the arrow of time and the nature of spacetime

Studies in History and Philosophy of Modern Physics 44 (2013) 242–262

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Studies in History and Philosophyof Modern Physics

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The arrow of time and the nature of spacetime

George Francis Rayner EllisMathematics Department, University of Cape Town, Rondebosch 7701, Cape Town, South Africa

a r t i c l e i n f o

Article history:Received 8 October 2011Received in revised form28 February 2013Accepted 11 June 2013Available online 24 July 2013

Keywords:Arrow of timeTop-down causationChronology ProtectionEvolving Block Universe

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This paper extends the work of a previous paper (Ellis, 2013) on the flow of time, to consider the origin ofthe arrow of time. It proposes that a ‘past condition’ cascades down from cosmological to micro scales,being realized in many microstructures and setting the arrow of time at the quantum level by top-downcausation. This physics arrow of time then propagates up, through underlying emergence of higher levelstructures, to geology, astronomy, engineering, and biology. The appropriate spacetime picture to viewall this is an Evolving Block Universe ‘EBU’, that recognizes the way the present is different from both thepast and the future. This essential difference is the ultimate reason the arrow of time has to be the wayit is.

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When citing this paper, please use the full journal title Studies in History and Philosophy of Modern Physics

1. Quantum theory and the arrow of time

The arrow of time problem is one of the major foundationalproblems in physics (Carroll, 2010; Davies, 1974; Ellis & Sciama,1972; Halliwell, Perez-Mercader, & Zurek, 1996; Price, 1996;Wheeler & Feynman, 1945; Zeh, 2007), because the one-way flowof time embodied in the second law of thermodynamics emergesfrom time-symmetric microphysics. This paper builds on a pre-vious paper (Ellis, 2011), where the emergence of higher levelstructures and their top-down influence on lower level structureswas considered to be a key factor in looking at the quantummeasurement problem and the nature of the classical quantumcut. This paper will propose that the same is true for the arrow oftime.

The paper is structured as follows: Section 2 sets up the basicideas underlying the rest of the paper: the basics of quantumtheory, and the ideas of bottom up and top down causation in thehierarchy of complexity, and makes the main proposal as tohow quantum theory underlies complex systems (Section 2.4.2).Section 3 discusses the arrow of time problem and variousproposals that have been made to solve it, focussing on the PastHypothesis in Section 3.3. Section 4 looks at the cosmologicalcontext and the various epochs relevant to discussing the issue ofthe flow of time, and how the basic cosmological direction of timeis set up. Section 5 proposes that the cosmological arrow of time

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cascades down to lower levels in the hierarchy, each level in turncommunicating the arrow to the next lower level. Section 6discusses how the arrow of time then flows up the hierarchy asemergent structures form out of lower level elements, indeedbeing taken for granted in this context. Section 7 considers howthe natural spacetime view to accommodate the experiencedongoing flow of time is an Emergent Block Universe that growswith time, (Ellis, 2006a) and where the past, present and futureare represented as having a quite different ontological characterfrom each other. This provides the ultimate rationale for the arrowof time (Section 8) and resolves potential problems arising fromthe possibility of closed timelike lines (Section 7.2). Thus theoverall picture that emerges is one of the arrow of time in physicalsystems being determined in a top-down manner, starting off froma special initial condition at the cosmological scale where thecosmological arrow time sets the basic direction of causation, butthen emerging in complex systems through bottom up causation(Section 8.1). The relation to state vector reduction is crucial;obviously the details of that relation are still to be resolved(Section 8.2).

2. Foundations

To set the scene, this section summarizes some foundational issuesdiscussed in more detail in Ellis (2011). It considers basics of quantumdynamics (Section 2.1), the hierarchy of structure (Section 2.2), andinterlevel relations in that structure (Section 2.3). It then puts the main

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Table 1The hierarchy of structure and causation for inanimate matter (left) and for life(right). For a more detailed description of this hierarchical structure, see http://www.mth.uct.ac.za/�ellis/cos0.html.

Level 10: Cosmology Sociology/Economics/PoliticsLevel 9: Astronomy PsychologyLevel 8: Space science PhysiologyLevel 7: Geology, Earth science Cell biologyLevel 6: Materials science BiochemistryLevel 5: Macro physics, Physical chemistry Organic chemistryLevel 4: Atomic physics Atomic physicsLevel 3: Nuclear physics Nuclear physicsLevel 2: Particle physics Particle physicsLevel 1: Fundamental theory Fundamental theory

G.F.R. Ellis / Studies in History and Philosophy of Modern Physics 44 (2013) 242–262 243

viewpoint underlying this and the companion paper (Section 2.4),emphasizing the interaction between bottom-up and top-downcausation as a key feature of physics.

2.1. Quantum dynamics

The basic postulate of quantum mechanics (Greenstein &Zajonc, 2006; Isham, 1995; Morrison, 1990; Rae, 1994) is that asystem is described by a state vector jψ⟩ that generically can bewritten as a linear combination of unit orthogonal basis vectors

jψ1⟩¼∑ncnjunðxÞ⟩; ð1Þ

where un are the eigenstates of some observable A (Isham, 1995,pp. 5–7). The evolution of the system can be described by a unitaryoperator U ðt2; t1Þ, and so evolves as

jψ2⟩¼ U ðt2; t1Þjψ1⟩ ð2Þhere U ðt2; t1Þ is the standard evolution operator, determined bythe evolution equation

iℏddt

jψ t⟩¼ H jψ t⟩ ð3Þ

the Hamiltonian H is time independent, U has the form (Isham,1995, pp. 102–103)

U ðt2; t1Þ ¼ e−ði=ℏÞH ðt2−t1Þ: ð4ÞAs well as the unitary evolution (2), measurements take place.Immediately after a measurement is made at a time t ¼ tn, therelevant part of the wavefunction is found to be in one of theeigenstates:

jψ2⟩¼ cN juNðxÞ⟩ ð5Þfor some specific index N. The data for totn do not determineeither N or cN, they only determine a probability for each possibleoutcome (5) through the fundamental equation

pN ¼ c2N ¼ ⟨eN jψ1⟩2: ð6Þ

One can think of this as due to the probabilistic time-irreversiblereduction of the wave function

jψ1⟩¼∑ncnjunðxÞ⟩ ⟶ jψ2⟩¼ cNuNðxÞ

Indeterminate Transition Determinateð7Þ

This is the event where the uncertainties of quantum theorybecome manifest (up to this time the evolution is determinateand time reversible). It will not be a unitary transformation (2)unless the initial state was already an eigenstate of A. Moregenerally, one has projection into a subspace of eigenvectors(Isham, 1995, p. 136; Wiseman & Milburn, 2010, pp. 10–12) or atransformation of density matrices (Isham, 1995, p. 137), or anyother of a large set of possibilities (Wiseman & Milburn, 2010,pp. 8–42), but the essential feature of non-unitary evolutionremains the core of the process.

The process (7) is where the time irreversibility, and hence thearrow of time, is manifested at the quantum level: The eigenstate(5) occurs at a later time than the superposition (1), and knowl-edge of the final state (5) does not determine the initial state (1);the values of the coefficients un have been lost. After collapse thedynamics will tend to cause new superpositions (1) to emergethrough the unitary process (2); then further effective non-unitarywave vector reduction events will produce eigenstates again.

There are other understandings of what happens when ameasurement takes place, such as the Everett many worlds theory(Wallace, 2002); however they have to lead to an effectivebehavior as outlined above, or they do not correspond toexperiment.

2.2. The context: the hierarchy of the structure

The context in which all this occurs is the hierarchy of structureand causation (Ellis, 2008, 2011, 2012). Table 1 gives a simplifiedrepresentation of this hierarchy of levels of reality as characterizedby corresponding academic subjects, with the natural sciences onthe left and the life sciences on the right. On both sides, each lowerlevel underlies what happens at each higher level in terms ofstructure and causation.

The ordering of the levels on the left is by scale, which is alsothe inverse of energy. However it also represents a putativelayering of emergent causation: For example one can claim thatparticle physics underlies nuclear physics in that nuclei are madeof combinations of quarks; nuclear physics underlies atomic physics, inthat atomic properties depend on nuclear properties; atoms underliemolecules which underlie the kinetic theory of gases; and so on; eachlevel emerges in this way from combinations of lower level entities. Inmany cases the relevant higher level variables are coarse-grainedlower level variables (Ellis, 2011).

In the case of the life sciences (Campbell & Reece, 2005),ordering is by physical scale (biochemistry underlies microbiology,which underlies cell biology for example) and timescale (interac-tions are much faster at lower levels) until the higher levels, wherethe nature of causation changes because of the emergence ofmind. Here it is commonly thought that psychology emerges fromphysiology (Nicholls, Martin, Wallace, & Fuchs, 2001), and societyfrom the interaction of individual minds. The relationshipsbetween the different levels are very different from one anotherin these cases; nevertheless the hierarchy as presented makessense as a hierarchy of emergent causal levels, each with its ownrelevant variables and effective laws of behavior (Anderson, 1972).

2.3. Interlevel relations

It is useful to characterize causation in this hierarchical contextas proceeding in both a bottom-up and a top-down manner (Ellis,2008; Ellis, Noble, & O'Connor, in press).

2.3.1. Bottom-up effectsHigher level structure emerges from combination of lower level

structural elements, for example molecules emerge from atoms(Feynman, Leighton, & Sands, 1963), with higher level dynamicsemerging from lower level dynamics through the effects of thelower level dynamics in the context of the higher level emergentstructure, for example molecular biology emerges from physics(Watson, 1970). Thus behavior on level X þ 1 emerges frombehavior on level X. Often there is coarse-graining of lower levelvariables (e.g. particle states) to give higher level variables (e.g.density and pressure) and effective emergent laws (e.g. the perfectgas laws) (Alonso & Finn, 1971), accompanied by a conversion of

t2

Gravity

t2

Same Same

t2

Cooler Hotter

t1

Fig. 1. The diffusion arrow of time: (a) Two compartments are separated by an opening, but it is closed by a slider. Gas is on the left, the right is empty. This is the preparedstarting state: it does not occur naturally (initial state). (b) Opening the aperture results in the gas spontaneously spreading to the other half, until equilibrium is reached andentropy is maximized. The arrow of time is evident in this flow, which is enabled by billions of state vector reduction events at the quantum level (because the outcome is awell-determined classical state). The transition from the first state to the second state can be used as an arrow of time detector (given photographs of state 1 and state 2, youcan reliably time order them). That deduction assumes no human intervention (in fact the improbable initial state was prepared by humans: another non-unitarytransformation) (final state). (c) If a gate only lets through high speed molecules, the second law compartment will become hotter than the first, in apparent contradiction with thesecond law. This is an example of creating order by a selection process (slower molecules are rejected) (final state with a gate that only lets through high speed molecules (Maxwell’sDemon)). (d) If gravity is turned on and the Jeans mass is attained in the right hand compartment, structure will spontaneously form. This is presumably in accord with the second lawof thermodynamics, but we have no definition of gravitational entropy that makes this good (final state with gravity (mass greater than Jeans mass)).

G.F.R. Ellis / Studies in History and Philosophy of Modern Physics 44 (2013) 242–262244

usable to non-usable energy when some energy is hidden in lowerlevel states, and hence not manipulable via higher level variables.

2.3.2. The emergence of higher level behaviorConsequent on bottom up causation, higher level behavior

emerges from that at the lower levels. Consider how higher levelbehavior relates to lower level behavior in two adjacent levels inthe hierarchy of complexity (Diagram 1).

Level Nþ1:
Initial state I Higher level theory T: ⇒ Final state F ⇑ Coarse grain ⇑
Level N:
Initial state i Lower level theory t: ⇒ Final state f
Diagram 1. The emergence of higher level behavior from lower leveltheory. Coarse-graining the action of the lower-level theory results inan effective higher level theory.

The dynamics of the lower level theory maps an initial state i toa final state f. Coarse graining the lower level variables, state icorresponds to the higher level state I and state f to the higherlevel state F; hence the lower level action t : i-f induces a higherlevel action T : I-F . A coherent higher level dynamics T emergesfrom the lower level action t if the same higher level action Tresults for all lower level states i that correspond to the samehigher level state I (Ellis, 2008), so defining an equivalence class oflower level states that give the same higher level action (Auletta,Ellis, & Jaeger, 2008) (if this is not the case, the lower leveldynamics does not induce a coherent higher level dynamics, asfor example in the case of a chaotic system). Then on coarsegraining (i.e. integrating out fine scale degrees of freedom), the

lower level action results in an emergent higher level dynamics:The effective theory at the higher level.

2.3.3. Top-down effectsOnce higher level structures have emerged, they then exert a

top-down influence on their components ‘whole-part constraint’by constraining the lower level dynamics (Ellis, 2011, 2012). Inaddition to bottom-up influences, contextual effects occur wherebythe upper levels influence what occurs at lower level by setting thecontext and boundary conditions for the lower level actions.

This can happen through setting boundary conditions oreffective potentials for the relevant variables (for example creatingelectronic band structures that determine how electrons flow in asolid), or by constraining lower level dynamics through structuralrelations (such as the wiring in a computer or synaptic connec-tions in a brain). This underlies the emergence of effective samelevel laws of behavior at higher levels (as in Diagram 1), enablingone to talk of the existence of higher level entities in their ownright (Ellis, 2008). It enables true complexity to emerge throughenabling feedback loops between higher and lower levels.

