Abstract
On the orthodox interpretation of bounce cosmologies, a preceding universe was compressed to a small size before “bouncing” to form the present expanding universe. William Lane Craig and James Sinclair have argued that the orthodox interpretation is incorrect if the entropy reaches a minimum at the bounce. In their view, the interface between universes represents the birth of two expanding universes, i.e., a “double bang” instead of a “big bounce”. Here, I reply to Craig and Sinclair in defense of the orthodox interpretation. Contrary to their interpretation, features of one universe explain features of the other universe and so must precede the other universe in time. Moreover, contrary to a crucial part of Craig and Sinclair’s interpretation, there are bounce cosmologies in which the thermodynamic arrow of time is continuous through the bounce even though the entropy is “reset” at the bounce.
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Notes
Though the Kal\(\overline{{a}}\)m argument, as stated, is theologically neutral, proponents of the argument can either add supplementary arguments for the conclusion that God was the cause of the universe or use the Kal\(\overline{{a}}\)m argument as one part of a cumulative case for God’s existence (Draper 2010).
In technical jargon, I am imagining a specific foliation of a Friedmann–Lemaître–Robertson–Walker (FLRW) space-time into space-like surfaces. Undoubtedly, calling this a “time-line” is an oversimplification, but I ask for the reader’s forgiveness for the sake of accessibility.
There are independent reasons to doubt that the BGV theorem tells us something significant about the origins of the totality of physical reality that I do not discuss here. Despite how the BGV theorem has sometimes been reported in the philosophy of religion literature, the BGV theorem is a result concerning the incompleteness of a congruence of time-like or null geodesics through a particular family of space-times as opposed to a more general result about the incompleteness of all of the time-like or null geodesics in a given space-time. Suppose that the average expansion rate along the time-like/null geodesics in the portion of space-time within our cosmological horizon is positive. If so, the BGV theorem tells us that those geodesics cannot be extended infinitely far into the past and remain within a classical space-time. Nonetheless, there could be time-like/null geodesics in regions beyond our cosmological horizon along which the average expansion rate is not positive. In that case, at least some time-like/null geodesics beyond our cosmological horizon could be extended infinitely far into the past. (That is, at least some time-like/null geodesics beyond our cosmological horizon could be complete, even if no geodesic within our cosmological horizon is complete, and the proper time measured along those geodesics could be infinite.) Guth has noted that eternally inflating models can lack a “unique beginning” and remain consistent with the theorem. Two time-like geodesics, along which the average expansion rate is positive and so cannot be extended infinitely far to the past, do not need to terminate at the same point or a common space-like surface. The theorem provides no upper bound to the lengths of all of the time-like/null geodesics within the space-times to which the theorem applies (Guth 2007, 6623). Andre Linde points out that, “If this upper bound does not exist, then eternal inflation is eternal not only in the future but also in the past.” As Linde continues, “at present we do not have any reason to believe that there was a single beginning of the evolution of the whole universe at some moment \(t = 0\), which was traditionally associated with the Big Bang” (Linde 2008, 17). Moreover, even if all of the time-like/null geodesics within a given space-time were incomplete, the conclusion that there is an absolute beginning for all time-like/null geodesics does not follow from the statement that every time-like/null geodesic has a beginning. General Relativistic space-times can be sufficiently heterogenous as to preclude the possibility of defining an absolute beginning. Contrary to the Kal\(\overline{{a}}\)m argument, the totality of physical reality would never have begun to exist.
Craig and Sinclair briefly discuss this matter, (wrongly) interpreting it as an objection to the theorem (see footnote 41 in Craig and Sinclair 2009, 142) and call the “objection” “misconstrued”. Craig and Sinclair go on to assert that if the universe is eternal then if “we look backward along the geodesic, it must extend to the infinite past if the universe is to be past eternal”. But this is false, as I’ve discussed; the point is that a space-time manifold can be geodesically incomplete—in the sense proved by the BGV theorem—without having an absolute beginning. Craig and Sinclair do admit that the BGV theorem is silent on what kind of singularity (or singularities) the metaverse contains.
I’ve assumed that heat, friction, sound production, and other dissipative processes are not part of Newtonian mechanics.
