Oliver Pooley (Oxford): Relativity, the Open Future, and the Passage of Time

Relativity, the Open Future, and the Passage of Time

oliver pooleyuniversity of oxford

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Oliver Pooley a University Lecturer in the Faculty of Philosophy at the University of Oxford and a Fellow and Tutor in Philosophy at Oriel College, Oxford. He works in the philosophy of physics and in metaphysics. Much of his research focuses on the nature of space, time and spacetime. Oliver read Physics and Philosophy at Balliol College, and took an MASt in Maths at St John’s College, Cambridge, before returning to Oxford to do graduate work in Philosophy. Before taking up his current position at Oriel, he was a British Academy Postdoctoral Fellow at Exeter College, Oxford.

e d i t o r i a l n o t e

The following paper is a draft version that can only be cited with the author’s permission. The final paper will be published in Proceedings of the Aristotelian Society, Issue No. 3, Volume CXIII (2013). Please visit the Society’s website for subscription information: www.aristoteliansociety.org.uk.

Relativity, the Open Future, and the Passage ofTime

Oliver Pooley,Oriel College, Oxford

�is is work in progress. Please do quote or cite this dra� without the permissionof the author. Comments gratefully received.

� �e ProjectI believe that the “block universe” picture of reality leaves nothing out, at least inits description of time. I take it to follow from this core B �eory commitmentthat time does not pass: there is no objective “�ow” of time itself. Defenders oftemporal passage typically suppose that the history of the world is partitioned intoa totally ordered sequence of instants, each corresponding in turn to the state of theuniverse as time unfolds. �e simultaneity surfaces of classical (i.e., non-relativistic)spacetimes admit such an interpretation, but relativistic physics is far less hospitable.Some relativistic spacetimes come with physically preferred foliations, but it is notstraightforward to connect these to traditional conceptions of temporal passage.

Does this mean that relativistic physics teaches us that time does not pass?Only if the B�eory, and views which understand passage in terms of a sequence ofspatially global Nows, together exhaust the options. A small number of philosophersand physicists have suggested that there is an overlooked alternative: one can respectthe spatiotemporal symmetries of relativistic physics and yet maintain that there ismore to time’s passage than the block universe picture permits (Stein, ��, ��;Shimony, ��; Sorkin, ��).

Although this paper is ultimately concerned with the viability of views of thiskind, much of it is devoted to pre-relativistic notions of passage. �e justi�cation isthat clarity about the classical case is a precondition for clarity about the relativisticcase. A�er setting out the core commitments of the B�eory, I consider A-theoretic

�

notions of temporal passage in the context of classical physics.� A key element of myinvestigation involves the distinction between certain familiar formal models andhow these models are supposed to express the central commitments of the defenderof real passage. In these terms, generalisation of A-theoretic views to relativityinvolves two stages: (�) identify relativistic analogues of the classical formal models;and (�) explain how such models articulate a non-B-theoretic view of time. �e�rst step is relatively straightforward. Generalising to relativity the kind of storiesthat can be told about pre-relativistic formal models is more di�cult.

� �e Block UniverseAs I am using the label, the B�eory of time involves two core commitments:

(a) Fundamentally speaking, all times are on a par.

(b) A complete description of reality can be given in tenseless terms.

�e best way to grasp the intended content of these claims is via spatial analogies.Consider, �rst, the spatial analogue of claim (a). Almost no one believes that anyparticular spatial location is ontologically, or metaphysically privileged. Spatialplaces di�er in all sorts of ways. One part of space currently contains me; anothersimilarly-shaped part is completely �lled with a portion of the Paci�c Ocean. Butthese di�erences do not make any particular place special in the relevant metaphys-ical, or ontological senses. Fundamentally speaking, they are all an equal part ofreality. In particular, our immediate spatial location, the spatial analogue of thepresent time, is not metaphysically special, whether by virtue of the possession ofsome peculiar property or otherwise.

�e spatial analogue of claim (b) is that a description of reality that includes thespatial disposition of all objects and events relative to one another (“object a is �vemetres from object b, in the direction de�ned by the line joining objects c and d”etc.) leaves nothing out, spatially speaking.� One does not need to further specifythat object a is here, or twenty metres to the le�. Such information simply servesto locate objects spatially relative to ourselves, and (barring global symmetries) isanyway deducible: if the view-from-nowhere characterisation of reality is reallycomplete, then one can in principle locate oneself and determine one’s orientationin the world so described.

�I will use the term “A-theoretic” (perhaps somewhat non-standardly) as a convenient label forany view of time according to which there is more to time than is captured by the B theorist’s blockuniverse.

�Spatial points might be amongst the objects in question, so this characterisation is neutralbetween substantivalist and relationalist views of space.

�

Consider, now, the present-tensed claim that you are reading. �is is a trueclaim—you are, in fact, reading now. It is also a fact that (as I write this) it is cloudy,i.e., cloudy here. �e B theorist claims that such facts are analogous. �eir truthmerely re�ects our spatiotemporal perspective. Just as the fact that it is cloudy hereis to be understood in terms of the spatially non-indexical fact that it is (on therelevant date) overcast at a latitude of ��.�°N and a longitude of ��.�°W, so too thefact that you are (now) reading is to be understood in terms of the tenseless claimthat you are (in the tenseless sense of “are”) reading at . . . on . . . (look at your watch,and �ll in the blanks). �e standard A-theoretic alternative denies this, claiming thatit is a rock-bottom fact (at least in its temporal aspects) that you are (present tensed)reading. On this view, a description of reality that includes the dated tenseless fact,but omits the tensed fact, leaves something out. What would seem to be missing isa speci�cation of which time is now, objectively speaking, for once that is given, thetensed facts follow from the dated, tenseless facts.

To treat time like space in the way just characterised clearly does not amountto treating time as “just another dimension.” �e minimal B�eory commitmentsare quite consistent with there being deep and important di�erences between timeand space. In particular, the traditional B-theoretic relation of “earlier than” hasno spatial analogue, and a B theorist might hold that fundamental distinctionsbetween time and space can be articulated in terms of notions that are not obviouslyspatiotemporal, for example, in terms of causation (as in, e.g., Mellor, ��).

For ease of exposition, my characterisation of the B �eory has employed arelativistically suspect notion. In spelling out the view, I made free use of talkof times (and of spatial places). According to relativity, at a fundamental levelthere simply are no times (or spatial places). �is means that claim (a) needs tobe revised, but its relativistic generalisation is straightforward. It is the thesis that,fundamentally speaking, all regions of spacetime are on a par, regardless of theparticularities of their extension in spatial, temporal and null directions. Claim (b)already makes perfectly sense in a relativistic context; the mathematical languageused to specify relativistic spacetime models is tenseless.� �e core of argumentsfrom relativity to the B�eory is that the classical A-theoretic alternatives to theblock universe do not generalise in the same way.

��is is not to say that relativity poses the B theorist no problems. Even those who do not taketense metaphysically seriously need to give an account of the truth conditions for tensed language(and of our ordinary talk of times) as used in a world that, fundamentally, does not contain times(see, e.g., Gibson and Pooley, ��, ��–�).

�

� Varieties of PassageIn denying that time passes according to the B �eory, I do not deny that manyordinary, everyday claims that use phrases like “time passes” or “the passage of time”can be true. With the passage of time, my views about the metaphysics of time have,I hope, become clearer and more articulate. And as I sit at my keyboard for minuteson end, not typing, I might be seized by the realization that time is passing while thepaper remains unwritten. But in cases such as these, the B theorist has no troublespelling out B-theoretic facts that are the objects of my attitudes.� �e B theory doesnot prohibit time’s passage in this subject-centered sense.

Relatively recently, several authors have insisted that time passes according tothe B�eory, even independently of the perspectives of subjects embedded withintime.� Dieks (��), for example, equates “temporal becoming” with the (non-perspectival) “successive coming into being of events.” �is sounds like it shouldbe incompatible with the B�eory, according to which all events tenselessly exist,each at their particular spatiotemporal location. �e block universe undergoes nochange, so how can some part of it “come into being”? Doesn’t that require a changein the block, from a state in which it did not have that event as a part, to a state inwhich it does? Dieks, however, is not proposing any such thing. Instead, he holdsthat an event’s coming into being is simply its happening (“what other coming intobeing could there be?”). He then notes that:

Since everything that happens is recorded in the block universe diagram,‘coming into being’ is also fully represented. . .�is proposal boils downto a de�ationary analysis of becoming: becoming is nothing but thehappening of events, in their temporal order. (Dieks, ��, ��–�)

If one wishes to label the successive occurrence of events “temporal passage”then, yes, time passes according to the B�eory. John Earman rightly labels thisa “thin and yawn-inducing” sense of passage (Earman, ��, ��). Its advocatesseem to be making heavy weather of facts that (almost) no one has ever denied. �equestion whether time really passes surely marks a central fault line in metaphysicaldebates over the nature of time. By pro�ering something less than becoming, andlabelling it “the passage of time”, one risks loosing a useful way of talking about anissue that genuinely divides B theorists from the majority of their opponents.

What, then, is the passage of time supposed to be? A central idea is ubiquitous:

��e B theorist’s story is likely to distinguish di�erent ways in which one can be acquainted withone and the same B-theoretic fact. Just as typical responses to Perry (��) do not posit fundamentalde se or spatially indexical facts, so it seem the analogous temporal phenomena do not requireprimitive de nunc facts. �is is the essential insight of the so-called new B�eory of time. For aB-theoretic account of the role tensed beliefs in timely action, see Torre (��).

�I have in mind, in particular, Dorato (��), Savitt (��) and Dieks (��).

�

Suppose we speak about something ‘becoming more past’ not onlywhen it moves from the comparatively near past to the comparativelydistant past, but also when it moves from the present to the past, fromthe future to the present, and from the comparatively distant future tothe comparatively near future. Whatever is happening, has happened,or will happen is all the time ‘becoming more past’ in this extendedsense; and just this is what we mean by the �ow or passage of time.(Prior, ��, �–�)

Time passes. Nothing fancy is meant by that. It is just the mundane factknown to us all that future events will become present and then dri� o�into the past. . .Time really passes. . .Our sense of passage is our largelypassive experience of a fact about the way time truly is, objectively. �efact of passage obtains independently of us. (Norton, ��, ��)�

�e B theorist denies that it is any kind of objective fact that, independentlyof the serial perspectives on the world of agents embedded in time, future eventsbecome present and then dri� o� into the past. Temporal passage in this sense isincompatible with the B�eory. And yet we do have a strong pre-theoretical incli-nation to judge that time passes, objectively, in just this sense. �e B theorist owesus an explanation of this inclination. Mention of our multiple, temporally-orderedperspectives on reality was intended to indicate the shape that many B theoristsbelieve the account must take. I do not think that a completely satisfactory versionof this story has yet been told, but I say no more about that project in this paper.From here on in, the focus is on views that seek to vindicate passage, rather than toexplain it away.

