5. l The arrow of time Précis. Many purported arrows do not establish asymmetries of time itself, but dynamical asymmetries can, with the help of the representation view. The ‘arrow of time’ is shorthand for time asymmetry. The idea appears to have been born in Trinity College, Cambridge in the early twentieth century. Trinity philosopher John McTaggart (1908, p.474) remarks on it in the conclusion to his famous discussion of the reality of time, asking, “what is that quality, and is it a greater amount of it which determines things to appear as later, and a lesser amount which determines them to appear as earlier, or is the reverse true?” The phrase itself was coined by Trinity physicist Arthur S. Eddington: “I shall use the phrase ‘time’s arrow’ to express this one-way property of time which has no analogue in space. It is a singularly interesting property from a philosophical standpoint.” (Eddington 1928, p.69) What is this ‘one-way property’ of time, and what evidence is there that time has it? Arrow Example Thermodynamic Dissipating gas Radiation Expanding electromagnetic waves Quantum measurement Dynamical state reduction Cosmological Cosmic expansion T-violation Kaon quark flavour mixing Spacetime Intrinsic temporal orientation Causal/Dependency Causes leading to effects Figure 5.1: Some arrows of time. 115 Figure 5.1 summarises a few existing responses.1 Some of these arrows may be related. Others are quite different from one another. And, there are likely many more! Caution is needed: the possibility of illusion looms. Many things presents the powerful appearance of being asymmetric in time to our naïve human senses, when in fact they are not. For example, a book will slide to a stop on a tabletop, but never the reverse: it does not spontaneously begin sliding! But when that experience is carefully described as dynamical systems, we find that it invariably omits degrees of freedom in a way that hides the true temporal symmetry. This chapter will present a framework for determining whether there is a phys- ical arrow of time. I begin by identifying some of the ways the arrow of time can fall short of establishing it. It can happen by resorting to heuristics, relying on boundary conditions, or by describing a physical system with missing information. Each can lead to a powerful explanation of our asymmetric experiences, but does not establish any asymmetry of time itself. My thesis is that the promising road to an arrow draws on the asymmetries of a dynamical theory. This possibility is afforded by a link between the symmetries of spacetime and those of a dynamical theory, which I call the representation view. In Section 5.1 will set out our aim, to determine what kind of evidence is require to establish a physical arrow of time. Section 5.2 will then review some prominent attempts that are widely agreed to fall short of a physical arrow of time, including radiation, thermodynamic, cosmological, quantum, and causal arrows. Section 5.3 presents what I take to be the royal road to a physical arrow, through the representation of time translations in a dynamical theory. Finally, I respond to a prominent critique of this approach to the arrow due to Huw Price in Section 5.4. 5.1 The arrow of time itself According to Price (2011, p.292), understanding the arrow of time requires answering a number of questions: “Is time anisotropic at all, and how could we tell if it is? What 1Eddington himself adopted a strategy similar to that of Boltzmann; for the latter, see Section 5.2.2. 116 could constitute good grounds for taking it to be so, and do we have such grounds?” I will attempt to these questions them in this chapter, would first like to highlight a general principle that they inspire: Spacetime-Evidential Link: An account of the asymmetries of spacetime must identify both a sense in which spacetime itself is asymmetric, as well as some plausible empirical evidence that supports that asymmetry. The difficulty we face in the special case of temporal symmetry is to link it to experience, so that we stand a chance of giving reasonable empirical evidence for or against it. In his seminal book Time’s Arrow and Archimedes’ point, Price (1996, p.16) characterises the difficulty in a picturesque way: think of time as a table that is asymmetric in shape. Whether or not ‘time itself’ is asymmetric is then a conceptually different question from whether the development of a state over time — analogous to the food and drinks forming the ‘content’ of the table — occurs in an asymmetric way, as in Figure 5.2. Figure 5.2: Price’s Table: the asymmetry of the table, like the asymmetry of time, is distinct from the asymmetric placement of its contents. The link that I propose involves a shift of