The sense in which quantum field theory is time reversal invariant is a subtle matter. Whenever we are provided with a microphysical description of the universe, in which we list sentences describing temporally sequential physical states, i.e., \(S \equiv \{S_1, S_2, \ldots , S_N\}\), an operation can be constructed that is said to produce the time reversal of S, i.e., \(S^* \equiv \{S_1^*, S_2^*, \ldots , S_N^*\}\). To say that the laws of physics are time reversal invariant is to say that both S and \(S^*\) are nomologically permissible sequences. Merely replacing t with \(-t\) in the equations of motion in fundamental physics does not suffice for time reversal. Instead, one must replace every charge with the opposite charge, replace every system with its mirror image, and replace every instance of t with \(-t\). That is, the fundamental laws respect the CPT symmmetry and not the T symmetry (Kobayashi and Maskawa 1973; Christenson et al. 1964). David Albert has argued that the fundamental laws have been known not to be literally reversible since the nineteenth century (Albert 2000, 21); for a reply, see Earman (2002). In any case, the dynamical asymmetries that are known to appear in the fundamental laws do not explain the macroscopic asymmetries that appear in thermodynamics or in the special sciences. In those cases, the best explanation for the temporal asymmetry is offered by the reduction of time asymmetric phenomena to time symmetric phenomena in statistical mechanics.
This needs to be qualified. For the purposes of this paper, we can understand entropy as the hypervolume of a phase space region. The entropy is defined as the sum (or the integral) of \(p_i \log (p_i)\), over the index i, where \(p_i\) is the probability of occupying the ith microstate. The entropy will only be the hypervolume of the phase space region of interest on the assumption that the appropriate probability distribution is uniform over the phase space regions of interest, e.g., the Liouville measure.
Though much of the argumentation that Craig and Sinclair offer in their (2009, 2012) concerns the Aguirre–Gratton model (2002, 2003), Craig and Sinclair draw conclusions which Craig and Sinclair take to apply to any cosmological model in which there is an interface at which the entropic arrow of time reverses direction, e.g., Craig and Sinclair (2009, 158).
Sean Carroll has offered a related argument. As Carroll points out, causal explanations typically depend upon the objects that stand in the explanation satisfying two conditions. First, that the objects obey the laws of physics and, second, a low entropy boundary condition in the past. The totality of physical reality is not an object within physical reality and there is no known collection of physical laws that could apply to physical reality, as a whole, as opposed to applying to all objects within physical reality. Moreover, there is no low entropy boundary condition beyond the totality of the physical world. Carroll concludes that we have no “right to demand some kind of external cause” for physical reality as a whole (Carroll and Craig 2016, 67–68); also see Carroll (2005, 2012).
Craig and Sinclair have objected that the ekpyrotic model is geodesically incomplete and, therefore, not past eternal (Craig and Sinclair 2009, 167–169). However, Ijjas and Steinhardt Ijjas and Steinhardt (2017, 2019) have recently proposed a new version of the ekpyrotic model. Steinhardt confirmed via correspondence that the new model can be made geodesically complete (per. corr. June 24, 2019).
In an argument that Steinhardt and Turok attribute to Richard Tolman the universe could not cycle through an eternity of contractions and expansions because entropy would build up in each cycle (Steinhardt and Turok 2007, 180–182). As Kragh (2009, 606) has pointed out, “Tolman did not actually conclude that there had been only a finite number of earlier cycles” and did not think thermodynamic considerations made a good case for the universe having begun at some finite time in the past. In fact, Tolman (1934, 486) argued the universe might well extend infinitely far into the past and infinitely far into the future. Nonetheless, as Steinhardt and Turok point out, the objection does not depend upon an increase in the total entropy; instead, the objection depends upon an increase in the entropy density in each cycle. For this reason, Tolman’s (2007, 192–193) argument is inapplicable to models in which the entropy becomes dilute or becomes hidden behind a horizon.
Two caveats are in order. First, the reader should not take the language of “beginning” too seriously. Smolin’s model is consistent with a metaverse with an indefinitely long history and does not require that the metaverse ever began to exist. Second, if we begin the model with \(n=1\) universes and the cosmological constant selected for the initial universe is too large, the one universe could accelerate apart without producing any black holes. Therefore, in order for Smolin’s model to work, either the iniital universe must be sufficiently improbable so to produce some population of black holes or else we should consider a situation in which we begin with multiple universes. Presumably, the most sensible possibility would be a metaverse that is eternal into the past so that there has always been some network of universes connected by black holes.
