OOne time, two times, or no time?
Christian W¨uthrich ∗ Forthcoming in Simone Gozzano, Eugenio Coccia, Rocco Ronchi, Alessandra Campo (eds.),
Einsteinand Bergson 100 Years Later: What is Time? , de Gruyter.
I dedicate this paper to the people of L’Aquila, who had to suffer through the catastrophicearthquake that hit their beautiful town in the early hours of 6 April 2009.May The Eagle rise again.
Abstract
Contemporary research programs in fundamental physics appear to suggest that there couldbe two (physical) times—or none at all. This essay articulates these possibilities in the con-text of quantum gravity, and in particular of cosmological models developed in an approachcalled ‘loop quantum gravity’, and explains how they could nevertheless underwrite ourmanifestly temporal world. A proper interpretation of these models requires a negotiationof an atemporal and a temporal sense of the emergence of (space)time.
The date of 6 April is of course noted for another event connected to this conference: the1922 Einstein-Bergson debate on the occasion of Einstein’s visit to the
Soci´et´e fran¸caise dePhilosophie in Paris. In his brief reply to Bergson, Einstein famously asserted that there wasno ‘philosopher’s time’, which he took to unduly reify aspects of our experienced time, i.e.,of the ‘psychologist’s time’. The trouble with the philosopher’s time was, for Einstein, that itcontradicted the ‘physicist’s time’: the philosopher’s time hypostatises the apparent simultaneityof distant events, while the physicist’s time, following the insights of special relativity, deniesthat these events are connected by an objective and absolute relation of simultaneity. At leastas expressed in these brief remarks, Einstein accepts the existence of (only) two times: that ofthe physicist and that of the psychologist.The reconciliation of what Wilfrid Sellars (1962) called the “scientific image”, which wouldinclude Einstein’s physicist’s time, with the “manifest image” and its psychologist’s time consti-tutes one of philosophy’s noblest— and most urgent—tasks. Einstein’s point—as I read him—isthat such a reconciliation does not require a tertium quid in the form of an extraneous structure tobe added to those required by the physics and the cognitive science involved in the phenomenol-ogy of temporality. Any such addition would be explanatorily idle at best and incoherent atworst. I tend to agree. ∗ This work was supported financially by the John Templeton Foundation (the views expressed are those of theauthors not necessarily those of the sponsors). Thanks to the audience at the conference ‘Einstein-Bergson 100years later: What is time?’, held at the University of L’Aquila on 4-6 April 2019. I warmly thank the organisersof this conference, Eugenio Coccia, Simone Gozzano, Rocco Ronchi, and last but not least Alessandra Campo,for their kind invitation. a r X i v : . [ phy s i c s . h i s t - ph ] S e p he debate between Einstein and Bergson turns on the then recently discovered relativityof simultaneity, a core element of Einstein’s theory of special relativity of 1905. With specialrelativity, physics had provided deep insights into the elusive nature of time, as it did on numerousother occasions. Our best current understanding of spacetime, and hence of (the physicist’s) time,derives from general relativity, a very successful theory of gravity and spacetime formulated byEinstein in 1915. Since general relativity presupposes that the matter which interacts withspacetime is correctly described by pre-quantum physics, however, it will eventually have to becorrected and replaced by a quantum theory of gravity. As quantum gravity remains beyondour empirical reach, physicists have developed diverging attempts to formulate such a theory.Despite their significant differences, many approaches suggest that space and time will not bepart of the fundamental furniture of the world, adumbrating the most radical revolution in ourunderstanding of time yet. Instead of being fundamental, space and time are emergent propertiesof the fundamentally non-spatiotemporal structures postulated by quantum gravity, very muchlike the solidity of a glass and the liquidity of the water it contains. Just as not any fusionwhatever of silicon dioxide combines to form glass or not any hydrogen dioxide forms liquidwater, the fundamental degrees of freedom may fail to coalesce into a spatiotemporal form, andthus will not give rise to anything like the space and time we know and love. Only a serendipitouscollective action of the fundamental degrees of freedom will form drops—or indeed an ocean—ofspacetime and thus deliver a world remotely like ours.If borne out, this leads to three scenarios. First, it may or, second, may not be that timeemerges from the fundamental structure. The third—tantalizing, but perhaps rather common—possibility is that a world contains both domains with and without emergent spacetime, renderingspacetime a ‘regional’ option. In fact, our universe seems to be just like that: apart from thewell-behaved spatiotemporal ‘phase’ we inhabit, it contains a very ‘early’ ‘epoch’, which shouldnot be expected to be spatiotemporal—the big bang. Quantum gravity suggests that ‘at’ or‘near’ the big bang, we find a non-spatiotemporal ‘phase’. If this is right, then it would suggestanother way in which time might emerge: as a ‘transition’ from an ‘earlier’ non-spatiotemporalto a ‘later’ spatiotemporal states of the universe. How should we categorize the emergence ofspacetime—or failure thereof—in these contexts? How can we conceive of a ‘transition’ fromtimelessness to the regular temporal evolution of our world? Answering these questions willbe necessary for an appreciation of the viability of the emergence of spacetime and thus of aquantum theory of gravity, particularly in a cosmological setting. And it constitutes a centralpiece of a reconciliation of the scientific with the manifest image. The aim of this essay is tostake out some first steps towards this goal. This paper offers an accessible presentation and elaboration of some central theses and ar-guments articulated in Huggett and W¨uthrich (2018). It presupposes no familiarity with eitherquantum gravity or the method of contemporary philosophy of physics. It starts