�is next three sections review three such views: presentism, the Moving Spot-light view, and the Growing Block view. �e examination of each builds on thepreceding discussion. In particular, a grasp how presentism vindicates temporalpassage is key to understanding the claims of the other two views. I then turn tothe possibility of “non-standard” variants of these theories, which give up the ideathat are absolute facts concerning how reality is. �e review of all these theorieslays the groundwork for discussion of views that link the passage of time to the ideathat the future is open. �e remainder of the paper then investigates the impactof relativity. Relativistic generalisations of the the Moving Spotlight view and theGrowing Block view have recently been considered by Skow (��) and by Earman(��) respectively. Reviewing their e�orts will pave the way for consideration ofrelativistic generalisations of views that link passage to the open future.

�For similar descriptions by critics of passage, see, e.g., Smart (��, ��) and Olson (��).

�

� Presentism and the Passage of TimePresentism is sometimes informally characterised as the view that only the presenttime exists.� It is true that the presentist, unlike the B theorist, does not believethat past or future times are distant parts of concrete reality. But this is not becausethey believe in the existence of just one of the B theorist’s many times, a single �-dimensional slice of the B theorist’s block. According to the presentist, the materialworld is extended in only three spatial dimensions, and not extended in a temporaldimension. �e world is not naturally characterised as the present time. A bettercharacterisation of presentism starts with the observation that, for the presentist,truth simpliciter is tensed. Truth simpliciter is just what is presently true. �e presenttruth about our three-dimensional world exhaustively characterises reality, and thisincludes how it was and how it will be, as well as how it presently is.

For the presentist, tensed facts are not reducible to how things tenselessly areat di�erent parts of a temporally extended reality. In fact, the presentist holdsthat the opposite is true. How things “tenselessly” are at past and future timesis to be analysed in terms of present, tensed truth. Times, including the present,are logical constructs that allow for a certain kind of elegant representation ofthe fundamental, tensed facts. It is a representation that allows for a certain kindof abstraction: one can represent the history of the world without representingwhich stage of that history the world has in fact reached. With presentist times sounderstood, we are at liberty to use time talk again.� It follows that the block universemodel, supplemented with an indication of which three dimensional subregion isthe present (together, perhaps, with a future-pointing arrow), exactly encodes thepresentist’s commitments. One need only be careful not to misinterpret di�erentparts of the block as corresponding to di�erent parts of a tenselessly existing concretereality. Instead they collectively represent all that is happening, has happened orwill happen.

A natural thought at this point is that something crucial is still missing fromthe picture. In what sense does it capture the passage of time? Doesn’t the pictureneed continual updating? Doesn’t the region of the block representing the presentneed to move up the block, in the direction of the arrow? Perhaps the single blockneeds to be replaced by uncountably many copies, each with the present di�erentlylocated, each representing the di�erent sets of tensed facts that hold as time passes.Here is how Kit Fine expresses the worry:

�e passage of time requires that the moments of time be successively

�A popular alternative, also in terms of existence, involves claims such as: necessarily, it is alwaystrue that only present objects exist (see, e.g., Markosian, ��). �is claim is intended to be implicitin the characterisation of presentism advocated in the text.

�A popular presentist move identi�es times with maximal propositions of a certain kind. Mycharacterisation of presentism is intended to be compatible with this ploy.

�

present and this appears to require more than the presentness of asingle moment of time. �e [presentist] at this point might appeal tothe fact that any particular future time t+ will be present and that anyparticular past time t− was present. However, the future presentness oft+ amounts to no more that t being present and t+ being later than t. . .We naturally read more into the [presentist]’s tense-logical pronounce-ments than they actually convey. But his conception of temporal reality,once it is seen for what it is, is as static or block-like as the [B theorist]’s,the only di�erence lying in the fact that his block has a privileged centre.(Fine, ��, ��)

�e worry is misplaced. To see why, we need to rehearse Prior’s account of presentistpassage.

Prior �rst glosses the passage of time in terms of events becoming ever morepast. He proceeds to unpack this in tense-theoretic terms. Part of the story involvesan eliminativism about events. Consider the recession into the past of Prior’s fallingout of a punt. According to Prior, to say that this event has so far receded �� yearsinto the past is to say nothing more than that Prior fell from a punt �� years ago.Or, in semi-regimented language using metric tense-operators, WAS��y(Prior isfalling out of a punt).� Prior contends that this last sentence is not about any objectsexcept him and the punt from which he fell: “there is no real reason to believe inthe existence either now or [��] years ago of a further object called ‘my falling outof a punt’” (Prior, ��, ��).

So far we have considered what the Priorian presentist means by apparent talkof events and their current pastness, or futurity. What about the changes in pastnessand futurity that are constitutive of time’s passage? Consider, again, Prior’s fall froma punt. It was, just a year ago, only �� years in the past. It is now no longer only ��years in the past. In another year, it will be the case that Prior fell from a punt ��years previously: the event will have receded a further year into the past. Letting ‘P’stand for “Prior is falling out of a punt,” in Prior’s operator-theoretic regimentation,one has:

WAS�y(WAS��y(P)) ∧ ¬WAS��y(P) ∧WAS��y(P) ∧WILL�y(WAS��y(P)). (�)

Can tense-theoretic claims like this really be all that the presentist needs toexpress the passage of time? Two re�ections might help to assuage doubts. First,compare these tensed claims to the presentist’s expression of everyday changes inordinary objects, for examplemy change in shape as I go from standing to sitting. Forthe presentist, this amounts to the fact that I am now sitting but that I was, a momentago, standing, and so not sitting. In other words, change involves conjunctions

�“WAS��y(P)” is to be read as “It was the case �� years ago that P” etc.

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of the form P ∧WASn(¬P). But, with a little rede�nition, this is exactly what wehave in the �rst two conjuncts in (�). �e combination of the tensed facts (i) that Iwas standing but (ii) that I am not now standing amounts to genuine change. Whyshould the same not be true of the combination of the tensed facts (i) that Prior’sfalling was �� years ago and (ii) that this falling is not now �� years ago?

�e second re�ection is this. If this second conjunction is genuine change, whatkind of change is it? What is changing? We have seen that, for Prior, it is not theevent of his falling, rather it is Prior himself.�� Now, going from being such that onefell �� years ago, to being such that one fell �� years ago might seem to be a veryanaemic kind of change. But this is exactly what one wants. It is exactly the changethat is due to no more than time’s passing. All robust intrinsic change could ceasebut, the defender of real passage thinks, things would continue to change merely bybecoming older, for time continues to pass.��

Let us returning to Fine’s charge that the presentism is as static and block-likeas the B�eory. One way to put the worry is that a block with a privileged centrerepresents only a proper sub-collection of the full set of tensed truths that obtainover time, as time passes. We have seen, however, that this set of present truthsexpresses genuine change. And, while some of these truths (like the facts that I’mnow sitting but that I was standing) concern ordinary change in ordinary objects,others express the very change we are concerned with, viz., change in what is true.�e present, tensed facts include, for example, facts to the e�ect that certain tensedpropositions are not now true but that they were or will be true.�� One simply cannotaccept all the present, tensed truths without accepting that what is true undergoesgenuine change.

Deng (��, ��–��) provides a sympathetic application of Fine’s criticism to

��Save for the fact that he no longer exists. How to treat apparently singular propositions aboutthings that no longer exist is a well known problem for presentism, but it is not per se a problem fortheir account of passage. Realistically, at any time there will always be numerous persisting thingsthen existing to which passage-related change can be ascribed. But even in a maximally evanescentworld that lacks persisting entities, one still has combinations of the formWAS(∃xFx) ∧ ¬Fx. ForPrior, this “quasi change” (which does not require a presently existing entity that once was F but isso no longer) is su�cient for passage.

��Note that Prior’s eliminativism about events turn out to be optional; one can combine aDavidsonian commitment to events with a Priorian account of passage. Regarding Prior’s fallingfrom a punt, one has combinations like the following: ¬∃x(x is a falling from a punt by Prior) andWAS(∃x(x is a falling from a punt by Prior)). �e identi�cation of genuine change goes throughas before, albeit without a subject of change. In fact, if unlike Prior but as entertained by Williamson(��), one accepts the tense-logical Barcan formula, then WAS(∃xFx) implies ∃xWAS(Fx), andone can provide something close to a literal rendition of the claim that events are forever recedinginto the past: (what were) past events still exist (now) and are continually changing. Something nowexists that was, �� years ago, a falling from a punt by Prior. And in a year’s time it will be �� yearssince this entity was a falling from a punt by Prior.

��See Markosian (��, ��) for a Prior-like cashing out of passage in closely related terms.

�

presentist accounts of passage of the kind considered here. Her conclusion is notthat time does not pass according to presentism, but that presentism is no better atcapturing passage than the B�eory. However, what we have seen is that presen-tism vindicates, (i) events objectively becoming ever more past, (ii) an absolutelyprivileged time and (iii) genuine change in which time is privileged. Together thesesurely correspond to a sense in which there is real passage, one that is (rightly!)missing from the B�eory.

� Passage and�e Moving Spotlight�e “moving” in “moving spotlight” signals that the view is supposed to capture theobjective passage of time. What is the view?

Here is Brad Skow’s description, in a paper concerned with its potential relativis-tic generalisation:

�e theory combines eternalism—the doctrine that past, present, andfuture times all exist—with “objective becoming.” �e claim that thereis objective becoming has two parts. First, facts about which timeis present are nonrelative. �at is, even if in some sense each timeis present relative to itself, only one time is absolutely present. . .Andsecond, which instant is absolutely present keeps changing. �e NOWmoves along the series of times from earlier times to later times. (Skow,��, ��)

In virtue of the �rst of these claims, the Moving Spotlight theory di�ers frompresentism. �e Moving Spotlight theorist holds that reality really is extended infour dimensions.

What sense can be made of the movement of the NOW along the time series? Afamiliar “two times” objection rears its head.�� Movement is just change in locationin some space with respect to time. But since the movement in question is thatof the NOW’s position in time, it seems that one needs to postulate an additionaltemporal dimension—supertime—with respect to which the NOW’s position inordinary time can be said to change. �e smooth movement of the NOW, frompast to future, is cashed out as the fact that, from the perspectives of ever Later��supertimes, the NOW is located at ever later times.

Skow thinks that the supertime picture is a useful aid to grasping the content ofthe theory rather than a reductio of it. He also insists that supertime is strictly a�ction. His o�cial story is spelled out in terms of primitive tense operators: “If it is

��See, e.g., Broad (��, ��) and, for the classic exposition, Smart (��).��I follow Skow’s convention of distinguishing relations that order supertimes from those that

order ordinary times by capitalising the terms for the former.