The key feature underlying topdown causation is the multiplerealizability of higher level states (Ellis, 2012). In a gas, many lowerlevel molecular states si correspond to a specific higher level stateS characterized by a temperature T, volume V, and pressure p.These are the effective macroscopic variables; one can ordinarilyonly access the gas by manipulating higher level variables, henceone cannot determine which specific lower level state si realizesthe chosen higher level state S. It does not matter which specificlower level realizes the higher level state, what matters is theequivalence class it belongs to; that is the real causally effectivevariable (Auletta et al., 2008). The number of lower level states


that correspond to a specific higher level state determines theentropy of that state (Penrose, 2011).

It should be emphasized here that these relations can occurbetween any two neighboring levels in the hierarchy; there may ormay not be a highest or lowest level. In Ellis (2011) I made the casethat top-down influences play a key role in the way quantum theoryworks, particularly as regards both decoherence and state pre-paration. This paper makes the case that top-down influences arealso key as regards the arrow of time.

2.3.4. Adaptive selectionAn important case of top down causation is adaptive selection

(Gell-Mann, 1994; Kauffman, 1993). Here, selection takes placefrom an ensemble of initial states to produce a restricted set offinal states that satisfies some selection criterion. Random varia-tion influences the outcome by providing a suite of states fromwhich selection is made in the context of both the selectioncriteria and the current environment (Holland, 1992). This is thebasic process whereby information that is relevant in a specificcontext (Roederer, 2005) is selected from a jumble of irrelevantstuff; the rest is discarded. This enables an apparent local violationof the second law of thermodynamics, as in the case of Maxwell'sDemon (Aharaonov & Rohrlich, 2005, pp. 4–6; Carroll, 2010,pp. 186–189, 196–199; Feynman et al., 1963, pp. 46–55; Leff &Rex, 1990) – who is indeed an adaptive selection agent, actingagainst local entropy growth by selecting high-energy moleculesfrom a stream with random velocities approaching a trap-doorbetween two compartments (Fig. 1c). The selection criterion is thethreshold velocity vc deciding if a molecule will be admitted intothe other partition or not. In quantum physics, such a processunderlies state vector preparation (Ellis, 2011; Isham, 1995).

2.4. The central proposal

Following Ellis (2012), a basic standpoint will be adopted(Section 2.4.1) and then a central proposal is made as the waythe different levels relate to each other (Section 2.4.2).

2.4.1. A basic standpointI adopt the following starting point for what follows:Basic premise: At the classical macroscopic level, Individual Events

Happen.Each aspect is important:Individual: Statistics is not enough. An ensemble of events is

made up of individual events. There is no ensemble if individualevents do not separately happen.

Events: Specific things occur. Universal laws describe multifoldpossibilities of what might happen, but we experience specificevents in our own particular history.

Happen: They occur in time. They are about to occur, they occur,then they have occurred. Uncertainty about what might occurchanges to the certainty of what has occurred.

Any theory adopted must recognize that this is the case.Theories that deal only with statistics of what happens areincomplete. How this classical behavior emerges from the under-lying quantum theory is of course in dispute; decades of non-realist interpretations of quantum mechanics claim that this is notthe way things are in the micro realm (Isham, 1995). Howeverwhatever the case is at the lower levels, in order to be consistentwith a huge amount of data, including our ability to successfullyperform experiments, there has to be some process which leads tothe emergence of macro realism where this basic premise is true,whether we understand that process of emergence or not.

I will take a particular position on this, associating such emer-gence of a classical realm with wave function collapse (Penrose,

1989a), but recognizing that there are other proposals that may work.Many of the considerations that follow will be unchanged whateverthat process is.

2.4.2. Proposal: nature of physical realityThe view proposed in Ellis (2011) on the nature of physical

reality is as follows.

1.
Combinatorial structure: Physical reality is made of linearlybehaving components combined in non-linear ways.
2.
Emergence: Higher level behavior emerges from this lower levelstructure.
3.
Contextuality: The way the lower level elements behave dependson the context in which they are imbedded.
4.
Quantum foundations: Quantum theory is the universal founda-tion of what happens, by applying locally to the lower level (verysmall scale) entities at all times and places.
5.
Quantum limitations: Linearity at higher (larger scale) levelscannot be assumed, it will be true only if it can be shown toemerge from the specific combination of lower level elements.
The same view will be adopted here.

3. The arrow of time

A key issue for fundamental physics is the determination of thearrow of time. Section 3.1 explains the problem, and Section 3.2considers basic approaches to its resolution: By coarse graining(Section 3.2.1), by statistical fluctuations (Section 3.2.3), by afoundational quantum arrow of time (Section 3.2.4), and by specialinitial conditions (Section 3.2.5). The latter seems to be the viableway to go, and Section 3.3 develops it in terms of the PastHypothesis – the idea that global conditions determine the arrowof time by top-down causation. Three possible interpretations ofthis idea are distinguished, and then pursued in the subsequentsections.

3.1. The issue

A crucial aspect of the relation between macro and microphysics is the origin of the arrow of time (Eddington, 1928,pp. 68–80): One of the major puzzles in physics. There is aprofound disjunction between macro and micro physics in thisregard.

At the macroscopic scale, the second law of thermodynamics isan unavoidable physical reality (Eddington, 1928): The entropy S ofisolated systems increases with time:

dS=dt≥0; ð8Þwith equality only in equilibrium cases. Irreversibility relentlesslyfollows. Examples of irreversibility are as follows:

�
gas in one half of a container spreading to fill the whole when apartition separating it from the other half is removed (Fig. 1(a)and (b)),
�
a glass falling off a table and smashing to pieces (Penrose,1989a, 2011),
�
water flowing downhill, � a block sliding on a plane and coming to a stop owing to
friction,
� a stone tossed into a lake and sending out waves along the
surface of the water,
� a radio signal or sound wave is received after it was sent, � a footprint left in the sea sand after you have walked past,

S


�
the moving finger writes and moves on, leaving its trace behind(Omar Khayam),
�
the progress of life from birth to death: The seven ages ofmankind (Shakespeare).
�
the evolution of life on earth: Once there was no life, nowthere is;
tt
� t1t2
0

Fig. 2. Loschmidt's paradox: Set initial conditions for a system time t0. Let timephysics evolve in the future direction of the time coordinate t, giving time t1.Entropy S(t) increases (Boltzmann's H-Theorem), so Sðt1Þ≥Sðt0Þ. But this does notaccount for the fact that the initial direction of time t was chosen arbitrarily. Thedynamics is time symmetric; it could equally set off in the opposite direction,giving development to time t2. Boltzmann's H-Theorem applies equally in thisdirection (set t↦−t, the proof is unchanged), implying Sðt2Þ≥Sðt0Þ. This is Losch-midt's paradox: the H-Theorem works in both directions of time (Penrose, 1989a).

the progression of structure growth in the universe: Once therewas no structure; now there is.

Thus the arrow of time and irreversibility of physical effect occurson all scales of the hierarchy, except perhaps the quantum scale,but it occurs there too if one accepts the reality of wave functioncollapse (7).

This irreversibility relates to loss of useable energy as thepassage of time occurs, and associated increase of disorder(Carroll, 2010, pp. 143–171). It is a core feature of thermodynamics(Fuchs, 1996) and physical chemistry (Atkins, 1994), and hence playsa crucial role in biology (Campbell & Reece, 2005, pp. 143–144),energy flows in ecosystems (Schneider & Sagan, 2005; Wikibooks,chap. 14), and energy needs of an industrial economy (Georgescu-Roegen, 1971).

At a microscopic level, with one caveat I attend to shortly, thebasic interaction equations for the four fundamental forces aretime symmetric, and so coarse graining them should lead to timesymmetric macroscopic laws. The unitary evolution described bythe quantum evolution equations also does not determine adirection of time, because the underlying unitary theory treatsthe future and past directions of time as equal. Specifically inEq. (4), one can make the swap t12t2 and get an identicalsolution to (4), but with the opposite arrow of time (let t↦~t≔−tand the solution will be identical to (4)). The same is famously trueof Feynman (1948) diagrams.

3.1.1. The micro–macro relationMacro effective laws are often determined by coarse-graining

micro laws (Section 2.3.2). The macro laws that emerge by coarsegraining should have the same time symmetry as the micro laws(simply reverse the arrow of time in the coarse graining process inDiagram 1). This is true even when we deduce higher levelequations for irreversible statistical behavior: There will be anequally good solution with the opposite direction of time. Hencethere is apparently a fundamental contradiction:

The macro behavior displays a time asymmetry that is notapparent in the fundamental equations out of which they emerge.

Thus for each solution of the equations of Newtonian dynamics,of Newtonian gravity, of electromagnetic theory, of special rela-tivity, there is a time reversed solution of the equations whereeverything happens in the opposite sense of time. In the case ofthe glass falling off a table and smashing to pieces, there is a timereversed solution where the pieces of the glass assemble them-selves into a whole glass and ascend back onto the table (Penrose,1989a). In the case of the water waves, there is a time reversedsolution where spherical incoming waves converge on a point andpull the stone back up out of the water (Zeh, 2007). But we neversee this happen in practice.

The basic problem: How does the macro theory determine whichis the future as opposed to the past, when this time asymmetry is notapparent in the underlying unitary theory? (Callender, 2011; Davies,1974; Ellis & Sciama, 1972; Halliwell et al., 1996; Zeh, 2007).

The caveat mentioned above is that there is a very weak timeasymmetry of weak interactions. However this seems too ineffec-tive to be the origin of the time asymmetry we see at macroscales:The weak interaction does not have enough purchase on the restof physics (indeed the time asymmetry is very difficult to detect).

3.2. Possible resolutions

This section considers in turn resolution by coarse graining, bystatistical fluctuations, by a foundational quantum arrow of time,and by special initial conditions. The first three are bottom-upapproaches that all cannot succeed because of Loschmidt's para-dox (Section 3.2.1). Hence a top down approach – special cosmo-logical initial conditions – has to be the way to go (Section 3.2.5).

3.2.1. Bottom-up resolution by coarse grainingNow an initial reaction is that coarse graining from micro to

macro scales results in an arrow of time, as shown beautifully byBoltzmann's H-Theorem (Zeh, 2007, pp. 43–48), resulting from thefact that random motions in phase space takes one from lessprobable to more probable regions of phase space (Carroll, 2010,pp. 172–174; Gemmer, Michel, & Mahler, 2004, pp. 43–47; Penrose,2004, pp. 686–696; Penrose, 2011, pp. 9–56). Hence one can showthat entropy increases to the future; the second law of thermo-dynamics at the macro level emerges from the coarse grainedunderlying micro theory. The quantum theory version of thisresult is the statement that the density matrix open systemevolves in a time asymmetric manner, leading to an increase inentropy (Breuer & Petruccione, 2006, pp. 123–125).

But this apparent appearance of an arrow of time from theunderlying theory is an illusion, as the underlying theory is timesymmetric, so there is no way that an arrow of time can emerge byany local coarse graining procedure. Indeed the derivation of theincrease of entropy in Boltzmann's H-Theorem applies equally toboth directions of time (swap t-−t, the same derivation stillholds). The same applies to any derivation from quantum fieldtheory, for example that given by Weinberg (1995).

This is Loschmidt's paradox (Penrose, 1989a, Fig 7.6; Penrose,2004, pp. 696–699; Penrose, 2011):

The H-theorem predicts that entropy will increase to both thefuture and the past.

(Fig. 2). The same will apply to the quantum theory derivation ofan increase of entropy through evolution of the density matrix(Breuer & Petruccione, 2006, pp. 123–125; Gemmer et al., 2004,pp. 38–42, 53–58): It cannot resolve where the arrow of timecomes from, or indeed why it is the same everywhere.

3.2.2. The locality issueThe latter is a key question for any local proposal for determin-

ing the arrow of time:

The arrow of time locality issue: If there is a purely localprocess for determining the arrow of time, why does it give thesame result everywhere?

1 Arrow of time arguments are notoriously tricky. If you consider the arrowgoing the other way, then this state would be the end, not the start.


Local determination has to arbitrarily choose one of the twodirections of time as the positive direction indicating the future;but as this decision is made locally, there is no reason whateverwhy it should be consistent globally. If it emerges locally, oppositearrows may be expected to occur in different places.

We are unaware of any contradiction as regards the direction ofthe arrow of time, either locally (time does not run backwardsanywhere on Earth) or astronomically (irreversible process indistant galaxies seem to run in the same direction of time as here,Rees, 1995). Some coordinating mechanism is called for to ensurethe arrow of time points in the same direction everywhere inour past.

3.2.3. Bottom-up resolution by statistical fluctuationsIt is often claimed that if one has an equilibrium state that lasts

an infinite time, fluctuations round equilibrium can lead to anystate whatever popping out of the vacuum just as a statisticalfluctuation, with associated emergence of a local arrow of time.This leads to Poincare's Eternal return (any state whatever that hasoccurred will eventually recur) and the Boltzmann Brain scenario:You can explain the existence of Boltzmann's brain not as a resultof evolution but just as an eventual inevitable result of statisticalfluctuations if an infinite amount of time is available (Carroll, 2010,pp. 201–227).

This is problematic in the real universe, as we have not hadstatistical equilibrium anywhere except for very short timescalesin very local contexts since the end of inflation. And it would alsorun into the locality issue (Section 3.2.2): Such fluctuations wouldbe local in space, and different arrows of time could emerge indifferent places. The context for relevance of these arguments hasnot occurred in the real universe: The argument does not takerealistic context into account (Feynman et al., 1963, p. 46; Zeh,2007, p. 42), except possibly in the far future of the universe if itexpands forever due to a cosmological constant (Carroll, 2010,pp. 313–314). One cannot explain the arrow of time we experienceat the present moment as being a consequence of statisticalfluctuations.

