properties; II. Change: Changei nvolvesi ncompatible properties; III. Persistence: Objects persist through change.
The tension is easilys een. If changei nvolvesi ncompatible properties,h ow can an object persist through change, givent hat it cannot instantiate incompatible properties?H erei sh ow Lewis phrases the point: Thingssomehow persist through time. When they do, they have some of their intrinsic properties temporarily. Fori nstance, shape: sometimes yous it,a nd then youa re bent; sometimes yous tand or lie, and then youa re straight.H ow can one and the same thing have two contrary intrinsic properties? ( Lewis 2002,1).
 These definitions raise an interestingquestion. Is it possible for apersistingobject to be locationallyf our-dimensional and mereologicallyt hree-dimensional?A nd conversely: is it possible for ap ersistingo bject to be locationallyt hree-dimensionala nd mereologicallyf our-dimensional?F or different answers, see Gilmore( 2006); Calosi and Fano (2015).  As imilar presentation is provided in Hinchliff (1996).

No Time for( No) Change
Each one of the metaphysics of persistenceIconsidered in §2provides aparticular answer to the puzzle of change.³² In what follows, let x be ap ersisting object that changes by instantiatingi ncompatible properties F 1 and F 2 . The Spatiotemporal solution has it thatspatiotemporalfacts mediate the instantiation of incompatible properties.This can be considered the relativistic (or spatiotemporal) counterpart of the more familiar solution offered in ac lassical setting-i. e., of the one accordingt owhich temporal (rather than spatiotemporal) facts mediate the instantiation of incompatible properties.S uch as olution comes in two variants,t hat are known as relativization and adverbialism,r espectively.Iwill present the more familiar temporal variant first as ap aradigmatic example.
Accordingt ot he relativization strategy-advocated, e. g., by Mellor (1998, 89 -93)-x has F 1 at t 1 ,t hat is, x bearst he F 1 relation to t 1 ,w hile it has F 2 at t 2 , that is, x bears the F 2 relation to t 2 . On the other hand,a ccordingt oadverbialism-defended, among others, in Lowe( 1988) and Haslanger (1989)³³-x has t 1 -ly F 1 and has t 2 -ly F 2 ,where t 1 -lya nd t 2 -lya ct as adverbialm odifiers.³⁴ This distinction within the spatiotemporal camp does not matter for the purpose of this paper.H ence, Iwill not geti nto the question whether the solution Ia mc onsidering should be advanced in line with the relativization strategyo ri nl ine with adverbialism. Moreg enerally, accordingt ot he Spatiotemporal solution: (16) Forany persisting object x and anytwo incompatible properties F 1 and F 2 ,if xchanges from having F 1 to having F 2 ,then there are distinct spacetime regions R 1 and R 2 which are parts of Path(x), such that x has property F 1 -at-R 1 and property F 2 -at-R 2 ,where "having F-at-R",issupposed to be neutral with respect to relativization and adverbialism.
 These solutions presuppose acertain metaphysicsoftime -that is assumed here as explicitly claimed in §1 -namelye ternalism, according to which there is no ontologicald istinction between the tenses. Hinchliffe (1996) criticizes this very point,i.e., the endorsementofaneternalist metaphysicsoftime. He claims that presentism, roughly the view that,strictlyspeaking, only the present is realp rovides ab etter solution to the puzzle of change.Iwill grant such ap oint. However,the tenability of apresentist metaphysicsoftime in arelativistic world is problematic to sayt he least.The literatureo nt he subject is literallyt oo vast to be mentioned.S ee also 13.  See Lewis (2002) for an argument to the effect that the two variants aren ot as different as they mays eem to be.  This formulation is an example of the classic "temporal variant" insofar as temporal instants mediatethe instantiation of properties.Inaspatiotemporal setting, temporal instants will be replaced by spacetime regions,a si n( 16). Gibson and Pooley (2006) provide the more straightforward example of the spatiotemporal solution in ar elativistics etting.³⁵ Thiss olution is advocated by three-dimensionalists.Anotable differenceb etween the temporala nd spatiotemporalvariant is thatt he latter seems more defective,a tl east in the formulation Ihavegiven at first; for it does not mention which spacetime regions mediate the instantiation of properties. Iwill engagewith this issue again in due time. The rival account is universallyk nown as the Temporal Parts solution: (17) Forany persisting object x and anytwo incompatible properties F 1 and F 2 ,if xchanges from having F 1 to having F 2 ,then thereare two distinct temporal parts y 1 and y 2 of x,e xactlyl ocated at distinct spacetime regions R 1 and R 2 which are part of Path(x), such that y 1 has F 1 and y 2 has F 2 .
Accordingtosuch asolution, apersisting object changes vicariously by having twod istinct temporalp arts that instantiate, respectively, one of the relevant (mutually) incompatible properties simpliciter. The passagefrom the temporal to the spatiotemporalcontext is less dramatic in the case of the Temporal Parts solution. For, in effect,this passagehas been alreadyaddressed in the definitionof the very notion of atemporal part.Needless to say, this is the solution that fourdimensionalists prefer.One of the most common objections against four-dimensionalismfocuses exactlyonthe Temporal Part solution to the puzzle of change. Sider (2001, 212-216) calls such an objection the "No-Change" argument.³⁶ The point can be traced backatleast to McTaggart (1927), but we can find classic variations on such an argument in Geach (1972, 304), Lombard(1986,108), Simons (1987, 135 -137), Mellor (1998, 89 -90).
