PX000 Foundations of Physics • Reading Task III • David Carvalho

The Cosmological Arrow of Time from BoundaryCondition Considerations

David Carvalho University of Warwick

David.Carvalho@warwick.ac.uk

November 5, 2012

Abstract

A paper is examined to produce a consistent account to explain the role of time in cosmological-like descriptions of the Universe. Considering the Space-Time as idealised by General Relativityallows a certain topology to be defined on the manifold Space-Time to produce equivalence classesof simultaneity - hypersurfaces and the idea of cosmic time, which can be assigned to a certaindifferentiable manifold if some conditions are met. A quick calculation is carried out to explainhow probable the existence of time-asymmetric universes is.

I. Aim / Main Considerations

The discussion of time is often forcefullylinked to the phenomenological nature of

entropy. The 2nd Law of Thermodynamics as-serts that the gradient of such function is alwayspositive. In this first paper, from Castagninoand Lombardi, the emphasis is put on the ge-ometry of space-time. The central feature isthat entropy does not offer a convincing plat-form to base hypothesis on, as it may well be aconsequence of the more general properties ofthe Universe. Moreover, entropy is the resultof an ensemble of compatible system configura-tions and not an intrinsic property of a point inspace-time. The authors argue that such a cos-mological arrow of time may be defined, uponverification of certain conditions.

II. Space-Time

General Relativity is used intensively through-out the paper to define the setting of how to de-scribe the Universe in cosmological terms. Space-Time is then a Riemannian manifold, the pair(M, g), with M a 4-dimensional differentiablemanifold and g a Lorentzian Metric such that∇g = 0. One of the starting equations whichcould model the Universe is the Einstein’s FieldEquation Rµν− 1

2gµνR−Λgµν = 8πGTµν , whereRµν is the Ricci Tensor, R the Ricci Scalar,g = gµν the Metric Tensor ,Λ the cosmologi-cal constant and Tµν the Energy-Momentum

Tensor. This equation is fundamental becauseit offers a relationship between the geometryof Space-Time and the distribution of Energy-Matter. The (Global) Arrow of Time is oftenthought to be implicit in the concept of time andis indeed verified in an expanding model. How-ever, more complex cosmological models, such asexpanding-contracting ones, such arrow ”points”in different directions at different sections of theevolution. The authors also stress the fact that,since time arises as one of the coordinates of a4-dimensional manifold, it may be that such anarrow is global or local (for which definitions aregiven) and this forces time orientability to be anecessary condition to make sense of global direc-tion of time. This property is observed indirectlyas the Friedmann-Lemaitre-Robertson-Walkermodel is astronomically observed.

III. Topological Considerationsabout Time

Many useful definitions can lead to useful condi-tions about time. If one defines the chronologicalfuture (past) I+(x) ( I−(x)) of a point x ∈ Mto be the set of all y ∈ M such that y can bereached from x by a future (past) directed time-like curve, and the causal future J+(x) (J−(x))to be the set of y ∈M that can be reached fromx by a future directed non-spacelike curve (time-like or lightlike), certain facts emerge. First,∀x ∈ M , I+ ∩ I− = ∅ - no timelike curves

1

PX000 Foundations of Physics • Reading Task III • David Carvalho

are possible. The causal condition is strongerthan the chronological condition, as it rules outnon-spacelike curves, with J+ ∩ J− = ∅. Manymore conditions can be imposed to avoid certainundesired curves. However, the stable causal-ity condition turns out to hold if and only if(M, g) possesses a global time function, a func-tion t|M → R such that ∇t is everywhere time-like. This allows splitting of Space-Time, as onecan construct a partition on the set of all eventsinto (equivalence) classes, such that every (dis-joint) classe is a spacelike hypersurface and thesecan be ordered in time. This equivalence de-fines simultaneity by assigning all simultaneousevents to a spacelike hypersurface. This processis known as Space-Time foliation. The authorsstress that these condition still cannot guaranteea physically meaningful well-defined notion ofsimultaneity. Amongst many choices of coordi-nate systems, a metric argument can be givenwhich defines a canonical choice such that a map-ping (τ, x1, x2, x3) 7→ x ∈M exists when a curve(x1, x2, x3) is given, τ increases along it and allits points have the same coordinates {xi}. Thiscondition is not still satisfactory, as the elapsedtime ∆τ =

∫ds =

∫ √g00(τ, x1, x2, x3)dτ and

so this time depends on the worldlines cho-sen. Cosmic Time can then be chosen suchthat g00 = g00(τ). Taking the proper time in-terval between two hypersurfaces τ = τ1 andτ = τ2 is invariant under the space of allworldline curves. Defining dt2 = g00(τ)dτ2

yields a metric ds2 = dt2 + hijdxidxj , with

hij = hij(t, x1, x2, x3). The existence of a cos-

mic time is a necessary condition to define atemporal sequence of instanteneous states of theuniverse. The parameter ”time” frequently usedto parametrise the evolution of the universe isonly valid if such time can be found for (M, g).Most modern accounts of cosmology do guar-antee such cosmic time to exist. It is normallydenoted by t in, for example, the Robertson-Walker metric.

IV. The Probability of aTime-Asymmetric Universe

A simple argument can be given to explainwhy a temporally asymmetric universe is prob-able. Consider a FLRW cosmological model,coupled with a scalar field φ(t). The Lagrangianis given by L = L(a, a, ψ, ψ) where a = a(t)is the scalar factor. The action S is thenS =

∫L(a, a, ψ, ψ)dtdxi. The functional depen-

dence between the dynamical variables are givenby a = f1(a, a, ψ, ψ) and ψ = f2(a, a, ψ, ψ).Consider now , from Einstein’s Field Equation,when µ = ν = 0. This allows variables to beexpressed of the other three. One can then thinkof a state of the universe as its associated pointin 3-dimensional phase space (ψ, a, ψ). As thetransformation t 7→ t+ T is invariant, fix t0 = 0and denote it henceforth by initial condition.Applying a Big Bang / Big Crunch model al-lows one to choose t0 to be time for which themaximum expansion occurs. Then a(0) = 0.As the constructive model is symmetric in time,so is a(t) and then a(t) = a(−t). ψ must alsobe symmetric - two situations emerge, with acharacteristic probability of temporal symmetry,starting from its initial condition:

• Even symmetry, and then ψ(t) = ψ(−t)and ψ(0) = 0.(ψ(0), 0, 0) is the only initialcondition leading to symmetric universes.

• Odd symmetry, and then ψ(t) = −ψ(−t)and ψ(0) = 0. (0, 0, ψ(0)) is the only ini-tial condition leading to symmetric uni-verses.

The space of all possible initial conditions is3-dimensional and the space of initial condi-tions that lead to a symmetric universe has is1-dimensional. Considering all solutions for eachinitial condition leading to a symmetric universe,the solutions space is 2-dimensional. One canthen infer that temporally symmetric universeshave zero probability regardless of possible uni-verses compatible with a Big Bang Big CrunchFLRW model.

2

PX000 Foundations of Physics • Reading Task III • David Carvalho

IV.References

[1] M. Castagnino, L. Lara, O. Lombardi The Arrow of Time in Cosmology http://philsci-archive.pitt.edu/800/1/TheArrowofT imeinCosmology.pdf

[2] M. Castagnino, L. Lara, O. Lombardi The cosmological origin of time-asymmetry arXiv:quant-ph/0211162

3