�

NOW time t, then to say that the NOWmoves from the past to the future is to saythat it was the case that a time before t was NOW, and it will be the case that a timea�er t [will be] NOW” (ibid.). Our review of presentism allows us to see that thismove indeed gives us genuine change in which time is NOW. Moreover it is a typeof change that can be combined with there being an absolute fact about which timeis NOW. So far, so good.

Let us consider in more detail how the o�cial tense-theoretic story and thesupertime metaphor are related. Suppose, again, that it is NOW time t and lethow things are from the perspective of supertime T correspond to how things aresimpliciter. What does it mean to say that WAS(a time earlier than t is NOW)?Skow suggests that this corresponds to the supertime claim that, at (i.e., from theperspective of) some supertime T ′ Earlier than T , some time t′ earlier than t isNOW.�� Someone familiar with typical model-theoretic treatments of tense logicmight be surprised at this. A�er all, the meanings of tense operators are normallyrelated to what is the case at ordinary times, not to what is the case at di�erentpoints in supertime. But Skow is absolutely right to set things up as he does. �eMoving Spotlight view’s points of supertime, not the ordinary times that it postulatesas distinct parts of concrete reality, play the role that ordinary times play for thepresentist.

�is fact highlights a deeply problematic feature of the view, at least in thisversion. �e eternalist thesis that past, present and future times exist is a tensedclaim. “Eternalism” is therefore apt, for the view involves the eternal persistence (inthe presentist’s sense of persistence) of all these times. And, in contrast to how pastand future objects are treated in Williamsonian presentism, these time are takento (always) be concrete parts of a (persisting) �-dimensional reality. �is realitychanges (in the presentist’s sense of change), but only by virtue of changes in whichpart of concrete reality is absolutely present.

We can now seewithwhat justi�cation theMoving Spotlight theory is sometimescalled a “hybrid view” of time. Appeal to primitive tense does not, a�er all, avoidthe two times problem. �ere are two times, it is just that one (ordinary time) isB-theoretic and the other (supertime) is A-theoretic. On the picture of reality beingo�ered, I exist (somehow or other) at many locations along a thin tube-like regionof a four dimensional block and, moreover, I always was so located and will alwayscontinue to be so. I would like to think that I (or some temporal part of me) islocated in a subregion of the block that is absolutely present. I would also like tothink this privileged location is where I am typing this sentence. But with what rightdo I assume either of these things?��

��Such supertime truth conditions for tensed statements are not meant as an analysis of tense.Rather, such claims are supposed to explicate supertime talk in terms of primitive tense.

��ere is now a small literature on this “how do I know that it is NOWnow?” objection, originally

��

�e Moving Spotlight theorist advocates a thoroughly tensed account of thechanges in which time is absolutely present, but what should they say about ordinarychanges in ordinary objects? What makes an ordinary utterance of “I sat down”true? �ere is always (i.e., from the perspective of every point in supertime) somepart of reality corresponding to the B theorist’s version of this change: some timeat which I am standing always exists, and some time at which I am sitting alwaysexists. Of course, it is almost never true (i.e., true from the perspective of almost nopoints in supertime) that either one of these bits of reality are part of the NOW. Isthe claim that I sat down (supposing I make this claim at a time to their future thatis absolutely present) about those bits of reality as they (present tense) are? �atmakes all ordinary talk in the past and future tense talk about how reality currentlyis.�� at seems undesirable. So, presumably, ordinary tensed talk is to be analysedin terms of the primitive tense operators used to state the theory. “I sat down,” forexample, might be held to be made true by the following tensed facts:��

WAS��m(t is present and I am standing at t) (�)WAS�m(t′ is present and I am sitting at t′). (�)

But in that case it is not clear what work the (present-tensed) existence of timesother than the time that is NOW is really doing. Reality is �-dimensional, but howmost of it is (rather than how it was or how will be) is irrelevant to our ordinarytensed talk. �e fact that how it is is just how it was or will be (except for whichtime is NOW) doesn’t make its current state any less redundant.

For reasons such as these, the Moving Spotlight theory is not a plausible meta-physical view. It succeeds, like presentism, in securing an absolute yet changing factof the matter about which time is present. In this context, however, its commitmentto the equal reality of past and future times is a �aw not a feature.

pressed by Bourne (��); Braddon-Mitchell (��).��e theory has the unfortunate feature that there are three fairly natural things that “presently”

in “how things are presently” could mean. (i) It could be an indexical device picking out thetemporal location of the utterance; (ii) it could be a way of referring to the region of reality that isabsolutely present; or (iii) it could appear redundantly: “how things are presently” might cover allof �-dimensional reality and just mean “how things are, simpliciter.” I am using “how things arecurrently” for how things are in this last sense, independent of their, or the utterance’s, relation tothe time that is absolutely present.

��Read “WAS��m(P)” as “It was the case ten minutes ago that P.” Are these ten ordinary minutes,or ten supertime minutes, and how the two are related? For discussion of these intricacies, see Skow(��).

��

� Passage and�e Growing BlockOur third and �nal traditional A�eory is the Growing Block view. To what extentdoes it vindicate the idea that events are forever becoming more past? Prior’s“becoming more past” covers, as well movement from the near past to the far past,movement from the far future to the near future, and from the near future to thepresent. �e Growing Block theory treats these three transitions very di�erently.

�e central primitive notion of the theory corresponds to an event’s passagefrom the future to the present. According to Broad (��), an event’s becomingpresent is exactly its coming into existence, a process he calls becoming. Before itcomes into being, the event simply does not exist. �is means that, on the theory,there is no literal passage of events in the far future to the nearer future, for thereare no future events around whose degree of futurity can be said to be changing.What about an event’s passage from the present to the past? Broad’s account of thisis highly revealing. It is held to be a genuine change, but not one involving anyintrinsic change in the event, and nor any change in the relations that the eventbears to given other events. Rather, the change is constituted by the coming intoexistence of new events, to which the now past event thereby comes to be related.In Broad’s words: “Nothing has happened to the present by becoming past exceptthat fresh slices of existence have been added to the total history of the world. �epast is thus as real as the present” (Broad, ��, ��).

�is last quotation highlights two aspects of the picture that deserve emphasis.First, past times are held to be on a par with the present, ontologically andmeta-physically. �e present is distinguished only by its being the “edge” of reality: thetime beyond which there are no further times. Second, since the sum total of realityis forever increasing via becoming, the time slices of the Growing Block, just likethe times of the Moving Spotlight view, persist. In fact, Broad seems to acknowledgethis explicitly. He writes: “there is no such thing as ceasing to exist; what has becomeexists henceforth for ever” (��, second emphasis mine).

�e parallels with the Moving Spotlight theory are clear. �e natural way tomake sense of talk of the Block’s growth in a way that avoids commitment to theexistence of the points of a second time dimension is in terms of primitive tense.�ere is an absolute fact about the extent of the sum total of reality, but this factchanges. �e sum total of reality was smaller; it will be larger. As with the MovingSpotlight theory, this manoeuvre does not avoid the postulation of two times. Just asin that case, we have one B-theoretic (truncated, but growing) dimension, and one A-theoretic dimension. And as before, the picture appears to be guilty of “spatializing”the former as a persisting dimension that, on closer analysis, does not connectdirectly with our ordinary tensed talk.

In fact, the the Growing Block analysis of ordinary tensed claims involves someinteresting twists. Recall that, in the Moving Spotlight theory, two natural choices

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involved (i) a B-theory style analysis involving how things are, absolutely speaking,at non-present times, or (ii) an analysis in terms of primitive tense involving howthings were or will be at these times, when these times were, or will be present. Atone level, Broad advocates something like (i) for past tensed claims, for he endorsesa B-theoretic analysis of qualitative change: he maintains that to say that a tra�clight changed from green to red, for example, is to say that a green stage of itshistory is adjacent to a red stage of its history and that the former is earlier thanthe latter. But at another level, his analysis really involves primitive tense, for hebelieves “earlier than” is ultimately to be analysed in terms of absolute becoming:“when we say that the red section precedes the green section, we mean that therewas a moment when the sum total of existence included the red event and did notinclude the green one, and that there was another moment at which the sum totalof existence included all that was included at the �rst moment and also the greenevent” (��, my emphasis of the past tense). �is analysis may not involve how thetra�c light was (rather than is), qualitatively speaking, but it does involve essentialreference to how the block was, and the then relative position of the event of thetra�c light’s being red within it.

Broad, unsurprisingly, treats the future tense very di�erently from the past tense.Given the non existence of the future, at least when the future tensed claim is present,a quasi B-theoretic analysis of the latter is not viable. Broad therefore proposes thatthe judgement expressed by “It will rain” (for example) be analysed as “�e sumtotal of existence will increase beyond what it is when the judgement is made, andsome part of what will become will be characterised by raininess,” admitting thatone cannot “analyse will away” (Broad, ��, ��).

�e resulting package is, I think, deeply unattractive. Spelling out the growth ofthe block in terms of primitive tense gives rise to a coherent view, but it is one thatis no more plausible than the Moving Spotlight view, for precisely parallel reasons.

� Non-standard A�eoriesOn the views so far considered there are supposed be absolute facts of the matterabout the way the world is. For the B �eorist, the absolute facts concern howthe world tenselessly is, from no temporal point of view. For the presentist, theabsolute facts concern how the world is presently (understood to include how itwas and how it will be). According to the Moving Spotlighter, there is an absolutefact about which of the many existent times is NOW. On the Growing Block view,there is an absolute fact about the extent of the sum total of reality. Non-standardA�eories give up the idea that there are absolute facts of the matter about the way

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the world is.�� e resulting views resemble the B�eory in that they deny that anyone time is absolutely privileged; they depart from the B�eory in upholding thenon-reducibility of tense.

One route to such a non-standard A theory starts with the presentist viewdiscussed above.�� e presentist takes the present facts to be the absolute facts.�ey postulated no other facts, but in terms of the present facts, they are able tosay what the facts have been and what they are going to be. On the correspondingnon-standard view, the facts that the presentist takes as absolute are reinterpretedas holding only relative to some particular time. Further, what facts hold relativeto past and future times is not taken to be reducible to what was and what will bethe case simpliciter. �e world is one way relative to one time; it is some other wayrelative to another time. �ere is no truth simpliciter to be had. Truth is taken to beessentially relative to times. �e present time only counts as the present relative tothe tensed facts we started with, the facts that hold relative to it. Every other time isequally present, relative to its own special collection of tensed facts.