in § § I apologise for the many scare quotes, but these are intended to serve as reminders that spatial and temporallocutions are not well-defined in this context. § § Space and time have long been used as convenient expedients to set apart material, physicalexistence from other forms of existence such as abstract, mental, or divine. According to thisvenerable standard of partitioning existence, all and only occupants of space and time enjoy physical existence. As Larry Sklar (1983, 45) exclaimed, a non-temporal, non-spatial world seems“devoid of real being altogether”. This is natural enough, as our physical world is manifestlyspatiotemporal, endowed with a time—with one time.It is precisely this commonplace which gets challenges in contemporary fundamental physics.General relativity (GR) is one of the most successful theories in the history of physics, apparentlycorrectly predicting phenomena such as the deflection of light near massive stars, gravitationaltime dilation, and the existence of black holes and gravitational waves, to name just a fewhighlights. It furnishes the basis of our vastly improved understanding of our cosmos and itsorigin and enables technology such as the global positioning system. GR interprets gravitationnot as a force as did Newton’s theory, but instead as encoded in the geometry of spacetime—a‘geometrization’—, the fusion of space and time necessitated already by special relativity. Moreparticularly, the strength of the gravitational field is given by the amount of curvature in thespacetime continuum, bending the worldlines of objects from stars and planets to humans andzebras to elementary particles in outer space as if the objects attracted one another. To date,no experiment or observation which directly contradicts GR is known.Despite its impressive palmar`es, GR cannot stand as the last word on gravitation and thuson the structure of spacetime. The reason is as simple as it is damning: GR assumes that matteris classical, i.e., as described by classical physics. But if the physics of the twentieth century hastaught us anything, then it is that matter cannot be so described: it is irreducibly quantum. Inthis sense, there is a plethora of experimental evidence invalidating GR, from simple double-slitexperiments to the Large Hadron Collider. Just as the location services in every smartphone relyon the insights from GR, its technology is built on the recognition of the deeply non-classicalproperties of its matter.In light of this, GR will have to be replaced by a quantum theory of gravity, by which Idesignate any theory that can combine the quantum effects of matter with the presence of (strong)gravitational fields. Such a theory is yet to be fully articulated, let alone empirically confirmed.Currently many approaches compete for attention, young talent, and funding. Among them,string theory is most prominent and presently the frontrunner in all three categories, but there isalso loop quantum gravity (LQG), non-commutative geometry, causal dynamical triangulation,causal set theory, asymptotic freedom, and many more. The foundational physical principles,the structures or ontologies, even the methods and ambitions differ widely across the field.Notwithstanding these differences between research programs, there is a recurring theme inthe field: many approaches to quantum gravity either presuppose or entail that fundamentally,there is no space or no time or neither of the two. This denial of space and time takes differ-ent forms in different programs, and certainly comes in different degrees. Regardless of thesevariations, the constancy with which the theme recurs is remarkable.This is the sense in which there may be no time.3efore we study a concrete instance of the disappearance of spacetime, let me remark ontwo generic philosophical challenges a physical theory without space and time in its ontologyfaces. First, an epistemological point. It may appear as if a non-spatiotemporal theory in thissense is empirically incoherent. A theory of physics is empirically incoherent just in case itstruth undermines the empirical justification for believing it to be true. A physical theory couldthus turn out to be empirically incoherent because it denies a necessary condition of empiricalconfirmation. A theory sans spacetime would thus be empirically incoherent since there could notbe, it seems, ‘local beables’ which manifest themselves in localized observables in space and time.Space and time and thus the possibility for material objects to occupy, or be located in, spaceand time appears to be a precondition for making and reporting observations. With space andtime inexistent, and thus this possibility denied, it appears impossible to make the observationsneeded to confirm the theory at stake and thus for justifying belief in it. Empirical incoherenceis not logical inconsistency: an empirically incoherent theory is certainly a logical possibility andpotentially even a nomological one. There is of course no guarantee that nature is so kind to usas to yield to scientific investigation; but empirical coherence is undoubtedly a presupposition ofscientific enquiry and its failure would be devastating to our attempts to discern nature’s deepstructure. So a physical theory better either be consistent with the fundamental existence ofspacetime or else identify the error of this argument.Second, a more metaphysical concern. By moving to a more fundamental theory, we may havegrown to expect that we thereby advance to the next level—at higher energies, or smaller spatialscales—of unveiling the constitution of (part of) our world. Thus, when we supplant GR with amore fundamental theory of spacetime, we would think that we will learn something about theconstitution of spacetime—and perhaps of matter, if the theory is unified enough to also be morefundamental than the standard model of particle physics. But constitution, or indeed mereology,appears to be inherently and ineliminably spatial, or perhaps spatiotemporal. How then coulda theory denying the fundamental existence of space or spacetime deliver a more fundamentaltheory of the constitution of anything? Clearly, such a theory better either be consistent withthe fundamental existence of spacetime or else deliver