3.2.4. Bottom up resolution by a foundational quantum arrow oftime

Could a resolution come from the local time asymmetry in statevector projection (7)? After all if quantum physics underlies allthe rest, perhaps the time asymmetry involved in (7) could bethe source for the rest, based on the way a local emergence ofclassicality works through collapse of the wave function(Aharaonov & Rohrlich, 2005, pp. 136–147, Penrose, 2004,pp. 527–530) and associated increase of entropy (Zeh, 1990). Butthen where does that quantum level time asymmetry come from?That depends on the resolution of the unresolved issue of statevector reduction. I will make the case that the nature of thisprocess is largely determined by local top-down effects (Section2.3.3) due to the specific nature of local physical structures (Ellis,2011). In that case, this asymmetry may be determined locally bytop-down causation, rather than being the source of the asym-metry. This conclusion is reinforced by the locality issue (Section3.2.2): In order to assign an arrow of time everywhere in aconsistent way, it has to be determined contextually throughsome coordinating mechanism ensuring it is the same everywherein a connected spacetime domain.

The likely solution is that resolution is by a top-down effect, thelocal time asymmetry of state vector reduction being based on atime asymmetry in the local detection environment, in turnfounded in conditions in the universe as a whole: The localenvironment too has to know which time direction to choose asthe future, else the set of local environments too will fall foul of

the locality issue. The issue arises for each level in the hierarchy ofcomplexity (Table 1): If it is not determined from below, it must bedetermined from above. Considering higher and higher levels, theanswer must lie at the top.

3.2.5. Resolution by large scale initial conditionsThe implication must be that the arrow of time results from

global environmental conditions, as it can't reliably emerge in aconsistent way from local physics that does not care about thedirection of time. Feynman stated in his lectures.

“So far as we know all the fundamental laws of physics, likeNewton's equations, are reversible. Then where does irreversibilitycome form? It comes from going from order to disorder, but we do notunderstand this till we know the origin of the order…for some reasonthe universe at one time had a very low entropy for its energycontent, and since then the entropy has increased. So that is the waytowards the future. That is the origin of all irreversibility, that is whatmakes the process of growth and decay, that makes us remember thepast and not the future…” (Feynman et al., 1963, pp. 46–48).

This fits into the fundamental nature of causality in thefollowing way: A key feature of causality as determined byphysical equations is that (Hartle, 2003; Zeh, 2007, pp. 1–3) theoutcome depends both on the equations plus the initial and/orfinal conditions. Hence

Broken symmetry: The solution of a set of equations will usuallynot exhibit the symmetry of the underlying theory (Anderson,1972). If there is no time asymmetry in the equations, it must lie inthe initial and/or final conditions.

Note that precisely because we are dealing with time asymmetry,we cannot assume that it is the initial conditions alone; inprinciple we may need to compare them to final conditions(Penrose, 1989a; Wheeler & Feynman, 1945). However I will makethe case below that we need to be concerned only with initialconditions, because the relevant context is that of an EvolvingBlock Universe (Section 7.1).

3.3. Top-down determination: the past hypothesis

Thus the only viable option seems to be the Past Hypothesis(Albert, 2000; Carroll, 2010, p. 176):

The direction of time must be derived by a top-down process fromcosmological to local scales.

It is strongly supported by the fact that the entropy of universecould have been much larger than it was (Carroll, 2010, pp. 345–346; Penrose, 1989a) because black holes could have had muchmore entropy (Carroll, 2010, pp. 299–302; Penrose, 2004, pp. 728–731). It started off in a very special state, characterized by the WeylCurvature Hypothesis (the universe is asymptotically conformallyflat at the big bang, Penrose, 2004, pp. 765–769), which wasrequired in order that inflation could start (Penrose, 1989b).

To investigate this further, I distinguish the following threepossible aspects:

�
AT1: Global time asymmetry. A difference in conditions at thestart and end of the universe (Ellis & Sciama, 1972; Feynman,Leighton, & Sands, 1964, pp. 28–36; Wheeler & Feynman, 1945).
�
AT2: Global past condition. Special conditions at the start, oncosmological scale: The expanding universe started in a speciallow entropy condition,1 which thus made it possible for it toevolve towards higher entropy states (Carroll, 2010; Penrose,

Figtimwhdireincrof t


2004, pp. 702–707; Penrose, 2011, pp. 57–136) and solvesLoschmidt's paradox because the global past condition cascadeddown to give a sequence of local past conditions.

�
AT3: An initial master arrow of time. The other arrows derivefrom the global master arrow of time resulting from the universe'searly expansion from an initial singularity in an Evolving BlockUniverse (Ellis, 2013). The arrow of time at the start is the timedirection pointing away from the initial singularity towards thegrowing boundary of spacetime; this then remains the direction oftime at all later times.
Such cosmological asymmetries provide a possible sourcedetermining why the local arrow of time is the way it is, by top-down causation from the global to the local direction of time (thelatter will therefore be the same everywhere, avoiding the arrowof time locality problem).

How are these different reasons related to each other? It mightseem that AT3 might be reducible to AT2; but this is not the case.AT3 is defined in the context of an Evolving Block Universe (Ellis,2013), where the flow of time would be determined as the timedirection leading away from the initial singularity, pointing fromthe start to the growing edge of spacetime, and so providing the‘master arrow’,which would cascade down to the other arrows asdiscussed below. If the start of the universe occurred in a veryinhomogeneous way, (AT2) would not be satisfied but (AT3) wouldstill set the direction of time. If the singularity was inhomoge-neous enough, the second law would not hold (entropy would notincrease in the future direction of time).

I will make the case that AT1 is not the way to go, because theproper context to view the situation is an evolving block universe(Ellis, 2013) (see Section 7), which rules this proposal out. Ratherthe direction of the flow of time is due AT3, which provides themaster direction of time. Then AT2 is required in order thatentropy increases as time flows. As Loschmidt's paradox makesclear, entropy can in principle increase in either direction of time;the master arrow AT3 makes the choice as to which direction isthe future, while AT2 makes sure entropy increase follows thatarrow. The other arrows of time (Davies, 1974; Ellis & Sciama, 1972;Halliwell et al., 1996; Zeh, 2007) (thermodynamic, electrodynamic,gravitational, quantum, and biological) all then follow.

In order to make this precise, I will distinguish between thedirection of time and the arrow of time as follows (Figure 3):

The direction of time is the cosmologically determined direc-tion in which time flows globally. It represents the way space-time is continuously increasing as an Evolving Block Universe.

The start: t = 0

The direction of time

The arrow of time

The present: t = t0

. 3. The direction of time: The direction of time is the cosmological direction ofe from the start of the universe to the present. It corresponds to the direction onich the Evolving Block Universe is growing. The arrow of time is the localction of time affecting local processes, so it is the direction in which entropy iseasing. The arrow of time at each time devolves from the global directionime.

By contrast

The arrow of time is the locally determined direction in whichtime flows at any time in the evolution of the universe. Itrepresents the way physics and biology manifest the flow of timelocally.

The proposal will be that

�
the flow of time, and hence the direction of time, is determinedby the cosmological master arrow of time AT3;
�
this then determines the arrow of time for local physicalprocesses by a top-down cascade in the hierarchy of physicalstructure, based on special cosmological initial conditions AT2;
�
this in turn determines the arrow of time for complex systemsand life by a bottom-up cascade in emergent systems.
4. The start and continuation of time

To look at this, we must have a reasonable model of cosmologyfrom some starting time through structure formation up to thepresent day (Dodelson, 2003; Ellis, Maartens, & MacCallum, 2011;Harrison, 2000; Silk, 2001). Section 4.1 shows how cosmic time isset up, Section 4.2 discusses the main relevant cosmic epochs sincethe start of inflation, and Section 4.3 the speculative pre-inflationpossibilities.

4.1. Cosmic time

The background model used in cosmological studies is theFriedmann–Lemaître–Robertson–Walker (FLRW) spacetime, givenin comoving coordinates by

ds2 ¼ −dt2 þ a2ðtÞ ds2 ð9Þwhere ds2 is a 3-space of constant curvature and a(t) the scalefactor (Ellis, 2006b; Ells et al., 2011; Hawking & Ellis, 1973).Perturbations around that model characterize how structure for-mation took place (Dodelson, 2003; Ells et al., 2011).

The dynamics of the FLRW model is governed by three inter-related equations. The energy-density conservation equation deter-mines the time evolution of the density ρðtÞ:

_ρ þ ðρþ p=c2Þ3 _aa¼ 0 ð10Þ

and so determines the evolution of the pressure p(t) through asuitable equation of state p¼ pðρÞ. Second, the scale factor a(t)obeys the Raychaudhuri equation

3€aa¼−

12κðρþ 3p=c2Þ þ Λ; ð11Þ

where κ is the gravitational constant and Λ the cosmologicalconstant. Third, the first integral of Eqs. (10) and (11) when _a≠0is the Friedmann equation

_a2

a2¼ κρ

3þ Λ

3−

ka2

; ð12Þ

where k is an integration constant related to the spatial curvature.Thus the cosmic time is the time parameter t that enters into theseequations determining the scale factor evolution. It is determinedby being the time parameter naturally appearing in the 1+3 covariantformulation of the Einstein Field equations in the cosmological context(Ells et al., 2011), which reduces to Eqs. (10)–(12) when specialized to aFLRW geometry. But Eqs. (10)–(12) are invariant under t-~t≔−t: if a(t)is a solution, so is if að~t Þ.

Classically, the universe began at a spacetime singularity(Hawking & Ellis, 1973), conventionally set to be t¼0. Cosmic timestarts at the creation of universe: Time came into being, it did not

Here and now

Fi+

Fi-

Finite Infinity

Visual Horizon

Inflation

Pre inflation

HBB era

LSS

Future light cone

Earth’s Word line

CMB Sphere

TIME

The present time

Origin event or process

Epoch 2: HBB

Epoch 1: Inflation

Epoch 0: Pre inflation

Epoch 3:Structure Growth Era

Epoch 4: Future

End event or process

Visible Universe

Past light cone

Matter Horizon

Fig. 4. Our cosmic context: A conformal diagram of the cosmic context for local existence. Time runs vertically, 2 space dimensions horizontally (one space dimension ishidden). Light rays travel at 451 to the vertical. The start of the universe is indicated at the bottom (this might possibly represent a start a finite time ago, or at minus infinity;conformal diagrams do not represent distance or proper time accurately). This is followed by a pre-inflation quantum gravity era, an inflationary era, and a Hot Big Bang(HBB) era, which ends at the surface of last scattering (LSS). The LSS marks the start of structure formation, which extends from the LSS to the present time. Structureformation may continue for some time to the future, but will eventually come to an end. The far future boundary of the universe may lie a finite time to the future, but moreprobably is an infinite proper time to the future. The Earth's world line is the vertical line at the center, with the present time “here and now” marked. Our past light coneextends down to the LSS, which it intersects in a 2-sphere; this is the set of events from which the 2.7 K cosmic microwave background (CMB) originated. We cannot see toearlier times because the universe was opaque in the HBB era; hence the matter world lines through this 2-sphere form our visual horizon (the surface in spacetimeseparating matter we have seen from that which we cannot detect by any electromagnetic radiation). For all practical purposes, “infinity” for local physics is a sphere of radius1 light year around the Earth. This is our “Finite Infinity”, its world tube surrounding our world-line in spacetime. Our future light cone intersects it in the 2-sphere Fiþ (ouroutgoing radiation sphere), and our past light cone intersects it in the 2-sphere Fi− (our incoming radiation sphere); this is our effective sky – every star and galaxy we see isan image on this sphere. On the Block Universe view of spacetime, everything here from the start to the finish exists as a single spacetime block, where all times are equal sothe present has no meaning. On the Emerging Block Universe (EBU) view, the present is the special time where, at this instant, the uncertain future is changing to thedetermined past. Hence on this view, the nature of existence is different in the past (below the surface labeled ‘The present time’ and in the future (above that surface). Theformer exists (as it has been determined), whereas the latter is presently only potential (so does not yet exist). As time progress, the present time moves up our world line, sothe past region of spacetime is continually getting bigger: spacetime is growing. In the far future, when everything has happened, the present will coincide with the futureboundary and the EBU will have evolved into a Final Block Universe.


exist before (insofar as that makes sense). At any small timet ¼ ε40, the arrow of time is defined to point from the singularityat t¼0 to the present time t ¼ ε (the future boundary of theevolving spacetime), because that is the direction of time in whichspacetime is increasing in the Evolving Block Universe (Ellis, 2013).Therefore

The Direction of Time: If the proper time τP along a fundamentalworld line from the initial singularity to the event P is greater thanthe proper time τQ along that world line from the initialsingularity to the event Q, then the direction of time is from Qto P.

After it has come into being there is no way it can reverse, becauseonce spacetime has come into existence, it can't disappear. Oncethe flow of time is established it just keeps rolling along,determining what happens according to (10)–(12) unless we reacha spacetime singularity in the future, when it comes to an end.