It seems one should be able to e�ect a similar trick startingwith other A-theoreticviews. For example, recall that the supertime model of the Moving Spotlight viewrepresents how reality is simpliciter as theway it is relative to some point in supertime,and it represents how reality was and will be (simpliciter) as ways it is relative toother, appropriately chosen points in supertime. One obtains the non-standardvariant simply by letting ordinary times, rather than points of supertime, be thatrelative to which reality is a certain way.�� One then decrees that the picture oneobtains is no longer merely a helpful metaphor. One can also make a similar movein the context of the Growing Block view.�� As of time t the sum total of existencehas a certain extent; as of time t′ the sum total of existence has a certain, greaterextent. Does it make sense to identify t and t′, as they occur in the characterisationof the time-relativity of these facts, with the parts of the block that are (then) itsleading edges? If not, what are they? I will not pursue these questions further, forI take the non-standard variant of the branching-time view discussed in the nextsection to be clearly preferable to the non-standard Growing Block.

Let us return to the non-standard view based on presentism. One might wonderwhether the (non-Williamsonian) presentist’s denial of the existence of non-presentthings (including past and future times) survives the transition. Relative to some

��Calling these views “non-standard” is an allusion to Kit Fine’s label “non-standard realism” (Fine,��). Non-standard views are closely related to semantic relativism. Whether one thinks that thisassociation helps to make non-standard views intelligible will, of course, depend on one’s view ofthe intelligibility of relativism.

��I take the result to correspond to Fine’s “external relativist” version of non-standard realismabout tense; see Fine (��, ��–��).

��is route is taken by Skow (��, ��–�).��is might be the right way to understand the version of the theory presented in Button (��).

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time in ��, Prior’s falling from a punt, and Prior himself, do not exist, but, onewants to say, it is true then that they existed. �e presentist takes how things arerelative to some time in �� to be how things are momentarily, but absolutely. Inthis context, it is not clear how one can secure these seemingly de re claims of pastexistence, without also being committed to the present existence of the entitiesin question (a bullet the Williamsonian bites). Are things any better on the non-standard view? A�er all, while it is not true, relative to some time in ��, that Priorexists, it is true relative to some time in �� that he does, and that he is falling froma punt.

A natural model of the non-standard view includes a representation of all times,and all the truths that are true relative to each time. Just as the presentist’s com-mitments could be modelled in terms of a block universe with a privileged centre,one might model the current non-standard view by an in�nite sequence of suchblocks, each corresponding to the facts as of some time (the time marked out asprivileged in the block). With this model in mind, it is tempting to suppose that theproblem which a�icted standard presentism has dissolved. Prior might not existrelative to ��, but the times relative to which he does exist are all an equal part ofthe model. So the model includes Prior as part of reality, for the past directed truthsthat hold relative to �� to be about. �is, though, would a mistake. It illegitimatelysupposes that the model provides an absolute, view-from-nowhen description ofreality. But according to the non-standard view being considered, there is no suchabsolute conception to be had. �ere are only time-relative ways that reality is. Oneof these ways corresponds to our current perspective on reality, according to whichPrior no longer exists. �e advocate of the non-standard view does not think thatthis perspective corresponds to how things are absolutely. But, unlike the B theorist,they deny that there is a perspective-free conception of reality in terms of which areductive explanation of this perspective can be given.

�e moral is that one should be wary of misinterpreting models of the non-standard view in B-theoretic terms. It should be kept in mind when addressingour next question: do non-standard views vindicate the passage of time? �e �rst,obvious, point to make is that everything the presentist (or the Moving Spotlighter)said was true absolutely remains true relative to a particular temporal perspective.And everything that the presentist maintained was always true remains true relativeto every temporal perspective. Since time passes according to the presentist, thesame holds true, as of any time, on the non-standard view. One of the view’s manyperspectives is supposed to be our perspective so we can truly say (now) that timepasses.

�is is to consider how things are from the perspective of each time. One mightalso consider how things vary across perspectives. I consider �rst the analogousvariation in the B�eory, which can then serve as a useful contrast to the variationinvolved in non-standardA theories. On the B�eory, events do not literally become

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ever more past. But nevertheless one can consider the perspectives of a sequenceof ever later times. In a sense, one can say of any given event that it becomes evermore past relative to such a sequence. But on the B�eory, all this amounts to isthat the event is located at an ever greater temporal distance from each time in thesequence. �is no more corresponds to the real passage of time than the analogousspatial truth that, relative to a sequence of locations ordered continuously by theirmutual spatial distances, the �rst element of the sequence is an ever greater spatialdistance from each subsequent members of the sequence.

Similarly, on the B �eory there is a sense in which each time counts as thepresent relative to itself. So even on the B�eory, “the present” can be said to changeits position in time relative to time. But again, the B-theoretic sense in which a timeis present to itself is exactly analogous to the sense in which each spatial locationcounts as “here” relative to itself. Being the centre of a perspectival representation isnot per se to be represented as special by that representation. In fact, this seems tobe a plausible way to think about the spatial origin of our visual representation ofspace. In one sense, vision represents our location di�erently from the other spatiallocations that it represents.�� But in another sense, even in visual experience, ourspatial location is not represented as privileged. It is presented visually merely as thespatial location at which we happen to be. We understand the perspectival nature ofour visual relationship to other places precisely in terms of our spatial relationshipto those places. Moreover, our being so related to them is something that is itselfrepresented in our spatially perspectival representation.

Consider, now, the analogous variations on the non-standard view. As oneconsiders ever later temporal perspectives, a given event does, literally, become evermore past. Similarly, as one considers ever later perspectives, later and later timesare distinguished as the present in a way that goes beyond the B-theoretic variationjust reviewed. �e way in which the tensed facts true relative to a particular timesingle out that time is supposed to be not reducible to tenseless relational facts about,e.g., the various B-relations that events stand in to that time, so a fortiori the tensedfacts cannot be facts about those relations.

Does this variation with temporal perspective provide us with a sense in whichthe non-standard view vindicates the passage of time? �ere is an obvious problemwith the suggestion that it does. �e variation is not itself a fact about how realityis. Our model of the view includes such variation but, as we saw earlier, featuresof the model that transcend what is true from each temporal perspective do notcorrespond to perspective-independent facts about reality. �ere are meant to beno such facts. �is issue recurs in the context of open future models of passage, so Ipostpone further discussion until the next section.

��In some sense it is not represented at all except as a limit.

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� Passage and the Open Future�e A-theoretic proposals of the previous sections are, inter alia, e�orts to supplymind-independent content to talk of events becoming more past, or of the presentmoving into the future. I have suggested that both fans and critics of objectivetemporal passage agree that this is what it takes to vindicate the idea that time reallypasses. �is claim, however, is misleading in one respect, for it fails to engage withanother set of ideas that are naturally associated with the belief that time passes.

�e basic idea, which perhaps goes back to Aristotle, is crisply summarised bythe cosmologist George Ellis:

�ings could have been di�erent, but second by second, one speci�cevolutionary history out of all the possibilities is chosen, takes place,and gets cast in stone. (Ellis, ��, ��–�)

�e viewhas two essential elements. First, there is the idea that the future is genuinelyopen: at any instant, there are several possible ways that the world might develop.Second, only one of these possibilities in fact happens: as time passes, exactly oneof the many possibilities becomes actuality, and the rest become mere might-have-beens.��

In order to bring the view into sharper focus, one needs to distinguish betweentwo things that can be meant by the claim that the future is open.�� First, supposethat the laws of nature are indeterministic in the sense that speci�cation of theworld’s history up to a certain time, together with those laws, does not �x all futurefacts.�� To say that the future is open might only be to say that the future is notnomologically determined in this sense. But that the past and present, together withthe laws, do not �x all future facts does not entail that there are no such facts. Intenseless terms, there can be a unique actual continuation of the world to the futureof some time t, but this continuation need not be the only one compatible with theactual laws and the way the world is up to and including t.

Several advocates of the open future (e.g., Geach, ��; McCall, ��; Barnesand Cameron, ��) claim that true openness requires, not just that the future notbe (nomologically) determined, but that it is also not be fully determinate. Andone popular way of cashing out what lack of determinateness means focuses on thestatus of future possibilities. When one hasmere nomological openness of the future,there are, as of a time, many possible futures compatible with the indeterministiclaws, but they are not all created equal. One amongst them corresponds to the actual

��For a similar characterisation, by a passage sceptic, see Lockwood (��, �). �e picture iscentral to the model of objective temporal passage advocated by McCall (��, ��, ��).

��For a more comprehensive review of the options, see Torre (��).��e relevant sense of determinism is the model-theoretic one pioneered by Montague and

re�ned and deployed by Lewis (��) and Earman (��).

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future. �e others are therefore ways the actual worldmight have been (consistentwith its past and the laws), but not genuine ways that itmight still be. For example,it cannot both be true that there will not be a sea battle tomorrow but that theremight be one (in the relevant, non-epistemic sense of might). �e open futureview now under consideration therefore insists that, for them all to be genuinepossibilities, none from amongst them is now singled out as what will take place.��is leads naturally to “branching time” models of reality: tree-like structures, thenodes of which correspond to spatially global instants. �e branches are intendedto represent the plurality of possibilities to the future of each of the nodes fromwhich they branch.

Given such a structure, there are many ways to construct semantics for tensed(andmodal) sentences relative to the structure’s instants. �e optionsmost in accordwith the intuitions that led us to this point are the ones that involve a failure ofbivalence for future contingents. Suppose, that, as of now, it is an open possibilitywhether there will be a sea battle tomorrow: in some possible futures such a battleoccurs, in others it does not. In such circumstances, the semantics should securethe truth of both of the following:

• �ere might be a sea battle tomorrow

• �ere might not be a sea battle tomorrow.

One might also reason that, whichever open possibility comes to pass, either therewill be such a battle or there won’t be. �at is,

• Either there will be a sea battle tomorrow or there won’t be

should also come out as true. However, since it is supposed to be genuinely unsettled,as of now, whether there will be a sea battle, neither of the following claims shouldcount as true:

• �ere will be a sea battle tomorrow

• �ere will not be a sea battle tomorrow.

For similar reasons, it seems that neither should count as false either. �e fact thatit is not now settled that tomorrow there will not be a sea battle, for example, can betaken to be a reason for denying that “there will be a sea battle tomorrow” is plainfalse. It turns out that there are relatively natural semantics for branching time thatsecure exactly these results.��

��MacFarlane (��, ��–�) o�ers an argument of this kind against views that seek to uphold thebivalence of future contingents. For criticism, see Torre (��, ��-�).

��e most well known are the supervaluationist semantics �rst proposed by�omason (��).For a recent discussion, see MacFarlane (��).

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�ere is a popular line of argument against this branching-time model of theopen future. �e basic idea of the objection is that there is nothing unsettled abouta branching reality. An omniscient being located at one of the nodes with branchingto their future, should not be uncertain about, or think it unsettled whether, therewill be a sea battle. �ere will be such a battle, for there is such a battle in at leastone of the future branches. �ere are also future branches in which there won’t besuch a battle. If the claim “the future contains a sea battle” does not come out astrue (and is also not false), this is only because of a presupposition failure. “�efuture” fails to refer because there are several futures (see, e.g., Lewis, ��, ��–�).

