Out of Nowhere: The emergence of spacetime from causal sets
CChapter 3: The emergence of spacetime from causal sets
Christian W¨uthrich and Nick Huggett ∗ Contents ∗ This is a chapter of the planned monograph
Out of Nowhere: The Emergence of Spacetime in QuantumTheories of Gravity , co-authored by Nick Huggett and Christian W¨uthrich and under contract with Oxford Uni-versity Press. More information at . The primary author of this chapter is ChristianW¨uthrich ([email protected]). This work was supported financially by the ACLS and the John Tem-pleton Foundation (the views expressed are those of the authors not necessarily those of the sponsors). We wishto thank Fay Dowker for correspondence. a r X i v : . [ phy s i c s . h i s t - ph ] S e p A statement of the problem
The conclusion of the last chapter denied that a non-spatiotemporal world is impossible: accord-ing to causal set theory (CST), it is physically (and hence presumably metaphysically) possiblethat the world is not spatiotemporal. After having introduced CST in the previous chapter andillustrating how at least space and arguably spacetime disappears in CST, we set out in thischapter to show how relativistic spacetime may emerge from a causal set, or ‘causet’.The immediate question before us concerns the relation between causets and spacetimes: howdo relativistic spacetimes emerge from the fundamental structures, i.e., causets, in CST? SinceCST, like any theory of quantum gravity, is supposed to be a theory of gravity more fundamentalthan GR, and since relative fundamentality in physics typically parallels scales of energy andsize, we are looking to understand the low-energy, large-scale limit of CST. Thus, we are atthe same time interested in the relationship between two theories—CST and GR in the presentcase—, as well as between the entities or structures postulated by these two theories—presently,causets and relativistic spacetimes. Whenever there is no danger of conflating the two—and thiswill normally be the case—, we will often not explicitly indicate which is the present topic. Aswe will mostly be concerned with the relationship between causets and relativistic spacetimes,we will be asking, more specifically, whether we can precisely state , i.e., with full mathematicaland philosophical rigour, the necessary ( §
2) and sufficient ( §
3) conditions for generic causets togive rise to physically reasonable relativistic spacetimes . Whatever the answer to this question,it better not merely deliver necessary and sufficient conditions for mathematical derivations,but supplement them with demonstrations of the ‘physical salience’ of these derivations. Theargument in § § §
4) and a (novel) form of non-locality implied by the discreteness of the fundamental structuretogether with the demand for (at least emergent) Lorentz symmetry ( § The restriction to physically reasonable spacetimes is supposed to leave room for CST to correctGR regarding which spacetimes are physically possible: GR is a notoriously permissive theory,and we would generally expect the more fundamental theory to rule out some relativistic space-times as physically impossible. If this were to happen, the question of what it is for a relativisticspacetime to be physically reasonable would find some serious traction. Before some theory ofquantum gravity is established though, answers to this question merely trade in human intuitionsand the prejudices of physicists. Since these preconceptions are all trained in our manifestly spa-tiotemporal world, relying on them in the quest for a fundamental theory of gravity becomesdeeply problematic. For an argument of how physicists’ intuitions about what it is for a spacetime to be physically reasonable canmislead us, e.g., in the case of singular or non-globally hyperbolic spacetimes, see Smeenk and W¨uthrich (2011),Manchak et al. (forthcoming), and Doboszewski (2017).
2n this sense, we will use the qualification of ‘physical reasonableness’ merely as a reminderthat we ought not to expect a quantum theory of gravity to return all sectors of Einstein’s fieldequation, i.e., all spacetimes which are admissible by GR’s lights. Thus, CST may only reproducesome, but not all, sectors of GR. For instance, as we have seen in chapter 2, the antisymmetryassumed in the partial ordering rules out causal loops at the fundamental level, and so mayprohibit spacetimes with closed timelike curves. Furthermore, models in GR with high energyor matter density may be eliminated by underlying ‘quantum effects’, some sectors of GR mayviolate energy conditions in ways inconsistent with CST.Conversely, we seek to understand how spacetime arises from generic causets. On the onehand, it would be unreasonable to expect that all causets give rise to a spacetime, or even justnearly so. There will be causets without anything like an appropriately spatiotemporal structure.That such ‘pathological’ cases will also satisfy the demands of the theory on a causet, and soqualify as physically possible, is one of the main reasons to consider the theory, in general, asnon-spatiotemporal. On the other hand, we would want ‘most’ causets, or at least ‘many’ ofthem to give rise to spacetimes, for otherwise we might ask ourselves why we were so incrediblylucky to be born in a spatiotemporal world when ‘most’ of them are not so. Clearly, more willneed to be said about this—and more will be said instantly—, but we hope that the intuitionbehind demanding genericity is reasonably clear for now.Thus, we wish to relate generic causets to physically reasonable spacetimes. More specifically,we wish to see how the properties and structure of a part of a causet can give rise to the geometricand topological structure of a region of a relativistic spacetime. The basic idea is captured infigure 1. Importantly, it needs to be demonstrated how the relationship between features of ) q ( { I p x n d g { R s ) = R ( Vp qR i ab g ; hMiÁ , hC Figure 1: The relata.causets and properties of spacetimes is not just mathematically definable, but physically salient.The method to accomplish this is not by staring at causets and ruminating over which of itsparts are physically salient; instead one identifies the physically relevant properties of relativisticspacetimes and determines how physical salience percolates down to the fundamental structure: But cf. W¨uthrich (forthcoming). Arguably, we couldn’t have been born in a world best described by a non-spatiotemporal causet, and so astraightforward anthropic argument should eliminate any astonishment as to why our world is spatiotemporal,given our existence. However, this leaves bewildering the issue of why the world came to be spatiotemporal inthe first place. ” elements, according toBombelli et al. (1987)). The reason for this demand is evident: if it is too small, the notionof what it is approximated by at large scales does not make sense. Thus, let us eliminate allcausets, which are not sufficiently large. Unfortunately, there is no hope that ‘sufficiently large’causets will generically give rise to spacetimes, due to the following problem: Problem 1.
Almost all ‘sufficiently large’ causets permitted by the kinematic axioms of chapter2 are so-called ‘Kleitman-Rothschild orders’, i.e., discrete partial orders of only three ‘layers’ or‘generations’ of elements. ‘Kleitman-Rothschild orders’ or
KR orders are partially ordered sets consisting in just threelayers of elements such that no chain as defined in definition 8 in chapter 2 consists in more thanthree elements, such that roughly half of the elements are in the middle layer and a quarter eachin the top and bottom layers. Furthermore, each element of the middle layer is related by theorder relation, on average, to half the elements of the top layer and to half the elements in thebottom layer. More precisely, Kleitman and Rothschild (1975) prove the following theorem:
Theorem 1 (Kleitman and Rothschild 1975) . Let P n denote the number of partial orders on aset of n elements. Let Q n denote a special case in which n -element partially ordered sets are KRorders. Then P n = (cid:18) O (cid:18) n (cid:19)(cid:19) Q n . In other words, in the limit of n → ∞ of n -element partially ordered sets, almost all of themare KR orders. Why is this a problem? Because KR orders, if considered cosmological causets,would represent highly non-locally connected ‘universes’ which ‘last’ for a lousy three Plancktimes, i.e., roughly 10 s, during which they first double in size and then shrink by the samefactor. If causets generically have the form of KR orders, then CST cannot offer a satisfactoryanswer to how they could give rise to anything like our world. It seems obvious that the vastmajority of even sufficiently large causets cannot give rise to anything like spacetime as we knowit. But how to tame the KR flood? Clearly, if the theory is conceived of as only comprising thekinematics described in the last chapter, it is too weak to clear out the KR weeds. Thus, whatis needed are additional, and sufficiently restrictive laws.
This is precisely the motivation behind introducing a ‘dynamics’:
Solution 1. ‘Dynamical’ rules should be appended to the kinematic axioms such that thosecausets that satisfy these additional axioms are not befallen by the KR catastrophe.
Accordingly, we impose ‘dynamical’ laws in order to avoid the KR catastrophe and to restrictthe vast set of kinematically possible causets to the physically reasonable models of the theory. This problem is what Smolin (2006, 211) calls the “inverse problem” and is often referred to as the “entropyproblem” in the literature (see, for example, Brightwell et al. (2009), Dribus (2017, 188), Surya (2019, 50)), in ref-erence to the fact from statistical physics that the large-scale behaviour of a physical system may be characterizedin terms of the multiplicities of microscopic states sharing their macroscopic behaviour. For details, see Kleitman and Rothschild (1975). See figure 9 in Surya (2019).