Poplawski’s model has an advantage over some other bounce cosmologies, because Poplawski’s model avoids one of the criticisms Craig and Sinclair leverage against bounce cosmologies. As Craig and Sinclair (2012, 111–112) argue, models in which a previous universe collapses to some minimum size before expanding into our universe needs to be carefully fine-tuned from eternity past in order to successful collapse to the minimum size. But, like Smolin’s model, Poplawski’s model involves the creation of offspring universes from black holes. For this reason, neither Smolin’s nor Poplawski’s models require such fine-tuning. Between the two, Poplawski’s model is more convincing because, unlike Smolin, Poplawski provides a mathematical model and a physical mechanism for the dynamical evolution of black holes within one universe into subsequent offspring universes.
For a reply to a related argument originally offered by Tolman, see footnote 10. As in Steinhardt and Turok’s model, in the offspring universe, the total entropy of the parent universe has become hidden behind a horizon and is not accessible to the offspring universe.
According to Craig and Sinclair (2012, 111), Penrose’s model is consistent with the BGV theorem because the average expansion rate of a cyclic universe is zero. This is incorrect. In the CCC, the expansion or contraction of the universe is not a well-defined notion for every period of the universe’s evolution. But, during those periods in which expansion/contraction are well-defined, the universe only expands and never contracts. The CCC is not singular because the CCC utilizes a space-time to which the BGV theorem cannot be applied.
This can be put more carefully. As I noted earlier in the paper, curvature singularities are not points that General Relativity includes in the space-time manifold. Therefore, one should not say that, in General Relativistic models, the universe began with a low entropy singularity. However, one can accurately say that, when the universe is reversed in time in General Relativistic models, space-time tends towards a low entropy singularity. If one chooses any arbitrarily small value \(\varepsilon > 0\) then there will exist some time t such that the scale factor \(a(t) < \varepsilon\). The low entropy singularity corresponds to the limit in which \(\varepsilon \rightarrow 0\).
I’m speaking loosely. In the absence of length and time scales, a single point and three dimensional space do differ, for example, in topological structure. A single point has the topology of \(\mathbb {R}^0\) while a three dimensional space has the topology of \(\mathbb {R}^3\). But note the qualifications that I made in footnote 16. For any \(a(t) > 0\), three dimensional slices of space-time have the topology of \(\mathbb {R}^3\). Penrose should be interpreted as arguing that when length loses its meaning, space-time loses length and time scales. For that reason, we can identify an arbitrarily “compressed” three dimensional space with an arbitrarily “expanded” three dimensional space.
For example, one of the processes Penrose (2012, 186–188) discusses for eliminating masses from the universe involves massive particles being swallowed by black holes and then the black holes undergoing a non-unitary decay process. The non-unitary decay process reduces entropy. Craig and Sinclair reply to this feature of CCC with the complaint that the late time evolution of the universe will be dominated by the entropy associated with the universe’s horizon. This entropy is far larger than the entropy reduced through the decay of black holes (Craig and Sinclair 2012, 120–121). While Penrose preempts this objection, Craig and Sinclair (wrongly) complain that Penrose only offers an instrumentalist interpretation of the entropy associated with the universe’s horizon. Importantly, Penrose provides a reply both to the realist and instrumentalist interpretations of the universe’s horizon entropy; on Penrose’s view, even if the entropy of the universe’s horizon is real, that entropy can be ignored because it plays no role in the universe’s dynamics (Penrose 2012, 202). Moreover, Craig and Sinclair are inconsistent in their interpretation of the entropy associated with the universe’s horizon. As Craig and Sinclair write in their (2009, 155), the universe’s horizon differs from the horizon of a black hole because the former should not (in their view) be understood as objectively real. But if the universe’s horizon is not objectively real, in what sense can the entropy associated with that horizon be understood as objectively real? In any case, the resolution of this debate is irrelevant for my purposes here because I am only concerned with how we ought to interpret CCC.
Penrose has been careful to argue that while the total entropy of the universe is reduced prior to the bounce through non-unitary processes (see footnote 19), a thermodynamic arrow of time is nonetheless preserved and never reverses direction (Penrose 2012, 175–190). For this reason, Penrose argues that the entropy reduction in CCC is not a violation of the second law of thermodynamics and CCC might not contradict ALP. Be this as it may, if we understand the entropic arrow solely in terms of the entropy gradient, then, according to CCC, there is an epoch in which the entropy gradient points contrary to the causal arrow of time.
Thanks to an anonymous referee for pointing this out.
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Linford, D. Big Bounce or Double Bang? A Reply to Craig and Sinclair on the Interpretation of Bounce Cosmologies. Erkenn 87, 1849–1871 (2022). https://doi.org/10.1007/s10670-020-00278-5
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DOI: https://doi.org/10.1007/s10670-020-00278-5