Before properlyengagingwith the argument,let me highlight (albeit briefly) its significance.³⁷ The argument is not confined to the classic formulations presented above. It is continuouslyd iscussed nowadays-see, among others, Kurtz (2006), Hinchliff (2006, Hales and Johnson (2007), Hawley (2015,1 0-11), and Skow (2015,2 4). Wasserman (2006,5 2) calls it "the most familiar objection to the temporalp arts approach".I ti sn ot onlyd iscussed, but also endorsed in,  They focus on particular properties,n amelys hape properties.  Fort he sakeo fp recision, this paper focuses onlyo nt hat part of the No-Change argument that Sider calls the argumentf roms patial analogy.  Iwill simplygrant that A-theorists could advance amoreradical argument to the effect that the Temporal Parts solution abolishes genuine change,for in their view the alleged "dynamical aspect" of change would be lost in such an account.Onthe other hand, it should be noted that, beginningwith Sider himself, the No-Change argument is not consideredasanargument against the B-theory of time per se,b ut rather as one against four-dimensionalism.
No Time for( No) Change e. g., Oderberg( 2004) and Alai (2016). Leadingf igures thath aver ecentlye ndorsed the argument include McCall and Lowe( 2009) ³⁸ and Simons (2014).³⁹ This should be enough to call for its thorough assessment.O nt op of that,a sI pointed out in the introduction, the discussion of the No-Changea rgument has an indirect significance,s ot os peak,i nt hat it will be pivotal to address some metaphysicallyc rucial relations between time, spacetime and changei n §4.I nt hats ection, Iw ill suggest thatr elativistic physics alreadyc ontained the seeds of av iew accordingt ow hich referencet od istinct earlier and later times is not necessary for change.
In discussing the argument,itwill be useful to take the lead from apassage of Geach (1972) thatc onveys several points that Iwill discuss in due course. To quote Geach at length: The view in which time is merely afourth dimensioninwhich things extend is in anyevent quiteuntenable.Onthis view the variation of apoker'stemperature with time would simply mean that therew ere different temperatures at different positions along the poker'st ime axis.B ut this,a sM cT aggartr emarked, would be no moreachange in temperaturet han av ariation of temperature alongt he poker'sl ength would be […]W et hus have av iew that reallyabolishes change,byreducingchange to amere variation of attributes between different parts of aw hole (Geach 1972,3 04).
From these words we can extract aq uick,p reliminary,d eliberatelyv ague, formulation of the No-Changea rgument: (P 1 )S patial variation is not genuine change  "Without enduring objects,there is no such thingasmotion. Asimulacrumofmotion can be created by filmingt he two hands coming together and rapidlyp rojectingt he stationary images on the wall. In this case nothingmoves" (McCall and Lowe2009,279). Though McCall and Lowe focus on motion, it is clear that their worry is much moreg eneral and extendst oa ny kind of change.  "We standardlyr egardacontinuant which has one property at one time and another property incompatible with the first,a tal ater time, as changing in this regard. We also standardly consider an occurrentwhich varies across its different times as precisely not changing, because the variation attaches to different (temporal) parts of the whole, not to the whole thing.S o, a party which starts quietlyand gets louder does not change in the literalsense, whereas aperson whoi sd ark-haired at twenty and white-haired at seventy does literallyc hange.P roponents of occurrents as the sole inhabitants of spatiotemporal nature often try to hijack the term "change" to covervariation, but Iagree with Geach and Dretskethat we should resist such attempts.Wedo not saythat the variation amongthe parts of ariver which is fast-flowingatits source and sluggish at its mouth constitutes ac hange,b ut as patial or geographical variation" (Simons 2014, 66 -67).
(P 2 )T he Temporal Part solution offers an account of changet hat is completely analogous to spatial variation⁴⁰ , ⁴¹ (C)T he Temporal Parts olution yields no genuinec hange( from P 1 ,P 2 ) The argument,when so looselya nd vaguelyp hrased,d oes not have much bite. It can be simplya nswered, it seems, by claiming that it is unclear why the analogybetween spatial variationand changeshould be thought of as problematic in the first place. Premise P 1 is in fact completelyun-argued for.Assuch, we could simply counter it without an argument,b yp ointing out,e .g., some cases in which we do seem to use "changel anguage" in the caseo fs patial variation. Sider (2001) mentions, for instance,t he case of ar oad that becomes-a typicale xample of "changel anguage"-narrow.I na lternative,t hink of ar iver that-we would say-"gets" deeper and deeper.