4y far the most popular proposal for causet dynamics is a law of sequential growth as introducedby Rideout and Sorkin (1999). The central idea of a sequential growth dynamics that a causet‘grows’ by the sequential addition of newly ‘born’ events one by one to the future of alreadyexisting events. Thus, what grows is the number of elements, and it is assumed that the ‘birthing’of new elements is a stochastic physical process in the following sense: the dynamics specifiestransition probabilities for evolving from one causet in Ω( n ) to another one in Ω( n + 1), whereΩ( n ) is the set of n -element causets. Since the growth always happens to the future of ‘already’existing events, the resulting causets are all finite towards the past or ‘past-finite’, in the sensethat they possess at least one minimal element. Only causets which could have been grown by aprocess consistent with the dynamical laws of sequential growth are then considered physicallyadmissible.What dynamical laws ought to be postulated? They should capture our best guesses as tothe conditions necessary to ‘produce’ spacetime. In particular, they should thus be ‘natural’—i.e., physically salient—principles, not just arbitrary mathematical rules. Thus, these naturalconditions encode the physical requirements on an acceptable dynamics. Rideout and Sorkin(1999) impose four such requirements, of which we give here their own gloss: Axiom 1 (Internal temporality) . “[E]ach element is born either to the future of, or unrelatedto, all existing elements; that is, no element can arise to the past of an existing elment.” (5) Axiom 2 (Discrete general covariance) . “[T]he ‘external’ time in which the causets grows... isnot meant to carry any physical information. We interpret this in the present context as beingthe condition that the net probability of forming any particular n -element causet C is independentof the order of birth we attribute to its elements.” (5f ) Axiom 3 (Bell causality) . “[E]vents occurring in some part of a causet C should be influencedonly by the portion of C lying in their past.” (6) Axiom 4 (Markov sum rule) . “[T]he sum of the full set of transition probabilities issuing froma given causet [is] unity.” (7) Axioms 1 and 2 are intended to jointly underwrite the idea central to classical sequentialgrowth that although there is a birth order in the way the dynamics is described, it and the ‘time’in which it plays out has no physical significance. One would expect axiom 3 to be violated ina quantum theory, which should naturally be non-local in that it admits Bell correlations, i.e.,correlations among spacelike related events. For a classical theory, it seems unproblematic,and indeed felicitous, to postulate axiom 3, despite its being earmarked for being dropped ina quantum theory. Finally, axiom 4, required for any Markov process, just innocently assumesthat for any given finite causet, there is exactly one way of possibly several in which it will infact grow. Following, Rideout and Sorkin (1999), let us call a dynamical rule which complieswith these four axioms a classical sequential growth dynamics .The four axioms can be thought to encapsulate the basic conditions any specific dynamiclaw must obey. Obviously, many dynamical laws specifying particular transition probabilitiesare conceivable. Yet a remarkable theorem by Rideout and Sorkin (1999) shows that if theclassical dynamics conforms to the four axioms 1–4, then the dynamics is sharply constrained.In particular, it must come from a class of dynamics of sequential growth known as ‘generalizedpercolation’. Since only the general properties of this class matter for our purposes, let usintroduce the simplest model of classical sequential growth which satisfies the Rideout-Sorkintheorem—namely, ‘transitive percolation’—a dynamics familiar in random graph theory. The technically more precise statements can be found in Rideout and Sorkin (1999, § III). To give just anexemple, internal temporality demands, as stated above, just that if x (cid:22) y , then label( x ) ≤ label( y ). , , , ... such that they are consistent with the causalorder, i.e., if x (cid:22) y , then label( x ) ≤ label( y ). The reverse implication does not hold becausethe dynamics at some label time may birth a spacelike-related event, not one for which x (cid:22) y .This is essentially the requirement of ‘internal temporality’ introduced below. We begin withthe causet’s ‘big bang’, the singleton set. Now when the second event is birthed, there are twopossibilities: either the second event (labelled ‘2’) causally succeeds the first event (labelled ‘1’),or it does not, i.e., 1 (cid:22) ¬ (1 (cid:22) p to thetwo events being causally linked and 1 − p to the two events not being causally linked. The sameholds for the third event, which has probability p of being causally linked to 1 and 1 − p of notbeing causally linked to 1, and probability p that it is linked to 2 and 1 − p that it is not linkedto 2.In general, an alternative way to conceive of the dynamics is that when a new causet with n +1 events comes into being, it chooses a previously existing causet of n events to be its ancestorwith a certain probability. Thus, one can think of transitive percolation as involving two stepsat each evolutionary stage from an n -element causet C n to an ( n + 1)-element causet C n +1 . First,select the subset of elements in C n which will have a direct causal link to the newborn element.Second, add all ancestors implied by transitivity. In this way, the dynamics enforces transitiveclosure, so if, e.g., 1 (cid:22) (cid:22)
3, then 1 (cid:22) C n , the probability ofit being part of a given set of ancestors is p . This determines the probability of the transitionfrom a given n -element causet to a given ( n + 1) − element causet. For instance, the transitionfrom a 2-chain (1 (cid:22)
2) to a 3-chain (1 (cid:22) (cid:22)
3) is the sum of the probability of picking bothelements 1 and 2 of the 2-chain as ancestors and the probability of picking only the causally laterelement 2 as ancestor, since in this case, the second step (to ensure transitive closure), adds thefirst element 1 again. Thus, the probability of this transition is p + p (1 − p ), which happens toadd up to p .A classical sequential growth dynamics can generally be though of as a ‘tree’ of permissibletransitions where each transition is assigned a probability consistent with the axioms above. Theresulting tree is itself a partial order, where the ordering relation is not a causal relation intrinsicto a world, but instead a relation of permissible sequential growth. Figure 2 shows the firstthree stages with the transition probabilities exemplified by those of transitive percolation. Infigure 2, the thicker arrow indicates that there are two ‘ways’ in which the transition from the2-antichain to the 3-element causet with a chain of two elements and an isolated third elementcan occur: the third element can either be to the causal future of 1, or to the causal future of2. Axiom 4 requires that the transition probability must count both these possibilities; hencethe factor of 2 in the transition probability. In turn, this naturally suggests that the dynamicspresupposes a non-structuralist metaphysics of elementary events according to which these eventshave a primitive identity—a ‘haecceity’, i.e., an identity which is independent of, and prior to,the causal relations they entertain to other events. That a careful consideration of the metaphysics of causets and their dynamics is needed canbe directly seen when we seek reassurance that transitive percolation satisfies axioms 1 through4. While axioms 1 and 3 are directly and unproblematically built in, it may appear as if axiom 2and axiom 4 stand in tension. To repeat, the transition probability from the 2-antichain to the This raises the same metaphysical issue we have already encountered in the discussion of ‘distinguishing’causets in § p (cid:54) = 0, this cannot be the case if we maintain the factor 2. But the contradictionis only apparent: in order to check axiom 4, we must add up the probabilities of all possibleways in which the 2-antichain could evolve (including the two ways in which it can evolve tothe 3-element causet of interest); in order to verify axiom 2, however, we only need to multiplythe probability to get to that 3-element causet from the same ur-causet along one path. Strictlyspeaking, there are three paths from the ur-causet to the 3-element causet at stake: one throughthe 2-chain, and two through the 2-antichain, and along all three paths, the probabilities mustfactor to the same product. And they do: they all factor to p (1 − p ) .There are further metaphysical assumptions built into the model of classical sequential growthdynamics. First, the probability of the ‘big bang’ causet, i.e, the minimal element of the partialorder depicted in figure 2—the ur-causet—, is assumed to be 1. In other words, the dynamicspresupposes that there is something rather than ‘nothing’: the completely empty world containingno basal element at all is not a physical possibility. Second, as already mentioned, all causetswhich comply with this dynamics are past-finite and grow indefinitely into the future, at leastas the transition probabilities are assumed to be non-zero.Third, as Rideout and Sorkin (1999) declare right at the outset, and as has been reaffirmedby leading causet theorists since, it is their hope that this dynamical ‘birthing’ of events, thissequentially growing ‘block universe’, will turn out to underwrite and give rise to the “phe-nomenological passage of time”, which “is taken to be a manifestation of this continuing growthof the causet” (2). This idea raises an immediate concern of conceptual inconsistency. On theone hand, physical time, and thus truly dynamical processes are supposed to only arise at theemergent level as an aspect of the emerging spacetime, and is at best only implicitly present inthe causal structure of the causet. On the other hand, the rhetoric is infested by temporal lo-cutions concerning the ‘growth’ of causets, the ‘birthing’ of events in ‘sequential’ order following7 ‘dynamical’ law, suggesting the presence of time in a way that is metaphysically prior to thecauset. Clearly, this tension has to be resolved, and the status of time will have to be settled.We will return to this and related metaphysical questions in section 4.Let us add a final, but crucial point concerning sequential growth dynamics as a physicaldynamical law for CST. Although it involves stochastic transitions between stages, there isnothing quantum in nature about the sequential growth dynamics—hence, it is known as classical sequential growth dynamics. Just as discreteness postulated in the kinematics of the theory doesnot make the theory a quantum theory, neither does the stochasticity introduced in the dynamics.Alas, this is largely the state of the art as the philosopher studying CST finds it. Although thereare incipient efforts to turn the theory into a full quantum theory such as those based on quantummeasure theory (Sorkin 1997a) or a quantization of classical sequential growth dynamics (Gudder2014), their incompleteness and inconclusiveness does not really permit a discussion of anythingbut the classical theory. Consequently, our conclusions are conditional on the proviso that thisis just the classical theory so far and so will have to remain tentative. Transitive percolation or something like it ought to be able to turn the KR tide, at least as longas the probability p does not vanish. Is adding a viable and physically motivated dynamical lawthus everything required to understand the emergence of spacetime from causets? A priori, no:resulting causet are generally still not ‘manifoldlike’ and so not yet candidates for giving rise tospacetime. Problem 2.
