Reassessing the Prospects for a Growing Block Model of the Universe

Abstract

Although C. D. Broad’s notion of Becoming has received a fair amount of attention in the philosophy‐of‐time literature, there are no serious attempts to show how to replace the standard ‘block’ spacetime models by models that are more congenial to Broad’s idea that the sum total of existence is continuously increased by Becoming or the coming into existence of events. In the Newtonian setting Broad‐type models can be constructed in a cheating fashion by starting with a Newtonian block model, carving chips off the block, and assembling the chips in an appropriately structured way. However, attempts to construct Broad‐type models in a non‐cheating fashion reveal a number of problematic aspects of Becoming that have not received adequate attention in the literature. The paper then turns to an assessment of the problem and prospects of adapting Becoming models to relativistic spacetimes. The results of the assessment differ in both minor and major ways from the ones in the extant literature. Finally, the paper describes how the causal set approach to quantum gravity promises to provide a mechanism for realizing Becoming, though the form of Becoming that emerges may not conform to any of the versions discussed in the philosophical literature.

Acknowledgements

I am grateful to Jeremy Butterfield, Craig Callender, Mauro Dorato, and Steve Savitt for helpful suggestions, but there is no implication that they agree with the opinions expressed herein. Thanks are also due to two anonymous referees for comments that led to substantial improvements.

Notes

[1] For a readable overview of ‘static’ vs. ‘dynamical’ conceptions of time and references to the literature, see Dainton (2001). The terminology is rather unfortunate. Those who defend the block model as an adequate representation of physical change, and as the basis for an explanation of our perceptions of change, will feel unfairly stigmatized by the ‘static’ label. However, some of the alternatives to the block model hardly seem to merit the label ‘dynamical’ (see Section 4). But these labels are so well entrenched it is pointless to try to displace them.

[2] As some commentators have noted, McTaggart would have done better to speak of A‐properties rather than the A‐series (what is the series and what is the ordering relation?). But the latter term has become standard in the literature, and I will continue to use it here.

[3] See the contributions to Oaklander and Smith (1994), and also Savitt (2001).

[4] The most sustained defence of the B‐series account of change is to be found in Mellor (1998). For an attempt to account for the phenomenology of temporality within a block universe setting, see Butterfield (1984).

[5] For an account Broad’s later views on Becoming, see Savitt (2002).

[6] It would be interesting to investigate the relation between the directionality of Becoming and the so‐called arrows of time discussed under the heading of the problem of the direction of time, but that is a project for another occasion. There is an obvious linkage to the so‐called electromagnetic arrow of time, since, presumably, the use of an advanced representation of the electromagnetic field is inappropriate if (as Broad maintains) future events are nonentities. For an overview of the issues involved with the electromagnetic arrow, see Earman (forthcoming).

[7] McTaggart died early in 1925.

[8] Indeed, I must confess: Ich bin ein Blockhead. But my purpose here is to give the Broadheads a run for their money.

[9] I will say that two spacetime models ⟨M, G ₁, G ₂, …, P ₁, P ₂, …⟩ and ⟨M′, G′₁,G′_2′,…, P′₁,P′₂,…⟩ are isomorphic iff there is a diffeomorphism d : M → M′ that is onto and d*G_i = G′ _i and d*P_j = P′_j for all i and j, where d* denotes the drag along by d. A diffeomorphism is a one–one map that preserves the differentiable structure. For sake of simplicity, assume that the manifolds are all C ^∞ and, thus, that d is C ^∞.

[10] The reader interested in the details can consult Earman (1989, ch. 2).

[11] Similarly, the relation ≺ on N defined by the condition that n ≺ n′ iff n can be isomorphically embedded as a proper submodel of n′. That ⟨M, G ₁, G ₂, …, P ₁, P ₂, …⟩ is isomorphically embeddible into ⟨M′, G′₁,G′_2′,…, P′₁,P′_2′,…⟩ means that there is a diffeomorphism d if M into M ′ such that d*G_i = G′_i and d*P_j = P′_j for all i and j.

[12] It would be suspicious to require that all of the elements of N are isomorphic to a chip off one and the same Newtonian block. This suspicion was avoided above by requiring only that each element of N is isomorphic to a chip off some Newtonian block or other. But the issue comes back later on in another form, namely, does the growing block model fix a unique block model? See below.

[13] I suspect that it can’t be done without introducing a fifth dimension and thereby reawakening the embarrassments attendant to it.

[14] I find Broad’s ordinary language analysis of this issue unpersuasive; see Broad (1959, 767–777).

[15] A Newtonian block model ⟨M, G ₁, G ₂, …, P ₁, P ₂, …⟩ is stationary if there is a one‐parameter family of diffeomorphisms d _λ, −∞ < λ < +∞, such that d _λ maps each plane of absolute simultaneity T = Δ to T = Δ + λ and d*_λ G_i = G_i and d*_λ P_j = P_j .

[16] Relationists will charge that the alleged indeterminism in the above example is only an illusion fostered by a substantivalist conception of spacetime as a non‐material medium in which physical objects are immersed. Formally the relationist claim would be that the block models N and N* of Figure 1 are merely different representations of the same physical situation. The connection between spacetime substantivalism and Becoming will come up again in the following section.

[17] I.e. a means of identifying time slices across elements of N. Then ≾ could be defined as follows: n ≾ n′ iff n can be isomorphically embedded as a submodel of n′ by a map that matches identical time slices. This relation is antisymmetric.

[18] Note that the ideal completion of a growing block models is not constructible by a Zorn’s lemma argument.