The cosmological direction of time AT3 is set by the start ofthe universe. There is no mechanism that can stop or reverse thecosmological flow of time, set by the start of the universe. It setsthe direction of flow of the time parameter t in the metric (9); timestarts at t¼0 and then increases monotonically along fundamen-tal world lines, being the parameter t occurring in the solutionfaðtÞ; ρðtÞ; pðtÞg to Eqs. (10)–(12).2

2 If we were perverse we could use the reverse time label where timeproceeded from t¼0 to values to0; but this is just a coordinate convention withno effect on the physics. It is psychologically sensible to assign the sign so that t

This is the master arrow of time AT3. The gravitational Eqs. (10)–(12) are time symmetric (because the Einstein equations are timesymmetric), but the actual universe had a start. This broke thetime symmetry and set the master arrow of time: The universe isexpanding, not contracting, because it started off from a zerovolume state. It had nowhere to grow but larger.

How this evolution actually occurs is determined by thechanging equation of state of the universe at different epochs(next section). When we take account of quantum gravity, thispicture is altered, and various options arise (Ellis, 2004, pp. 16–18);these are discussed in Section 4.3. However the universe will stillset a unique monotonically increasing time for all epochs after theend of the quantum gravity epoch.

4.2. The cosmic epochs

The basic dynamics of cosmology to the present time can beregarded as having five phases (Dodelson, 2003, pp. 1–20),3

summarized in Fig. 4:

�

(foopro‘ear

des

Epoch 0: Pre-inflationary era. Any quantum gravity era thatmight precede inflation. The dynamics at this time is hypothe-tical: We do not know what happened then (there may or maynot have been an actual physical start to the universe).

tnote continued)ceeds to positive values. This sets the convention for the concepts ‘later’ andlier’ in the usual way.3 See (Ellis, 2006b, Sections 2.6–2.8) for a conveniently accessible shortcription.


�
Epoch 1: Inflationary era. A very brief period of exponentialexpansion, ending at reheating and conversion of the inflatonfield to radiation, marking the start of the Hot Big Bang era.Inflation is the time when quantum perturbations arose thatprovided the seeds for structure formation at the end of the HotBig Bang era.
�
Epoch 2: Hot Bang era. An epoch of radiation and matter inquasi-equilibrium, up to the time of decoupling of matter andradiation at the Last Scattering Surface ‘LSS’. This epoch includesbaryosynthesis, nucleosynthesis, and the transition from aradiation dominated to matter dominated expansion. Theuniverse was opaque up to the end of this era.
�
Epoch 3: Structure formation era. The epoch from the LSS to thepresent day. The universe became transparent, matter andradiation decoupled leading to the universe being permeatedby Cosmic Black Body Radiation (CBR), and structure formationcommenced, leading to the existence of large scale structures,galaxies, stars, and planets. At a late time (close to the presentday) dark energy started to dominate the dynamics, leading toa speed-up of the expansion of the universe.
�
Epoch 4: The future. The epoch from the present day on, eitheran unending expansion (the most likely option), or recollapseto a future singularity; which is the case depends on para-meters and physics that are not well known.
The dynamical behavior is different in each epoch. To a goodapproximation we can represent the inflationary era, radiationdominated era, and matter dominated era as

Inflation: p¼ ρc2⇒aðtÞ ¼ a0 eHðt−t0Þ ð13Þ

Radiation dominated: p¼ 13ρc

2⇒aðtÞ ¼ a1ðt−t1Þ1=2 ð14Þ

Matter dominated : p¼ 0⇒aðtÞ ¼ a2ðt−t2Þ2=3 ð15ÞDark energy started to be significant late in Epoch 3, altering (15) abit at late times, but it only alters the big picture significantly inthe future era (Epoch 4), where it suggests there will be no finalsingularity: Eq. (13) will hold again and expansion will last for ever(but the physics is uncertain: Other options are possible).

4.3. The speculative pre-inflationary era

What previous mechanism could lead to a universe satisfyingthe past condition? Why is the universe an almost-FLRW universewhen it emerges from the inflationary era? What set the condi-tions before inflation such that inflation could start?

We are in a domain of untestable speculation here, and thepossibilities we can imagine certainly won't encompass all thatmight have been the case. Still it's fun to speculate. The problem isto find a way that generates a smooth start to the universe, giventhat inflation can't do so for generic initial conditions (Penrose,1989a, 1989b). Amongst the possibilities are,

�
The smooth universe machine. Whatever creates the universemakes a nice smooth universe: It works in a uniform way at allemergent spacetime events. This was the assumption everyonemade from the 1930s until the 1970s (Harrison, 2000).
�
The multiple universe machine. Whatever creates the universekeeps doing it, with variation. This includes the chaotic infla-tion scenario. Both arrows of time can occur in differentdomains (Carroll, 2010, pp. 359–364, 371–372; Carroll & Tam,2010).
�
The bounce machine. The present epoch of the universe resultedfrom some kind of bounce or rebirth from a previous era thatsets up the special state needed (Carroll, 2010, pp. 349–353;Penrose, 2011).
�
The emergent universe machine. Whatever creates the universemakes an emergent universe that spends a long time in an almoststatic state, with compact spatial sections (Ellis & Maartens, 2004;Ellis, Murugan, & Tsagas, 2004; Mulryne, Tavakol, Lidsey, & Ellis,2005). The previous state does not matter as there is plenty oftime for matter to come into equilibrium: Causal effects can travelround the universe thousands of times.
�
The quantum gravity machine. Whatever creates the universedoes so in a quantum gravity era where causality has not yetemerged, for example there is a spacetime foam, so causalrestrictions don't yet exist. Horizons emerge later. Anotheroption is the causal set approach to quantum gravity wherespacetime emerges from a discrete basic structure (Henson,2006). This is in accord with the view put here, as it allows agrowing universe like the Evolving Block Universe.
Whatever happened in this era is the ultimate source of the arrowof time, but the physics is completely uncertain, as is the overallcontext. I will here assume that whatever was needed happened,and led to a suitable start of inflation.

5. The descent of time: contextual effects

The basic idea in this section is that the expansion of universedetermines the arrow of time at the natural sciences hierarchy soas to concur with the direction of time set by the expansion of theuniverse AT3. The arrow of time ripples down from higher tolower levels in the physical hierarchy, as a consequence of thespecial global initial conditions AT2; this process sets up similarspeciality conditions at smaller scales in the hierarchy.

Crucially, at the beginning of the HBB era the universe wasexpanding and cooling, not vice versa. This sets the arrow of timefor local physics, by passing a variable (the global temperatureT(t) of matter and radiation), determined by the global expansionhistory (Section 4.2) to local physical systems. T(t) is decreasing ast increases; this is what determines the local arrow of time at eachlower level. It gets communicated to all the lower level physicsbecause they are systems imbedded in a heat bath with decreasingtemperature. Initially, it is a heat bath created by being immersedin the expanding cosmic fluid; later after decoupling, it is aradiative heat bath. Additionally matter is moving apart andthinning out: So density decreases as time progresses, withcomoving world lines moving even further apart, constrainingcausal processes.

This occurs in the inflationary epoch (Section 5.1), the Hot BigBang era (Section 5.2), and the astronomical epoch (Section 5.3).This determines the local thermodynamic arrow of time (Section5.4) and the radiative arrow of time (Section 5.5). To examine thelatter clearly, it is useful to introduce the idea of a finite infinity forisolated systems (Section 5.5.2). Finally the arrow of time cascadesdown from local systems to microsystems (Section 5.6).

5.1. Epoch 1: inflation

This exponentially expanding era (13) is driven by a scalar fieldcalled the inflaton (Silk, 2001, pp. 115–123). It has three effects asfollows:

�
The exponential expansion causes initial inhomogeneities andcurvature to decay away (Dodelson, 2003, pp. 144–155).
�
Quantum fluctuations generate tensor perturbations that resultin gravitational waves (Dodelson, 2003, pp. 155–162).
�
Quantum fluctuations generate scalar perturbations that resultin density inhomogeneities that later on are the seeds for largescale structure formation (Dodelson, 2003, pp. 162–173).


These effects all evolve in the forward direction of time thatunderlies the expansion occurring then (Dodelson, 2003, Figs. 6.7& 6.8). It is this expansion and arrow of time that sets the contextfor these important physical effects, being an example of AT3.However the inflation would not start if the universe at thebeginning of the inflationary epoch was not of limited anisotropy(Rothman & Ellis, 1986) and inhomogeneity (Penrose, 1989b) (anexample of AT2).

Inflation ends at reheating, when the inflaton gets converted toradiation. This sets the almost homogeneous initial conditions forthe next stage.

5.2. Epoch 2: the Hot Big Bang era

In the Hot Big Bang epoch, there was a heat bath with matter,photons and neutrinos (Dodelson, 2003, pp. 40–46) mainly inequilibrium at a temperature T(t) that decreases with time. Theearly part of this epoch is radiation dominated, but it becomesmatter dominated before last scattering (Dodelson, 2003, pp. 50–51).In the early radiation dominated era, the scale factor goes as (14) andthe temperature varies as (Dodelson, 2003, pp. 4–5)

TðtÞ ¼ aðt0ÞaðtÞ T0∝

T0

ðt−t1Þ1=2; ð16Þ

decreasing as t increases. This sets up a context which provides thearrow of time for local reactions: The time symmetry is broken by thesteady drop in temperature of the heat bath that is the time-changingenvironment for local reactions such as nucleosynthesis.

The expansion of an equilibrium hot equilibrium mixture ofparticles and radiation is time reversible (pair production andannihilation, element formation and decomposition balance) untilreaction thresholds are passed so some of these reactions ceaseand leave behind out of equilibrium decay products, characterizingirreversible behavior at those times. The major such non-equilibrium features are as follows:

�

the

Baryogenesis processes and the development of an asymmetrybetween particles and antiparticles (Riotto, 1998; Silk, 2001,pp. 134–137; Turner & Schramm, 1979).

�
Neutrino decoupling, neutron decoupling, and the formation oflight elements at the time of nucleosynthesis (Dodelson, 2003,pp. 9–12, 62–70; Silk, 2001, pp. 140–143).
�
Recombination of electrons and protons into neutral hydrogen,resulting in decoupling of matter and radiation (Dodelson, 2003,pp. 70–73; Silk, 2001, pp. 162–163), so determining the LSS andoriginating the CBR (photons stream freely from then on).4
�
Possibly, production of dark matter (Dodelson, 2003, pp. 73–78).
These irreversible processes can all be described by suitableversions of the Boltzmann equation (Dodelson, 2003, pp. 59–62,84–113). A crucial feature is

�
Growth of perturbations as different comoving scales leave and re-enter the Hubble horizon (Dodelson, 2003, pp. 180–213), andbaryon-acoustic oscillations take place (Dodelson, 2003, pp.224–230), accompanied by diffusion on some scales (Dodelson,2003, pp. 230–234) and radiative damping of shorter wavelengths(Silk, 2001, pp. 176–180).
Reversible and irreversible processes in this period have their timearrow set by the expansion of universe which determines howcontext changes: A time dependent heat bath where the

4 There is an earlier process of helium decoupling, which is not as importantrmodynamically.

temperature decreases as a result of AT2. Initial conditions werespecial, which also plays a role: The expansion rate and hence lightelement production would be different in very anisotropic orinhomogeneous cosmologies, so this is also an example of AT2.

Overall, this epoch serves to prepare special conditions on theLSS, homogeneous to one part in 10−5, which marks the end of thisepoch when Trad≃4000 K.

5.3. Epoch 3: the astronomical arrow of time

In this epoch, matter and radiation are initially decoupled, sothey are no longer in thermal equilibrium. Radiation pressure(which previously led to the baryon acoustic oscillations) nolonger resists gravitational collapse, and structure formation cancommence.

Structure formation takes place spontaneously through grav-itational instability (Saslaw, 1987, chap. 21). An initially uncorre-lated system develops correlations through gravitationalattraction: “Gravitational graininess initiates clustering” (Saslaw,1987, pp. 158–162). There is no arrow of time in the underlyingtime-symmetric Newtonian gravitational law

md2xi

dt2¼−∇iΦ; ∇2Φ¼ 4πGρ ð17Þ

(which derives in the appropriate limit from the Einstein Fieldequations). The process attains an arrow of time in the expandinguniverse context because of the change of equation of state at theLSS: Pressure forces that resisted collapse melt away, and structureformation begins from the tiny density inhomogeneities presenton the LSS. The arrow of time is then provided by the context ofcosmic evolution, communicated from the global scale to localscales by passing down the expansion parameter H(t) whichoccurs in the perturbation equations (Ellis & Bruni, 1989).

Structure forms spontaneously through a bottom up processdominated by cold dark matter (Rees, 1995, pp. 23–35, 39–45; Silk,2001, 183–186); this process apparently violates the second law ofthermodynamics as often stated (Carroll, 2010, pp. 295–299; Ellis,1995; Penrose, 1989a); see Fig. 1d. The entropy law somehowmustbe consistent, but we do not at present have a viable definition ofgravitational entropy for such situations. In any case it is clearthat the process requires special smooth initial conditions sothat structure can form: If black holes were already presenteverywhere, there would be no possibility of further structureformation (Penrose, 1989a).