The conditions stated so far may not guarantee that causets give rise to manifoldsof reasonably low dimensionality (i.e., significantly lower than ) or that it has Lorentziansignature or a reasonable causal structure. The discrete structures of causets and continuum Lorentzian spacetimes are mathematicallyrather different structures, as already noted. But in order for a causet to give rise to somethingthat is at some scales well described by a relativistic spacetime, these structures cannot be toodifferent. Thus, the second necessary condition after the requirement that candidate causets be‘sufficiently large’ is that they are ‘manifoldlike’:
Solution 2.
Impose conditions such that, generically, causets (cid:104)C , ≺(cid:105) are ‘manifoldlike’. But what is it for a causet to be ‘manifoldlike’ in the required sense? It means that it standsin an appropriate relation to spacetimes:
Definition 1.
A causet (cid:104)C , ≺(cid:105) is manifoldlike just in case it is ‘well approximated’ by a relativisticspacetime (cid:104)M , g ab (cid:105) . If a causet is manifoldlike, we also say that the associated spacetime (cid:104)M , g (cid:105) approximates thecauset. Clearly, this definition, and hence solution 2, can only get any traction once we spell outwhat is meant by ‘well approximated’. To provide a rigorous understanding of what it would befor a causet and a spacetime to stand in the required relation turns out to be challenging and isthe subject of ongoing research. There are at least two ways of approaching the problem. First,one could understand both causets and (distinguishing) Lorentzian manifolds as ‘causal measurespaces’ and use a so-called
Gromov-Hausdorff function d GH ( · , · ) (or the ‘Gromov-Wassersteinfunction’ if we have a probability measure) to give a measure of the closeness or similaritybetween such spaces. Such a distance function relating any two causets, any two Lorentzian This line is pursued by Bombelli et al. (2012), although that paper has been withdrawn and never beenfollowed up, as far as we can tell. Sam Fletcher has given talks about this in 2013 developing this ansatz, but, toour knowledge, has not published this material. ) q ( { I p x n d g { R s ) = R ( Vp qR i ab g ; hMiÁ , hC Figure 3: Faithful embedding (Figure from Lam and W¨uthrich 2018).spacetimes, or any causet and any Lorentzian spacetime in terms of their relevant similaritycould provide a rigorous way to spell out the idea of ‘approximation’.But worked out in much more detail is a second approach, using the notion of a ‘faithfulembedding’ of a causet into a spacetime. The task here is to find a map ϕ : C → M such thatthe image looks like a relativistic spacetime, as shown in figure 3. More precisely, a spacetimeapproximates a causet if there exists a faithful embedding of the causet into the spacetime in thefollowing sense:
Definition 2.
A causet (cid:104)C , ≺(cid:105) is said to approximate just in case there exists a ‘faithful embed-ding’, i.e., a injective map ϕ : C → M such that1. the causal relations are preserved, i.e. ∀ a, b ∈ C , a ≺ b iff φ ( a ) ∈ J − ( φ ( b )) ;2. ϕ ( C ) is a ‘uniformly distributed’ set of points in M ;3. (cid:104)M , g ab (cid:105) does not have ‘structure’ at scales below the mean point spacing. Let us discuss the three conditions in turn.The first condition requires that the causal structures of the causet and the spacetime towhich it gives rise are isomorphic. This requirement captures the idea that causal structure isfundamental and ‘percolates’ up the scales in a precise sense. This demand seems acceptable fora fundamentally classical theory, such as the version of CST discussed here. Once this theoryis replaced by a quantum CST, we cannot reasonably uphold this demand as we should expectquantum ‘fluctuations’ of the causal structure rather than some determinate causal structureand so no untainted emergence of that structure.The second condition expresses the expectation that regions of causets of a similar numberof elements give rise to spacetime regions of correspondingly similar volume. Thus, the mappingcannot be such that the image points of the causet elements are too dense and too sparse in thespacetime, resulting in a more or less uniform distribution. The intuition behind this conditionis thus clear, although its articulation obviously needs to be precisified. More precisely, the causet and the set of images of its events in the spacetime. One way in which it needs to be precisified is that the required uniformity must be with respect to thespacetime volume measure of (cid:104)M , g (cid:105) . Such a sprinkling selects events in Minkowski spacetime at the requireddensity uniformly and at random and then imposes the unique partial ordering among themwhich is induced by the causal structure of Minkowski spacetime. Thus, Minkowski spacetime’scausal structure is inherited by the causet and the first condition of a faithful embedding issatisfied by construction, as are the second and third. Notice that the way the relationship between fundamental causets and emergent spacetimesas assumed in this ‘sprinkling’ is really the wrong way around: we started out with the spacetimegiven, and then sprinkled events into it and connected those which were causally related as givenby the causal structure of the spacetime. Surely, this is putting the cart before the horse: thefundamental causet is prior to the relativistic spacetime, which is merely derivative. In thissense, what we ultimately want to understand is how a generic causet can give rise to the sortsof relativistic spacetimes we believe to be physically realistic.What remains unclear to date is what conditions need to be imposed in solution 2 in orderfor the abiding causets to be manifoldlike. In CST, it is hoped that the dynamical conditionsimposed to solve problem 1 double up to also solve problem 2. In this sense, one would needto postulate one set of natural conditions or physically meaningful axioms, which solve bothproblems at once. Computer simulations fuel the hope that this may indeed be the case byproviding evidence, e.g., that a significant class of dynamical laws closely related to transitivepercolation yields a cyclically collapsing and re-expanding universe in which each era containsphases of exponential expansion approximated by de Sitter spacetime. The restriction to manifoldlike causets is surely a necessary condition for there to be any way inwhich a causet lends itself to the emergence of spacetime. But this in itself does not solve thechallenge: GR is a notoriously permissive theory in that the number of spacetimes which satisfythe basic requirements of physical possibility imposed by GR, such as that the spacetime be The image points of the causets in the spacetime cannot stand in a ‘regular’, lattice-like structure since thiswould violate Lorentz invariance (see § For a useful illustration of Poisson sprinkling, see Dowker (2013, figure 1) and Surya (2019, §
3, particularlyfigure 6). See Ahmed and Rideout (2010). The class of dynamical laws is that of so-called ‘originary percolation’, whichis transitive percolation where the possibility of an element being born unrelated to any prior element is excluded.
Problem 3.
Given just how many ‘unphysical’ spacetimes are physically possible according toGR, causets may generically give rise to ‘unphysical’ relativistic spacetimes.
One might argue that this is not really a problem of CST, or indeed of any quantum theory ofgravity, since it is already at the level of GR that we are faced with rampant unphysical solutionsadmitted by the theory. Although this is undisputedly the case, we could, and perhaps should,invert this reasoning: it is precisely because
GR is too permissive that we are looking to the morefundamental theory for guidance and hope that it will rule out many or most of these unphysicalsolutions. So this problem would be solved by the following solution:
Solution 3.
Impose conditions such that qualifying causets are approximated by ‘physically rea-sonable’ spacetimes.
Evidently, this solution could only be implemented once we specify what a ‘physically rea-sonable’ spacetime, a locution on which we certainly have an intuitive grasp, but which turnsout to resist systematic and satisfactory treatment. Although a full explication is unlikely tocome forward, we might hope to make some progress. For instance, one might think that thedemand encoded in the kinematic axiom that the causal order be antisymmetric (cf. chapter2) ascertains that closed timelike curves cannot arise in the emerging spacetime. However, thisimplication only holds under some interpretations of the theory and must ultimately await itsfull articulation (W¨uthrich forthcoming). Consequently, although the problem is real and thesolution appears promising, its further elaboration is the task for another day.Just as problem 2, it is hoped that no additional conditions are needed and that the axiomspostulated above would also suffice to generically deliver physically reasonable spacetime, what-ever that is supposed to mean. But there is another reason why we would not want to imposeany additional axioms just because they enforce our intuitions as to which spacetimes we taketo be physically reasonable: part of the point of formulating a theory more fundamental thanGR is for us to learn from it in which ways GR is false and needs to be corrected. In particular,this may apply to the range of physical possibilities which a fundamental theory more curtailmuch more restrictively than notoriously permissive GR. The point, however, would be to learnthis from the fundamental theory, rather than not to conversely constrain that theory by thebrute force of our antecedent intuitions. Thus, we take it that we have excellent reasons to rejectproblem 3 as something we need to attend to at this point, all the while keeping in mind theissue as deserving revisiting once the theory is more fully understood.In other words, the project is to determine, and to some extent to guess , what the fundamen-tal physics is which will produce structures which are generically approximated by relativisticspacetimes.But we are not quite done. Even if we thus arrive at the point at which judiciously chosencausets generically give rise to physically reasonable relativistic spacetimes, one further necessarycondition must be imposed. A necessary condition for the spacetime to emerge is surely that itsupervenes on the fundamental structure, to use the jargon of philosophers. In other words, therecannot be a difference at the level of the spacetime without there also being a difference at the See Doboszewski (2017) and Manchak (forthcoming) for further discussion of the notion of ‘physically rea-sonable’ spacetimes.