[19] It is wholly implausible that by ‘there is’ Broad means ‘there is at present’, since this would turn his statement into the triviality that there is at present no event to which a present event is temporally precedent. Broad believed that one happy consequence of his doctrine of the literal unreality of the future was that it explained the ‘impossibility of absolutely certain knowledge about the future’ (Broad 1923, 73). His claim was that we can have certain knowledge of an event only if we are directly acquainted with it. But ‘we can be directly acquainted only with something, not with a mere non‐entity. On our view we cannot stand in the relation of direct acquaintance to future events, for the same reason which prevents us from robbing a Highlander of his breeks [viz. there are none]’ (79). One might object that the same result can be obtained by accepting the reality of the future while requiring that direct acquaintance requires a causal connection and that there is no backward causation.

[20] That a relativistic spacetime M, g_ab is time orientable means that there exists an everywhere‐defined continuous timelike vector field. Such a field induces a continuous division of the lobes of the null cones at every point of M into two classes. The choice of one of these classes as the ‘future’ lobes constitutes a time orientation. Block models face the problem of providing a basis for this choice, which is an aspect of ‘the problem of the direction of time’.

[21] These and other causality requirements for relativistic spacetimes are discussed in Hawking and Ellis (1973) and Wald (1994).

[22] For example, the Einstein static cosmological model is unstable; see Hawking and Ellis (1972, sec. 5.3).

[23] The most famous member of this class is the Wheeler–Feynman time symmetric, action‐at‐a‐distance version of classical relativistic electrodynamics; see Wheeler and Feynman (1945, 1949). There are good reasons for discounting such theories on grounds of empirical adequacy. But they do serve to make the point that there is no a priori guarantee that future‐truncated models are compatible with relativistic theories.

[24] More precisely, M, g_ab satisfies strong causality iff for any point p ∈ M and any open neighborhood N of p, there exists a neighborhood N′ of p contained in N such that any causal curve that leaves N′ never reenters it. If it is required only that for any point p ∈ M there exist an open neighborhood N of p such that any causal curve that leaves N never reenters it, then one gets the weaker requirement of future distinguishing (see below).

[25] See Arntzenius and Maudlin (2005) and Earman et al. (forthcoming).

[26] The underdetermination issue will rear its head in the next section.

[27] The necessary and sufficient condition for V^a to be hypersurface orthogonal is that it be irrotational, i.e. V_[a;b] = 0.

[28] See note 27.

[29] A relativistic spacetime M, g_ab is globally hyperbolic iff it is strongly causal and, for all p, q ∈ M, J⁻ (p) ∩ J ⁺(q), compact. A spacetime is globally hyperbolic in this sense iff it admits Cauchy surfaces.

[30] Here are some of the points on which I disagree with Tooley. 1) Analyzing our commonsensical judgments (or even worse, judgments tutored by analytical metaphysicians) is not a fruitful way to get at the deep structure of the world. 2) Causation is a folk notion, and trying to use it in interpreting theories of physics produces more mischief than enlightenment. 3) There aren’t two versions of STR, one which posits that the one‐way speed of light is the same in all directions and in all frames and one which doesn’t. In fact there isn’t any substantive issue here, since the effects in question are coordinate‐dependent and since STR can be stated in a completely coordinate‐independent fashion. 4) Understanding the Einstein‐Rosen‐Podolsky‐Bell correlations does not require an Absolute Frame or Absolute Simultaneity.

[31] For example, Wald (1994, 360) writes, ‘[I]f ϕ : M → M is a diffeomorphism, then (M, g_ab ) and (M; ϕ*g_ab ) represent the same physical spacetime’.

[32] For an overview of the issues involved, see Stachel (1993).

[33] ‘A relative lapse of time … if any meaning at all can be given to this phrase, would certainly be something entirely different from the lapse of time in he ordinary sense, which means change in the existing. The concept of existence, however, cannot be relativized without destroying its meaning completely … A lapse of time … which is not a lapse in some definite way seems to me as absurd as a colored object which has no definite colors’ (1949, 558–5).

[34] But at least the stipulation works. For the worldline‐relativized version of Becoming discussed in the next section, the stipulation may not work.

[35] Note that the sense of immortality involved here does not entail that the observer lives for an infinite amount of proper time but only that she goes on living as long as the spacetime structure permits. An endpoint p of a worldline γ need not be a point on γ ; rather the condition for being an endpoint is that any open neighborhood of the point intersects γ.

[36] For a relativistic block spacetime M, g_ab the event horizon of a timelike curve γ can be defined as the boundary of J ⁻ (γ) in M. In the example illustrated in Figure 2, the event horizon of consists of the null line L.

[37] See Hawking and Ellis (1973, sec. 5.2) for a description of this spacetime.

[38] For an overview of the evidence for and interpretations of accelerating expansion, see Carroll (2004).

[39] This is a slight modification of Malament’s (1977) definition, which uses the chronological past instead of causal past.

[40] In addition to the Markov condition, three other conditions, called internal temporality, discrete general covariance, and Bell causality are imposed. The first two of this trio will come into play below.

[41] If C is not regarded as a subset of C ^′ (as would be the case if C _∞ is composed of isomorphism classes of finite causets), then C ↝ C′ can be defined to hold iff there is an isomorphic imbedding ι : C → C ′ such that C ′ = ι(C) ∪ {d}. This introduces an element of ‘gauge freedom’ into the description of the transition from C to C′, since two such embeddings are regarded as physically equivalent if they are related by an automorphism of the child C′. Naturally, the transition probabilities for transitions associated with equivalent embeddings are required to be the same.

[42] See the discussions of ‘discrete general covariance’ in Rideout and Sorkin (1999) and Varadarajan and Rideout (2006).

[43] Sometimes called possibilism in the literature.

[44] For attempts to resist deflation, see Sider (2001, 2006). I think that resistance is futile if the block model is accepted.