The Fundamental Arrow of Time in Classical Cosmology

نویسنده

  • D. Wohlfarth
چکیده

The aim of this paper is to show that a new understanding of fundamentality (section 2), can be applied successfully in classical cosmology and is able to improve a fundamental understanding of cosmological time asymmetries. In the introduction (section 1), I refer to various views from different literature. I begin by arguing against various arguments, provided in favour of the view that the directedness of time is not a fundamental property of nature, (see for example Price [(1996)]) in section 1.1. Next, in section 1.2, I refer to some suggestions for defining the arrow of time in cosmology via the behaviour of entropy. In addition, I argue that the various types of such approaches cannot explain the occurrence of a fundamental arrow of time in cosmology; I believe that this argument is mostly acknowledged in the literature. Moreover, in section 1.3, I broaden my scope in order to formulate approaches in the context of theories of a multiverse and hyperbolic curved spacetimes as well. After that overview of different approaches, I provide a definition of my understanding of fundamentality (section 2). To verify the effectiveness of this new understanding, I present an example to show that it can be applied in classical cosmology in sections 3.1–3.5. Finally, in section 4 I conclude by regarding the effects of adding the new understanding of fundamentality to classical cosmology. 1 Views from different literature Before discussing the reports of various literatures, it is necessary to clarify two important points: First, the used terminology regarding the terms ‘time asymmetry’, ‘directedness of time’, ‘arrow of time’ and ‘time direction’ and second, my understanding of the terms E-Mail: [email protected] ‘fundamental’ or ‘non-fundamental’ in the context of the terms ‘time asymmetry’, ‘directedness of time’, ‘arrow of time’ and ‘time direction’. In this paper, I will understand the term ‘time asymmetry’ as the most general term. An equation, a function, a physical model or a whole physical theory, will be called ‘time asymmetric’, if the change of the sign of the time coordinate(s) in the equation, function, physical model or theory provides some sort of change in the equation, the value of the function or the physical content of the model or theory. Furthermore, the term ‘directedness of time’ is used to describe the particular situation, in which the time coordinate(s) under consideration, in the context of an equation, a function, a physical model or a theory, behaves asymmetrically regarding the change of the sign of the time coordinate(s) on every time point. Thus, a time asymmetry can provide the directedness of a time coordinate in a given context of an equation, a function, a physical model or a theory, if the coordinate, in this context, behaves asymmetrically on every time point. Additionally, I will use the term ‘arrow of time’ to describe the situation that, in the context of a physical model or theory, the time coordinate(s) in the theory has the property of directedness. Thus, the directedness of time provides an arrow of time, if the theoretical or formal context in which the directedness of time is given is a whole physical model or a theory. The term ‘time direction’ is used if, in a given context of an equation, a function, a physical model or a whole theory, the directedness of time is given and additionally the sign of the time coordinate(s) is fixed for intrinsically rezones. This means that the time direction is not provided by time directedness or the arrow of time alone. The arrow of time is given by the directedness of time but the orientation of the arrow is not fixed by that property. (For a deeper motivation and consideration of the used terminology regarding the term ‘time direction’ see for example Earmann [(1974)] or Price [(2011)]). In this paper I will focus on the arrow of time in the sense from above and only on the property of the directedness of time. But if, for example, the cosmic time coordinate contains an arrow of time, it is possible that, even if this coordinate has ‘only’ the property of time directedness, the physical and inter-theoretical connections to other fields in physics, apart from cosmology, yields a time direction of other time coordinates (for example proper times). But this is not in the scope of this paper and a sufficient analysis of this possibility would require an analysis of the inter-theoretical connection between the directedness of the cosmic time coordinate and the proper time coordinates in other fields of modern physics. So, I shall come to the first sketched understanding of fundamentality. In section 2, I precisely define fundamentality through mathematical conditions and the associated motivation. Before moving ahead, I shall explain which properties, according to me, an arrow of time and a time asymmetry should possess for it to be called fundamental. First, a fundamental arrow of time should be based on the fundamental properties of the theory, which is assumed as fundamental in a given physical description of nature. Second, a fundamental arrow of time and a fundamental time asymmetry should be global, i.e. at least in our particular spacetime, the arrow should not change its (perhaps conventional chosen) direction. Thus, on the basis of these properties, I shall define fundamentality through mathematical conditions in section 2. But, before, I will argue that the prominent accounts, at least those that I am aware of, cannot satisfy the mentioned conditions for a fundamental time arrow. 1.1 The time-symmetric view In this subsection, I briefly examine the suggestions of Price ([1996]) regarding time symmetry because the arguments that favour a time-symmetric view do not appear plausible or compelling; thus, the search for a fundamental arrow of time is not doomed to fail, at least not owing to the arguments made by Price. Price starts his cosmological analysis with the observation that the early state of the universe is special in one interesting way: the universe near the Big Bang is smooth (Price [1996]). A matter distribution consisting of black holes would be much more likely than a smooth 2 The following argumentation is focused on an interpretation of Price [(1996)] in which his arguments are considered to be arguments for a time symmetric understanding of cosmology. If the arguments from Price [(1996)] are considered ‘only’ as arguments against some wellknown statistical attempts to the cosmological time arrow and not for a time symmetric understanding, my view is in perfect agreement with Price [(1996)]. 