Star formation takes place leading to ignition of nuclear fusionand nucleosynthesis in stellar interiors (Silk, 2001, pp. 301–331,339–345). The CBR is now collision free (Silk, 2001, pp. 75–81), soits temperature drops as the inverse of the scale factor (Dodelson,2003, p. 5):

TradnðtÞ ¼aLSSaðtÞ 4000 K: ð18Þ

This is much lower than the stellar temperatures, so the stars canfunction in thermodynamic terms (they can get rid of heat byradiation to the sky). The many irreversible astrophysical pro-cesses (Rees, 1995; Shklovskii, 1978) leading to the evolution ofstars (Silk, 2001, pp. 202–204, 229–231, 288–298) and galaxies andgalaxy clusters (Silk, 2001, pp. 187–202, 323–327) are based onbottom-up emergence of effective thermodynamical behavior, butthis is possible only because of the non-equilibrium context set bythe early universe according to the cosmological master arrow oftime. The lowness of the Sun's entropy (remoteness from thermalequilibrium) is because of the uniformity of the gas from which theSun has gravitationally condensed (Penrose, 2004, pp. 705–707).


Conclusion: Structure formation takes place by irreversibleprocesses starting in the inflationary era, resulting in fluctuationson the LSS that irreversibly lead to stars, galaxies, and planets afterdecoupling. The arrow of time for these processes derives from thecosmological master arrow of time (Section 5.2).

5.4. Thermodynamic arrow of time: local systems

For local systems on Earth, the arrow of time is apparent in thediffusion equation and in local physical interactions in machines,plants, animals, ecosystems, and the biosphere as a whole (Zeh,2007, pp. 39–84). This is all possible because we live in a non-equilibrium local environment, which is due to the larger astro-nomical environment (Section 5.3).

Bright Sun plus dark night sky: The Sun is a radiation sourcewhich is a hot spot in an otherwise cold background sky. Becauseof their higher energy, there are many photons coming in from theSun than those reradiated in the infrared to the sky, since the totalenergy carried in is the same as that going out (Carroll, 2010,pp. 191–194; Penrose, 1989a,p. 415; Penrose, 2004, pp. 705–707).The radiation heat balance equation for received solar short waveradiation (Monteith, 1973, pp. 60–61) leads to overall annual, dailyand instantaneous heat balances (Monteith, 1973, pp. 71–77) dueto the properties of incoming solar radiation (Monteith, 1973,pp. 23–58) and radiative properties of natural materials (Monteith,1973, pp. 60–71). This leads to the heat balance equations foranimals that enables life to function (Monteith, 1973, pp. 150–170).This is all possible because the sky acts as a heat sink for theemitted long wavelength radiation.

The reason the sky can act as a heat sink is the modern versionof Olber's paradox (why the sky is dark at night? Harrison, 1987;Harrison, 2000, pp. 248–265). The sky is dark because the universeis expanding (Harrison, 2000, pp. 491–506; Silk, 2001, pp. 55–58),so by (18), the Cosmic Background Radiation has cooled from itstemperature of 4000 K at the LSS to the present day CBR back-ground temperature of 2.75 (Harrison, 2000, pp. 339–349; Zeh,2007, pp. 26–27). Star formation since decoupling has made anegligible further contribution: It has led only to an effectivetemperature of 3 K also. This would not be the case if there were aforest of stars covering the whole sky (Harrison, 2000, Fig. 12.1,p. 250), when every line of sight would intersect a star and thetemperature on Earth would be the same as on the surface of astar. Thus we are in a thermal bath at 3 K as a result of expansionof universe and its subsequent thermal history, resulting both incooling of the CBR to 2.73 K and in stars only covering a very smallfraction of the sky. This astronomical context underlies the localthermodynamic arrow of time.

Example: Broken glass. A classic example is a glass falling froma table and lying shattered on the floor (Penrose, 1989a, pp. 397–399). Because the underlying micro-dynamics is time-reversible,in principle it can be put together again by just reversing thedirection of motion of all the molecules of the glass and in the airand the floor: It should then jump back onto the table andreconstitute itself. But this never spontaneously happens. Why isthe one a natural event and the other not?

It does not spontaneously reverse and reconstitute itselfbecause this is fantastically improbable: The asymmetric increaseof entropy, due to coarse graining, prevents this (Penrose, 1989a,pp. 391–449); and that is because of special conditions withcorrelations (the crystal structure) in the initial state that don'toccur in the final state.

5.5. Isolated systems and the radiative arrow of time

There is a further important issue: The relation between theradiative arrow of time and the thermodynamic arrow of time

(Ellis & Sciama, 1972). Consider first water waves spreading out,consequent on a stone being thrown into a pond. In principle,because of the time reversible microphysics, one can reverse thedirection of time to see the waves focus in and make the stone popout of water. In practice this can't be done. Again, we have aresolution by asymmetric correlations: Typical incoming wavesare not correlated, but the outgoing waves are (they diverge fromone point). Thus the arrow of time is reflected in the asymmetry ofcorrelations in the future relative to the past.

But is this asymmetry a cause or an effect? Suppose we don'twant to talk about the future: Can we just talk about special initialconditions in the past? Yes, this should be possible: All we need isthe structure of phase space (Penrose, 1989a, pp. 402–408) plusspecial conditions in the past (Penrose, 1989a, pp. 415–447). Onedoes not need a future condition. But one does need the pastcondition on the relevant scale (it is needed on the scale of stars forthe astrophysical arrow of time, but on the scale of molecules forthe arrow of time in water waves). Hence we need a

Local Past Condition (LPC): special initial data occurred at therelevant scale for the phenomenon considered.

The initial conditions that lead to structure are less likely thanthose that don't, but they did indeed occur in the past. This LPCapplies at the scale relevant to broken glasses and unscramblingeggs because it has cascaded down from the cosmological scale tothe local scale. This incoming and outgoing asymmetry applies towater waves, sound waves in the air, and elastic waves in solids. Itapplies on the astronomical scale to supernova explosions: Onecan in principle reverse the direction of time to see the outgoingradiation focus in and the supernova reassemble; in practice thiscannot happen because of the very special initial conditionsrequired for the time reverse motions to do this.

5.5.1. Electromagnetic wavesThe same issue arises for electromagnetic radiation, indeed this

is the relevant case as regards the heat from the Sun because thatarrives as radiant energy. Why does it come from the past nullcone rather than the future null cone, given that Maxwell's theoryis time symmetric? And why do radio signals arrive after they aresent, rather than before? The answer is similar to that for acousticwaves: There is a fundamental difference between incoming andoutgoing electromagnetic radiation, in terms of coherence on thefuture null cone as compared with the past. But then why is thatso? It derives from cosmic initial conditions, cascading down fromlarger to smaller scales. To look at this properly, we need to beclearer on the spacetime domains involved.

5.5.2. Isolated systems: the relevant domainsWe need to consider an effectively isolated system, such as the

Solar System (see Fig. 4, which is in conformal coordinates, withmatter world lines vertical lines). The Earth's world line is at thecenter, the event ‘here and now’ is where the present timeintersects our world line. The incoming light cone (to the past)intersects the LSS on a 2-sphere, which I call the CMB sphere,because this is the part of the LSS that emitted the radiation wetoday measure as 2.73 K Cosmic Black Body Radiation (the CMBsphere has been mapped in detail by the WMAP and Plancksatellites). The Visual Horizon is formed by the matter world linesthrough the CMB sphere (we cannot see any matter further out byany electromagnetic radiation). Hence the whole visible universelies between our world line and the visual horizon, from the LSStill today. On these scales we can extrapolate to the future, butwith increasing uncertainty we further extrapolate towards thefinal future events.


To examine the relation between incoming and outgoingradiation, one can use the idea of Finite Infinity F i (Ellis, 1984).We surround the system S of interest by a 2-sphere F i of radius RF i

such that it is at infinity for all practical purposes: Spacetime isalmost flat there, because it is so far away from the source at thecenter, but it is not so far out that the gravitational field of otherneighboring objects is significant. For the Solar System, Rf is about1 light year; for the Galaxy, 1 Mpc. The world tube marked out byF i is shown in Fig. 3; we can examine the interaction of the localsystem with the rest of the universe by considering incoming andoutgoing matter and radiation crossing this world tube.

The intersections of our past light cone C− with F i give a2-sphere C− a distance RF i away which is our effective sky; allincoming radiation crosses C−. Similarly the 2-sphere Cþ definedby the intersection of our future light cone with F i is our futuresky; all outgoing radiation crosses Cþ. The arrow of time has twoaspects. First, it lies in the difference between data on Cþ, whichhas high correlations with our position due to outgoing signalsfrom the Earth, whereas that on C− does not have time-reversedsimilar correlated incoming signals focussed on the Earth. Second,it lies in the fact that the amount of incoming radiation on C− isvery low; this is the dark night sky condition mentioned in theprevious section. It is in effect the Sommerfeld incoming radiationcondition (Zeh, 2007, p. 23).

The reason there is little radiation coming in on C−, and that it isuncorrelated to a high degree, is two-fold. Firstly, there is acontribution from the radiation temperature on the CMB sphereon the LSS, which is almost Gaussian, and then is diluted by thecosmic expansion from 4000 K to 2.75 K (see above). Second, theintervening matter between the LSS C− and us is almost isotropicwhen averaged on a large enough scale, and luminous mattercovers a rather small fraction of the sky (we do not see a forest ofstars densely covering the whole sky); hence we receive ratherlittle light from all this clustered matter (stars in our galaxy, allother galaxies, QSOs, etc.).

But how does this relate to solutions of Maxwell's equations interms of advanced and retarded Green's functions (Zeh, 2007,pp. 16–38)? And why does matter here, and the interveningmatter, emit radiation to the future rather than the past? This isallowed by thermodynamic constraints on the emission processes;but this does not by itself explain the electrodynamic arrow oftime. Why does a shaken electron radiate into the future and notthe past? We need a condition where the waves generated by asource are only waves that go outward, so only the outgoing wavesolution makes physical sense (Feynman et al., 1964, pp. 20–14).I suggest that the reason is that only the past Green's function canbe used in such calculations, because we live in an Evolving BlockUniverse (EBU): We can't integrate a Green's function over a futuredomain that does not yet exist. This is discussed in Section 7.The Local Past condition LPC is needed on these scales so thatthermodynamic and associated electrodynamic processes can takeplace in the forward direction of time; but the foundationalcosmological arrow of time AT3 in the EBU is the ultimate reasonthat times goes to the future and not the past. Actually it defineswhat is the future direction of time.

5.5.3. Incoming and outgoing matterAs well as radiation, an isolated system is subject to incoming

and outgoing matter. There are two aspects here. First, the matterthat made up the solar system and nearby other systems – indeedthe matter out of which we are made – originated within ourmatter horizon (Ellis & Stoeger, 2009), a sphere with comovingradius of about 2 Mpc, lying between F i and the visual horizon(Fig. 4). The intersection of the matter horizon with the LSS (seeEllis & Stoeger, 2009 for a discussion and detailed spacetime

diagram) is the domain where we require special past conditionsto be true in order that the Solar System can arise from astro-physical processes with the forward arrow of time. This is the LPCfor the Solar System.

Second, there may be incoming particles (cosmic rays, blackholes, asteroids, comets) crossing F i since the Solar Systemformed and impacting life on Earth. These too must be of lowintensity in order that local equilibrium can be established andlocal thermodynamic processes proceed unhindered. The LocalPast Conditions on the LSS in the domain close to the matterhorizon will ensure this to be true. We assume this for example inexperiments at particle colliders such as the LHC: If there was ahuge flux of incoming cosmic rays, we would not be able to doexperiments such as at the LHC. Local thermodynamics canproceed as usual because we are indeed an effectively isolatedsystem. The universe does not interfere with our local affairs: Acase of top-down non-interference that could have been otherwise:There could have been massive gravitational waves or streams ofblack holes coming in and interfering with local conditions, as wellas high energy photons. Isolated systems are necessary for life(Ellis, 2006b, Section 9.1.3), and the cosmic context sets this localcontext up suitably.

5.6. Micro systems: quantum arrow of time

The same kinds of considerations hold for everyday physics andlocal quantum systems. They work in the forward direction of timebecause of the non-equilibrium local context inherited from thehigher level Solar System context. If we were in a higher tem-perature heat bath, there would be different outcomes.

The quantum arrow of time (Zeh, 2007, pp. 85–134) shouldfollow from the local context, because for example the way wavefunction collapse occurs in detectors (based on the photoelectriceffect) is due to the local physical context (Ellis, 2011). That contextincludes the local thermodynamic arrow of time. So for exampleone can ask why photodiodes or chlorophyll in plant leaves don'tbehave reversibly: Why does a plant or a CCD not emit light ratherthan absorbing it? The answer must be that they involve specialstructures that create thresholds that general non-unitary beha-vior, and the specific arrow of time that occurs are set by preparedinitial conditions in the physical apparatus, that make the detectorwork in the one direction of time, not the other, in the localcontext discussed above. This is related to the general feature ofanisotropic spatial structures plus special initial conditions.

The derived quantum arrow of time is synchronized with theoverall system through contextual effects, and in particular deco-herence and the Lindblad master equation inherit their arrow oftime from the environment. Each is not time reversible becausethe environment is in a non-equilibrium state (Section 5) and alocal asymmetry condition LPC applies on micro scales: As in thecase of reconstructing a supernova, one could not easily reversethe chemical reactions in photosynthesis, as this would requireimprobable coordination of incoming entities.

6. The ascent of time: emergent structures

The existence of time, and the direction of the arrow of time, istaken for granted in applied physics, engineering, biology, geology,and astrophysics. It is assumed as a ground rule that the arrow oftime exists and runs unceasingly according to the second law.Given the arrow of time problem as set out above, how does thiscome about?