This would undermine P 1 .Yet the No-Changeargument can be givenamuch strongerf ormulation. In fact,Iwill myself propose two different strongerv ersions. They are strongeri nt hat P 1 is not left un-argued, but rather is supported by two different arguments. Iw ill start from the following passageo fM ellor: "Change,o bviouslyi fv aguely, is somethingh avingaproperty at one time and not at another.M orep recisely, it has at hing having incompatible properties,l iked ifferent temperatureso rB -places, at different B-times" (Mellor 1998, 70,i talics added).⁴² Three crucial elements are mentioned in the passagejust quoted: (i) incompatible properties, (ii) athing having them, and (iii) different times. Iwill simply take for grantedt hat changed oes indeed require incompatible properties as I have been doing from the beginning.The other two elements can be used to for-  CompareSider (2001,214): "the differencebetween merelyspatial variation and four-dimensional change is vanishinglysmall",and Kurtz 2006,5:"the perdurantist takeschange over time to be analogous to change over space".  Notethat all the variants of the No-Changeargument that will be explored in the rest of the paper endorse P 2 .P 2 -which is in fact the coreofthe objection and is takenalmost verbatim from e. g., McTaggart (1927); Geach (1972); Sider (2001) and Simons (2014)-to mention afew-just says that "changethrough time is analogous to variation to space".Thus, it should not be confused with the controversial claim according to which "time is analogous to space"-see e. g., Skow (2007). To appreciatet hat these questions areo rthogonal consider yeta nother case, i. e., the modal case. One can maintain that the modal dimension is not analogous to the temporal onee. g.,b yh olding modal ersatzism on the one hand and eternalism on the other-and yeti nsist that (persistence) and change through both the modal and temporal dimensionsare importantly analogous,e.g., by endorsingcounterpart theory.For the record,Ido not want to commit myself to the endorsemento fs uch theses.  See also Mellor 1998, 87. No Time for( No) Change mulate strongerv ariants of the No-Changea rgument.The first one strengthens Mellor'sn otion of at hing having incompatible properties-i. e., element (ii)-via the claim thatt he incompatible properties must be exemplified by the very same thing,where "the very same thing" is understood in terms of strict numerical identity. The argument can be formulated as follows: (P 3 )E xemplification of incompatible properties by the very same entity is necessary for change(or,equivalently: if something counts as achangefrom F 1 to F 2 then we must have:( i) F 1 (x), (ii) F 2 (y), (iii) x = y)⁴³ (P 4 )E xemplification of incompatible properties by the very samee ntity is not necessary for spatial variation⁴⁴ (P 5 )S patial variation is not genuine change( from P 3 ,P 4 ) (P 6 )T he Temporal Parts solution offers an account of changethat is completely analogous to spatial variation (C)T he Temporal Parts solution yields no genuine change( from P 5 ,P 6 ) P 5 -P 6 /C is actuallyt he very sameN o-Changea rgument we started with. But this formulation is better in that P 5 ( P 1 )isallegedlysupported by an argument, so that it will not do to simplyc hallengei t, as Ih aved one previously, without thereby challengingt he argument supportingi t. In other words. Just pointing out spatial analogues like the riverand the road in which we invoke changelanguagetodescribe the situation at hand is not enough. One has to undermine either P 3 or P 4 .This is exactlyw hat Iw ill do.
The problem with such an argument is that,a tl east in this explicit formulation,i ts implyb egs the question against four-dimensionalists. P 3 is basically an explicit denial of the Temporal Parts solution. Let me expand. It begs the question against four-dimensionalism insofar as it requires that the relation holding between the bearer(s) of incompatible properties should be numerical identity,whereas the Temporal Parts solution entails thatt hose bearers are numerically distinct. No wonder the conclusion follows. So, either there is an onquestion begging argument in favoro fP 3 or the four-dimensionalist can simply discard it.A nd as am atter of fact,s he should.⁴⁵  Alai (2016) explicitlye ndorses such ar econstruction.  As am atter of fact,i ti se ven ruled out by spatial variation as we understood it so far.  Someone might try to resist the question-beggingcharge by insistingt hat even four-dimensionalists can sayt hat the very same four-dimensional object instantiates "being F 1 -at-R 1 " and "being-F 2 -at-R 2 ".N ow,i fb eing F 1 -at-R 1 and being-F 2 -at-R 2 aret akent ob em onadic properties it might be argued that these aren ot incompatible properties.T he incompatible properties are F 1 and F 2 simpliciter. Iw ill put forwardt he same argument when consideringa nother possible Actually, there might be another wayf or four-dimensionalists to resist the argument without charging three-dimensionalists of question-begging.S he could remind her opponent that she could adopt aversion of de-re spatiotemporalp redication that mimics the counterpart-theoretic account of de-re modal predication. Some background details are in order here: Kripke (1972) famously puts forward what has become famousinthe literature as the "Humphrey objection" against counterpart theory,aspresented, e. g., in Lewis(1968). Accordingto the counterpart analysis of de-re modal talk ap roposition such as "Humphrey could have won the election" is true iff thereisapossible world in which acounterpart of Humphrey (exists and) did win the election. Kripke'scomplaint is that while Hubert Humphrey cares very much that he might have won the 1968 U.S. presidential election,h e" could not care less whether someone else, no matter how much resemblingh im, would have been victorious in another possible world" (1972,45). Lewis (1986,1 96) has as imple and effective answer,t hough: accordingt oc ounterpart theory,t he modal property of possiblyw inning is the property of having ac ounterpart who wins. Humphrey has ac ounterpart that wins, and so Humphrey himself has the right modal property,e xactlyb ecause he has the right counterpart. Now,f our-dimensionalists might want to sayt hat accordingt ot heir metaphysics the spatiotemporal property of being F 1 at,s ay,region R 1 is the property of having at emporal part exactlylocated at R 1 that is F 1 (the same goes, mutatis mutandis,f or F 2 and R 2 ). The four-dimensional object itself has the right spatiotemporalp roperties exactly because it has the right temporal parts. If this analysis is found compelling, the four-dimensionalist could then insist that the Temporal Parts solution and spatial variation are not analogous in arelevant respect: the four-dimensional object itself has the right sort of spatiotemporalproperties whereas this is not so in the case of spatial variation. Hence, the No-Changeobjection loses its force.
replyonb ehalf of the four-dimensionalist.Alternatively,itm ight be insistedthat the verysame four-dimensional object bears the relation F 1 to R 1 and the relation F 2 to R 2 . Iamnot surewhether four-dimensionalists would have anyr eason to sayt hat,a nd therei sa tl east one reason why they should not.One of the (alleged) advantageo ft he Temporal Parts solution is that it allows F 1 and F 2 to be truly monadic properties. In the end, this is what the argument from temporary intrinsics is cruciallya bout.H ence what the four-dimensionalist should sayi st hat,a tb est, the same four-dimensional object bears the relation F 1 *a tR 1 and F 2 *a tR 2 .And it bears those relations because it has distinct temporal parts that instantiate F 1 and F 2 . The point remains that the trulyi ncompatible monadicp roperties are F 1 and F 2 ,a nd these are had by numerically distinct temporal parts of the four-dimensional object.