�e objection presupposes a B-theoretic interpretation of branching-time mod-els, in which the block universe is replaced by a “block multiverse.” Whether such apicture vindicates the intuition that the future is open is controversial. Many agreewith Lewis that it does not.�� e A-theorist, however, need not worry how thisdispute is resolved, for, from their perspective, the objection gets things back tofront. On the view we are exploring one starts with the intuition that, as of sometime, the future is open. �is is taken to mean that (i) there are several ways thatthing might happen, and (ii) nothing in reality singles out (as of that time) one ofthese possibilities as the way things will actually be. On the intended interpretation,branching-time structures are introduced as a means to represent these purportedfacts, and to see whether a consistent formal theory incorporating them can bedevised. Against this background, one is simply not permitted to reinterpret thebranches relative to some time as several equally real futures, rather than as severalequally real possible ways that the single future might turn out.��

It is time to confront an issue that has been lurking in the background. Recallthat the view summarised in the quotation from Ellis has two components. �eopen future was the �rst. �e second was the idea that, as time passes, just one of theseveral possibilities for each moment obtains. As of now, no future is distinguished

��e block multiverse picture was once routinely associated with the Everettian interpretationof quantum mechanics (EQM). Something like the Lewis objection lies behind the argument inGreaves (��) that there can be no “subjective uncertainty” in an Everettian world. Recently, therehas been a subtle shi� in the position of some advocates of subjective uncertainty in EQM. It is nowclaimed that the theory’s fundamental ontology is equally consistent with a diverging (rather thanbranching) picture of quasi-classical worlds, and that this picture is to be preferred for precisely thekind of reasons alluded to above; see Saunders (��) and Wilson (��).

��Here are two possible sources of confusion. First, the defender of the open future is likely toinsist both that (a) “in reality” there are “real” future possibilities and that (b) these future possibilitiesare all “equally” real. As we have seen, the point of this insistence is to di�erentiate the view fromadvocacy of a merely nomologically open future. Second, the models represent all events in thepossible futures as temporally related to the instant from which they branch. But of course eventsthat might occur in the future are represented as temporally related to the present: if they occur, theywill be some temporal distance from the present. It does not follow from this (for the A-theorist)that there (tenselessly) is both a sea battle �� hours from now and (in some other realm of reality)an absence of such a battle �� hours from now.

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but, with the passage of time, onewill come to be. How is this idea to be incorporatedinto branching-time models?

At one level, what is required would seem to be straightforward. A path througha branching-time structure corresponds to a single determinate course of events.For the open-future A theorist, such a path corresponds to a possible view from the“end of time”: a possible way for the entire history of the world to have unfolded. Picksuch a path through a given branching-time structure. Its linearly ordered instantsde�ne a linearly ordered sequence of subtrees of our original tree, with eachmemberof the sequence a proper substructure of the preceding ones. Each instant from thechosen history is the privileged node of the corresponding subtree, where it featuresas the �rst instant of branching. With this machinery in place, the natural thought isthat the passage of time is to be represented, not within a single branching structure,but by a sequence of the kind just described. As time passes, successive elements ofthe sequence represent how reality is. Moving along the sequence corresponds totracing an upward path through the original tree. Moment a�er moment, one ofthe many possibilities for each successive time is chosen. But there always remainsbranching to the future. No element of the sequence corresponds to the “view fromthe end of time.” �at exists only as the ideal limit of the sequence as a whole.��

Something very like this model of temporal passage was proposed by StorrsMcCall (��). �e picture is, I think, very suggestive, but it invites the by nowfamiliar two-times objection. Time seems to be doubly represented, �rst by thesequence of ever shrinking trees and then by the temporal dimension within eachtree.�anks to our review of traditional A�eories, we are well placed to identity themost plausible A-theoretic responses, and to note some new twists that modellingthe open future brings.

To my mind the least plausible option, which I take to be McCall’s view, is thenatural analogue of the Moving Spotlight and Growing Block views. As on the blockmultiverse view, reality as a whole is a branching entity whose individual branchesare extended in four dimensions. As time passes, this reality changes. “Branchattrition” occurs as more and more parts of it go out of existence. As on the MovingSpotlight view, future times are as real as past times. As on the Growing Block view,the present time is metaphysically on a par with other times. It is distinguished onlyin terms of its location relative to structurally de�nable features of reality. Ratherthan being the bleeding edge, it is the surface at which branching begins.��

What sense can be made of branch attrition? As on standard versions of theGrowing Block and the Moving Spotlight, one can think of a single element of

��Earman (��) makes an analogous point concerning models of the Growing Block view.��is is to be faithful to McCall’s view, and to interpret distinct branches as distinct realms of

reality. �ere is also a linear time interpretation involving branching only at the level of courses ofmaterial events occurring within a four-dimensional block. �e present time is then metaphysicallydistinguished, as the frontier between determinateness and indeterminateness in reality.

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the sequence as corresponding to how things are absolutely. Change in branchingis then understood in primitively tensed terms: other elements of the sequencerepresent how things were, and how they will be. Such a combination of A- andB-theoretic faces the usual dilemma when one comes to interpretting ordinary talkin the past and future tense. And now there is an additional problem. How thingswill be at some time t, as judged from within a branching structure correspondingto an earlier time, does not match how the branching structure is as of t. Lookingin the future direction within the earlier branching structure, one concludes thereis no fact of the matter whether there will be a sea battle tomorrow. Looking at howbranching reality will be tomorrow, one concludes (say) that there will be a battle.

One way to e�ect a reconciliation is to look to the branching structure whenconsidering what facts are settled, and to look to future elements of the sequencewhen considering ordinary, amodal future tensed claims. �e branching structureas of t < t′ tells us that, as of t, it is not settled whether there will be a sea battle at t′.�e branching structure as of t′ tells us that, as of t, there will be a sea battle. �ishighlights that there is, in fact, a natural correspondence between a sequence ofbranching structures of the kind we have been considering and a single branchingstructure that includes a “thin red line”: a highlighted route through the structureintended to represent the way things actually turn out.

�is is not a route that the open-future A theorist should take. We should notloose sight of the fact that, prior to our seeking to articulate the view in terms ofan appropriate model, there seemed to be no incompatibility between insistingthat, as of now, it is unsettled, and hence neither true nor false, that there will bea sea battle tomorrow, but that, as time passes, things get settled one way or theother. �at just one of the possibilities will be actualised with the passage of timeshould not lead us to revise our original assessment that, as of now, there is nofact of the matter concerning which possibility that will be. A sequence of eversmaller branching structures seems like an attempt to do justice to both featuresof the view. �e parity of the branches in each element of the sequence respectsthe �rst; branch attrition respects the second. What we have seen is that a view likeMcCall’s, that interprets the sequence as snapshots of an ever shrinking branchingreality, has trouble resisting interpreting tensed claims made as of some element ofthe sequence in terms of how things are relative to other elements of the sequence.We do better to reject the hybrid interpretation of the model. Concrete reality doesnot branch. �e branching structures are simply ways of representing the particularpattern of tensed claims that the preferred semantics for such structures generates.�e open-future A theorist should take such tensed claims as basic.

�ere are two kinds of such a view to consider: an analogue (or version) ofpresentism, and a non-standard view. �e presentist variant claims that just oneelement of the sequence corresponds to a complete catalogue of the absolute facts.Strictly speaking, therefore, the other elements of the sequence cannot be required.

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�is mirrors our earlier claim that, when modelling the presentist view in terms ofa block universe with a privileged centre, one does not need additional blocks inorder to show that the privileged centre moves forward in time. �is fact is alreadyencoded in the original block. Does this remains true once we have branching?

�e key thing to note is that, while future elements of the sequence cannot beread o� from earlier elements, that the future facts will correspond to one suchelement can. Amongst the tensed facts that a branching structure encodes is thatexactly one amongst the possibilities open at that time will occur. Here is one wayto visualise the point: while a given branching structure (absent a thin red line)does not encode a single sequence of the kind we have been considering, it doesencode that the future tensed facts that hold at later and later times correspondto some such sequence. Given that it is not now determined how things will turnout, one might think that the lack of a preferred sequence is exactly as it should be.Passage is accommodated in this model just as it is in standard presentism. But theview adds the further feature that, as time passes, what once was unsettled becomessettled: there might be a sea battle or there might not, and either there will be oneor there won’t.

�e model of the non-standard variant of the view does involve a particularsequence. Each element of it represents the irreducibly tensed facts that hold as ofsome time. �is might seem to give us a more explicit representation of once openpossibilities being settled by the passage of time: what is indeterminate as of t issettled in such-and-such a way as of t′. But, as mentioned when the non-standardview was discussed earlier, it is not clear that this use of the model is legitimate. �esequence of trees does not represent how reality is absolutely, as conceived from noparticular temporal point of view.

Just as the tensed facts that hold as of some time are not reducible to tenselessfacts, there is no need for them to be deducible from the tensed facts that hold as ofother times. �is means that the view is not a�icted by the dilemma that plaguedthe McCall-style interpretation of the model. As of t, it is neither true nor falsethat there will be a sea battle at t′. As of t′, it is true that a sea battle is raging. �isseems to be exactly what one needs if one is to capture the motivating idea withwhich we began this section. In fact, for exactly this reason, it might seem thatthis open-future version of non-standard A theory better captures the passage oftime than a version in which the tensed facts as of one time can be read o� fromthose that hold at another. In the latter case, it is hard to see what the insistencethat such facts are not reducible comes to, for there is a unique representation ofreality—the block universe—from which the perspectival facts can be derived. �einsistence that this representation does not correspond to how things are absolutelylooks like arbitrary stipulation. �is is no longer true of the open-future model. �eprimordial branching-structure captures only how things might turn out, not howthey will turn out. �e block universe history that constitutes the ideal limit of the

��

sequence of the model’s branching structures not only does not correspond to thefacts as of any time (the end of time is never reached), it also, when interpreted asrepresenting the absolute facts, misrepresents as determinate future facts that aregenuinely unsettled.

I therefore take the combination of objective temporal passage and the branching-time conception of the open future to be a way that the A theorist can drive a wedgebetween their view and the B�eory. Of the various A-theoretic view surveyed upto this point, I take presentism, and some variant of non-standard A�eory to bethe most attractive vehicles for this combination. It will not come as much surprisethat, in the face of relativity, only one of these views remains a going concern.