The suggestion here will be that the arrow of time that exists atthe lower levels (because of the suitable context, as discussed inthe previous section) propagates up to higher levels through the


process of creation of emergent structures. This is a bottom upprocess from lower to higher levels. There are specific emergentmechanisms that enable this to happen, with three related crucialcomponents: Arrow of time detectors (Section 6.1), rate of timemeasurers (Section 6.2), and flow of time recorders (Section 6.3).These are what make the flow of time real. I look at them in turn.

6.1. Arrow of time detectors

If we take series of pictures of objects such as a breaking glassor an exploding supernova, we can discriminate the future fromthe past by just looking at them: We can order the picturesappropriately and determine the arrow of time. But these aren'tregular occurrences that can be used generically to determine thedirection of time; there are more systematic ways of doing this.

Generically, a cause precedes its effects; how does one harnessthis to show which way time is going? The basic principle is

A spatial asymmetry is converted into a time asymmetrythrough suitable environmental and initial conditions.The requirements are structures with suitable spatial asymme-try, plus special initial conditions.

This is a form of top-down action, both due to the existence ofemergent structures, and the link in to the LPC discussed abovethrough the requirement of special initial conditions. Specific casesshow how this works out in detail.

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Figtimthastarwo

Downhill flow: A rock spontaneously falls downhill, not up;water naturally flows downstream, not up (Fig. 5). This appliesto any energy gradient: Exothermic reactions take place spon-taneously in chemistry (hence the danger of fires and the need

Time T1

gravity gravity

Time T2

. 5. An arrow of time detector. A wheel is clamped in a static state for an extended periode (because there is no dynamics!). The slope establishes a spatial asymmetry. At time T1 itn time T1: the rolling down the slope (to the right) establishes which is the past and whit instant, the motion would have been time symmetric (it could have rolled up to an instauld have been determined by the dynamics.

for fire prevention services); electric currents flow from thenegative to the positive terminal of a battery, and in electricaland electronic circuits, currents flow towards ground. A naturalexample is lightning which goes spontaneously to the ground(Feynman et al., 1963, pp. 9-2 to 9-7). In electronics, forms ofelectric current include the flow of electrons through resistors orthrough the vacuum in a vacuum tube, the flow of ions inside abattery or a neuron, and the flow of holes within a semiconduc-tor; each flow one way as time progresses. A reversed arrow oftime would reverse their spatial direction of movement.Biology is crucially based on the absorption of high energymaterials and excretion of low energy waste, the central featurebeing cellular respiration, based on glycolysis, the citric acid cycle,and oxidation (Campbell & Reece, 2005, pp. 160–178). Each ofthese processes has a forward direction of time that arises out ofthe underlying physics plus the local context, and so acts as anarrow of time detector.

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Ratchets: A mechanical ratchet turns one way because of a pawland ratchet mechanism permitting motion in one directiononly as time progresses (Fig. 5), thus it is a mechanism fordetecting the direction of time as contained in Newton's law ofmotion. How does it do it? – it is a mechanism designed to doso! There is no direction of time in Newton's law of motionitself, but there is in the special solution of the law of motionthat describes the ratchet: A case of spontaneous symmetrybreaking, converting spatial asymmetry into time asymmetry.It can only do so when suitable environmental conditions aresatisfied: In general Brownian motion takes place, and the pawlcan jump out of ratchet allowing it to fluctuate back: When thepawl and wheel are both at the same temperature, the motionis reversible (Feynman et al., 1963, pp. 46-1 to 46-9). At lowertemperatures it is a disguised form of asymmetric sawtooth
Time T1

Time T2

Time

Start Distance

future

on a downhill slope; during that time dynamics does not distinguish an arrow ofis released. The time T2 when it has rolled some distance down the slope is laterch is the future direction of time. If the wheel had not been clamped before thentaneous stationary state at time T1 and then down again) and no arrow of time

gravity One-way motion

RATCHET

Fig. 6. An arrow of time detector. A ratchet wheel is constrained to rotate only oneway by a pawl held down by a spring or by gravity. Its one way motion


potential, where diffusion extracts the direction of time fromthe spatial direction provided by the sawtooth.Engineering applications include ratchet wrenches and screw-drivers, turnstiles, and hoists. Ratchets are a key mechanism inmicrobiology: Serreli, Lee, Kay, and Leigh (2007) describes amolecular information ratchet, Roeling et al. (2011) describeorganic electronic ratchets doing work, and Kulic, Mani,Mohrbach, Thaokar, and Mahadevan (2009) describe botanicalratchets. Brownian ratchets work by inscribing and erasing anasymmetric potential which induces a directed motion of aparticle. Molecular motors are based on biological ratchetsLacoste and Mallick, 2010, and work by hydrolyzing ATP alonga polar filament.

characterizes the future time occurring in the Newtonian equations of motion. Its
� environment must be cool.
Rectifiers are devices that make currents flow in only onedirection. If you reverse the direction of time, it will go theother way. This works thermodynamically in the case of avacuum tube rectifier (emission of electrons at a hot electrodeand reception at a cooler one). It works by adaptive selectionthrough a detailed physical structure of the rectifier, in the caseof solid-state rectifiers, taking advantage of diffusion currents.In a p–n junction with forward bias, the electrostatic potentialin the n-region is lowered relative to p-side, increasing thediffusion current; the pair current is unchanged. The diffusioncurrent exceeds the pair current and there is a net current fromthe p-side to the n-side; however with a reverse bias this doesnot happen, hence the junction acts as a rectifier (Durrant,2000, p. 997). A mechanical example is a one-way valve in awater system, with a ball and spring in a water outlet into acontainer. Water flows in only through this valve, not out.

�
A crucial example of rectifiers in biology is ion channels(Campbell & Reece, 2005, pp. 133–136, 1017–1025; Kandel,Schwartz, & Jessell, 2000, pp. 105–124; Rhoades & Pflanzer,1996, pp. 219–226). These are pore-forming proteins thatestablish and control a voltage gradient across the plasmamembrane of cells, thereby allowing the one-way flow of ionsdown their electrochemical gradient. They occur in the mem-branes that surround all biological cells, for example potassiumion channels (the hERG channel) mediates a delayed rectifiercurrent (IKr) that conducts potassium (K+) ions out of themuscle cells of the heart (Trudeau, Warmke, Ganetzky, &Robertson, 1995). All such rectifier functions are based ondetailed biological mechanisms (e.g. Goettig, Groll, Kim,Huber, & Brandstetter, 2002). These underlie active transportsystems in the cell that act like Maxwell's Demon in creating agradient of Kþ and Naþ across a cell wall (Lehninger, 1973,pp. 191–206).
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Filters are devices that select some components of a mixture orensemble from others, discarding those not selected. The arrowof time is revealed by the process of selection where a subset ofthe whole emerges as the output (Section 2.3.4). Running it inreverse would generate more states from less, but the finalstate does not have the information needed to tell what theincoming state was. Examples are polarizers (Ellis, 2011),wavelength filters in optics due to selective absorbtion basedon the crystal structure of an optical medium, and tunableradio receivers based on resonance properties of AC circuits(Hughes, 2008, pp. 299–320). Electrical filters can be low-pass,high-pass, or passband filters (Hughes, 2008, pp. 359–385). Inbiology, excretory processes in organisms are based on a series offilter mechanisms involving selectively permeable membranes andselective reabsorption (Campbell & Reece, 2005, p. 929).
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Diffusion is a basic physical process that detects the direction oftime: Macro-properties of gases naturally smooth out in thefuture, not the past (Fig. 1). Thus diffusion is migration ofmatter down a concentration gradient (Atkins, 1994, p. 818).Similarly migration of energy down a temperature gradient
underlies thermal conduction, migration of electrical chargedown a potential gradient underlies electrical conduction, andmigration of linear momentum down a velocity gradientgenerates viscosity (Atkins, 1994, p. 818). This is a crucialprocess in chemistry (Atkins, 1994, pp. 817–830, 846–856)and thermal physics (Fuchs, 1996, pp. 649–654).

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Physiological processes often involve diffusion between com-partments and through membranes, the direction of timedetermining the direction of diffusion as an emergent propertyof the lower level dynamics (Rhoades & Pflanzer, 1996, pp. 116–129, Riggs, 1963, pp. 168–220). Diffusion plays a key role incapillary systems (Rhoades & Pflanzer, 1996, pp. 590–592),hormone transport (Rhoades & Pflanzer, 1996, pp. 373–374),and lungs (Rhoades & Pflanzer, 1996, pp. 618–619), where itdetermines action of an anesthetic gas (Riggs, 1963, pp. 312–333). Diffusion is crucial at the synapses connecting neurons inthe brain (Campbell & Reece, 2005, pp. 139–146, Rhoades &Pflanzer, 1996, pp. 113–116).
In each case, the macro context acts down on the micro level toinduce time asymmetric behavior arising from spatial gradients(either horizontal, so unaffected by gravity, or with a verticalcomponent, so gravitationally influenced). The macro leveldynamics due to inhomogeneities at that level acts down on themicro level to create micro differences (the temperature of afalling rock is hotter at bottom of hill than at top) (Fig. 6).

6.2. Rate of time measurers: clocks and ages

Clocks are rate of instruments that reliably measure the rate ofprogress of time, converting it to a linear scale; an integrated clockreading gives an age. Translating it to digital form will alwaysinvolve non-unitary events.

Some clocks are the direction of time detectors in addition (anyof the arrow of time detection processes mentioned above can beused as a clock, if it's behavior is regular enough). Many clocks donot measure the direction of time but merely the number ofintervals between two events, independent of whether thoseintervals run from a : -b or are reversed to run from b : -a.These kinds of clocks are represented by solutions which doexhibit the time-symmetry of the underlying laws of motion.A light clock is one such example, a ratchet clock is not.

6.2.1. Reversible clocksClocks that are not the direction of time detectors are as

follows.Distance traveled at reliable speed measures time. An example is

an analog clock that rotates its hands at a constant speed throughan electric motor drive. The non-unitary part of the process occurs


when a dial reading is noted at a specific time by some observer.A sundial is a projection to Earth of such a reliable motion (albeitnon-constant: Seasonally dependent but predictable correctionsmust be made). Some forms of clock use the invariance of thespeed of light to provide a fundamental basis for timekeeping.A simple example is a light clock consisting of an emitter and twomirrors kept a fixed distance apart; the ‘ticks’ of the clock are thereflection events at one end. The non-unitary part of the process isthe reflection events.

Counter for repetitive processes: Most current clocks work bycounting cycles of some reliable periodic process, like the swing ofa pendulum, the cycles of a balance wheel, or the vibration of aquartz crystal. A reader or latch records the cycle, and this analogto digital transformation involves thresholds and so is non-unitary.A computer utilizes a circuit that emits a series of pulses with aprecise width and a precise interval between consecutive pulses,made by an oscillator and latch (a circuit that remembers previousvalues) (Tanenbaum, 1990, pp. 98–103). This is of course a classicaldevice; with an adder (Tanenbaum, 1990, pp. 96–98), it detects thedirection of time and measures the progression of time. It must bebased on repeated wave function projection at the quantum levelin the transistor gates that underlie its operation (Tanenbaum,1990, pp. 76–86).

6.2.2. Irreversible clocksClocks that are the direction of time detectors are as follows.Dating: One can estimate ages of objects by examining histor-

ical records, for example in astronomy, archaeology, and geology.Accurate age measurements come from integrating clock readings.

Reliable decay processes: Radioactive materials spontaneouslydecay; the future direction of time is that choice where theamount of radioactivity is less in the future than the past(Feynman et al., 1963, 5-3 to 5-5). This provides centrally impor-tant radiocarbon methods of dating in archaeology (Wood, 2005,pp. 31–33). This is a form of the emptying reservoir methodmentioned above; state preparation took place in the supernovaexplosion that created the radioactive elements.

Flow in and out of a container: One of the oldest methods is acontainer with a steady inflow and/or outflow of some quantity,and suitable calibration. It requires a prepared initial state (full orempty) and identifiable final state, when it will require a reset tothe initial condition; this is the non-unitary part of the process.Examples are water clocks (filled to the brim and then steadilyemptying) and sand funnels (egg timers). One can conversely havea water container that is steadily filled up, and then flushed whenfull. In electrical circuits, charging and discharging a capacitor isthe equivalent (Hughes, 2008, pp. 93–94, 108–114).

Reliable growth processes can also be used for dating. The classiccase (apart from horse's teeth!) is dendrochronology: That isdating by counting tree rings (Wood, 2005, p. 34). Actually thisis really a process of recording the annual cycles of the Earth'smotion around the Sun using the tree's developmental processesas the recorder. In astronomy, stellar ages can be determined viastellar evolution theory and observations of the distribution in theHertzsprung–Russell diagram of cluster stars (Shklovskii, 1978,pp. 189–193).

All of these measuring processes result as emergent propertiesof the underlying physics that at some point involves non-unitaryevolution (without this feature, the flow of time would be evidentin the system dynamics but it could not be recorded).

6.3. Flow of time recorders: records of the past and memory

We are aware of the flow of time because of the existence ofrecords of the past. These are of two types:

Passive records: These are physical records of what happenedin the past, such as primordial element abundances, geologicalstrata, palaeomagnetic records, fossils, the genetic code, thenature of biological species, vegetation patterns in the country-side, buildings and infrastructure in cities, and so on. This isdata which can be used to provide information about the past,when we relate them to some theoretical model.

These data have not been laid down for some purpose; they arejust there as remnants of past events. They have not been indexedor classified, but we are able to interrogate them and use them todetermine past history, for example in astronomy (Shklovskii,1978; Silk, 2001), geology, and biological evolution (Wood, 2005,pp. 24–35).