No Time for( No) Change
This response has some appeal. Yetitshould be noted that the argument appeals to spatiotemporalproperties of "being F 1 at R 1 " and "being F 2 at R 2 ",rather than F 1 and F 2 simpliciter. And it might be argued thatthe spatiotemporal properties are not incompatible after all. Whateverthe fate of this de-re spatiotemporal predication, it remains the fact that the very formulation of the previous version of the No-changea rgument,b egst he question against the four-dimensionalist. In the light of this,w ec an safelyc onclude that the three-dimensionalist has failed so far in formulating at hreatening variant of the No-Changeo bjection.
(Un)fortunately, this is not the end of the story.For yetanother variant of the No-Changeargument can be givenbyfocusing on element (iii) from Mellor'spassage, namely the reference to distinct times. This is, as far as Ican see, the most interesting version of the No-Changeo bjection and the one that has the most wide-rangingi mplications.Iwillt herefore conclude this section by spelling out such av ariant in some detail.
We seem to have as trongp re-theoretical intuition that changer equires referencetodistinct earlier and later times, if not the passageoftime.⁴⁶ The passages from Geach and Mellor Iconsidered explicitlymention that reference. And the relevant examples are legion. One of the most commonlyq uoted passages that make such aplatitude explicit is arguablyinGödel (1949,558): "Changebecomes possibleo nlyt hrough the lapse of time".I fs o, the following argument can be put forward: (P 7 )R eferencet od istinct earlier and later timesi sn ecessary for change (P 8 )R eferencetodistinct earlier and later times is not necessary for spatial variation (P 9 )S patial variation is not genuine change( from P 7 ,P 8 ) (P 10 )T he Temporal Parts solution offers an account of changethatiscompletely analogous to spatial variation (C)T he Temporal Parts solution yields no genuine change( from P 9 ,P 10 )

4S patiotemporal Change
This section contains athorough discussion of the argument Ijust presented and of its metaphysical consequences. What has been said about the overall dialectic of the argument P 3 -P 6 /C applies here as well. Iw ill eventuallya rgue thatb oth  Presentists,orA -theorists moregenerally, would probablydisagreeinthat they will have an account of change in terms of tense operators that does not refert od istinct earlier and later times.H owever,a si ti se xplicit in §1, this papera ssumes some sort of B-theoreticf ramework.
three and four-dimensionalists should give up on P 7 and on our strong pre-theoretical intuition about the need to refer to distinct times in order to have genuine change. Or,a tl east,t hey should seriously consider this option.B ut before that,Iwill give another argument on behalf of the four-dimensionalist; the argument consists in rejectingP 10 in this precise context. Iw ill cast the argument in purelyt emporal terms first.T his contravenes both the letter and the spirit of the conclusiono f § 1. Yett his needst ob ed one, for the sake of perspicuity.
As Ijust pointed out,when the No-Changeargument is presented as Idid in P 7 -P 10 /C the four-dimensionalist could, and should, simplyd enyP 10 at first.The argument,s op hrased, is essentiallya na rgument by analogy. But consider the necessary requirement for changep resented in P 7 -i. e., the reference to distinct earlier and latert imes. Spatial variation and the Temporal Parts solution are not analogous in that respect. While it is true that referencet od istinct and later timesisnot necessary for spatial variation, reference to twodistinct temporalparts is necessary when providingaTemporal Parts solution to the puzzle of change. And this reference is an indirect reference to twodistinct and later times. To appreciatethis point,recall one of the most influential definitions of temporal part in the literature-the one in Sider (2001, 59). That definition is cast in purely temporalt erms: x is an (instantaneous) temporalp art of y at t =( i) x exists at t but only at t;(ii) x is part of y at t;(iii) x overlaps everything thatispart of y at t. It follows immediatelyfrom clause (i)-an uncontroversial one-that referencetotwo distinct temporalp arts entails ar eferencet ot wo distinct times. This should be enough to warrant the rejection of P 10 on the part of the four-dimensionalist.
This argument is however not immediatelya vailable to those who want to take the hints from Relativity at face value and thus aim to talk about spatiotemporal facts of different sorts. In this case, the spatiotemporalf acts behind the temporalfacts mentioned either explicitlyorimplicitly in the previous argument should be made clear.First of all (and naturallyenough), we should replacethe definitiono fat emporalp art in the argument with the one giveni n( 11), that mentionssolely spacetime regions that are the exactlocations of those temporal parts. But what about the requirementmentioned in P 7 /P 8 -i. e., the reference to two distinct times? Call it the "Distinct Times Requirement",DTR.This leadsto the following DTR-related question (DTRQ): DTRQ: What are the spatiotemporal facts behind DTR?
The problem of spellingout aprecise answer to DTRQ constitutes the main driving forcebehind the arguments in this section. Beforeg etting into details Ishall just give the bare skeleton of the overall argument.Isuggest thatt here are at least twow ayst oa nswer DTRQ,which Is halll abel Weak DTR and Strong DTR No Time for( No) Change answers. Is hall arguet hat (i) if the Weak DTR answer is endorsed then there is no No-Changeo bjection against four-dimensionalism and, (ii) if the Strong DTR answer is endorsed, then the Three-dimensionalist is in no better predicament than the Four-dimensionalist,b ecause premise P 7 of the No-Changea rgument abovef ails in both of their metaphysical pictures,r espectively.T his is enough to show that there is no No-Changeproblem thattroubles the four-dimensionalist alone. The argument in (ii) suggests ag eneral problem for persistencet heories. Buildingo nm yc ase against the No-Changea rgument,Iwill then set forth yeta nother argument for the claim thatt here is indeed some pressure to abandon P 7 anyway,t hus acceptingametaphysics of changet hatd oes not require reference to distinct times. This, Ic ontend, is as uggestion thats hould be explored further-and that, Is uspect,d eserves an independent scrutiny.