an account of constitution and mereologywhich is not implicitly spatial.We will return to these two challenges in the next section. For the remainder of this section,let us look at loop quantum gravity (LQG) to discover one way in which space and time maybe absent in a theory of quantum gravity. LQG starts out from GR as our most successfulcurrent theory of gravity (and of spacetime), and attempts to transform it into a theory ofquantum gravity by applying a ‘quantization’, i.e., an almost algorithmic procedure which turnsa classical theory into a quantum one. There are several such quantization recipes available,and just as with cookbook recipes, none of them guarantees success. LQG uses the so-called‘canonical’ quantization, a procedure which has successfully been applied in other instances,such as the quantization of electrodynamics. Omitting the technical details, what matters forpresent purposes is that this recipe requires, as a first step, that GR be recast in a particularform, the so-called ‘Hamiltonian’ form. In this form, the physics at stake is captured in terms ofa spatially extended physical system which evolves over time according to dynamical equations.This is perfectly adequate, indeed apt, for many phenomena we may be interested in.However, it appears inept for relativistic physics in which space and time have been fused intoone, disregarding GR’s central lesson that there is no absolute ‘time’ external to spacetime itself Huggett and W¨uthrich (2013) applied the concept of empirical incoherence to quantum gravity. See Yates(forthcoming) for a clear recent presentation. See Le Bihan (2018) for a discussion of this point. This material draws on Huggett and W¨uthrich (forthcoming), as well as on Huggett et al. (2013, §
2) for theproblem of time and on W¨uthrich (2017) for the various aspects of the disappearance of spacetime in LQG. Thesesources go into significantly more technical details, which I try to avoid presently.
4n which ‘space’ might evolve. This disregard shows in two ways. First, relativistic spacetimeswith an ‘unusual’ topology which does not permit a consistent (however non-unique) split ofspacetime into space and time are simply declared unphysical and thus omitted. Second, thedynamical equation of GR, Einstein’s field equation, is not formally equivalent to the dynamicalequation of the Hamiltonian form. In order to establish this equivalence, additional equationscalled ‘constraint equations’ must be imposed on the Hamiltonian formulation. Such equationsalways ‘constrain’ some function to be zero, hence their name. One of the constraint equationsdeserves our attention: the Hamiltonian constraint equation. This equation is remarkable in thatthe function constrained to vanish is the so-called ‘Hamiltonian function’, which in Hamiltonianmechanics serves to generate the system’s dynamics. Since the classical Hamiltonian function is thus constrained to vanish, or the quantum Hamil-tonian operator annihilates the physical states of the system, it appears as if there cannot beany non-trivial dynamics. Furthermore, in the equations we find no quantity which seems tocorrespond to a physical time, no ‘ t ’. In Hamiltonian GR, and any canonical quantization builton it, all physical quantities are constrained to remain constant over a time that does not appearto exist. This is the ‘problem of time’. Naturally, this is considered a problem since our worldis manifestly imbued with “blooming, buzzing confusion”, thoroughly awash in constant changewhich occurs over physically real, measurable—indeed experienced—time. That the resultingquantum theory of gravity appears to deny the fundamental existence of either time or changethus stands in crass tension with our manifest picture of the world. Consequently, even some ofthe founding fathers of LQG have paid considerable attention to resolving this issue either bysketching how to recover time from the fundamentally non-temporal structure or by rejectingthe above rough argument to its unwelcome conclusion. LQG remains a work in progress. The canonical quantization program turns out to lead tohopeless technical challenges. The partial glance at the fundamental structures it permits de-spite its unfinished status unveils the following. The (provisional) ‘Hilbert space’ of the system—roughly, the space of its possible quantum states—admits a basis of states with a natural geomet-ric interpretation. In other words, the theory as it is known to date states that the fundamentalstructure (the ‘gravitational field’) is generically in a ‘superposition’—roughly, a state of ‘simul-taneous combination’—of these geometrically interpretable base states. These base states aredubbed ‘spin network states’ and can be represented by abstract labelled graphs (see e.g. figure1 in W¨uthrich 2017).Spin network states afford a straightforward geometric interpretation—as they are eigenstatesof geometrically interpretable operators defined on this Hilbert space. This natural interpretationsees them as discrete, combinatorial structures consisting of grains of space or spacetime con-nected by adjacency relations. Let us unpack this a bit. First, the structure is often interpretedin the literature (e.g. in Rovelli 2004, § §
6) to be spatial (rather than spatiotemporal) inthe sense that it is the structure which one expects to give rise to space (rather than spacetime),connected to the fact that the Hilbert space is not the definite one just yet. The correctness ofthis interpretation, however, is far from obvious, and depends on one’s perspective on the roleand fate of time and, relatedly, on how to complete the research program of LQG. I will returnto this point below.Second, the structure is discrete or ‘chunky’, giving us a sense that space(time) consists ininscissible atoms of space or spacetime, rather than in an ever more finely divisible continuum. Via ‘Hamilton’s equations’ in the classical theory. In the quantum theory, the Hamiltonian operator resultsin zero when acting on quantum states. Apologies to the reader for all these homonyms, but at least it shouldnot be too difficult to guess the name of the Irish mathematician who developed the formalism. See, again, Huggett et al. (2013, §