Problem 4.
A given causet might be approximated by multiple relativistic spacetimes.
This would indeed be a problem, since in this case, the fundamental structure would notuniquely determine the emergent large-scale structure. As Sorkin (2005, 313) puts it: “Implicitin the idea of a manifold approximating a causet is that the former is relatively unique; for ifany two very different manifolds could approximate the same [causet] C , we’d have no objectiveway to understand why we observe one particular spacetime and not some very different one.”The implicit condition then is something like this: Solution 4.
If a causet is approximated by a spacetime, then the approximating spacetime is‘approximately unique’.
Following Sorkin, this has become known as the ‘Hauptvermutung’, or principal conjecture.Of course, this is but a template to fill in the specifics of the solution. In particular, what itmeans for the spacetime to be ‘approximately unique’ will have to be specified. It should alsobe pointed out that it is hoped, or conjectured, that the satisfaction of the earlier conditionsguarantees this solution such that no additional condition ought to be imposed. In other words,the hope is that the Hauptvermutung can be established as a theorem of the theory as articulatedso far and need not be added as an independent stipulation. In fact, to show that this is the caseis the overarching goal of the research program (at least at the classical level).To date, the Hauptvermutung remains an unproven conjecture. In order to have any hope ofproving it, a more precisely formulated proposition will be needed. Just how to accomplish thatwill depend on the chosen notion of ‘approximation’. Above, I identified two paths: one basedon the Gromov-Hausdorff measure of similarity, and another in terms of a faithful embeddingof causets into spacetimes. On the first research programme, the Hauptvermutung could beprecisified as follows:
Solution 5 (Hauptvermutung, Gromov-Hausdorff version) . If there exist two spacetimes (cid:104)M , g ab (cid:105) and (cid:104)M (cid:48) , g (cid:48) ab (cid:105) such that d GH ( C , M ) < (cid:15) and d GH ( C , M (cid:48) ) < (cid:15) for some causet C , then d GH ( M , M (cid:48) ) < (cid:15) . Although the choice of some particular (cid:15) ∈ R + is rather arbitrary, something like the abovecondition could encapsulate the idea that a causet is approximated by a spacetime in an ‘approx-imately unique’ sense. If, on the other hand, we followed the more standard path of articulatingthe relevant sense of approximation in terms of faithful embeddings, then the Hauptvermutungmight be expressed differently: Solution 6 (Hauptvermutung, standard version) . If there exist two spacetimes (cid:104)M , g ab (cid:105) and (cid:104)M (cid:48) , g (cid:48) ab (cid:105) , which approximate a given causet C in that there exist faithful embeddings ϕ : C → M and ϕ (cid:48) : C → M (cid:48) , then they are ‘approximately isometric’.
It might appear as if not much has been accomplished by replacing solution 4 by solution 6,since we basically replaced the problematically vague notion of ‘approximate uniqueness’ with anapparently equally problematically vague notion of ‘approximate isometry’. But we have madereal progress: the first stab at solving problem 4 just offered a general template of what, in rathergeneral terms, would be needed to solve the problem, whereas the much more precise solution 6reduces the problem to one of defining and defending a measure of approximate isometry between12pacetimes. This is by no means a trivial problem, as is witnessed by the fact that it has sofar resisted resolution. Such a resolution would presumably require that we identify the salientgeometric properties of spacetime, introduce measures of similarity along the dimensions of theselected properties, and compound these measures into a score for approximate isometry. (Exact)isometry does not require any of the above; however, if the isometry is no longer ‘exact’, and thusthe metrics compared no longer identical in the relevant sense, the notion of (exact) isometry getsno traction, and the salient geometric features with respect to which the equivalence is supposedto be determined must be selected. We shall leave it at this, as the point of this section is to sketch, only in general terms, whatwould be required in order to establish the emergence of spacetime in CST. Now suppose that thecauset research programme offered satisfactory accounts of how to fill in the details in the abovesketch and could thus be said to have fulfilled all necessary conditions listed. Of course, merelysatisfying some necessary conditions gives us not guarantee that the work is jointly sufficient.And at the end of the day, we will want the assurance that the challenge of the emergence ofspacetime is fully met, and this guarantee is only forthcoming if we convince ourselves that thenecessary conditions we have discharged are also jointly sufficient.
Before we enter the fray of the debate regarding the sufficiency of these conditions, let us pauseto consider what would have been accomplished had we successfully discharged all necessaryconditions. In fact, in this case, we could generically ‘derive’—i.e., systematically relate withfull mathematical rigour—relativistic spacetimes from causets. No mean feat. But would thisaccomplishment conclude the project? Would we be done? Not according to the r´esistance ,which believes that the conditions listed in the previous section are merely necessary but notjointly sufficient for the emergence of relativistic spacetimes from fundamental causets.The r´esistance has different cells, which may be differently motivated. A first cell consistsin primitive ontologists who insist on fundamental local (and so a fortiori localizable ) beablesin spacetime. According to this view, for a theory to qualify as a candidate physical theorydescribing a sector of our physical world and thereby accounting for some of our empirical dataor, more broadly, for aspects of human experience, the theory must postulate an ontology ofentities populating regions of spacetime. If the entities posited by a theory are not localizable,we seem to be losing a convenient way of parsing that which exists into a plurality of distinctentities: location offers a straightforward criterion for individuating objects. Although the loss ofthis criterion is already adumbrated by the non-locality of quantum physics, without spacetimeat our side, how are we to dissect that which exists fundamentally into distinct and separateentities? Clearly, there are alternative ways of distinguishing entities, and so the criterion ofspatiotemporal location commands no necessity. Admittedly though, for these alternative criteriato get traction at the fundamental level of a theory of quantum gravity may be tricky, and soa structuralist or monist stance may most naturally fit these cases. Even so, fundamentalspatiotemporality is not necessary for parsing an ontology. Although the realist motivation ofthis camp is laudable, the particular form of realism demanded thus comes with an unduly narrow Two spacetimes (cid:104)M , g ab (cid:105) and (cid:104)M (cid:48) , g (cid:48) ab (cid:105) are (exactly) isometric just in case there exists a diffeomorphism φ : M → M (cid:48) such that φ ∗ ( g ab ) = g (cid:48) ab . This definition delivers the identity “in the relevant sense” invoked in themain text. Cf. W¨uthrich (2012), who argues for a structuralist interpretation of CST, and discusses the challenges ofsuch an approach. anti-Pythagoreans who fear a lossof the venerable distinction between physical stuff and abstracta, and object to the idea that thephysical world is fundamentally mathematical or abstract by its nature. The distinction betweenconcrete physical entities and abstract entities is thought to be threatened by the presumednon-spatiotemporality of a putative ontology of quantum gravity. On a standard demarcation,the concrete entities are taken to be those in spacetime, and the abstract ones those which arenot. This criterion of demarcating the concrete from the abstract seems to suggest that thefundamental, non-spatiotemporal structures which make up our natural world must be abstract,perhaps mathematical. Alternatively, concrete physical entities have been characterized as thoseengaging in the causal commerce of the world. Worries of circularity concerning this criterionaside, attempts to explicate a notion of causal efficacy in the absence of spacetime may bethwarted by insurmountable difficulties (Lam and Esfeld 2013, § sans spacetime deny the very conditions necessary for theirempirical confirmation, and in this sense be ‘empirically incoherent’ (Huggett and W¨uthrich2013). Although empirical incoherence does not amount to any kind of impossibility, it would bemost unfortunate for empirical science if we lived in a world in which the necessary conditionsfor such an enterprise could not be set in place.We believe that all these cells of the r´esistance can be put to rest if it could only be estab-lished that spacetime emerged in the appropriate limit or at the requisite scales. Showing howspacetime emerges would involve demonstrating how macroscopic (or indeed most microscopic)objects have location and are thus localizable. In this sense, to the extent to which the demandof primitive ontologists was reasonable, it would be met. Pythagoreanism would be averted, be-cause although the fundamental ontology would neither be spatiotemporal nor directly causallyefficacious, the fundamental structures would have been shown to directly connect to the spa-tiotemporal structures we associate with physical being. Presumably, the emergent spacetimewould be shown to ontologically depend on these fundamental structures, thus endowing thelatter with concreteness in a ‘top-down’ manner (Huggett and W¨uthrich 2013, 283f). Finally,the only condition necessary to evade the threat of empirical coherence is that the conditions forempirical confirmation are set in place at the scales of the human scientist, not at the level offundamental ontology. Establishing that at human scales, the world is indeed spatiotemporal toa very good approximation circumvents the menace of empirical incoherence, at least insofar itwas motivated by the apparent absence of spacetime.In order to establish the emergence of spacetime, it would evidently not be enough to satisfymerely necessary, but jointly insufficient conditions. But the three resisting concerns listed abovemay guide us in what it would take to appease them and thus arguably arrive at jointly sufficientconditions. Collectively, they suggest that in order to accomplish this, we need to focus on thefunctions spacetime plays in structuring our ontology, distinguish its elements from abstracta,14nd enabling empirical confirmation, among others. Thus, spacetime should be analysed interms of the functions it performs to other ends, rather than for its own sake. Spacetime is asspacetime does, to invoke the motif in Lam and W¨uthrich (2018), although this need not beinterpreted in ontologically thick ways. In this spirit, establishing the emergence of spacetimeinvolves showing how the fundamental may instantiate these functions or roles in favourablecircumstances. This ‘spacetime functionalism’ we have already encountered in chapter 1 assertsthat once the functional roles of spacetime have been identified and the fundamental physicsshown to fulfil these roles, the emergence of spacetime has been fully established with full physicalsalience and no work remains to be completed in this respect. How would one implement a functionalist programme for CST? The first step (SF1) of spacetimefunctionalism as articulated in § § § § § Now that we have sketched CST, illustrated how spacetime disappears in it, and sketched howit re-emerges, let us finish the discussion by addressing two points of considerable philosophicalinterest which come up in CST and in its conception of the emergence of spacetime: the possibilityof a relativistically invariant passage of time or becoming, and the appearance of highly non-localbehavior in this classical, relativistic, but discrete theory. We save the second point for the nextsection ( §
5) and turn to the first issue, the metaphysics of time based on CST. Both pointsaptly illustrate the philosophical and conceptual issues that arise in developing a new quantumtheory of gravity, and how spacetime emerges from it. The emergence of spacetime in quantumgravity is inextricably entangled with deep, and unavoidable, philosophical questions.