To getinto the dialectic Ijust sketched, consider first,e.g., two distinct achronal temporalparts x 1 and x 2 of an object y,thatare exactlylocated at maximal subregions of y'spath-i. e., R 1 and R 2 respectively.Aweak answer to DTRQ has it that the spatiotemporalfacts behindDTR are the ones captured in the following claim, that, as Is aid, Il abel Weak DTR: (18) There exist two points p 1 in R 1 and p 2 in R 2 such that the vector connecting them is causal, i. e., time-likeo rn ull.⁴⁷ In other words, according to Weak DTRt he existenceo fa ny twoc ausallys eparated points on R 1 and R 2 is enough to ensure ar eferencet od istinct and later times. It can be easilys een that if (18) does provide as atisfactory answer to DTRQ,i tf ollows from the definitions Ip rovided thatr eferencet od istinct times is entailed by the reference to two distinct temporalparts.Thissimplyfollows from the fact that distinct temporalparts are exactlylocated at distinct maximal subregions of ap ersisting object'sp ath. Thus, a spatiotemporal/relativistic friendly variant of the argument Is tarted with in this section is available to four-dimensionalists. They could-and simply should-reject P 10 .S o, in the end, there is no No-Changea rgument with Weak DTR. The problem with this replyi st hat (18) is far toow eakt oc laim rights to be an adequate answer to DTRQ.Ors os hould the three-dimensionalist contend at first.That answer,she should go on, should be much stronger.Infact, it should  Those whow ish to restrict their attention to material objects and want to claim that only fermions can be constituents of material objects,could replace causal with time-like. Aphoton, whose path is such that every two points in it aren ull-separated, is aboson, and thus will not count as ac onstituent of anym aterial object givent his account. be something like the following,which, for obvious reasons, Ilabel StrongD TR: (19) Forevery point p 1 in R 1 thereisapoint p 2 in R 2 such that the vector connecting them is causal.⁴⁸ GivenS TRONG DTR, the right wayt oe nsureareference to distinct and later times is to claim that for everyp oint in R 1 therei sa tl east one point on R 2 that is causallys eparated from it.Clearlye nough, (19) entails (18), but the converse does not hold. This is why ( 19) is much stronger. Whydoes this answer to DTRQ give the three-dimensionalist an advantage? The reason is thatn othing in the definitions we have givensofar guarantees that (19) is met by different exact locationso fd ifferent temporalp arts of four-dimensionalo bjects. Fori ne ffect, exact locationso ft emporal parts can overlap, and overlap entails the failure of (19).
Let us see the point in some details. Once again, consider the previous case, that is, consider two distinct achronal temporal parts x 1 and x 2 of y,t hat are exactlylocated at maximal subregions of y'spath, namely R 1 and R 2, and suppose, furthermore, that R 1 and R 2 do overlap. Since they overlap, therei sa tl east a point-call it p 1 ,that is part of both.Both R 1 and R 2 are achronal regions by definition of an achronal temporalpart.Thus, in particular,for R 2 we will have that: (20) Forevery two distinct points p, p 2 in R 2 the vector connectingthemisspacelike.
This holds for p 1 as well, which is, recall, part of both R 1 and R 2 . Thisj ust means that therei sa tl east ap oint in R 1 ,n amely p 1 thati ss pace-like separated from every other point in R 2 . Hence, if (20) holds, then (19) fails. Andi f ( 19) is the right answer to DTRQ,then three-dimensionalists have indeed aN o-Changea rgument against four-dimensionalism. Thisi saf airlyi nteresting argument indeed. Foritcruciallydepends on the fact that exact locations of achronal temporalp arts overlap. This happens onlyi nt he relativistic case. In fact,i nt he classicalcasedisjointnessofdistinct instantaneous temporal parts is guaranteed by,e.g., clause (i) in Sider'sd efinition of temporalp art above. Thus, if successful, the argument would give three-dimensionalism at rulyr elativistic argument against four-dimensionalism, contrary to the widespread agreement on the idea  See the previous footnote.

No Time for (No) Change
that relativistic physics,i fi td oes not support directly, at least favors af our-dimensional metaphysics.⁴⁹ But,o nce again, this is not the end of the story.I nf act,i nt he following I shall arguet hati f ( 19) is the right answer to DTRQ then three-dimensionalism has no advantage over four-dimensionalism. As Ip ointed out,a ccordingt ot he spatiotemporal solution to the puzzle of changet hat three-dimensionalists prefer,spacetime regions mediate the instantiation of properties.But (16), the rigorous formulation of the Spatiotemporal solution,fails to tell us which regions these are. Consider at hree-dimensional object.A rbitrarinessc onsiderations seem to favort he following Exact LocationP rinciple: (21) If athree-dimensional object x has property F-at-R then x is exactly located at R.