Memory records: These are physical records that are a mean-ingful distillation of what happened, somehow indexed inrelation to some classification scheme, so that they can berecovered. This is information that can be used for some usefulpurpose (Roederer, 2005).

In both cases, laying down these records of events that havehappened involves a physical process with a definite physicaloutcome which is then stable over some length of time.

6.3.1. RecordingSome physical property is set at a specific value by a local

physical process, and then stays at the value because a thresholdhas to be surmounted to reset it. The arrow of time comes in thatwhere the record did not initially exist but does at later times.

The kinds of properties that serve as the physical substrate atthe lower levels include

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Magnetic states of a magnetizable medium, based on the mag-netization properties of specific materials (Feynman et al., 1964,36-1 to 37-5).
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Coded surface properties of a medium: Binary data is stored inpits on the surface of CD ROMs (Tanenbaum, 1990, pp. 52–54),based on the stability of material properties. One can alsoinclude in this category, writing and printing on paper.
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Electric circuit states: Binary data is stored in high/low voltagesin specific circuit elements in some memory array, based onthe stability of electronic circuit states (Hughes, 2008,pp. 544–554).
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Biomolecule structure: The coding patterns in DNA molecules isa record of evolutionary history, and can be used for dating andcladistic analysis (Wood, 2005, pp. 21, 51–52). This is based onthe reliability of the DNA copying process (Campbell & Reece,2005, pp. 293–308).
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Pattern of connectivity and activation in network connections:The specific pattern of connections and their relative strengthsin neural networks records short term and long term memories(Kandel et al., 2000, pp. 1227–1246). Short term data can alsobe stored in as activation patterns such as synchronized activityin brain circuits (Buzsaki, 2006, pp. 136–174).
Generically, recording takes place through emergence of anyidentifiable structure that is stable over a relevant time scale.

6.3.2. RememberingThis is some process whereby the stored record is interrogated:

Examples are DNA epigenetic processes, reading of memory incomputer systems, and recalling memories in one's mind. This is asopposed to interrogating a record, where the record is analyzedrather than being read, as in the cases of geology and archaeology.For memory to be useful, there has to be some kind of indexing


system of what is stored: It is no good storing information, if youdon't know where it is.

The process of indexing needs a sorting and classificationsystem, which will generally involve a modular hierarchicalstructure used in the classification of folders and files and inassigning names or other identifiers to them as well as to storedentities. In a computer it is implemented at the lower level byassociation of names with specific memory addresses(Tanenbaum, 1990, pp. 40–44). The index is itself a further formof memory.

6.3.3. DeletingBecause of the finite capacity of memory and the ongoing influx

of new information, generically one needs some kind of resetprocess that wipes out old memory to create space for newinformation to be stored. This is state preparation for the nextround of remembering. It will be a non-unitary process, indeed itis precisely here so that irreversibility occurs and entropy isgenerated, as shown by Landauer (1961) and Bennett (2003).The arrow of time comes in that the record that initially existeddoes not exist at later times.

However you don't delete all that is in memory: Selectivedeletion takes place, because one selects what is deleted and whatis kept by deleting unwanted files, e-mails, and so on. this istherefore a form of adaptive selection: The creating of usefulinformation deleting that which is not useful in relation to someclassification system and guiding purpose. What is kept is deter-mined by the user's purpose: This is top-down causation from theuser's purpose to the electrons in the computer memory system.The arrow of time is involved in the transformation of randomrecords to useful data.

6.3.4. State vector reductionThe crucial feature of all of this in relation to quantum physics

is that every recording event, reading event, and deletion eventinvolves effective collapse of the wavefunction, because a recallableclassical record is laid down and has a definite state. Quantumuncertainty makes way to classical definiteness as each such eventtakes place. This is happening all the time everywhere as passiverecords are laid down, as well as when memories are recorded,read, and deleted. The outcome is no longer a quantum super-position: It is a definite classical outcome. If this was not the case,specific memories could not be recalled.

6.4. The ascent of time: emergent properties

The proposal now is that the passage of time, which happens inevents such as those just discussed, ripples up from the lowerlevels to the higher levels through developmental processes thatdepend on the lower level arrow of time and therefore embodythem in higher level processes.

The process of emergence builds an arrow of time into eachhigher emergent level N þ 1 because it is imbedded in the nextlower level N, through the process whereby coherent lower leveldynamics leads to emergence of coherent higher level dynamics(Diagram 1). If you reverse the arrow of time at the lower level inDiagram 1, it will result in a reversed arrow of time in the higherlevel. For example, if the future direction of time is built intomachine language level in a computer, so a program at that levelruns in the positive direction of time t when the computer is run,the same will be true at all the emergent abstract machine levelsin the computer (Tanenbaum, 1990); for example a Java virtualmachine running on top of the machine language will have thesame arrow of time.

The basic ways the lower level processes embody the arrow oftime has been discussed above, they include

�
Things naturally fall downhill. � Electric currents flow from positive to negative potentials. � Waves convey information as they spread out from their sources. � Energy changes from useful to useless forms as dissipative
processes take place.
� Diffusion spreads heat and matter out from their sources.
These embody an arrow of time: For example given a definition ofpositive (+) and negative (−) potentials, the current flows from the+ to the − terminal in the future direction of time. If the directions oftime were reversed, the flow would be the other way.

These effects at the basic physics level then affect processes inapplied physics, chemistry, all forms of engineering, geology,planetary science, and astronomy, as well as in microbiology,physiology, developmental processes, psychological processes,and evolutionary history, leading to similar effects due to the flowof time. Indeed it is because this happens reliably as we have ourbasic concepts of cause and effect, the latter always occurring afterthe former. The whole idea of causation is premised on thisproperty.

6.4.1. Local physics and technologyPhysicists, chemists, and engineers can assume that the second

law of thermodynamics holds on macro scales with the forwarddirection of time. This becomes a basic feature of their analyses(Fuchs, 1996), involving viscosity, the production of heat, dissipa-tion, and entropy production. It affects entities such as engines,refrigerators, heat exchangers, as well as chemical reactors. Parti-cularly important is the way separation and purification processesunderlie our technological capabilities by enabling us to obtainspecific chemical elements and compounds as needed – anothercase of adaptive selection (Section 2.3.4), locally going against thegrain of the second law at the expense of the environment. Thespecific processes that enable these non-unitary effects aredetailed in Henley, DSeader, and Roper (2011).

Molecular and solid state systems inherit the arrow of timefrom their constituents (current flows, rectifiers, gates), leading toelectrical and electronic systems (Hughes, 2008), computers(Tanenbaum, 1990), and nanotechnology devices (Ziman, 1979).The flow of time in the response of the components at each level isused by the designers in making time-responsive higher circuitelements; they are all classical elements, so at their operationemerges out of state-vector reduction at the micro level.

6.4.2. Geological and astrophysical arrow of timeDiffusion plays a crucial role in the environment, where it

relates to pollution hazards in the ocean, atmosphere, rivers, andlakes (Csanady, 1973).

It occurs in astrophysics, where the Fokker–Planck equationimplies continuous production of entropy (Saslaw, 1987, chap. 4).One-way processes based on the underlying radiative processesand diffusion are important in stars, galaxies, radio sources, QSOs,and so on (Rees, 1995; Shklovskii, 1978). In particular supernovaeare irreversible processes: Gravity creates order (attractor ofdynamic system), direction comes from initial conditions, SNexplosion creates disordered state irreversibly. This is crucial forthe formation of planets round second generation stars that canbecome homes for life (Silk, 2001, pp. 345–350).

6.4.3. Biological arrow of timeBiology takes the arrow of time in lower level processes for

granted. These time asymmetric processes, e.g. cell processes,

Not yet in existence

Present

Start Past

Fig. 7. The direction of time. The direction of time results from the fact that at anyspecific time t0, spacetime exists in the past (tot0) but not in the future (t4 t0).The direction of time – the direction in which spacetime is growing – points for thestart of the universe to the growing boundary at the present.


synaptic process, propagate their arrow of time up the hierarchy(Campbell & Reece, 2005) to the macro states. Thus diffusionacross synaptic cleft, propagation of action potential from dendriteto axons, the rectifying action of voltage-activated channels under-lying the nerve impulse, lead to time asymmetric brain processes.Micro asymmetry in these processes results in emergent timeasymmetry in macro events in the brain: Emergent structures inbiology inherit their arrow of time from the underlying modulesand their non-equilibrium interactions.

At the community level, crop ecology depends on the one-wayprocess of leaf photosynthesis (Loomis & Coonor, 1992, pp. 257–288) and its consequences for plant growth. Dissipative forces inthe interaction of components must be modeled in looking atenergy flows in ecosystems (Odum, 1972) and in physical pro-cesses such as weathering, erosion, and deposition (White,Mottershead, & Harrison, 1984, pp. 225–249, 274–280). A thermo-dynamically based arrow of time is involved, starting at the levelof grains of sand and dust particles, and cascading up to weath-ering of mountains and global spread of pollution.

As for evolution, there is a time asymmetry in the Darwinianevolutionary process: Initial conditions on earth were very simplein biological terms, so there was no way but up! At a macro level,fitness flux characterizes the process of natural selection(Mustonen & Lässig, 2010) and satisfies a theorem that showsexistence of a fitness flux even in a non-equilibrium stationarystate. As always, adaptive selection is a non-unitary process. Insuch processes, energy rate density serves as a plausible measureof complexity (Chaisson, 1998, 2010), but the process is notnecessarily always up: Evolution is an imperfect ratchet.

6.4.4. Experience and memoryThe process of experiencing and remembering is based on

underlying time-asymmetric synaptic processes (Kandel et al.,2000) that lead to the experience of subjective time (Gray, 2011).

Overall, lower level entities experience an arrow of timebecause of their context; as they act together to create higherlevel entities, they inevitably export this arrow of time into thehigher level structures that emerge.

7. The nature of spacetime

The passage of time is crucial for the understanding of physicalreality presented here: Not just as a subjective phenomenonrelated to the mind, but as an objective phenomenon related tophysical processes occurring from the very early universe to thepresent day. The best spacetime model for what occurs is anevolving block universe (Section 7.1), increasing with time fromthe start of time until the end of time. This provides the ultimatesource of the microphysics arrow of time (Section 8), as well as asolid reason for preserving causality by preventing existence ofclosed timelike lines (Section 7.2).

7.1. The Evolving Block Universe

The passage of time is a real physical process, as exemplifiedin all the cases discussed above. Our spacetime picture shouldadequately reflect this fact.

The nature of existence is different in the past and in the future.Becoming has meaning (Eddington, 1928, pp. 94–110). Differentontologies apply in the past and future, as well as differentepistemologies.

One can express this essential feature by viewing spacetime asan Evolving Block Universe (EBU) (Ellis, 2008; Tooley, 2000). In sucha view the present is different from the past and the future; this isrepresented by an emergent spacetime which grows with time,

the present separating the past (which exists) from the future,which does not yet exist and so does not have the sameontological status. The past is the set of events that have happenedand so are determined and definite; the future is a set ofpossibilities that have not yet happened. The present separatesthem, and the passage of time is the continual progression bywhich the indeterminate becomes determinate. This is not derivedfrom the physics equations, but postulated independently as theway they function.

It must be emphasized here that it is not just the contents ofspacetime that are determined as time evolves; the spacetimestructure itself also is only definite once events have taken place.For example quantum fluctuations determined the spacetimeinhomogeneities at the end of inflation (Ellis, 2008); hence theywere intrinsically unpredictable; the outcome was only deter-mined as it happened. The part of spacetime that exists at anyinstant is the past part of spacetime, which continually grows. Thisis the Evolving Block Universe. The future is a possibility space,waiting to be realized. It does not yet exist, although it is notgeneric: There are a restricted set of possibilities that can emergefrom any specific present day state. Classically they would bedetermined, but irreducible quantum uncertainty prevents uniquepredictions (Ellis, 2011; Section 2.1). This is where the essentialdifference between the future and the past arises.

Thus in Fig. 7, the time up to the present has happened, andeverything to the past of the present is determined. The time tothe future of the present has yet to occur; what will happen thereis not yet determined. The present time is a unique aspect ofspacetime at each instant, forming the future boundary of space-time, and it keeps moving up as time elapses. It is determined byintegrating proper time along the fundamental world lines definedby being Ricci eigenlines (Ellis, 2013), because causation takesplace along timelike lines, not spacelike surfaces; this gives theusual surfaces of constant time in a Robertson–Walker spacetime.The relativity of simultaneity is a psychological construct that isirrelevant to physical processes, and so that issue has no physicalimport (Ellis, 2013).

It is the viewpoint of this paper that at a micro level, timepasses as effective wave function collapse takes place: Theindefinite future changes to the determined past as this happens(equation (7)). Alternative views abound, but this is the view Ipropose here; whatever underlying theory one may have, this isthe effective theory that must emerge. This is possibly bestrepresented as a Crystalizing Block Universe (CBU) when we takequantum effects such as entanglement in time into account (Ellis &Rothman, 2010). When one coarse grains the local micro timedetermined in this way, related to physical processes by (3), it willlead to the macro time parameter.