Contemporary physics is notoriously hostile to a what philosopher have come to call ‘A-theoretic’metaphysics of time, i.e., a metaphysics of time which fundamentally includes an element ofbecoming or dynamical passage, an aspect of time captured in our language by the use of tenses. An important example of an A-theoretic metaphysics is presentism , according to which onlypresent objects and events exist, but in a dynamically updated way, such that we arrive at ametaphysics of a dynamical succession of ‘nows’. Another example is the growing block view ,according to which only past and present objects and events exist such that the sum total ofexistence continually grows by including ever more slices of existence.CST promises to reverse that verdict against A-theories: its advocates have argued that theirframework is consistent with a fundamental notion of ‘becoming’. Relevantly, the dynamical‘growth’ of causets we introduced in § This section is based on work previously published in W¨uthrich and Callender (2017). As opposed to a ‘B-theoretic’ metaphysics of time, which does not fundamentally include such elements ofbecoming, but instead seeks to explain them as emergent or illusory phenomena. One might not believe that our intuitive notion of time needs or deserves rescuing,but there is no denying that if this claim is correct it would have significant consequences for thephilosophy of time. Specifically, it may underwrite a ‘growing block’ model of the metaphysicsof time, as John Earman (2008) has speculated.As argued in W¨uthrich and Callender (2017), we believe that the introduction of CST and itsdynamics does not ultimately change the fundamental dilemma any fan of becoming or passageconfronts when facing relativistic physics, even though some novel aspects arise, mainly due tothe discreteness of causets. The dilemma is the following: any metaphysics of time including afundamental sense of becoming or passage either answers to their fan’s explanatory demands oris compatible with relativistic physics, but not both . To illustrate this with an example, suppose one thought, as does the presentist, that in orderfor there to be becoming or passage, there needs to be a (dynamically changing) present, a‘now’, identified in the fundamental structures of the world. In the context of special relativity,with its relativity of simultaneity, one might introduce a foliation of spacetime into spacelikehypersurfaces totally ordered by ‘time’. Presumably, that would answer to the presentist’s notionof a (spatially extended) present and of becoming, but at the price of introducing structure notinvariant under automorphisms of Minkowski spacetime and hence arguably violating specialrelativity. Conversely, the present can be identified with invariant structures such as a singleevent or the surface of an event’s past lightcone, and successive presents as a set of events ona worldline or as a set of past lightcones totally ordered by inclusion, respectively, but suchstructures will have radically different properties from those ordinarily attributed to the presentby those seeking to save it (see Callender (2000) and W¨uthrich (2013)).Returning to CST, a natural transposition of the idea of a foliation of spacetime into space-like hypersurfaces is to partition a causet into maximal antichains, as considered in § § §
3) argue in more detail. They conclude that, at least atthe kinematical level, CST embraces the dilemma and in fact makes it more rigorous. But thatmay not be all that surprising; after all, the heart of the idea that CST rescues becoming involvestaking sequential growth seriously: becoming is embodied in the ‘birthing’ of new elements, andso in the theory’s dynamics.Although we are interested in becoming, we should immediately remark that sequentialgrowth is certainly compatible with a tenseless or block picture of time. In mathematics astochastic process is defined as a triad of a sample space, a sigma algebra on that space, anda probability measure whose domain is the sigma algebra. Transition probabilities are viewedmerely as the materials from which this triad is built. In the case at hand, the sample spaceis the set Ω = Ω( ∞ ) of past-finite and future-infinite labeled causets that have been ‘run toinfinity’. The ‘dynamics’ is given by the probability measure constructed from the transitionprobabilities; for details, see Brightwell et al. (2003). On this picture, the theory consists simplyof a space of complete histories with a probability measure over them. Cf. e.g. Dowker (2003, 38). For other important expressions of the claim, see Sorkin (2007); Dowker (2014,2020). For earlier articulations of the same dillemma, see Callender (2000) and W¨uthrich (2013). This interpretation corresponds to Huggett’s first option (2014, 16), which is fully B-theoretic. When weconsider ‘taking growth seriously’, we mean to essentially follow the second route he offers: augmenting the causalstructure with an additional, but gauge-invariant, dynamics. .2 Taking growth seriously However, let’s take the growth seriously. There are different extents to which this can be done.At a more modest level, and consistent with explicit pronouncements by advocates of causetbecoming, we can articulate a localized, observer-dependent form of becoming. Here, the idea isthat becoming occurs not in an objective, global manner, but instead with respect to an observersituated within the world that becomes. The only facts of the matter concerning becoming arelocal, and are experienced by individual observers as they inch toward the future. In Sorkin’swords, which are worth quoting in full,[o]ur ‘now’ is (approximately) local and if we ask whether a distant event spaceliketo us has or has not happened yet, this question lacks intuitive sense. But the‘opponents of becoming’ seem not to content themselves with the experience of a‘situated observer’. They want to imagine themselves as a ‘super observer’, whowould take in all of existence at a glance. The supposition of such an observer would lead to a distinguished ‘slicing’ of the causet, contradicting the principle that such aslicing lacks objective meaning (‘covariance’). (2007, 158)According to Sorkin, instead of “super observers”, we have an “asynchronous multiplicity of‘nows’ ”. It seems fairly straightforward that a perfectly analogous kind of becoming can be hadin the context of Minkowski spacetime. Indeed, ‘past lightcone becoming’, in the sense of Stein(1991), and ‘worldline becoming’, as articulated by Clifton and Hogarth (1995), both satisfythe bill. Furthermore, past lightcone becoming and worldline becoming are also available ingeneral-relativistic spacetimes, as they do not depend on the spacetime admitting a foliation.Although Sorkin himself remains uncommitted concerning whether the analogy holds, FayDowker (2014, 2020) rejects it, arguing that ‘asynchronous becoming’ is not compatible withGR, but only with a dynamics like the one provided by the classical sequential growth. Shediscusses ‘hypersurface becoming’ in GR—which does of course depend on the spacetime admit-ting a foliation—and rejects it for its obvious violation of general covariance. She is clear thathypersurface becoming is but one way to implement becoming in the context of GR, but doesunfortunately not discuss alternatives. In particular, she does not discuss worldline becoming orpast lightcone becoming, which both seem much more promising analogues of the asynchronousbecoming identified by Sorkin in CST. She maintains that what is needed for there to be be-coming is not the mere existence of events, but a process she term terms their occurrence (2014,22). She claims that spacetime events do not ‘occur’ in GR in this special sense, and so thereis no genuine form of becoming possible in GR. Against this, we note firstly that (an importantsubsector of) general relativity certainly can be described in a ‘dynamical’ manner via its many‘3+1’ formulations. To make her objection, Dowker would first need to elaborate the reasonswhy a 3+1 dynamics does not provide the ‘occurrence’ she desires. Of course, we admit that ourretort here lacks full generality.Furthermore, we note here a possible tension. If occurrence is simply a label for some eventsfrom the perspective of other events, then there is no problem—but then we note that such labelscan be given consistently in GR too. But if occurrence implies something metaphysically meaty,such as existence or determinateness—then there is a possible tension between occurrence andthe local becoming envisioned by Sorkin and Dowker. If spacetime events that are spacelikerelated do not exist for each other, for instance, then that is a radical fragmentation of reality. Cf. also Arageorgis (2016) who makes a similar point. Cf. Wald (1984, Ch. 10). In Pooley’s view (2013, 358n), dynamical classical sequential growth should best be interpreted as a “non-standard A Theory” in the sense of Fine (2005), i.e., as giving up “the idea that there are absolute facts of thematter about the way the world is.” (2013, 334) x in a relativistic spacetime.Any dynamics distinguishing a particular label order will be non-relativistic. Not wanting thedynamics to distinguish a particular label (‘coordinatization’), the authors impose discrete gen-eral covariance , i.e., axiom 2 on the dynamics. This is a form of label invariance. As statedabove, the idea is that the probability of any particular causet arising should be independent ofthe path to get to that causet. In fact, returning to figure 2, there could be no physical fact ofthe matter whether the 3-element causet in the middle of the top row grew via a stage consistingin a 2-chain or a 2-antichain (see also figure 2 in W¨uthrich and Callender 2017). In the 3-elementcauset, the event with another event in its past (labelled ‘ a ’ in figure 2) and the causally isolatedevent (labelled ‘ b ’) are spacelike related. Consequently, there is not fact of the matter whichof the two came into being first and which one second. To say which one happened ‘first’ isto invoke non-relativistic concepts. It is therefore hard to understand how there can be growthhappening in time.Seeing the difficulty here, John Earman (2008) suggests a kind of philosophical addition tocausets, one where we imagine that ‘actuality’ does take one path or another. With such a hiddenvariable moving up the causet, we do regain a notion of becoming. But as Aristidis Arageorgis(2016) rightly points out, such a move really flies in the face of the normal interpretation ofthese labels as pure gauge. The natural