2) and references therein. Cf., e.g., Rovelli (2020),Smolin (2013). quantumsuperposition of these spin network states. Thus, the structure cannot generically be interpretedto have a particular, determinate geometry in the sense of a determinate number of atomsof determinate sizes with determinate facts of the matter which ones among them are adjacent.Generically, the structures are in some state of combination of networks with different determinategeometries. The sense in which the generic states does not have a determinate geometry is thusanalogous to the sense in which a generic electron does not have a determinate spin in a givendirection, or, perhaps more accurately, a generic quantum electromagnetic field does not containa determinate number of photons with a determinate direction of polarization.Finally, to repeat, LQG has not so far delivered a final and complete theory—it remains aresearch program under construction. The pi`ece de r´esistance turns out to be the Hamiltonianconstraint equation. Although known in formal outline, the construction of a concrete Hamil-tonian operator, let alone the full solution of the resulting equation prove to present formidabletechnical and conceptual challenges that have resisted convincing and widely accepted resolutionto date. These obstacles have not halted the program; instead, physicists have developed twopromising work-arounds, to be discussed in the next section.
The first strategy to circumvent the stumbling block of the Hamiltonian constraint equation is toswitch half way through from the canonical quantization recipe to the so-called ‘covariant’ one,which seeks a path integral formulation of the theory. This move is based on the insight thatat least for simple quantum systems, the two recipes deliver equivalent quantum theories. It ishoped—though of course not proved—that this will be the case here too. This covariant versionof LQG takes spin network states as ‘initial’ and ‘final’ states and seeks to express the dynamicalcontent of the theory in terms of transition amplitudes (and hence probabilities) for such pairsof states. Despite the importance of this approach in LQG, we will not further discuss it here. The second strategy simplifies considerably the systems studied to render the technical dif-ficulties manageable, hoping that the lessons learned from such reduced systems transpose intothe more generic context. Loop quantum cosmology (LQC) does just this: prior to quantization,it reduces the space of admissible models by imposing an additional condition. This conditiondemands that the model be precisely spatially isotropic and homogeneous, thus reducing thenumber of degrees of freedom required to account for the system.Requiring complete spatial isotropy and homogeneity is precisely what is done in moderncosmology, and hence LQC is thought to capture the ‘cosmological’ sector of LQG, i.e., thosemodels which may in fact describe the large-scale structure of the universe over the course of itshistory. The so-called ‘Cosmological Principle’ requires that both the spacetime structure andthe matter distribution are spatially isotropic around us, i.e., exhibiting the same properties inall directions around us. This principle is reasonably well confirmed by the distribution of thecosmic microwave background radiation and the large-scale distribution of luminous matter ingalaxies. However, isotropy around us does not imply either isotropy around any other point inspace or full spatial homogeneity, i.e., the sameness of properties everywhere in space. For this, See Rovelli and Vidotto (2015) for an excellent introduction. See Bojowald (2008) and Bojowald (2010, ch. 4) for a popular introduction. a ( t ), which is a function of time t (and so presupposes a ‘cosmic time’ t , which is, however, available in FLRW spacetimes). Imposing this symmetry (of isotropyand homogeneity) thus leads to a significant simplification of the system studied and makesthe attendant mathematical difficulties more manageable. This simplification occurs at theclassical level, i.e., before quantization is attempted. In the full theory, the spin networks aregenerally large, complicated, and irregular, without recurrence or pattern. In contrast to this, thesymmetry-reduced quantum configurations of LQC turn out to be highly regular. Consequently,they can be represented by lattice graphs with straight edges of the same length resulting insurfaces of an area of the square of this base length.Again, these surfaces are interpreted to represent the universe spatially, with the scale factorgiving its ‘size’. In the quantum theory, there is an operator corresponding to this scale factor.When applied to the spin network states, one can determine the value the scale factor takesfor these states, finding that it can be zero or assume one of a (quasi-)discrete set of values. Classically, the FLRW models are ‘singular’, i.e., all worldlines of physical objects are past-finite.This is naturally interpreted as the universe having a finite age. One might think that there isthus a time before which there was no time. However, the past-finiteness of the universe doesnot imply that there was a first (instantaneous) moment; rather, one should think of the cosmictime t as assuming positive values strictly larger than 0. Just as there is no smallest positivereal number, there was no first moment in time in those models. The moment t = 0 is notpart of these models, and so corresponds to no physically existing time. The past-finitenessof all worldlines is a sign that the FLRW spacetimes are singular ‘there’, i.e., they are notmathematically well-defined. This non-place, or the behaviour of the universe in its infancy right‘after it’, is called ‘the big bang’.Another sign of singular behaviour at the origin is the diverging (scalar) curvature ‘there’.Tracking the relative size of the universe backward in cosmic time it becomes smaller and smaller.As the curvature scales inversely with the scale factor, it grows beyond any bounds as we approach t = 0. This divergence at the big bang is often seen as a failure of GR itself, as are similarsingularities in relativistic models (such as those found in black holes). Peter Bergmann (1980,156) articulated what I take to be a rather common view among physicists:[Singularities] are