� �ree Ways to Reconcile Passage and RelativityGlobal instants play a fundamental role in all of the A �eories reviewed so far.Presentism takes the world to be extended in only three, spatial dimensions. Inorder to interpret a spacetime model as a representation of presentist reality, oneneeds to foliate it by a family of �-dimensional instants and indicate which corre-sponds to the current time. �e Moving Spotlight view embraces a �-dimensionalreality, but singles out a �-dimensional subregion as metaphysically privileged. �eGrowing Block view’s �-dimensional reality is truncated to the future, bounded by a�-dimensional, spatially extended surface. �e nodes of the branching-time modelsrepresent �-dimensional, spatially extended global instants.

Minkowski spacetime, the spacetime of special relativity, lacks such structure.While somemodels of general relativity come with physically preferred foliationsby sequences of �-dimensional spacelike hypersurfaces, the physical characteristicsof such surfaces do not mark them out as obvious candidates for the privilegedsurfaces of a classical picture of temporal passage.�� Moreover, the local physics(which presumably is the physics in terms of which we should seek to understandour temporal experience of the world) is as blind to these privileged surfaces as it isin special relativity. In fact, the local physics just is the physics of special relativity.

In the face of these facts, there would seem to be three distinct routes by whichthe passage of time might be reconciled with relativistic physics. �e �rst is toadvocate a version of the de�ationary analysis of passage mentioned at the start ofsection �, as do Dieks (��) and Savitt (��). Earman’s assessment remains theright one. Such proposals are no more compelling in the context of relativity thanin pre-relativistic physics.

�e second option is argue that a model of passage that requires a preferred setof global Nows is compatible with relativity when the latter is correctly understood.

��For a review of some of the obstacles to interpreting such foliations in presentist terms, seeWüthrich (��, §�).

��

Here is not the place to review the sizeable literature on this possibility, but I wantto highlight one cost of this route.�� Its advocates face a dilemma: either theyinterpret the spatiotemporal structure of relativistic spacetime at face value, or theyare committed to spatiotemporal facts that go beyond this structure. Neither choicelooks attractive.

According to typical versions of the second choice (e.g., neo-Lorentzian inter-pretations of special relativity), the spacetime metric systematically misrepresentsthe true spatial and temporal distances between events. But it is the spatial andtemporal distances of the spacetime metric that correspond to the measurements ofphysical rods and clocks. Given this, it might seem preferable to adopt a literalisticattitude to the spacetime metric and simple superadd a metaphysically preferredfoliation in terms of which passage is to be understood. But this choice also has itscosts.

Suppose, for example, that some momentary event e is presently occurring andthat some other event e′ will occur. Suppose that the Minkowski metric representsthe spatiotemporal distance between e and e′ as being, say, two minutes. So far, sogood. But now consider the set of events that will co-occur with e′ according to thesuperadded foliation. Most of these will not occur two minutes a�er e accordingto the spacetime metric. Many will occur a much shorter temporal distance a�ere, e.g., some small fraction of a second. �at may seem strange but it is perhaps aconsequence that the A theorist can live with. Temporal distance, as measured byclocks and as encoded by the metric, is no longer a measure of a single distancebetween successive sets of co-occurring events. In short, it is no longer a measureof the passage of time. �e situation, however, is worse than I have so far let on. Notall events co-occurring with e′ happen some temporal distance a�er e. �ere willbe many events co-occurring with e′, much further from it spatially, that, accordingto the metric, occur some spatial distance from e. �at is, they happen a�er e(according to the A theorist’s conception of passage) but they lie at no temporaldistance from e (including the zero distance). �e view may not be incoherent, butit is very strange indeed. We have seen enough to motivate consideration of thethird and �nal option. Is it possible to generalise the models of the previous sectionsin order to obtain a genuinely A-theoretic view that does without global nows?

Something like this task has been undertaken recently by Earman (��) andSkow (��). Earman is concerned with the Growing Block view and Skow with theMoving Spotlight. �ese were the views that looked least attractive in our reviewof classical models of passage, and their shortcomings carry over to the relativisticdomain. Nonetheless, it is worth reviewing Earman’s and Skow’s e�orts, for theycan serve as a template for the generalisation of the branching-time models of theprevious section.

��For a systematic defence of this option, see Zimmerman (��, §�).

��

�� e Relativistic Growing BlockIn order to appreciate Earman’s generalisation of Growing Block models to rela-tivistic physics one �rst needs to quickly rehearse his de�nition of classical models.He de�nes these as follows. Let N = �M ,G�,G�, . . . , P�, P�, . . .� be a spacetimemodel of some Newtonian theory. M is a four-dimensional manifold representingspacetime. G�,G�, . . . are �elds de�ned on it representing the standard spatiotempo-ral structures of Galilean spacetime. P�, P�, . . . are �elds representing the materialcontent of the model. One can de�ne a time function T ∶ M → R that encodes thesimultaneity structure and temporal metric of the model. In terms ofN , Earmande�nes future-truncated models, NT≤∆, by deleting from the spacetime manifoldofN all the points p such that T(p) > ∆, −∞ < ∆ < +∞, and then restricting thegeometric and matter �elds Gi and Pj of N to the truncated manifold (Earman,��, ��). One can then characterise a model of the Growing Block view as a pairB = �N, ��. N is a set such that, for someN , each element is isomorphic toNT≤∆for some ∆. �e relation � is de�ned via the condition that for any n,n′ ∈N,n � n′i� n can be isomorphically embedded as a submodel of n′. For B to be an allowedmodel, � should be a total order. It is to be interpreted as “contains at least as muchexistence as.”

N, ordered by �, thus provides us with a sequence of the kind familiar from ourearlier discussion. In principle, there are two ways of interpreting it as representingan A-theoretic reality. �e “standard” way interprets one of the elements ofN ascorresponding to how reality is, absolutely speaking. “Earlier” and “later” elementsof the sequence then represent how realitywas and how itwill be.�e “non-standard”way seeks to interpret each element as a representation of how reality is as of sometime, where the time-relative facts are held to be not further reducible to facts thathold absolutely.

So much for the Newtonian case. At some level of abstraction, the possiblegeneralisations to relativistic physics are straightforward. One replaces N witha (non-extendible, orientable) spacetime model R = �M , gab , P�, P�, . . .� of somerelativistic theory. Earman then considers two options, which he labels hypersurfacebecoming and worldline becoming.

Hypersurface becoming requires that R admit a global time function, i.e., afunction t ∶ M → R such that for any p, q ∈ M where p is in the chronological pastof q according to the spacetime metric gab, t(p) < t(q). �e construction of whatEarman calls a model of “hypersurface becoming” then parallels the Newtoniancase. One considers pairs of the form B(R, t) = �{Rt≤δ ∶ l < δ < u}, ��. As before,Rt≤δ is the future-truncated model one obtains fromR be deleting all points p ofM such that t(p) > δ and then restricting the �elds ofR to result. l and u are thelower and upper bounds of the range of t. �e relation � is de�ned via the condition:Rt≤δ �Rt≤δ′ i� δ ≤ δ′.

��

So far our construction has paralleled the Newtonian case too closely. �eelements of {Rt≤δ ∶ l < δ < u} are totally ordered by the relation �. �is means thatthe A theorist can apply whichever was their preferred interpretation of the classicalgrowing block model directly to the relativistic model, but they are le� confrontingthe thorny issue of which of the uncountably many time functions compatible witha given spacetimeR corresponds to the surfaces of real becoming. �is is a variantof the second route to reconciling passage with relativity, a route we have disavowed.We need, instead, to generalise the model to one that does not single out a preferredfamily of global Nows. �e natural move is to consider the set R of all possiblefuture-truncations of R associated with every possible time function on R. Wecan then de�ne a relation � on this set in the obvious way. Since two distinct timefunctions can share the same level surfaces, the required conditions is that, for allr, r′ ∈R, r � r′ i� there is some time function t onR such that r is isomorphic toRt≤δ , r′ is isomorphic toRt≤δ′ , and δ ≤ δ′.��

We now confront an instance of the de�ning feature of the type of relativisticmodels to be considered in the remainder of this paper. �e relation � is a partialorder, not a total order. How does this key di�erence with the classical case a�ectthe type of interpretation that the A theorist is able to give of the model? In theclassical case there were two options to consider: standard A�eory, which takesa single element of the relevant set as a representation of how reality is absolutelyspeaking, and the non-standard option, which treats each element of the set equallyas corresponding to a representation of how reality is as of (in the classical case) atime.

�e �rst of these options looks like a non-starter. Suppose one took an elementr ∈R isomorphic toRt≤δ , for some value δ of some time function t, as correspond-ing to how reality is absolutely.�� is might seem already to give up on our aim ofan A theoretic view without global Nows. But, so far we only have one global Now,not a whole sequence, so let us bracket this objection and move on. With this choiceof r as representing the absolute facts, it might seem as if one can straightforwardlyinterpret any r′ � r as corresponding to how reality will be. But even this much isnot straightforward, for we have to decide what to say about two elements r� andr� ofR that both correspond to extensions of r but which are not comparable by�. Can one maintain that reality will be both as r� represents it as being and as r�represents it as being? Essentially the same problem becomes far more acute whenone considers elements ofR that are incomparable (according to �) with r itself.Let r′ be such an element that does not include the here and now (which, I assume,is part of the futuremost boundary of r). r′ cannot represent how things will be

��e resultingmodel is similar but not identical to what Earman calls a “super Broad hypersurfaceBecoming model” (Earman, ��, ��).

��Presumably one would like the here and now (i.e., your reading this sentence) to be located in aregion of spacetime somewhere on the surface t = δ.

��

unless the existence of what is happening right here and now can come to cease tobe. But nor can it represent how things were unless something that had come to behas now come to cease to be.

�e moral is that, if one maintains that a unique element ofR corresponds tohow things are absolutely, there does not seem to be an appropriate way to treat allthe other elements of the set. Can non-standard A theory do any better? On such aview, the problem of what to make of two incomparable elements of theR from theperspective of a third does not arise (at least o�cially), for, on the non-standardview, one does not look to other elements of the model in order to deduce the tensedfacts that hold relative to a given element. Such facts are supposed to be representedby the element itself. Even so, the prospects for the view are not much better. Oneproblem concerns the nature of the perspective relative to which the irreduciblyperspectival facts are supposed to obtain. Spacelike hypersurfaces of relativisticspacetimes simply are not naturally interpreted as things with respect to whichreality might be a certain way. Our here and now is a subregion of uncountablymany such surfaces if it is a subregion of any. We enjoy a particular spatiotemporalperspective on reality, but it is not a perspective that naturally extends to any one ofthe encompassing hypersurfaces.