7.1.1. The preferred world linesThe past/future cut continually changes with time, at any

specific time defining the present, so it is fundamental to physical


processes; how is it determined? Physical processes are based ontimelike and null worldlines rather than spacelike surfaces, so thisprocess of becoming determinate happens along preferred time-like worldlines in spacetime, as a function of proper times alongthose world lines since the beginning of the universe, augmentedby processes along null geodesics(in the real universe, timelikeeffects dominate at most places at both early and late times). Themetric function determines proper time for an observer along eachobserver's world line, as a line integral along their timelike worldlines – this is a basic feature of general relativity (Hawking & Ellis,1973). This provides the proper time parameter τ at each eventthat determines the rate at which physical processes happenthrough local dynamical equations such as the Schrödinger equa-tion (3) at the micro level, and the Maxwell equations and Einsteinfield equations in the 1+3 covariant form (Ells et al., 2011) at themacro level.

The proposal is that on a large scale, what matters is theaverage motion of all matter present in an averaging volume,which determines the average 4-velocity of matter in the universe(Ellis, 1971), so this is what determines the communal cosmolo-gical time. The surfaces of constant time SðτÞ will be determined bythe integral of proper time τ along the timelike eigenvectors of thetotal matter stress tensor Tab from the start of the universe to thepresent time (Ellis, 2013). Through the Einstein equations(Hawking & Ellis, 1973), these curves, representing the averagemotion of matter at each event determined on a suitable averagingscale, will be the timelike eigencurves of the Ricci tensor Rab,which will be uniquely defined in any realistic spacetime (the realuniverse is not a de Sitter or Anti de Sitter spacetime).5

This construction non-locally defines a unique surface Sðτ0Þ ‘thepresent’ where the transition event is taking place at any specifictime τ0; spacetime is defined for 0oτoτ0, but not for τ4τ0. Thesesurfaces are derivative rather than fundamental: As indicated, theessential physical processes take place along timelike world lines.These surfaces of transition need not be instantaneous for thepreferred world lines, and are not even necessarily spacelike. Theirexistence breaks both Lorentz invariance and general covariance;that is fine, this is an unsurprising case of a broken symmetry, asoccurs in any specific realistic solution of the field equations (forexample any perturbed Friedmann–Lemaître spacetime, Dodelson,2003; Ells et al., 2011). The existence of physical objects is relatedto conservation relations between entities at successive times,entailing continuity of existence between correlated sets of prop-erties as time progresses.

The quantum measurement process (i.e. effective projection ofthe wave function to an eigenstate with specific values for therelevant variables) is associated with specific local physical entitiessuch as a particle detector or photon detector or rhodopsinmolecules (Ellis, 2011), for these determine what happens. Thuswe may expect that the relevant world lines for the quantum toclassical transition (superpositions changing to eigenstates) will befixed by the local motion of matter: On a small scale, that of adetection apparatus; on a larger scale, the average motion ofmatter in a large averaging volume.

7.1.2. ProposalIn summary, at any instant the ontological nature of the past,

present, and future is fundamentally different. This determines thedirection in which time flows.

5 There are no preferred worldlines or space sections in special relativity, so theidea does not work in that context; this has been used as an argument against theEBU idea. But it is general relativity that determines the spacetime in the realuniverse via the Einstein Field equations, so this special relativity indeterminacy isnot the physically relevant case and the argument does not apply.

�
Viewpoint: The evolving nature of space time. The proposal isthat spacetime is an Evolving Block Universe, where the essen-tial difference between the past (it exists) and future (it doesnot yet exist) generates a time asymmetry in all local physicalprocesses and so creates the direction of time (Fig. 7). Space-time starts at the beginning of the universe and then growssteadily until the end of time; this direction then cascadesdown to determine the arrow of time in local systems (Fig. 4).
It has been suggested to me that this EBU proposal is a philoso-phical position. I disagree: It is a selection principle for viablecosmologically relevant space times, in much the same way thatone insists that the speed of sound vs must be less than the speedof light c in any viable solution. It has mathematical outcomes, asexplained above: At any specific time τ0, spacetime is defined for0oτ≤τ0, but not for τ4τ0; hence any integral for fields orradiation on the surface Sðτ0Þ can only range over values τoτ0.This excludes advanced solutions of the wave equation for anyvariable, and only the retarded Feynman propagator (Feynman &Hibbs, 1965) will make physical sense, because you can't integrateover the future domain if it does not yet exist.

The way this time asymmetry “reaches down” to the quantummeasurement process and the state preparation process is still tobe clarified. The working hypothesis is that it must do so,determining the local quantum arrow of time locally in eachdomain in such a way that they do indeed add up to a coherentglobal arrow of time. This is clearly a speculation, but it sets apossible agenda for investigation.

7.1.3. The contrary viewThis view is of course contrary to that expressed by some

philosophers (e.g. Price, 1996) and by many quantum physicists,particularly related to the idea of the wave function of the universeand the Wheeler–de Witt equation (e.g. Barbour, 1999). In Ellis(2011), I claimed that the basis for believing in that approach is noton a solid footing. In brief, the argument is, We have no evidence thatthe universe as a whole behaves as a Hamiltonian system. Indeed,because the behavior of the universe as a whole emerges from theconjunction in complex configurations of the behavior of its compo-nents, it is likely that this is not true, except perhaps at the veryearliest times before complex configurations existed (Ellis, 2011).

This counter viewpoint is put in many articles and books,stating that every event in the past and future is implicit in thecurrent moment, because that is what the equations say; eithertime does not exist, or it does not flow (Barbour, 1999; Davies,2012; Price, 1996).

But the question is which equations, and when are theyapplicable, and what is their context of application? As empha-sized so well by Eddington (1928, pp. 246–260), our mathematicalequations representing the behavior of macro objects are highlyabstracted version of reality, leaving almost all the complexitiesout. The case made in Ellis (2011) is that when true complexity istaken into account, the unitary equations leading to the view thattime is an illusion are generically not applicable except to isolatedmicro components of the whole; Ellis (2013) shows an alternativecoordinate system where the Hamiltonian does not vanish. Thecounter viewpoint expressed often supposes a determinism of thefuture that is not realized in practice, denying the applicability ofquantum uncertainty to the real universe. But that uncertainty is awell-established fact (Feynman, Leighton, & Sands, 1965; Greenstein &Zajonc, 2006), which can have macroscopic consequences in themacro world, as is demonstrated by the historic process of structureformation resulting from quantum fluctuations during the inflationaryera (Dodelson, 2003). These inhomogeneities were not determineduntil the relevant quantum fluctuations had occurred, and then


become crystalized in classical fluctuations; and they were unpredict-able, even in principle.

7.2. Closed timelike lines

A longstanding problem for general relativity theory is thatclosed timelike lines can occur in exact solutions of the EinsteinField equations with reasonable matter content, as shownfamously in the static rotating Gödel solution (Hawking & Ellis,1973). This opens up the possibility of many paradoxes, such askilling your own grandparents before you were born and socreating causally untenable situations.

It has been hypothesized that a Chronology Protection Conjec-ture (Hawking, 1992) would prevent this happening. Variousarguments have been given in its support (Visser, 2002), but thisremains an ad hoc condition added on as an extra requirement onsolutions of the field equations, which do not by themselves givethe needed protection.

The EBU automatically provides such protection (Ellis, 2013),because creating closed timelike lines in this context requires thedetermined part of spacetime intruding on regions that have alreadybeen fixed. But the evolving spacetime regions can never intrude intothe completed past domains and so create closed timelike lines,because to do so would require the fundamental world lines tointersect each other; and that would create a spacetime singularity,because they are the timelike eigenvectors of the Ricci tensor, and inthe real universe, there is always matter or radiation present. Theextension of time cannot be continued beyond such singularities,because they are the boundary of spacetime (Ellis, 2012).

Causality: The existence of closed timelike lines (Carroll, 2010,pp. 93–116) is prevented, because if the fundamental world linesintersect, a spacetime singularity occurs (Hawking & Ellis, 1973):The worldlines are incomplete in the future, time comes to an endthere, and no “Grandfather Paradox” can occur.

8. The arrow of time

In an Evolving Block Universe, where the flow of time is real,one cannot resolve the arrow of time problem through the ideaAT1: There are different conditions in the far future and the farpast. It does not apply, because that cannot be applied if thefuture does not yet exist. The solution is rather the combinationof AT3, setting the master direction of time at the cosmologicalscale, in combination with the speciality condition AT3, whichvalidates the second law of thermodynamics. It propagates downto give an arrow of time at each lower level by setting specialenvironmental contexts at each level, and then propagates up inemergent structures, to give effective time asymmetric laws ateach level.

Together these create the EBU where the arrow of time is builtin to the fact that the past has taken place, and the future is yet tocome; the past exists as what has happened, the future as(restricted) potentialities

�
Only radiation from the past can affect us now, as only the pasthas happened. Radiative energy arrives here and now from thepast null cone, not the future null cone (Fig. 4).
�
Only the retarded Feynman Green's function makes physicalsense, because only the past can send causal influences to us.This solves the issue of the local electromagnetic arrow of time.
�
The matter that exists here and now was created by nucleo-synthesis in the past (Fig. 4). It bears in its very existencea record of the events of cosmological and stellar nucle-
osynthesis. Future potentialities are unable to influence us inthis way.

�
We can influence the future by changing conditions as to whatwill happen then; we cannot do so for the past, as it has alreadyoccurred. The relevant wave function has already collapsed anddelivered a specific result.
�
Overall, the micro laws of physics are time symmetric, forexample Feynman diagrams can work in both directions intime, but the context in which they operate (the EBU) is not.Thus their outcome of necessity has a determinate arrow oftime, which underlies the very concept of causation as weknow it. If this was not so, cause and effect would not bedistinguishable.
8.1. The top-down and bottom up cascades

The overall picture that emerges is shown in Diagram 2.

TheArrowofTime

Cosmology
Brain, Society Top-down effects ⇓ ⇑ Bottom-up effects Non-equilibrium environment ⇒ Molecular processes Top-down effects ⇓ ⇑ Bottom-up effects Quantum Theory ⇒ Quantum Theory
Diagram 2. Contextual determination of the arrow of time cascadesdown from cosmology to the underlying micro processes, on thenatural sciences side, and then up to the brain and society, on thehuman sciences side.

In summary:

�
Spacetime is an evolving block universe, which grows as timeevolves. This fundamental arrow of time was set at the start ofthe universe.
�
The observable part of the universe started off in a special statewhich allowed structure formation to take place and entropyto grow.
�
The arrow of time cascades down from cosmology to thequantum level (top down effects) and then cascades up inbiological systems (emergence effects), overall enabled by theexpanding universe context leading to a dark night sky allow-ing local non-equilibrium processes to occur.
�
There are an array of technological and biological mechanismsthat can detect the direction of time, measure time at variouslevels of precision, and record the passage of time in physicallyembodied memories.
�
These are irreversible processes that occur at the classical level,even when they have a quantum origin such as a tunnelingprocess, and so at a foundational level must be based on thetime-irreversible quantum measurement process.
�
In conceptual terms they are the way the arrow of timeparameter t in the basic equations of physics (the Dirac andSchröedinger equations (3), Maxwell's equations and Ein-stein's equations on the 1+3 covariant formulation, Ellset al., 2011) is realized and determines the rate of physicalprocesses and hence the way time emerges in relation tophysical objects.
�
Each of these processes is enabled by top-down action takingplace in suitable emergent local structural contexts, providedby molecular or solid-state structures. These effects could notoccur in a purely bottom-up way.


8.2. A contextual view of the arrow of time

This paper has extended the broad framework of Ellis (2011) tolook in detail at the issue of the arrow of time. It has made the casethat this is best looked at in terms of the hierarchy of complexity(Table 1), where both bottom-up and top-down causation occur.Detailed examples have been given of how this works out in termsof arrow of time detectors, clocks, and records of past events. AT3sets the master arrow of time. The EBU starts at the beginning oftime, the future direction of time is that direction in whichspacetime is growing.

The Arrow of Time: On the view presented here, the ultimateresolution of the Arrow of Time issue is provided by the fact welive in an Evolving Block Universe starting from an initialsingularity. Only the past can influence us, because the futuredoes not yet exist, so it cannot causally affect us.

It then cascades down in physical systems, allowing entropy togrow because of the past condition AT2, and the up in biologicalsystems, allowing complexity to emerge because adaptive selec-tion takes place. But these are not the basic sources of the arrowsat each level of the hierarchy: They are effects of thefundamental cause.

Key to this is the time-asymmetry of the quantum measure-ment process, which I suggest emerges in a contextual way.

Firstly, a detection process depends on setting the detector intoa ground state before detection takes place (analogously to theway computer memories have to be notionally cleared before acalculation can begin). This is an asymmetric adaptive selectionprocess, because what is needed is kept and what is not needed isdiscarded, whereby any possible initial state of the detector isreduced to a starting state, thereby decreasing entropy. It will beimplemented as part of the detector design.

Secondly, one might suggest that the asymmetry of the collapsemay derive from the fact that the future does not yet exist in a EBU(Section 7), and this is the time-asymmetric context in which localany physical apparatus or other context leads to a constrained setof outcomes by their specific construction. There does not seem tobe any other plausible way to relate the global cosmological arrowof time to the local arrow of time involved in collapse of the wavefunction. How this happens needs to be elucidated, as part of theinvestigation of how state vector collapse takes place as a contextuallydependent process in specific physical contexts (Ellis, 2011).

Acknowledgments

I thank Max Tegmark and Anthony Aguirre for organizing avery useful meeting of the FQXI Institute on The Nature of Time,the participants at that meeting for many interesting presenta-tions, and Paul Davies for very helpful discussions. I thank theNational Research Foundation (South Africa) and the University ofCape Town for support, and the referee for a careful reading of thepaper that has led to significant improvement.

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the arrow of time and the nature of spacetime

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