intolerable from the point of view of classical field theory becausea singular region represents a breakdown of the postulated laws of nature. I thinkone can turn this argument around and say that a theory that involves singularitiesand involves them unavoidably, moreover, carries within itself the seeds of its owndestruction. Although the order of symmetry reduction and quantization should in principle not matter. The precise sense of ‘discreteness’ in play here is subtle. For more on this, see W¨uthrich (2006, 123). In scare quotes because this is not part of the model and so not a physically existing location in spacetime.The singularity is a ‘global’ property of the spacetime. the quantum operator corresponding to the scalar curvature iswell-defined at what would classically be the big bang. More specifically, the curvature in themodel of LQC increases as we trace it back through smaller and smaller scale factors, becomesrather large and peaks at a small (but non-zero) scale factor, before it decreases again to bezero for the zero-size universe at the quantum big bang (see figure 10 in W¨uthrich 2006). Thequantum model does not exhibit a curvature singularity.These are merely kinematic facts and do not rely on any ‘dynamics’. The ‘dynamical’ Hamil-tonian constraint equation—and that was to a significant degree the point of the program ofLQC—simplifies significantly and can be solved explicitly. Quantum states of the universe whichnot only satisfy the kinematical constraints, but also this dynamical one are the truly physicalstates according to the theory. Since these states are the possible states of a universe throughoutits history, giving rise (it is hoped) to something like the four-dimensional cosmological modelsof GR, these states ought to be considered extended in ‘time’.We will return in § atemporal formof emergence of spacetime: spacetime ontologically depends on more fundamental, ultimatelynon-spatiotemporal degrees of freedom.This atemporal form of emergence is central in the dissolution of the threat of empiricalincoherence as articulated in the previous section. By establishing that spacetime emerges athuman scales, empirical evidence can unproblematically come in a spatiotemporal form. Theadditional demand that the spatiotemporality be fundamental cannot be justified. Just as itis not necessary for measuring the temperature of a gas that this temperature somehow be afundamental properties of the constituents of the gas, it is in no way required that the spa-tiotemporality be fundamental for the evidence collected in observations and experiments to bemanifestly spatial and temporal. The menace of empirical incoherence is thus averted. As forthe concern regarding the constitution of spacetime, I have recommended elsewhere that a overlyspatial or spatiotemporal concept of constitution be replaced by a functionalist understandingof spacetime (Lam and W¨uthrich 2018).To return to the emergence of spcetime, in the cosmological models of LQC, or at least inmore realistic ones with many more degrees of freedom, there arises now a second, temporal , sensein which spacetime ought to emerge. To obtain a more faithful description of the physics of theearliest universe is, together with a deeper understanding of black holes, among the primaryobjectives of quantum gravity. Given the strength of the quantum effects in the very earlyuniverse, we do not expect the classical description of a smooth spacetime to remain applicable.Rather, the universe was born out of a deeply quantum gravitational Ursuppe . That this
Ursuppe is not spatiotemporal, or at least not ‘temporal’, is further supported by recent findings (Brahma2020) according to which the fundamental structures in the very early universe are ‘spatial’, ifanything, rather than spatiotemporal: there appears to be a ‘signature change’ from the usualLorentzian signature to a Euclidean signature. A ‘Lorentzian’ signature is characteristic of arelativistic spacetime with one dimension of time and 3 (or n ) dimensions of space, whereas a‘Euclidean’ signature indicates a purely spatial structure. Thus, time truly disappears around For the caveats, see W¨uthrich (2006, § Ursuppe .If this is correct, then we cannot hope that the physics at that early stage is orderly andspatiotemporal. This is of course again precisely because quantum gravity is generically non-spatiotemporal. In sum, we should expect that cosmological models based on LQG encompassdifferent ‘phases’: an ‘earlier’, non-spatiotemporal, quantum-gravitational phase, as well as a‘later’ phase, for which the classical spacetime description offered by GR delivers a valid approx-imation. Thus, it looks as if these models ought to contain a ‘process’ of emergence, a ‘transition’from the first phase to the second, which involves the (at least approximate) emergence of space-time. This is the sense in which there must be a temporal form of emergence spacetime: spacetimearises as an effect of an earlier state, i.e., it depends, perhaps causally, on ‘prior’ states of affairs.Thus, we are faced with two distinct notions of emergence and, correspondingly, two waysin which time comes to be. These two notions answer to two distinct, but equally important,questions. First, how can classical relativistic spacetime (or, ultimately, the space and time ofour experience) be grounded in a fundamental structure which is non-spatiotemporal? This isarguably the most urgent philosophical question arising in the context of quantum gravity andis extensively addressed in the literature. I sketched only some central points pertaining to thisfirst question here.Our focus here is the second question: how can the (fundamentally non-spatiotemporal)universe ‘evolve’ from a non-spatiotemporal ‘phase’ to a spatiotemporal one; i.e., how can therebe such a ‘process’ or, more fundamentally, a ‘change’ with time? In fact, how could we possiblyorder the phases into a ‘before’ and ‘after’ ? This is the sense in which our world may contain two ‘times’: the emergent temporal aspectof effective spacetime and the temporal aspect of the emergence of spacetime from the
Ursuppe itself. How can we make sense of this puzzling situation? Let us outline a physical interpretationof the models of LQC.