suggestion, espoused by (almost all?) philosophers ofphysics, is then that the above non-dynamic interpretation in terms of a block universe is bestbecause it does not ask us to imagine that one event came first.Perhaps the sensible reaction to this problem is to abandon the hope that the classical se-quential growth dynamics does produce a novel sense of becoming. Still, we are tempted to presson. The intuition motivating us is as follows. True, the dynamics is written in terms of a choiceof label, but we know that a consistent gauge invariant dynamics exists ‘beneath’ this dynamics.In fact, rewriting the theory in terms of a probability measure space, as indicated above, onecan quotient out under relabellings to arrive at a label-invariant measure space (for constructionand details, see Brightwell et al. 2003). And one thing that we know is gauge invariant is thenumber of elements in any causet. Focusing just on these and ignoring any labeling, we do havetransitions from C n to C n +1 and so on. There is gauge-invariant growth.The problem is that we are generally prohibited from saying exactly what elements exist atany stage of growth. Take the case of the spacelike related events above. The world grows from C to C to C . That’s gauge invariant. We just cannot say—not due to ignorance, but becausethere is no fact of the matter—whether C is the 2-chain or the 2-antichain. causet reality doesnot contain this information. There simply is no determinate fact as to whether C in the 2-chainor the 2-antichain; but there is a determinate fact that it contains one of them. If it is coherent,therefore, to speak of a causet having a certain number of elements but without saying what thoseelements are, then classical sequential growth dynamics does permit a new kind of—admittedly Cf. also Butterfield (2007, 859f). C world like? It does not havethe 2-chain and the 2-antichain in it (that’s C ), nor does it have neither the 2-chain nor the2-antichain in it (that’s C ). The world determinately has the 2-chain or the 2-antichain in it,but it does not have determinately the 2-chain or determinately the 2-antichain. ‘Determinately’cannot penetrate inside the disjunction. Notice that this feature is a hallmark of vagueness or ofmetaphysical indeterminacy more generally. Without going into any details of the vast literatureon vagueness, let us note that there is a lively dispute over whether there can be ontologicalvagueness. The causet program, interpreted as we have here, supplies a possible model of aworld that is ontologically vague. Further discussion of this model seems to us worthwhile.We would like to point out that Ted Sider (2003) has supplied arguments that existencecannot be vague. That existence cannot be vague or indeterminate was a central assumption ofhis argument to four-dimensionalism in his (2001). In fact, he asserts (2001, 135) that anyonewho accepts the premise that existence cannot be vague is committed to four-dimensionalism,the thesis that objects persist by having temporal parts. To the extent to which many advocatesof becoming reject four-dimensionalism anyway, they would thus be open to embrace ontologicalindeterminacy even if Sider’s arguments of 2001 and 2003 were successful. And they may wellnot be: one of them, for instance, is based on the claim that it cannot be vague how many thingsthere are in a finite world (2001, 136f). Obviously, a defender of observer-independent becomingin CST may agree that it is at no moment vague how many events there exist, but neverthelessdisagree that existence cannot be vague. Thus, we may have ontological indeterminacy withoutvagueness in the cardinality of the (finite) set of all existing objects. One may be worried that on this notion of becoming in CST, no event in a future-infinite causetmay ever be determinate until future infinity is reached, at which point everything snaps intodeterminate existence. This worry is particularly pressing as realistic causets are often takento be future-infinite. So does any event ever get determinate at any finite stage of becoming?In general, yes. One way to see this is by way of example. As it turns out, causets based ontransitive percolation in general have many ‘posts’, where a post is an event that is comparableto every other event, i.e., an event that either is causally preceded by or causally precedes everyother event in the causet. Rideout and Sorkin interpret the resulting cosmological model asone in which “the universe cycles endlessly through phases of expansion, stasis, and contraction[...] back down to a single element.” (1999, 024002-4) Consider the situation as depictedin figure 4. There is a post, p , such that N events causally precede p , while all the others—potentially infinitely many—are causally preceded by p . At stage N −
1, shown on the left,there exist N − p except those three events whichimmediately precede p , shown in black, must have determinately come to be. Of the threeimmediate predecessors, shown in grey to indicate their indeterminate status, two must exist;however, it is indeterminate which two of the three exist. At the prior stage N −
2, the grey setof events existing indeterminately would have extended one ‘generation’ further back, as it couldbe that two comparable events are the last ones to come to be before the post becomes. At thenext stage, stage N , N events exist and it is determinate that all ancestors of p exist. There isno ontological indeterminacy at this stage. Event p has not yet come to be at either stage andis thus shown in white. At stage N + 1, not shown in figure 4, event p determinately comes into Cf. also Bollob´as and Brightwell (1997). vents{3 N pp post : N at stage : {1 N at stage Figure 4: Becoming at post p (Figure from W¨uthrich and Callender 2017).existence. At stage N + 2, one of the two immediate successor to p exists, but it is indeterminatewhich one. And so on.One may object that this interpretation of the dynamics of a future-infinite causet presupposesa given final state toward which the causet evolves, and thus involve a teleological element. Eventhough everything in the preceding paragraph is true under the supposition that the final causetis the one represented in figure 4, the objection goes, at stage N it is not yet determined that p is a post, as there could have been other events spacelike-related to p . Given that it is thusindeterminate whether p is indeed a post, and since this is the case for all events at finite stages,no events can thus snap into determinate existence at any finite stage of the dynamical growthprocess.First, it should be noted that even if this objection succeeds, it is still the case that it isobjectively and determinately the case that at each stage, one event comes into being and thatthus the cardinality of the sum total of existence grows. Although the ontological indeterminacyremains maximal, there is a weak sense in which there is objective, observer-independent becom-ing. Second, if the causet does indeed not ‘tend’ to some particular future-infinite causet, thenall existence would always be altogether indeterminate (except for the cardinality). There wouldbe no fact of the matter, ever, i.e., at any finite stage, of how the future will be, or indeed ofhow anything ever is. If this is the right way to think about the metaphysics of the dynamics ofCST, we are left with a wildly indeterminate picture. Third, it should be noted that the math-ematics of the dynamics is only well-defined in the infinite limit; in particular, for there to be awell-defined probability measure on Ω, we must take Ω = Ω( ∞ ) (Sorkin 2007, 160n; Arageorgis2016, § C that C determinately is one way rather than the other.Second, note that as a causet grows, events that were once spacelike to the causet mightacquire timelike links to future events. If we regard the growth of a new timelike link to aspacelike event as making the spacelike event determinate, modulo the above type of vagueness,21hen this is a way future becoming can make events past. That is, there is a literal sense inwhich one can say that ‘the past isn’t what it used to be’. Having said that, there is a relativisticanalogue in the growth of past lightcones, which also come to include formerly spacelike-relatedevents as one moves ‘up’ along a worldline.Finally, although we don’t have space to discuss it here, note that despite appearances tran-sitive percolation is perfectly time reversal invariant. This allows the construction of an evenmore exotic temporal metaphysics. If we relax the assumption that events can only be born tothe future of existing events, then it is possible to have percolation—and hence becoming—goingboth to the future and past. Choose a here-now as the original point. Then it is possible tomodify the theory so that the world becomes in both directions, future and past. Of course,similarly, we could have a causet that is future-finite and only grows into the past, and thus ispast-infinite.In sum, then, does CST rescue temporal becoming? At the kinematical level (not muchdiscussed here, but see W¨uthrich and Callender 2017), CST does offer new twists in dealingwith time and relativity, but the basic contours of the relativistic challenge remains. Seriousconstraints also threaten becoming if we take the time in CST’s dynamics seriously too. Here,however, if one is open to the costs of a sufficiently radical metaphysics, we maintain that thereis a novel and exotic type of objective, observer-independent temporal becoming possible. Itshould be noted, again, that all this remains purely classical and so subject to potentially radicalchange in a future quantum theory of causets. So far, it is really just a classical, discrete, anddynamically stochastic theory. The discreteness of its structure has another unusual consequence:a classical form of non-locality. As we have discussed in chapter 2, the fundamental discreteness of causets has certain concep-tual and technical advantages, as it promises to eliminate the nasty ultraviolet divergencies ofquantum field theory (QFT) and the singularities of GR. However, it seems to also come at aprice: it is incompatible with the demand for Lorentz symmetry and the expectation of a rea-sonably local physics. It seems as if one can have any two of these three features—discreteness,Lorentz symmetry, and locality—in a theory, but not all three, as was already noted by Moore(1988) within half a year of the publication of the founding document of the causet program(Bombelli et al. 1987). There is of course a sense in which no discrete structure can be