In other words: the spacetime regions that mediate the instantiation of properties in the Spatiotemporal solution to the puzzle of changeare the exactlocations of objects. By 'arbitrarinessconsideration' Imean the following.Any choice of R beside one of the exactl ocations of the three-dimensional object in question seems quite arbitrary.⁵⁰ Consider anyo ther region R 1 -d istinct from anyo ft he exact locations of the object.Why R 1 instead of, say, ar egion thath as R 1 as a proper part?W hy that,i nsteado faregion thati saproper part of R 1 ? Ia dmit that it would be better to have astrongerargument in favorof(21). Yetinthe absence of anyother plausible candidate, Icontend that it is at least the most natural choice. Note that some philosophers that are sympathetic with three-dimensionalism, most notablyG ibson and Pooley (2006,1 64),⁵¹ endorse such a principle explicitly, albeit with no argument. Let us go now back to the definition of locational three-dimensionalo bject in §2,i.e., definition (9). As we saw, it follows from thatdefinition that three-dimensional objects are exactlyl ocated at different achronal proper parts of the object'sp ath. But the definitioni ni tself does not suffice to single out which ones among the many. Following Gilmore (2006), this problem is sometimes called in the literature the location question (LQ). Roughly, the question is the following:  See Balashov (1999); Balashov (2010); Calosi (2015); Gilmore( 2006).  It might be thought that therei saless arbitrary suggestion: the entires pacetime. Yett his would contradict (16).  Here is arelevant quote: "The endurantist should hold that persistingobjects do not (in general) instantiateproperties simpliciter,but rather only relativetoparticularspacetime regions,viz. their locations" (Gibson and Pooley2 006,164). LQ: Let x be apersisting object.Which subregions of its path is x exactlylocated at? Gilmore (2006) classifies the answers that athree-dimensionalist mayprovide to LQ as either overlap or non-overlap answers. According to the latter,exact locations of three-dimensional objects do not overlap one another,whereas according to the former they do. Gilmore convincingly argues thatany non-overlap answer would not do in relativistic spacetimes.A samattero ff act,i ti sv irtually universallyh eld that,whatever the answer to LQ might be, it is an overlap answer.⁵² Actually, overlapping of exact locations can be regarded as the hallmark of the passage from ac lassical to ar elativistic setting.
We thus have the following:three-dimensional objects instantiateincompatible properties at their exact locations, and their exact locationscan overlapone another.F urthermore, it follows from the definition of at hree-dimensional object that these overlappinge xact locations are achronal. Hence, the situation three-dimensionalists are in is exactlyt he same situation thatw as supposedt o spell trouble for four-dimensionalists in the first place.
Af urther wayt od escribe the predicament is the following.The No-Change objectiona gainst four-dimensionalism thatIam consideringc ruciallyd epends on the fact that different temporal parts that instantiate incompatible properties overlap one another. But the same argument would concern the three-dimensionalist as well, insofar as the relevant spacetime regions thatm ediate the instantiation of incompatible properties-i. e., the exact locations of the three-dimensional object-overlap one another.
This leads to the following conclusion: if (19) is the right answer to DTRQ, then the No-Change argument P 7 -P 10 /C cuts both ways. Beforem oving on to sum up the overall dialectic of the No-Changea rgument,i ti sw orth spending some time on aquestion that naturallyarises in this context.AsIalreadypointed out,the previous argument depends cruciallyonthe fact thatexact locations of achronal temporalp arts on the one hand,a nd exactl ocations of persisting three-dimensional objectsonthe other, overlap one another. Such exact locations turn out to be the major actors-sot os peak-in the metaphysics of change. This raises an interesting question -one that Is hall label the possibility of overlapping change question (POCQ):  The fact that LQ is such ad ifficultq uestion for three-dimensionalist to answer,whereas it has asimple answer for four-dimensionalists -x is exactlylocated at the onlyimproper subregion of its path, namelythe path itself-can be takenasastartingpoint for an argument favoringfourdimensionalism. See Gilmore( 2006).

POCQ: Is changep ossible at overlapping regions?
Now,e ither change is possible at overlappingr egions,o ri tf ails to be; no third possibilityisgiven. In fact,anumber of arguments seem to favorthe former option. First,i fc hangei snot possiblea to verlapping regions,then we should not count relativistic length contraction as ac hange.⁵³ Yeti td oes seem to have all the credentials to be counted as one. As am atter of fact,a ccording to an influential, though highlycontroversial, explanation of relativistic length contraction -the one that is labeled dynamical explanation and was advocatedi nB rown (2005) -the phenomenon is as genuineachangea sa ny one we are familiar with.⁵⁴ One needsnot relyonBrown'sdynamicalexplanation. Consider the account of relativisticlength-contraction threeand four-dimensionalists are likelytogive. Three-dimensionalists should claim that they measure different lengths of the same rodatdifferent spacetime regions that are both among the exactlocations of the rod. Four-dimensionalists on the other hand should claim thatt hey measure different temporal parts of different lengths of the same four-dimensional rod. Upon inspection, this fits exactlythe template for ageneral case of change.
Asecond argument to the sameend is more general and-Ithink-the most effective one available. Overlap is not transitive.N ow consider,f or instance, three distinct exactl ocations of at hree-dimensional object⁵⁵ R 1 , R 2 , R 3 such that R 1 overlaps R 2 , R 2 overlaps R 3 ,b ut R 1 and R 3 do not overlap, as in Fig.1. If changeisnot possible between R 1 and R 2 -that is, x, by hypothesis cannot have incompatible properties at R 1 andR 2 -because they overlap, then x has the same  This is because length contraction can be understood as the differenceinthe rod'slength at different exact locations that overlap each other,where "different lengths" aret aken to be mutuallyincompatible properties.This formulation implicitlyassumesathree-dimensional ontology,though: four-dimensionalists should give ad ifferent account of length contraction, perhaps along the lines of footnote5 6. However,the point about overlap and the incompatibility of different length-properties would arise in this contexta sw ell.  Idonot aim to subscribe to Brown'sinterpretation here.Asamatter of fact,Ibelievethat,if possible, we should stick to the so-called "geometric explanation".T he point has no consequencef or the sake of the dialectic heret hough -hence, Iw ill not pursue it anyf urther.F or ac ritique of Brown'sa pproach see Norton (2008).  The argument applies, mutatis mutandis,tofour-dimensionalobjects as well, as can be seen by just phrasing it in terms of achronal temporal parts exactlylocated at R 1 , R 2 and R 3 . However,I will focus on the argument as appliedtothree-dimensional objects because Ifind it moreinteresting -in particular,given the wayitconnects to the answers to LQ that Idiscuss in this paper. The formulation of the parallel argument that can be raised about four-dimensional objects is left to the reader.