It would hardly be coherent if there were indeed two distinct notions of physical time, oneatemporally emerging from fundamental physics, and another one temporally emerging fromearlier physics. If there were in fact two times, it would be surprising if the time of our experience,which ultimately arises from the fundamental structures on which the physics in our experientialvicinity ontologically depends, and the time of the cosmos, born in the big bang, were to coincide.Let us address this threat of incoherence by considering the physical interpretation of the simplestmodel of LQC and then by ruminating, somewhat speculatively, on extending these lessons tomore realistic models. But first, let us prepare the ground for these discussions.The propaedeutic remark concerns the standard cosmological model of GR, the FLRW space-time we have already encountered above, and how to think about it. We have already statedthat it contains a singularity, the ‘big bang’, marking a ‘beginning’ of the universe. The waycosmologists interpret cosmic history like this is by starting out—‘in thought’—from the presentstate of the universe, and to calculate things backward in time, reaching ever earlier times. Inother words, the direction of the evolution of the universe as we theorize about it is opposite tothe direction of the actual, physical evolution theorized about. Given that we live today, cosmol-ogists, like Schiller’s “Universalhistoriker” (1789, 127f), have no choice but to infer from what ispresently the case to what must have transpired before, at earlier and ever earlier times. There The same problem arises not only in LQC, but also in string cosmology (Veneziano 2004;Huggett and W¨uthrich 2018) and in Oriti’s ‘geometrogenesis’ (Oriti forthcoming).
9s excellent evidence, indirect though it may be, that the classical relativistic model of cosmologyoffers a surprisingly accurate description of the cosmos for most of the times it assumes to exist,retraced backward by almost 13.8 billion years. As already stated, the model is past finite andis singular.Let us now analyze in more detail the simplest model of LQC, i.e., the one with perfectsymmetry. The highly regular lattice graphs of varying size are taken to represent the state ofthe universe at a cosmic time. The Hamiltonian constraint equation delivers the ‘dynamics’ inthe sense that it mandates how a set of them can be knit them together into an ordered sequence.The resulting fabric—the maximal family of compossible ‘instantaneous’ states—represents theLQC-equivalent of a four-dimensional spacetime, i.e., the universe throughout its entire history.Physically, such a family can be thought of as an ‘evolving quantum geometry’. In this simplest,altogether isotropic case, the family members ‘vary’ in size in the sense that the quantum propertycorresponding to the scale factor differs.For the fully isotropic case, the Hamiltonian constraint equation becomes substantially sim-pler, and in fact turns into a difference equation, rather than a differential equation, as is theusual case for dynamical equations. This is not so surprising, given that the ‘time’ LQC usesto stitch together the ‘instantaneous’ or ‘momentary’ states is discrete, rather than continuous.Differentiation, i.e., building derivatives, requires a backdrop of continuous variation, and henceof a continuum. For a discrete set, we are left with differences, however small they may be.it should be emphasized that the fact that time is discrete does not imply that there are onlyfinitely many ‘instants’ of time; rather, their number is countably infinite.As a curious aside, it should be mentioned that this difference equation does not give amomentary state at a time as a function of another momentary state at another time. Instead,it fixes a momentary state as a function of two other momentary states at different times one unitof time apart. Thus, in order to fully specify initial conditions, we need to specify all momentarystates in a (closed) unit interval.If the conditions for an interval are thus given, the simplified Hamiltonian constraint equationdetermines the states at all other times, and thus the entire time-ordered family of momentarystates (barring the vanishing of the coefficients in the equation). The hope voiced above thatquantum effects may wash out the dynamical singularity of the classical model is indeed borneout, at least almost. If, in Schillerian fashion, we evolve a later interval of ‘initial conditions’backwards in time using the Hamiltonian constraint equation (which does not care about thetemporal direction), we find that the evolution continues beyond what classically was the bigbang. Beyond the big bang, there is a mirror universe very much like the one we know and loveon ‘this side’ of the big bang.What is the physics of this newly found realm beyond the big bang? Although parallel und inmany ways similar, the two sides are not exactly identical. In the simple model, the orientation ofspace turns out to be inverted between the two sides (Bojowald 2010, 113ff). In more complicatedmodels, one would in addition expect the probabilistic quantum fluctuations to differ.How do the two realms connect with one another at the big bang? In the standard interpre-tation, offered e.g. in Bojowald (2008) and Bojowald (2010, ch. 4), the model is one in which alarge universe shrinks and ultimately collapses to zero size before it rapidly expands again. Fromslightly more complicated physics, it is believed that the universe heats up before its collapse,only to cool down again as it