Lorentzinvariant, as Lorentz transformations are only defined for continuum spacetimes. However, evenis the fundamental structure is not Lorentz invariant, one would not want to give up Lorentzsymmetry at the emergent level, at least to an extremely close approximation. This places avery strong constraint on both the spacetime structures and the dynamics of matter that canemerge from causets. As discreteness is built into the DNA of the causet program, it seems asif CST is committed to a form of non-locality, at least at the fundamental level. But our worldseems manifestly local at all observed scales—apart from quantum non-locality of course—, andso local physics better emerge from the fundamental physics of causets. How should we understand the claim that the demand for Lorentz symmetry and the discretenessof causets entail a form of non-locality of the physics? The discreteness of causets in built in from The best references explaining this resulting non-locality in the context of CST are Sorkin (2009a) and Dowker(2011). and so the demand is on empirically rather secure grounds.In order to recognize the non-locality of a Lorentz-invariant, discrete structure, let us returnto the standard way of conceiving of the relation between causets and spacetime as depicted infigure 3, prototypically captured by a Poisson sprinkling of elements into Minkowski spacetime(as in § n elements into a spacetime region of spacetime volume V depends onlyon the density ρ of the sprinkling and V : P ( n ) = ( ρV ) n e − ρV n ! . (1)Since ρ is fixed, this probability is manifestly invariant under all transformations preservingspacetime volume, such as Lorentz transformations (Henson 2012, § ρ of the sprinkling is a fundamental constant of the theory, but is naturally setat something like the Planck scale. As basically noted by Moore (1988), the spacetime volumeof a hyperboloid shell of any finite thickness (defined spatiotemporally by an invariant interval∆ s ) at a given spatiotemporal distance in the past from any event p is infinite. The greyed outregion in figure 5 represents such a hyperboloid shell to the past of an event p . Thus, whateverthe fixed, constant density ρ is, the number of elements in the hyperboloid shell diverges. If p rq Figure 5: There are infinitely many events in the thickened past hyperboloid of event p .the hyperboloid shell is, for example, at a Planck interval to the past of p , this means that p will have infinitely many predecessor elements a Planck distance in its past. In fact, whatever In the sense that the bounds for violating Lorentz symmetry are extremely tight (Will 2014, § p ’s past lightconewhich consists of points less than the chosen interval in the past (i.e., the past lightcone of p minus the hyperboloid shell and its past) is also infinite and so contains infinitely many sprinkledelements. Thus, within any fixed spacetime interval to the past of an element of a causet beingapproximated by a Minkowski spacetime, there are necessarily an infinite number of elements.In this sense, there is no nearest neighbour to the past of any given event in a sprinkling ofMinkowski spacetime. For any event sprinkled close to p in p ’s past, there is always anotherevent sprinkled to the past of p , which is even closer to p (Bombelli et al. 2009). The same holdsfor immediate neighbours to the future. This is a form of non-locality in that in any inertialframe, events arbitrarily far away spatially from p (in that frame) will be arbitrarily close to p in terms of the invariant spacetime interval.Inevitably, it thus seems as if any element of any causet being approximated by Minkowskispacetime must have an infinite number of ‘immediate’ predecessors and an infinite number of‘immediate’ successors. As Moore (1988, 655) concludes, it seems as if discrete ‘spacetime’“has the nasty property that every point is influenced by an infinity of ‘nearest neighbors’ which,in a given frame, are arbitrarily far back in time.”In their reply, Bombelli et al. (1988) offer two lines of defence, both of which are sound in ourview. First, they point out that this is really the natural discrete analogue of the fact that anevent in a Lorentz manifold has neighbourhoods which converge to its lightcones as well, and so should be reflected by causets. In this sense, in a continuum spacetime, events spatially arbitrarilyfar away from p in a given frame are also arbitrarily close to p as measured by the spacetimeinterval. Perhaps the non-locality inherent in causets is really not all that troubling after all.Second, they point out that (general-)relativistic spacetimes are only locally Lorentz invariantand that in important cases such as the Friedmann-Lemaˆıtre-Robertson-Walker spacetimes, thepast lightcones of each event has a finite volume and so a sprinkling would suggest only a finitenumber of events to the past in a discrete analogue. In fact, they assert, this will be so in thegeneral situation using a ‘sum-over-histories’ approach, which they prefer.This latter point deserves some unpacking. In general, Lorentz invariance will not turn outto be a global symmetry of emergent spacetime. In this sense, spacetime curvature will limitLorentz symmetry, restricting the number of nearest neighbours of an element in a underlyingdiscrete structure to a finite number. However, unless the spacetime curvature is rather large,i.e., the radius of curvature comparable to the Planck length, the number of nearest neighboursof any element will still be very large. Hence, fundamental physics according to CST cannot be‘local’ in that the physics at an element of the causet depends only on that of a neatly confined,finite (and somewhat ‘small’) set of elements.A remark before we proceed. This non-locality becomes apparent only once we considerembeddings of causets into Minkowski spacetime and countenance the symmetries of that space-time. One might thus be tempted to think that the non-locality is somehow an artefact of thisconception of the relation between causets and spacetimes, an artefact which might disappear ifwe replace the Poisson sprinkling of events into Minkowski spacetime with a appropriate formof relating the two. But this would be too quick: the argument shows, quite generally, that acauset whose elements do not have a very large number of immediate neighbours cannot possibly ‘Nearest’ as measured in terms of the invariant spacetime interval ∆ s . We are not interested in spacelike relations between events, even though the same applies mutatis mutandis for spacelike related neighbours. What is an ‘immediate’ or ‘nearest’ neighbour? In the fundamental theory, it is defined as being directlyrelated. In Minkowski spacetime, the sprinkled events are ‘immediate’ or ‘nearest’ neighbours if the Alexandrovinterval defined by the two events does not contain another event. The Alexandrov interval is defined as thenon-empty intersection of the causal future of one event and the causal past of the other. n of elements. Assuming that it is the dynamics which tames the KR catastrophe(see § § n elements grows as e n / ,their contributions to the partition function risk being outcrowded by those of non-manifold-liketerms. Although this is of course not a conclusive argument, Benincasa and Dowker thus donot expect a local dynamics to be able to account for the manifold-likeness and thus not be asolution in the sense of § This hope will be fulfilled if there exists an intermediate scale at which effective physics isapproximately local. In the previous subsection, we have seen that the physics of CST mustbe non-local at the fundamental level. Due to the presence of spacetime curvature, the globalstructure of spacetime will not be Minkowskian, and hence there is no requirement that thephysics be Lorentz-invariant at cosmological scales, and it may well be local. However, to repeat,physics must definitely be Lorentz-invariant across a wide range between the fundamental and thecosmological scales. Thus, whatever the fundamental physics, and in particular the fundamentaldynamics, it must give rise to quasi-local physics, which is Lorentz-invariant to an extremelyclose approximation. What exacerbates the situation, as argues Sorkin (2009a, 28), is that asthe non-local couplings by far outnumber the local ones, the non-locality cannot be entirelyrestricted to the fundamental level.However, as Sorkin goes on to explain, there is promising evidence that the emerging physicsis reasonably local. He considers (in § φ in two dimensions with the standard equation of motion, i.e., (cid:3) φ =0, where ‘ (cid:3) ’ is the d’Alembertian operator. It is the significant achievement of this paperto propose a ‘discretized’ version of a two-dimensional d’Alembertian in terms native to thefundamental causet (“fully intrinsic”, in Sorkin’s words 2009a, 33). This d’Alembertian is Assuming that (cid:3) acts linearly on the field φ , as Sorkin does, the d’Alembertian for a causet (cid:104) C, ≺(cid:105) can beexpressed as a suitable matrix B xy , where x and y range over the elements of the causet. Sorkin also requiresthat B is ‘retarded’ or ‘causal’ in that B xy = 0 in case x is spacelike to, or precedes, x . Using a series of informedguesses based on methods in theoretical physics and computer simulations, Sorkin (2009b, 33) proposes that Their operator iseffectively local as well, courtesy of the non-local contributions to magically cancel out, leavingjust ‘local’ contributions. Here, ‘local’ refers to the frame defined by φ itself, i.e., to the framein which φ varies ‘slowly’. It turns out that the operator is effectively local in this frame, withnon-local contributions suppressed. Although it must be clearly stated that none of this suffices for anything like a definitiveverdict on the matter, these preliminary results amount to a ‘proof of concept’, as Benincasa andDowker (2010, 4) insist, in that they establish “the mutual compatibility of Lorentz invariance,fundamental spacetime discreteness and approximate locality”.One upshot of this incipient research is also that the fundamental non-locality in CST cannotbe completely tamed. Nor should it be: the proposed physics should ultimately have empiricallydetectable consequences, after all. As the theory delivers Lorentz-invariant mesoscopic physicsby construction, no violation of Lorentz symmetry should be expected in the approach. Thus,the phenomenology must come from elsewhere, and the non-local discreteness seems a promisingsource. We will close the chapter by discussing two such potential effects: swerving particles andthe cosmological constant.