A desire to do better justice to the local nature of our spatiotemporal perspec-tive can be used to motivate the second of Earman’s two options, viz., worldlinebecoming. One starts again with an inextentible relativistic spacetime modelR.��One then considers a past and future endless timelike curve γ inR. In now familiarnotation, one can represent a corresponding Growing Block model based on thiscurve as B(R, γ) = �{RJ−(p) ∶ p ∈ γ}, ��. RJ−(p) is obtained by deleting all thepoints ofR not in the causal past of the point p and restricting the �elds ofR to theresult. �e relation � is de�ned via the condition: RJ−(p) �RJ−(r) i� J−(p) ⊆ J−(r).Once again, the elements of {RJ−(p) ∶ p ∈ γ} are totally ordered by the relation �.�e properly relativistic model we desire considers all possible worldlines or, moresimply, all points ofR. �e resulting model isB(R) = �{RJ−(p) ∶ p ∈ M}, ��, with� de�ned exactly as before.�� It is a partial order on {RJ−(p) ∶ p ∈ M}.

As before we have two potential A theoretic interpretations of this model toconsider: the standard and non-standard variants. �e standard variant wouldseem to su�er from most of the problems that a�icted the standard interpretationof the hypersurface-based model. In addition, it displays an additional peculiaritythat is surely decisively problematic. Suppose one takes RJ−(p) as representativeof how reality is absolutely. What’s so special about p? Presumably you hope thatp is (roughly speaking) the (i.e., your) here and now. What’s so special about

��One no longer requires thatR admit a time function. It is enough for the construction to workthatR be causal-past distinguishing. See Earman (��, ��-�) for the relevant de�nitions.

��e model is closely related (though not identical) to what Earman calls a “super worldlinebecoming model” (Earman, ��, ��).

��

you? We therefore do better to consider the viability of a non-standard A theoreticinterpretation. Here we face once again the general unsuitability of the GrowingBlock model as something that might underpin a non-standard view. RJ−(p) looksadequate to representing irreducible past-tensed facts that hold as of spacetimepoint p, but what do we want to say about tensed claims, made as of p concerningthe future, or the elsewhere?�� It is hard to resist the temptation to look to otherelements of the model as encoding these, but that way lies many of the problemsthat plague the standard interpretation.

�� e Relativistic Moving Spotlight�oroughly relativistic Growing Block models do not look like promising materialsfor the would-be A theorist. Let us turn, instead, to Skow’s suggested generalisationof the Moving Spotlight view. Skow motivates his proposal via the supertime repre-sentation of the classical view. �e following is a natural “constraint law” describinghow supertime intervals and time intervals should mesh:

If p and q are points in supertime, and p is r units Later than q, thenthe time that is NOW from the perspective of p is r units later than thetime that is NOW from the perspective of q. (Skow, ��, ��–�)

Skow asks how this should be generalised when one replaces Galilean spacetime,with its unique family of global instants, withMinkowski spacetime. With supertimestill in place, one is a�er something of the form:

If p and q are points in supertime, and p is r units Later than q, thenthe BLANK-� from the perspective of p is BLANK-� than the BLANK-�from the perspective of q.

Where “BLANK-� holds the place for the kind of region that is “lit up” from perspec-tives in supertime, and BLANK-� holds the place for the relation that those regionsstand in” (Skow, ��, ��). As Skow notes, the structure of relativistic spacetimesprovide us with no natural way to �ll in these blanks. His “solution” is to replacethe perspectives of the points of supertime with those of the points of “Minkowskisuperspacetime”. From each such perspective, just one point of ordinary spacetimeis “lit up” as PRESENT. One can then state natural constraint laws, including, forexample:

If p and q are points in superspacetime that are Timelike related, and pis to the Future of q (that is, lies in the Future Light Cone of q), then

��e elsewhere of a point in a relativistic spacetime is the set of points spacelike related to it. I.e.,the set of points neither in nor on either its past or future lightcones.

��

the point that is PRESENT from the perspective of p is timelike relatedto and to the future of the point that is PRESENT from the perspectiveof q. (Skow, ��, ��)

Skow claims that this relativisticmodel vindicates passage, for the relativistic PRESENTcan be said to move just as much as the NOW of the classical theory:

Just as, as one moved from Earlier to Later points in supertime, onesaw the NOWmove from earlier to later times, so as one moves fromEarlier to Later points along any Timelike curve in superspacetime,one will see the PRESENT move from earlier to later points along acorresponding timelike curve in spacetime. (Skow, ��, ��)

At this point, the reader is likely to recall Skow’s insistence, when discussing theclassical model, that supertime was just a metaphor. According to the o�cial theory,the perspective of exactly one point in supertime corresponds to the absolute facts.�e perspectives of other points in supertime are representations of facts o�ciallyspelled out in terms of primitive tense operators. It is really these tensed facts,understood as absolute facts, that secure the “movement” of theNOW.One thereforewants to know: what is the o�cial story for which Minkowski superspacetimeprovides a metaphor? Skow declines to answer. A�er speculating that it might bepossible to spell it out in terms of “primitive tense-like operators that are adaptedto the structure of relativistic spacetime,” he excuses himself from doing so bysuggesting that the result would not be worth the e�ort because “the presentationof the theory using superspacetime is easier to understand” (Skow, ��, ��–�).

�is does not seem good enough. �e issue is not whether a story in terms ofrelativistic tense operators might be more perspicuous than the superspacetimemetaphor. �e issue is whether such a coherent story can be told at all. Can wemaintain that the facts associated with a particular perspective in superspacetime(however these are to be rendered in terms of relativistic tense operators) correspondto the absolute facts? �e absolute privileging of (not just the now but) the here andnow that this involves seems unacceptable. While it might be natural to think ofourselves as (momentarily) metaphysically special compared to the contents of pastregions of spacetime, we do not think of ourselves as metaphysical special comparedto, say, the inhabitants of the other side of the Earth. �e classical A theorist can alsoplay down the lack of egalitarianism by insisting that past times have beenNOWandfuture times will be NOW. On the most obvious ways of reconciling tensed claimswith relativistic spacetime structure, the relativisticMoving Spotlighter cannot eventclaim this of spacetime regions in our elsewhere. �ese never have been and neverwill be PRESENT.�e best one can say of them is that it will be the case that theyhave been PRESENT (cf. Putnam, ��, ��).

��

�e supertime metaphor and (presumably) the superspacetime metaphor aresupposed to explicate versions of the Moving Spotlight view conceived of as variantsof standard A �eories that involve absolute (tensed) facts. �e di�culties justreviewed suggest that embracing the non-standard route is the relativistic A theorist’sbest option.�� Rather than explore the consequences of this move in the unattractiveframework of the Moving Spotlight, I wish to introduce the natural relativisticgeneralisations of the classical branching-time models of passage.

�� Branching Spacetimes and the Passage of TimeRecall the distinction, central to di�erentiating a mere nomologically open fu-ture from the notion modelled by branching-time structures, between facts beingundetermined and their being indeterminate. �is contrast has also featured in dis-cussion of the compatibility of becoming and relativistic physics. Nicholas Maxwelldistinguishes between what he calls “predicative probabilism” and “ontologicalprobabilism.” �e former is essentially the Montague–Lewis–Earman notion ofindeterminism, combined with a unique actual history. �e latter asserts that “thebasic laws are probabilistic and that the future is now in reality open with manyontologically real alternative possibilities whereas the past is not” (Maxwell, ��, ��,original emphasis). It therefore involves a commitment to the open future in thesense of Section �. Maxwell’s central contention is that relativity and ontologicalprobabilism are incompatible. His paper is the main focus of Stein’s criticisms inthe latter’s “On Relativity�eory and the Openness of the Future” (Stein, ��). Inthis article, Stein defends the viability of a notion of becoming that he originallyarticulated in response to Putnam’s and Rietdijk’s arguments from relativity to theblock universe view (Rietdijk, ��; Putnam, ��; Stein, ��).

In a nutshell, Stein’s relativistic notion of becoming is this: all and only thoseevents on or in the past lightcone of a spacetime point p have become determinateas of p (see, e.g., Stein, ��, ��). It is not evident from this characterisation alonethat Stein is o�ering something that goes beyond a de�ationary notion of becomingof the kind o�ered recently by Dieks and Savitt.�� However, it is clear elsewhere thatStein is interested in the viability of a position that is distinct from the block universeview. For example, deliberately quoting Maxwell’s terminology, he claims to haveargued that “special relativity is perfectly compatible (in general) with ‘ontologicalprobabilism’” (Stein, ��, ��). How does the idea that an event has become as of p

��In more recent work, this is also Skow’s view.��For example, Stein claims that “the leading principle” that justi�es the use of “becoming” in

a relativistic setting is: “At a space-time point a there can be cognizance of—or information orin�uence propagated from—only such events as occur at points in the past of a” (Stein, ��, ��).�is principle is something that B theorists need have no trouble in accepting.

��

i� it is in the past lightcone of p achieve this?On the open-future view of passage outlined in Section �, to say that an event

has become determinate is to say that it is no longer one of several equally realalternative possibilities. So, to say that only events in the past lightcone have become,as of some spacetime point p, is to say that, while there is a unique fact of the matterconcerning what has occurred in all regions to the past of p, there are (as of p) aplurality of possibilities open for regions of spacetime to the absolute future of pand in its elsewhere. �is suggests that a �rst step towards a relativistic version of theopen-future view of passage should be a relativistic generalisation of branching-timemodels to structures that encodes this pattern of relational indeterminacy.

One type of “relativistic” generalisation of branching-time structures has beenby Nuel Belnap, who called the result branching space-times. �ey have since beenextensively studied by Belnap himself,�omas Müller and Tomasz Placek, amongstothers.�� Branching-time (BT) models involve a set of global instants (or, better,spatially global, instantaneous possibilities) partially ordered by a relation <, whichone can read as “is in the causal past of.” Belnap’s generalisation involves replacingglobal instantaneous possibilities with possible point events. A branching space-time (BST) model is set OW of such possible point events partially ordered by arelation < which retains the meaning “is in the causal past of.” (“OW” stands for“Our World.”)

In both BT and BST models, a history is a subset that corresponds to a sin-gle, complete courses of compatible events. In a classical BT model, histories aresimply maximal totally ordered subsets of the model. In contrast, maximal totallyordered subsets of OW are something like inextendible wordlines: maximal chainsof causally-related point events lying within histories. �e key to identifying thehistories of a BSTmodel is the notion of two compatible possible events, for historiesjust are maximal sets of compatible events. In the case of BT models, two events arecompatible if they are part of the same global instant, or are parts of causally relatedinstants. With the generalisation from instants to point events, we need to allowthat distinct events can be compatible even though they are not comparable by therelation <. Belnap’s solution is to classify two events as compatible if there is someevent which includes both of them in its past.�� One therefore has a distinctionamongst pairs of events incomparable by < between those that are “spacelike related”(those which jointly occur in some histories) and those that are incompatible (thosewhich jointly occur in no history). �e histories of OW can then be de�ned asmaximal directed subsets of OW . (A subset E of OW is directed i�, for any elements

��e seminal work is Belnap (��), which, incidentally, cites Stein (��) in its opening remarks.It exists in a slightly updated form as Belnap (��). Belnap (��) contains concise pointers to someof the recent literature.