properties at R 1 and R 2 . The same goes for R 2 and R 3 . But then it follows that x has the sameproperties at R 1 and at R 3 -regions that do not overlap.⁵⁶ No matter what region(s) we consider,the argument will run the same way. But then, givent he assumptions, how can x changep roperties at all?I no ther words: suppose changei sn ot possiblea to verlappingr egions.T hen, givent hato verlap is not transitive,i ti sd ifficultt os ee how at hree-dimensional object could changea t all. Hence, Icontend that we should answer PCQ in the affirmative and conclude that changea to verlappingr egions is indeed possible.
There might be two ways the three-dimensionalist can replyatthis point.According to the first,d espite the fact that three-dimensionalo bjectsa re exactly located at overlappingr egions,t hey changeo nlya tn on-overlappingo nes. This would amount to saying that,f or anyt wo non-overlappingr egions such that the object changes its properties at thoser egions,t here is no other region that overlaps the two that counts as an exact location of that object.B ut this just means to have aN on-Overlap answer to LQ -an option that wasa lreadyr uled out.A ccordingtothe second, it onlymakes sense to ask whether changeoccurs along at hing'sp ath when the path is foliated, which requires that every leaf is mereologicallyd isjoint from every other.This is allegedlyc ompatible with saying,e .g., that a3 Dt hing is exactlyl ocated at overlappingr egions:f or it only says that some of these regions playarole in allowing genuine change.
The problem with this proposal is that the foliation of the path that is doing most of the metaphysical work here is nothing but areference frame in disguise. Indeed, in non-pathological spacetimes,such afoliation could easilybeextended to construct such af rame. To appreciatet he point,c onsider the following.  This is actuallytoo strong. Change requires incompatible properties (or so Iassumed). From the mere hypothesis that change is not possible at overlapping regions it does not follow that x has all the same properties at those regions.I tm ight very well have different properties, albeit not incompatible ones. This "variation" in properties would however not counta sg enuine change.
Take the foliation of the path. Embed every mereologicallyd isjointl eaf of that foliation into ag loballyu nextended space-likeh ypersurface. Then construct an entiref amilyo fs pace-like hypersurfaces that are everywherep arallel to the ones yous tarted with. This familyo fh ypersurfaces counts as af oliation of the entire spacetime. Af rame of reference is then obtained by taking at ime-like line that is orthogonal to anyhypersurface in the global foliation. This construction should be slightlym odified if we want to consider non-flat foliations.The problem with this proposal is that it would run against the spirit of the conclusion of §1. Remember Gibson and Pooley'swords: "no one should attach significance to properties of objects thata re essentiallyd efined in terms of canonical frames of reference".⁵⁷ Where does thatleave us?W esaw thatthere is an interesting No-Changeargument P 7 -P 10 /C that builds upon the intuitiveclaim that "changerequires reference to distinct times".Iconsidered several ways to account for such ar equirement.I ft he requirement is accounted for in terms of what Il abeled Weak DTR, then, Ia rgued, four-dimensionalism is not touched by such an argument.F ourdimensionalists should reject P 10 . If it is accounted for in terms of Strong DTR, then four-dimensionalism does indeedf ace it,b ut so does three-dimensionalism. In fact,s upposew ed oi nsist that DTRs hould be understood via Strong DTR.N ow,t he following two alternativesp resent themselves. Either both three-dimensionalsists and four-dimensionalsists raise up their arms and claim that they cannot account for genuine change, insofar as their metaphysics cannot meet DTR, or they claim that their solution does indeedaccount for genuine change, but changed oes not require satisfying DTR. And this amounts to abandon P 7 ,f or P 7 exactlys ayst hat satisfying DTRi sn ecessary to account for genuine change. Indeed, Iwill just assume thatthe latter is the wayboth parties should go.That is to saythat,should three and four-dimensionalists be confront- If change is possible at overlappingr egions,F our-dimensionalists betterh aveo verlapping temporal parts.I ng eneral, this follows fromt he very definition of general temporal part,a sa sum of achronal ones,g iven that parthoode ntails overlap.I ti si ndeed orthodoxy that four-dimensional objectsh avet emporal parts at all maximal sliceso ft heir path. Gibson and Pooley write: "[F]rom ar elativistic point of view,t he assumption that ap erduringo bject has parts at every proper subregion of its worldtube is overwhelminglyn atural. Call this the doctrine of arbitrarys patiotemporal parts. From ar elativistic point of view,i ts hould be as tartingp oint, not somethingt hat falls out from af rame-relative generalization of the non-relativistic notion of atemporal part together with unrestricted composition. Indeed, from the relativistic perspective,the existenceofspecifically 'temporal' parts of an object does not even warrant comment" (Gibson and Pooley2006,162). On topofthat,the No-Changeargument we areconsideringturns exactlyo n4 Do bjects havingo verlappingt emporal parts.I ft his not grantedt herei sn oN o-Change argument to begin with. ed with an argument to the point that StrongD TR is the right answer to DTRQ, they should simply discard P 7 .Iwill put forward as ketch of such an argument myself.