re-expands after the big bang. Physical time runs unidirectionallyfrom negative infinity to positive infinity all the way through what classically was the big bang. Physicists also consider slightly more complicated models with a small amount of anisotropy. As by this point in the essay, the reader should be sufficiently ‘scared’, I now unceremoniously drop the scarequotes around ‘instantaneous’, ‘momentary’, ‘precede’, ‘before’ and similar expressions. See W¨uthrich (2006, § If the universewere perfectly homogeneous, we would not be here to discuss its origin! A more realistic modelwill thus have to admit anisotropies and inhomogeneities and exhibit more irregular geometries.If the larger program of LQG is right, then space and time are not directly implemented at thefundamental level, as we have seen above. Thus, given that around the big bang, we will be inthe deep quantum-gravitational regime (the reader is reminded, e.g., of the signature change inthat epoch), we cannot expect that a simple scalar field will play the role of cosmic clock, as itdoes in the simple model of LQC with perfect symmetry. If we extrapolate the physics backwardsto earlier and earlier times, at some time very soon after the big bang (usually given as aroundthe ‘Planck time’ 10 − seconds after the big bang) we arrive at the deep quantum-gravitationalregime, where quantum fluctuations are believed to have been so strong that spacetime in theusual sense evaporates.The philosophically acute reader will naturally ask what it could possibly mean that thequantum-gravitational period lasted for something like 10 − seconds when there is no physicallymeaningful notion of time present during that epoch. The short answer is nothing . In the absenceof (fundamental or emergent) time, it is simply meaningless to quantify a duration—includingsetting that duration to zero. More importantly for our purposes, in the absence of a timeticking continuously and uninterruptedly through the period, it is not meaningful to speak of aunified physical process running from an earlier (pre-big-bang) state to a later (post-big-bang)one, particularly not if there is a signature change.However, there is a way something like this scenario would be the correct interpretation, eventhough the way of referring to it as a single, unified process is strictly incoherent, just as stating To put this number in context, the closest star from the Sun, Proxima Centauri, is just over 4 light yearsaway, the centre of the Milky Way about 26,000 light years, and the closest major (spiral) galaxy, Andromeda,about 2.5 million light years. As is suggested by Huggett and W¨uthrich (2018, 1201f). − seconds. Although durations are meaningless,there is a sense in which that epoch occurred before our era, i.e., is in our past (rather than ourfuture or in no temporal relation to us). Locally, presently, we do have a physical (space)timewith a well-defined temporal direction, owing to the second law of thermodynamics. Our futureis distinct from our past in myriad ways. And if we extrapolate our local time and its directionbeyond the scope of its proper applicability, we recognize that the atemporal phase at the bigbang is in our past, before our current era. Even if by itself timeless, it is thus meaningful tosay that the big bang is in the past relative to our local determination of the direction of time.Hence, it occurred ‘before’ our era.This extrapolation of local time and its direction has two immediate consequences relevantfor the purposes of this paper. First, it unifies the two times identified in §
3. The sense in whichthe big-bang epoch precedes ours is fully due to an extrapolation of our local, emergent time andits arrow. In other words, the temporal emergence of time derives from the atemporal one. Thelatter is thus prior to the former. In this way, there is an unambiguous and unproblematic way inwhich we can order the cosmic epoch into a ‘before’ and an ‘after’. Even so, it should be clear thatthere is no process in the strict sense from a non-spatiotemporal phase to a spatiotemporal one.If the sense in which the big-bang epoch, i.e., the quantum-gravitational regime, precedes our eraderives from the (atemporal) emergence of spacetime and the extrapolation of thermodynamicprocesses in our local region, then there is just one ‘time’, not two.Second, it also implies that the standard interpretation of the bounce may not be completelymeaningless. As we already noted, the big bang took place in our past. But similarly, andcompletely independently, denizens of the other universe may extrapolate their local time andits direction and determine that there exists a ‘big crunch’ to their future. They would judgethe event not a big bang, but a big crunch, since they would be observing that their universecontracts in their local direction of time, and might consequently await their fate with trepidation.Although there would thus not exist a continuous, unified physical process as usually described inthe standard scenario, there would be two universes, one of which is contracting to an ever hotterand denser state, and the other expanding and cooling, in their respective (extrapolated) localdirections of time . Although they would be separated from one another in that there is no sharedsense of spacetime between them, they would nevertheless form a single physical world, connectedby the quantum-gravitational physics at the big bang. To repeat, these connections wouldneither be spatial nor temporal, but they would most definitely be physical, thus challenging thevenerable standard of physical existence mentioned at the outset. What their exact nature turnsout to be is for a fundamental theory of quantum gravity to describe—in our case, for LQG.However, the absence of a direct temporal relation through the big bang opens up the pos-sibility of an altogether different interpretation: instead of a big bounce from a contracting toan expanding universe, we witness the birth of twin universes from the same quantum