focus in the description of time, from a temporal ‘axis’ or coordinate variable to the more structural, functionalist perspective of time translations. Price himself often adopts e former, writing for example that, “the contents of the block universe appear to be arranged asymmetrically with respect to the temporal axis” (Price 1996, p.17). But, as I argued in Chapter2, time is much more than that: it has rich relational, topological, and other structural properties described 117 by a structure called the time translations, with elements of the form t time shift by two hours, rather than time coordinates like t two o’clock. I will correspondingly interpret the statement that ‘time has an arrow’ to mean that the structure of time encoded by the translations T has an asymmetry, in that t 7! −t is not an automorphism of time translations. And, to replace Price’s notion of the ‘contents’ of time, I will refer to a representation of time translations ' : T ! Aut¹Mº amongst the automorphisms of a state space M, which determines a notion of time evolution among physical states. As I argued in Chapter4, this is part of what it means to be a ‘dynamical’ theory, called the representation view. As a result, the very meaning of a dynamical theory encodes the structure of time translations and their symmetries. Price’s table is not quite the right analogy for the representation view. Given a representation of time translations, the structure of time is projected down onto state space through like the shadow of a tabletop on the floor (Figure 5.3). This suggests that, by studying dynamical asymmetries in a representation of time translations, it is possible to infer an asymmetry or arrow of time itself. That is the approach that I will argue for in this chapter. I will discuss its formulation in detail in Section 5.3. But let me begin with a review of some competing ‘arrows’ of time. Figure 5.3: The structure of time is projected onto a state space by a representation, the way the structure of a table is projected onto the floor by its shadow. 118 5.2 Arrows that misfire We begin with a brief discussion of five well-known approaches to time asymmetry: radiation, thermodynamics, cosmology, quantum measurement, and causation. I will argue that, at least in their current formulations, these approaches do not yet past muster as accounts of the arrow of time, for one of the following two reasons: 1. Heuristics: making essential use of an informal extra-theoretical judgement, which is not explained in any well-supported physics; or 2. Boundary conditions: identifying an asymmetry with a particular temporal devel- opment, by postulating special initial or boundary conditions that force it. My discussion of these particular arrows will in some respects refine the discussion of Price (1996), with whom I agree about these arrows. However, I will diverge from Price’s account in Section 5.3. 5.2.1 The radiation arrow An oscillating charge is associated with an outgoing shell of electromagnetic radiation, which expands with phase velocity equal to the speed of light.2 The phenomenon is analogous to the circular ripples of a water wave when a stone is dropped in a pond. However, like the ripples in a pond, we never seem to observe this radiation in the reverse form of an inward-collapsing shell, as illustrated in Figure 5.4. Is this an arrow of time? The question was discussed in a correspondence between Planck (1897) and Boltzmann (1897), and then more famously in a debate between Ritz (1908) and Ein- stein (1909) in the journal Physikalische Zeitschrift. It has recently generated renewed philosophical interest.3. The phenomenon is particularly puzzling because the laws of 2Thomson (1907, p.217) gave an early derivation of this result, which he identified as analogous to the mechanism producing “Röntgen radiation” or X-rays. 3Popper (1958) seems to have arrived at a similar idea independently. Davies (1977, Chapter 5), Zeh (2007, Chapter 2), and Price (1996, Chapter 3) give classic discussions. Compare also the Ritz-like position of Frisch (2000, 2005, 2006) to the responses of North (2003), Price (2006) and Earman (2011). 119 - Figure 5.4: An expanding radiation shell (left) is easily produced from an oscillating charge, while a collapsing one (right) is not. electromagnetism are time reversal invariant: both the outgoing radiation wave and its time reverse are possible! Ritz argued that the observed asymmetry is due to an additional time asymmetric law of nature, while Einstein maintained that it is just a matter of special boundary conditions: roughly speaking, one would have to begin with a ring of charges oscillating in perfect unison in order to produce a collapsing circular wave, just like a water wave.