Before turning to that, however,Iwill consider yetanother possibilityonbehalf of three-dimensionalists. Forthere might be away to recover the asymmetry about the No-Changea rgument-i. e., the dialectic situation that made their view, as opposedt othe Four-dimensionalist's, immunet ot he argument.T he strategywould be that of providingananswer to DTRQ thatwould be in between Weak DTR and StrongD TR.T he in-between answer should be such that( i) three-dimensionalists could build aNo-Changeargument in terms of such an answer,a nd (ii) that variant of the No-Changea rgument affects four-dimensionalism alone. Iknow of no suggestion likethis in the literature, so Iwill just present it as achallenge. It is up to three-dimensionalists to come up with an interesting proposal in this sense. However,i tw ould not be enough for them to provide an in-between answer thats uits their aim -i. e., that satisfies (i) and (ii). They also should effectivelyargue that (iii) such an answer is the best waytospellout the intuitiveD TR requirement in P 7 .
Having said that,Iwill now set forth the final-yetIam afraid not fullyfledged-argument of the paper.Answers to DTRQ provide different ways to account for the intuitive claim that changer equires reference to two distinct times-i. e., the DTRr equirement as mentioned in premise P 7 of the No-Changea rgument at the end of §3-in ways that take seriouslyt he suggestions that come from Relativity.N ow,i nt he light of the above, there seem to be two options. On the one hand, it can be insisted that Weak DTR is the right waytoconstrue the intuition behindD TR.I fs o, Ih avea rgued, botht hreea nd four-dimensionalism can stick to the DTRrequirement.This is because the No-Changeargument is not athreat. On the other hand,itcan be insisted thatStrongDTR is the right waytogo. But then, Ihaveargued, it turnsout that boththe metaphysics of persistenceIhave considered have to abandon P 7 ,a nd hence reject the DTRr equirement itself. What horn should one take?
In what follows, Iwill suggest that there is some pressuretotake the second, but Is hall admit (once again) that Ih ere simplyg esture towards an argument rather than providing af ully-fledgedo ne.
Ia lreadya rgued that changea to verlapping regions is possible. Regions R 1 , R 2 can overlap in manydifferent ways.One such wayisdepicted in Fig. 2below.⁵⁸ Itakeitthatthe intuition behind the DTRrequirement is the following:when an object changes from having F 1 at t 1 to having F 2 at t 2 , t 2 is later than t 1 . But sup- The first to bringa ttention to this point was Gilmore( 2006). See also Balashov (2014b).

No Time for (No) Change
pose the persisting object x depicted in Fig. 2changes from having F 1 at R 1 to having F 2 at R 2 . Can we still hang on to the intuition behind DTR? It seems that we should give anegative answer to this question, for there are points p 1 and q 1 on R 1 and points p 2 and q 2 on R 2 such that the time-likev ector p 1 p 2 is future directed, whereas q 1 q 2 is past directed.⁵⁹ In otherw ords, there are parts of R 1 that strictly temporally precede parts of R 2 (e. g., p 1 strictlyt emporallyp recedes p 2 ), whereas there are parts of R 2 that strictly temporally precede parts of R 1 (e. g., q 2 strictly precedes p 2 ).
Givenall this, Icontend that there is some pressuretoclaim thatW eak DTR is tooweakananswer to DTRQ.F or in effect,Weak DTR is not strongenough to rule out possibilitiessuch as the one depicted in Fig. 2. And, crucially, the intuitive pressurefor DTRdemands exactlythat possibilitiessuch as that one have to be ruled out. On the other hand, StrongDTR does fill the bill. Hence, Iargue, we should insist that this is the right answer to DTR-the right way, that is, to capture the DTRr equirement.
And this leads to my final point.Ihave argued that if StrongDTR is indeed the right answer to DTRQ,itfollows that,ifboth three-dimensionalism and fourdimentionalism have anyright to claim that they provide an account of genuine change, then they both have to abandon P 7 and claim-contra both our pre-theoretic intuitions and awidespread philosophicalagreement -that reference to distinct times is not necessaryf or change. Now,this maysoundcontroversial and counterintuitive.Yet there is no need to gettoo alarmed.Physicists and philosophers of physicshavealreadyset forth  Admittedly, this restricts our attention to so-called "time-orientable spacetimes".This is no reasont og et alarmed though:f or in effect, temporal orientability is avery weak condition. Any manifold with aL orentzian metric that admits the definitiono facontinuous time-like vector field is, as am atter of fact,time-orientable. independent suggestions to the effect thatc hanged oes not requiret ime at all. Earman (2002) suggests that,ifwelook at the deep structure of GTR,wewill indeed find cases of temporally-independent change. Rovelli (2011) develops aformalismf or aQ uantum theory of gravity in which dependent and independent variables are treated in one and the same waya nd time does not playa ny role; such af ormalismi sn onetheless supposedt oa ccount for change. Huggett and Wüthrich (2013) consider different candidates for theories of Quantum Gravity (Loop Quantum Gravity,S tringt heory and Causal Set-theory) wheret he fundamental ontology does not include spacetimea nd yeti ts items do undergo significant changes. If the arguments in this paper are right,t hen the seeds of such an intriguing and deep suggestion werealreadyinclassicalrelativistic physics,o rb etter,a tt he intersection between relativistic physics and (relativistic) metaphysics.
In effect,one might find the arguments from Quantum Gravitymore compelling than the ones from classic relativisticphysics.Iwould actuallyagree. But it is still interesting that something that will become clear onlyi nl ater theories was somehow alreadyh idden in classical Relativity theory.T he conclusion all these considerations points at is that spatiotemporalvariation is changeenough. And it is all around us.