Ursuppe ,which are both expanding in their local futures. , Instead of one single process ‘through’ thebig bang era, there are two parallel birthing processes from which a spacetime (each) emerges.Time arises twice over, separately and independently in each of the twin branches, though pre-sumably in each case as the result of the propitious coalescence of a large number of fundamental,but individually non-spatiotemporal degrees of freedom.More generally, given that for spacetime to emerge from the fundamental degrees of free- See W¨uthrich (2006, 132f) for a first articulation of this idea in the context of LQC, which has been suggestedto me by Carlo Rovelli. This articulation is deepened in Huggett and W¨uthrich (2018, § § One might ask why two universes, why not three, or four, or infinitely many? While these may all be optionsin other models, the mathematics of the simple LQC model naturally suggests (just) two. structure ), the conditions must be just right,as indicated in §
2. Just as H O molecules must be in the right phase for the water to flow asa liquid, the spin networks must be in a sufficiently ‘geometric’ state for spacetime to emerge(W¨uthrich 2017). And just as it is possible for water to be (near the phase line) in a state of aninhomogeneous mixture of liquid and gas ‘pockets’, one and the same spin network may containgeometric (and hence spatiotemporal) and non-spatiotemporal ‘regions’. This was the third sce-nario described on page 2. The result would be distinct, isolated islands of spacetime emergingfrom an ocean that is the fundamental structure. Generically, and unlike in the scenario of thetwin birth sketched above where both branches would be connected with one another througha joint structure in their respective local pasts, it would not be the case that these spacetimepockets stand in any meaningful spatiotemporal relation to one another.Whether the big bounce or the twin birth scenario obtains in more realistic models of LQC—and whether the loop program is onto a true quantum theory of gravity for that matter—remainsto be seen. What I hope to have shown, however, is that the joint appearance of atemporal andtemporal forms of the emergence of spacetime can be reconciled in a coherent interpretation ofthese models.
Leaving aside the psychologist’s time, Einstein identified two distinct notions of time in his debatewith Bergson: the time of the physicist and the time of the philosopher. Whether or not wefollow Einstein in denying the existence of the philosopher’s time, how can we be sure that therewill be exactly one kind of physical time? Contemporary research into fusing quantum physicswith relativity theory into a quantum theory of gravity suggests that there may not be a physicaltime, at least at the fundamental level of existence. Thus, although Einstein took it for grantedthat there was a physical time, physics itself may end up eliminating time from its fundamentalontology. Does this mean that there may be not time at all?It does. If LQG or a similarly non-spatiotemporal theory turns out to be the correct theoryof quantum gravity, then it is physically possible that the fundamental structures do not conspireto form space and time; instead, they form a world devoid of space and time and so very muchunlike ours. However, any such theory will have to contain models, which give rise to somethinglike the relativistic spacetimes which accurately describe physical aspects of the actual world.In this sense, any such theory must permit the emergence of (space)time under the appropriatecircumstances. In other words, it must bring forth the physicist’s time.In cosmological models based on LQG and other theories of quantum gravity, it appears asif the classically singular big bang is replaced by a quantum foam, which dissolves time (andperhaps space). Therefore, in those models, there must also be a temporal sense in which timeemerges from something that is not (yet) temporal. But if time emerges twice over, how dowe not end up with two separate notions of time? There is only one resulting (space)time,so whatever these relations of emergence may be, they cannot lead to distinct times. In fact,the local physical time of our era is grounded in fundamental, intrinsically non-spatiotemporalstructure, making the atemporal emergence the primary notion. The way in which time emergestemporally is derivative on our local emergent time. This secured the coherence of physicaltime in light of its potential dual emergence, but also opened up the possibility of a differentinterpretation of the big bang as the birth of twin universes, rather than a big bounce.Although prima facie more secure than the psychologist’s and particularly the philosopher’stime, the physicist’s time remains elusive, potentially ineffable in principle, and perhaps foreverbeyond our ken. However, I hope to have sketched how its disappearance from the fundamental13ntology and its attendant emergence at different scales is a coherent possibility, even in casevarious forms of emergence come together in producing time.
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