��ismove onlyworks because branching spacetimes theory rules out the possibility of “backwardbranching” by �at.

��

e�, e� ∈ E, there is some element e� of E such that e� ≤ e� and e� ≤ e�.)One can therefore think of both BT and BST models as certain kinds of sets of

overlapping histories. �ey di�er in terms of the pattern of overlap. In the former,histories branch at global instants. In the latter, histories branch at one or morespace-like related point events. �is means that for BST models, but not for BTmodels, if h�, h� and h� are three mutually overlapping histories, then it is possiblethat h� ∩ h� � h� ∩ h� and h� ∩ h� � h� ∩ h�.

In order to make contact with Stein’s constraint on relativistic becoming, weneed to be able to say when two incompatible possible events count as di�erentpossibilities for one and the same location in spacetime. Whether two incompatibleevents are collocated is not, in general, de�ned in Belnap’s BST framework, but it issomething that can be de�ned for speci�c classes of models. In particular, Placekand Belnap (��) have recently described a class of BST models the histories ofwhich are isomorphic to Minkowski spacetime.�� If one considers an element ofsuch a model, i.e., a possible event e, occurring at some particular spacetime pointp, then all the histories in which e occurs overlap in the past of p. However, forspacetime locations q to the future or in the elsewhere of p, one will, in general,have two or more incompatible events located at q which share a history with e.In other words, amongst the models of Belnap’s BST theory, there are structuresthat would appear to give a precise expression to the kind of relativistic “ontologicalprobabilism” that Stein seems to have had in mind. From here on, my discussion isimplicitly restricted to BST models of this type.

Relativistic ontological probabilism, however, does not by itself constitute arelativistic theory of becoming. Recall that the theory of Section � had two elements:(i) genuine openness that (ii) was settled with the passage of time. So far we haveconsidered only the relativistic generalisation of (i). Just as classical BT modelshave a natural “block multiverse” interpretation, so do BST models. In fact, it isbecause of the possibility of such a “relativistic block multiverse” that Everettianquantum mechanics can evade the troubles that plague other realist interpretationsof quantum theory (such as collapse theories or Bohmian mechanics), and secure astraightforward reconciliation between quantummechanics and relativity. Whetherthis essentially B-theoretic interpretation of BST models can underwrite genuinerelational indeterminateness, it surely no more involves the objective passage oftime than its classical analogue. In order to vindicate real temporal passage, oneneeds to provide an A-theoretic interpretation of the model according to which, astime passes, what was indeterminate becomes determinate.

In the classical case, this was achieved by considering the sequence of eversmaller branching structures that one obtains from a given BT structure, W , by

��Other examples of “Minkowskian Branching Structures” had previously been constructed byMüller (��) and by Wroński and Placek (��).

��

selecting from it a single history, h. One obtains a unique set of sets of histories inW ,namely, {H(m) ∶ m ∈ h}, that are totally ordered by the relation of subsethood. (H(m)labels the set of histories in W that contain instant m.)�� One can do the exactlyparallel thing to a BST model �OW , <�, i.e., one can consider the set {H(e) ∶ e ∈ h}de�ned by some history h in OW .�� As the previous discussion might have led oneto expect, ⊆ is only a partial order on this set. Its elements are natural relativisticanalogues of the elements of a classical branching-time model of passage. AdaptingEarman’s notation, one might write B(OW , h) = �{H(e) ∶ e ∈ h}, ��, where � isnow de�ned via: H(e) � H(e′) i� H(e′) ⊆ H(e).

Note that variant models can be constructed by choosing di�erent types ofsubregions in h. �e set {H(e) ∶ e ∈ h} embodies the choice of individual spacetimepoints as the relativistic heirs to the present. �emodel is therefore the BST analogueof (the generalisation of) Earman’s worldline becoming models, and of Skow’srelativistic moving spotlight. One could, instead, focus on “slices” of h, maximalsets of spacelike related events in h. �e resulting set of subsets of OW is {H(E) ∶E is a slice of h}. It too will be partially ordered by the relation of subsethood. Itis the natural analogue of (the generalisation) of Earman’s hypersurface becomingmodels.

We can now consider whether such models admit of a plausible A-theoreticinterpretation, and thereby allow for the possibility of time’s really passing in arelativistic world without global Nows. As before we have two options to consider:the analogues of the standard and non-standard classical views. �e former takesexactly one element of the model—a set of histories de�ned in terms of their inclu-sion of some particular event e—as representative of the absolute facts. In otherwords, the facts as of some particular event e are taken as the absolute facts.

�e classical analogue of the view relied on a variant of the presentist accountof passage. In spelling this out, we had to deal with a delicate issue: future indeter-minacy meant that what the absolute facts were going to be could not be read-o�from the (current) absolute facts. �erefore a particular choice of “future” elementsin the original model was not really justi�ed. Ultimately this was not problematic,

��is set of histories is not, strictly, a substructure of the original. Here and in what follows, oneshould really be considering the union of such a set of histories, structured by the restriction of <.

��ere has been surprisingly little discussion of this kind of construction in the context ofbranching spacetimes. As far as I am aware, something similar has only been considered by Placek(��). S̆varný (��) has recently investigated whether BST models can accommodate the “�owof time,” but the approach he adopts does not appear to be an analogue of the one taken here.McCall intended his branch-attrition model of temporal passage to be compatible with relativity,and in Appendix � to McCall (��), he provides a “frame-invariant” characterisation of a relativisticbranching structure in a manner that owes much to Belnap’s. However, his characterisation of thebranch attrition is always done in frame-relative terms. How distinct frame-relative descriptions areto be understood as di�erent descriptions of a single underlying objective process is not explicitlyaddressed.

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because amongst the current facts were facts to the e�ect that future indeterminacywas later going to be resolved (one way or another). One could do justice to theidea that, as time passes, open possibilities are going to be settled without a modelthat includes how they are going to be settled.

In order for a similar story to be viable in the context of the relativistic model,at least two things are required. First, the tensed facts as of a spacetime point, inter-preted as absolute facts, should underwrite a relativistic analogue of the presentist’saccount of passage. Second, these tensed facts must include facts to the e�ect that“as time passes” the current openness concerning the future (and the elsewhere) willbe (or will have been) settled one way or the other.

I leave unresolved these intriguing issues, for even if successful on this front,the view is untenable, for the same reason that the corresponding interpretationsof relativistic versions of the Growing Block view and the Moving Spotlight viewfounder. It is simply not plausible to take as absolute facts, facts that correspondto the perspective of a spacetime region that is both spatially as well as temporallylocal. I want to round up by considering what I take to be the most promising way toreconcile becoming with relativity, namely, non-standardA-theoretic interpretationsof our BST-based models.

Consider the model B(OW , h) = �{H(e) ∶ e ∈ h}, ��. According to a non-standard A-theoretic interpretation of this model each element of {H(e) ∶ e ∈ h}represents the facts that hold as of some spacetime point. H(e), for example, encodesthe facts that hold as of the spacetime location of the event e. Even though suchfacts are the facts that hold as of some spacetime point, they are not supposed tobe reducible to further facts that hold absolutely. �ese facts display a particularpattern of indeterminacy. As of some point p, what happens outside of p’s casualpast is indeterminate. But, as of every point, including all points outside of p’scausal past, what happens at that point is determinate. Despite not being inter-deducible, the sets of perspectival facts in this network mesh in the obvious ways.What is happening at q, as of q, will be among the things that might happen (ormight later have happened) as of points not in the causal future of q. �e model isinequivalent to a single BST model. Facts that are indeterminate as of earlier pointsin spacetime are settled as of later points. �e model is also inequivalent to thepreferred history it encodes, or to a BSTmodel that includes a “thin red line,” at leastas the latter is normally understood. In both of these models, indeterminate factsare misrepresented as determinate. �e particular pattern of perspective-relativefacts that the model encodes cannot be understood as reducible to a B-theoreticreality corresponding to either a block universe or a block multiverse. �e modeltherefore constitutes an apparently coherent, thoroughly relativistic A-theoreticalternative to the the B�eory.

Does this mean it vindicates the objective passage of time? In the classical ana-logue of the model, one could traced through a unique, totally ordered sequence of

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temporal perspectives, and see facts once open became settled. �e spatiotemporalperspectives of the relativistic model are only partially ordered. One can consider amaximal totally ordered subset of them, corresponding to a maximal chain of eventsin the model’s preferred history. According to such a sequence, the tide of becominghas the shape of a past lightcone that moves up the privileged worldline. Doessome aspect of such “worldline-dependent becoming” correspond to somethingobjective? Can one see di�erent such sequences as “gauge equivalent”? Are they justdi�erent ways of representing the same underlying passage of time?

It is standard in cases where gauge equivalence is postulated to demand somekind of characterisation of the gauge-invariant reality that gauge-related descriptionsdi�erently represent. �e prospects for providing something of this sort look betterif we change the model, from one based on spacetime points, to one based on slicesthrough our preferred history. Ironically, it is a model involving the analogue ofglobal spacelike hypersurfaces that best represents local becoming, conceived ofas transition from the indeterminate to the determinate. �e reason is that, asone shi�s from “perspective” to “perspective” along some maximal totally orderedsequence of elements from the set {H(E) ∶ E is a slice of h}, the resulting change inwhat is determinate is not spread out over a past lightcone, but is spatiotemporallylocal.Let S� and S� be two arbitrary, maximal totally ordered subsets of {H(E) ∶E is a slice of h}. One might seek to characterise their gauge-invariant content asfollows. Very crudely, maximality ensures that for any event e in h, one can �ndshort enough stages of both sequences S� and S� where pretty much all that happensis a transition from e’s potentiality to its actuality. Transitions like this are obviouscandidates for the objective local becoming that both sequences represent. Suchsequences can di�er over whether this transition happened “before” or “a�er” thebecoming de�nite of some other event spacelike related to e. But since the eventsare spacelike related, there is no fact of the matter concerning which occurred �rst.

AcknowledgementsRelated material was previously presented in St Andrews, London, Birmingham,Bristol, Warwick, Oxford, Bonn, Nottingham, Florence, Dubrovnik, San Diego,Geneva, Dublin and London, Ontario. I am grateful to numerous members of thoseaudiences for useful comments. �e passage of time has, unfortunately, loosenedmy grip on who deserves to be individually thanked for discussions and feedback.A proper subset includes: John Norton, Tomasz Placek,�omas Müller, AndrewBacon, Steven Savitt, Richard Arthur, Stephan Torre, Antony Eagle, Natalia Deng,Brad Skow, Matt Farr, Craig Callender and Carl Hoefer. Initial work for this paperwas supported during ��–�� by a Philip Leverhulme Prize.

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Oliver Pooley (Oxford): Relativity, the Open Future, and the Passage of Time

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