An issue wide open for CST is how causets interact with matter, e.g., with particles or non-gravitational fields. The d’Alembertian constructed in Benincasa and Dowker (2010) and relatedresearch efforts paint a way in which one might see fields propagate on causets. How should weput matter ‘on’ causets even kinematically, prior to giving it the proper dynamics? In the simplecase of a (real-valued) classical scalar field, one just sticks a number on each element of thecauset. More precisely, a scalar field on a causet is represented by a map from the causet to thefield R of real numbers. The ‘dynamics’ then gives us a rule restricting the relations among thesenumbers. Vector fields, complex-valued fields, quantum fields—although more complicated—willintuitively be constructed along similar lines, mutatis mutandis.What about particles? Restricting ourselves to the simpler case of point particles, a particlewill presumably trace out a timelike path on a causet. In other words, its worldline will consistin a set of pairwise causally related elements of the causet, i.e., on a chain of causet elements.Any chain in a causet is a kinematically possible worldline for a point particle. Surely, we willwant to restrict the physically possible trajectories to some subset of such worldlines. Thus, we (ignoring a constant scale factor) B xy = − : x = y , − , x (cid:54) = y for n ( x, y ) = 0 , , , respectively , , where n ( x, y )is the cardinality of the order interval (cid:104) y, x (cid:105) = { z ∈ C | y ≺ z ≺ x } . This d’Alembertian is local inthat the physics at events removed by more than two ‘ticks’ has no influence. However, it is still non-local in thatevents which are immediately causally connected will be arbitrarily far away spatially in some frames. Glaser and Surya (2013) propose an order-theoretic characterization of local neighbourhoods in manifoldlikecausets, thus providing a frame for this as well as further evidence. We thank Fay Dowker for private correspondence on this point. the basic idea is that the continuation whichbest preserves linear momentum (in the frame in which the three-momentum going ‘into’ theevent vanishes) is the dynamically chosen one. In general, the three-momentum going ‘forward’,although minimized, will not be zero. In this sense, the particle appears to ‘swerve’ away fromthe geodesic, apparently undergoing a random acceleration. Since this swerving diverges fromthe expectation of strictly geodesic motion, it would be in principle an empirically detectablesignature of CST.That the effect size would be rather small compared to observationally accessible scales is justa practical problem. There are also more principled reasons to be critical of the model. First, theparticles are not modelled in a credible manner: realistic particles will not be point particles, andbe quantum in nature, so that the classical, deterministic rule above seems ill-fitted. Second, themodel makes ineliminable use of Minkowski spacetime to deliver a standard of inertial motion.Surely, the details of the particle motion should not depend on the embedding of the causet intoa spacetime, but be intrinsically defined in the causet. Third, and perhaps most damning, thereis no sense in which the dynamics includes a backreaction of the matter content on the causet,as we would expect from GR. There are ways to overcome some of these difficulties (see e.g.Philpott et al. 2009), but research continues. Λ Let us end with a brief discussion of CST’s most suggestive ‘prediction’, viz., that the cosmologicalconstant Λ has a small, but non-zero positive value of the order of 10 − in natural units. The‘prediction’ earns its scare quotes due to its being a heuristic, and certainly defeasible argument,rather than a tight quantitative calculation. But it clearly is a prediction in that it has beenmade prior to the relevant observations, and not a mere ex post facto construction of a just-sojustification. And this makes it rather intriguing. The prediction is due to Rafael Sorkin andappears to date from the late 1980s, roughly a decade before the discovery of the accelerating ofthe universe’s expansion in 1998 by Saul Perlmutter, Adam Riess and Brian Schmidt and theirteams. The first brief published version of the argument can be found in the last paragraph ofSorkin (1991), a published version of the talk given a year earlier, in 1990. We here mostly followSorkin (1997b, § V of spacetime. Thus, itoffers an explanatory paradigm for a small positive Λ (or for the accelerated expansion), whichdiffers interestingly from the standard account invoking dark matter. A useful illustration can be found in figure 1 in Dowker (2011). See also Henson (2009, § N of causet elements. Earlier ( § N is a (in fact, the ) generally covariant quantity in classical sequential growthdynamics. Thus, N is the measure of the spacetime volume V of emergent spacetime. As asecond ingredient, the argument uses the idea (borrowed from unimodular gravity) that thecosmological constant Λ is in some sense conjugate to V . In a loose analogy to the energy-timeuncertainty, with Λ an ‘energy’ and V a ‘time’, we have ∆ V ∆Λ ∼ N is ameasure of V , the latter suffers from Poisson fluctuations in N , with a typical magnitude of √ N .Consequently, for a fixed N in the fundamental causet, V will only be fixed up to fluctuationsof magnitude √ N ∼ √ V . Thus, the resulting uncertainty of V is given as ∆ V ∼ √ V . Puttingthis together, we obtain for the uncertainty of Λ:∆Λ ∼ √ V . (2)Assuming that Λ fluctuates about zero (by whatever mechanism), then what we observe todayis only this fluctuation, i.e., Λ ∼ V − / . Further assuming that V is a four-dimensional ‘Hubblesphere’, we replace V by the fourth power of the Hubble radius H − , where H is the Hubbleconstant, and thus come to expect Λ ∼ H , (3)which turns out to be of the order of the 10 − in natural units mentioned above for present-dayvalues. This is remarkably close to values derived from astronomical observations.A few remarks in closing. First, it should be noted that (3) holds for any cosmic epoch.Unlike for other approaches where Λ can vanish during some epochs, it is thus ‘ever-present’here. Second, given the immense advances in observational cosmology, the heuristic argumentpresented above, although suggestive, will need to be supplemented by physically more rigorousand quantitatively more precise models based on CST in order to fulfil the explanatory promiseissued by the above argument. Ahmed et al. (2004) offer the beginning of a more sophisticatedanalysis; other models exist. Third, just as we should be used to by now, both the heuristicargument reproduced here and the more sophisticated model in Ahmed et al. (2004) and else-where derive the effect from purely classical fluctuations. Of course, in order to deliver a fullyquantum theory of gravity, these considerations cannot be but first steps towards the full theory.Nevertheless, the prediction of the everpresent Λ on the basis of such simple and straightforwardideas connected to CST surely is intriguing, particularly also in light of the explanatory plightof standard approaches to account for the acceleration of the universe’s expansion. We could have cast the issue of the emergence of spacetime in CST in terms of Pythagoreanism (asdiscussed in chapter 1), the view according to which all being is mathematical, formal, or abstract.The worry would be that if our physical world is not fundamentally spatiotemporal, then thePythagoreans were right and there could not exist something physical, material, or concrete. Butthis implication does not hold: the world according to CST is quite clearly non-spatiotemporal inits fundamental nature (as we have argued in chapter 2), and yet, space, time, and perhaps evenmaterial objects of our manifest world emerge, or at least can emerge, as we hope to have sketchedin this chapter. We have articulated what we take the task of establishing this to consist in, andwe have invoked spacetime functionalism in order to reject further (and impossible) tasks suchas deriving the qualitative nature of spacetime from mathematical structures. With spacetime28unctionalists, we maintain that a functional reduction of spacetime and its roles in other theoriesand in their empirical confirmation exhausts the task to be completed. It is by means of thisfunctional reduction, and by means of it alone, that the mathematical derivations involved inthe task obtain physical salience.Apart from this central task of establishing the emergence of spacetime as sketched in § §
3, we have also addressed foundational and philosophical issues that arise in the context of theemergence of spacetime in CST. We discussed the consequences of CST and its dynamics neededto regain spacetime for the metaphysics of time ( §
4) and the non-locality inherent in CST andthe related questions of how to incorporate matter into the framework and thereby identify itsempirical signatures ( § References
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