3 However, on this point, a crucial question arises that is important for understanding Price’s suggestions: It is not clear what the word ‘early’ (which Price uses many times; see, for example, Price 1996 pp. 79 and 80) means in this context. Price does not seem to refer to a proper time period with respect to the constituents of the universe when he uses the phrase ‘early state of the universe’. Instead, it seems that he refers to a) cosmic time or b) the geometrical fact that these states of the universe are ‘near’ (according to a metric such as the FRW metric) the Big Bang. To make this interpretation of Price plausible, we need a definition of the distance between a spacetime point p (or the singularity) and a threedimensional spacetime region R (a state of the three-dimensional universe). However, there are many possible ways in which we could define this distance, for example, taking the nearest spacetime point p’ of a region R and calculating the distance between p and p’ according to the metric. Of course, we can also construct many definitions of distance for this task. Such definitions would be more or less plausible, but they all define what it means to say that ‘a spacetime region is near a spacetime point or the Big Bang’. It seems fair to assume that Price refers to the most acceptable meaning of ‘early’. That is, it seems most unproblematic to define his use of the term ‘early’ as ‘near the Big Bang according to a metric and a plausible definition of distance between a spacetime point p and a spacetime region R’. distribution if classical gravity is the dominant force, as assumed in classical cosmology. Thus, according to classical thermodynamics, Price argues that this fact shows that the ‘early’ universe has very low entropy. Now, Price ([1996], p. 78) continues to argue that entropic behaviours (as well as all other properties), cannot provide a fundamental time arrow in classical cosmology. His arguments are based on the fact that all statistical considerations, which in fact yield the second law of thermodynamics, are also valid in the reverse time order. Thus, the time asymmetry of entropic behaviour is based on the initial low-entropy conditions of the Big Bang (see also Albert [2000]). Therefore, Price ([1996], pp. 81–99) argues that according to entropic behaviour (or other statistical reasoning’s), a closed universe with low-entropy boundary conditions in the ‘future’ is time symmetric and the low entropy boundary in the ‘further’ is as likely as the one in the ‘past’, which seem to be given in our actual world. Additionally (see Price [1996], pp. 95–96), Price argues that we are not concerned with whether our particular spacetime is closed or open, but with whether a closed spacetime with symmetric boundary conditions is possible given the laws of classical cosmology. ‘This point [the possibility of open spacetime geometry] is an interesting one, but it should not be overrated. For one thing, if we are interested in whether the Gold universe [a kind of a time-symmetric closed spacetime] is a coherent possibility, the issue as to whether the actual universe recollapses is rather peripheral. [...] Of course, if we could show that a recollapsing universe is impossible, given the laws of physics as we know them, the situation would be rather different’ (Price [1996], p. 95). Thus, Price seems to argue that if the existence of such a universe is possible given the laws of classical cosmology, there is no reason to assume that time is directed in a fundamental sense. Even if our particular spacetime would have boundaries that yield an entropic time asymmetry; this would not affect a fundamental time direction because the direction, in such a spacetime, would be given by (perhaps accidental) boundary conditions. On this point it seems necessary to mention that, perhaps, Price [(1996)] can be understood in a weaker sense (see also footnote 1). In my description of Price arguments above, I had assumed that his claim is to argue for the implausibility of a fundamental time asymmetry in the theory of classical cosmology. If, in contrast to that assumption, the main claim of the consideration is to argue against statistical methods for defining a cosmological arrow or time, than his claim is in perfect agreement with my view. The conflict only arises if the arguments from Price [(1996)] are taking as arguments against the plausibility of a fundamental time arrow in general. Regarding this interpretation of Price my critique can be outlined very briefly. Price mainly argues that in a possible closed spacetime, there would be no physical parameter that distinguishes a Big Bang from a Big Crunch (and thus could be used to define an time arrow). This is grounded on the assumption that the scale factor (Price calls it the radius of the universe), which behaves symmetrically in a closed spacetime, is the only fundamental property to distinguish time directions. If this assumption would be true, the possibility of closed spacetimes, given general relativity, would in fact show that no fundamental property can be used to distinguish between a ‘Big Bang’ and a ‘Big Crunch’. Thus, time would be symmetric in the physical description. But his crucial assumption, which according to me is wrong, seems to be that the scale factor (or the radius) is the only basic property that could be used to distinguish between the two singularities in a closed spacetime, given the standard theories of classical cosmology (see Price ([1996] pp. 86-111). Thus, I will argue that this assumption is not plausible and other geometrical properties (not statistical considerations) of spacetime apart from the scale factor must be considered for defining a fundamental time direction. Thus, I conclude that if it is possible to define spacetime properties that are as basic as the scale factor but independent of them, Price’s arguments are no longer plausible, and thus, the cosmological time arrow could be fundamental. Nevertheless, this brief investigation was necessary to show that the search for a fundamental time arrow is not doomed to fail on the grounds of Price’s arguments, if there are other fundamental properties of spacetime apart from the scale factor. But this, in fact, seems the case according to modern cosmological models. I will come back to this point more precise in section 3. However, before I define ‘fundamentality’ with respect to time asymmetries, I briefly examine some approaches to the arrow of time in classical cosmology in order to show that the suggestions that I am aware of cannot define a fundamental time arrow in classical cosmology. 1.2 Entropy-based approaches In this subsection I shall show that various entropy-based approaches cannot define a fundamental arrow of time in classical cosmology, that is, the time directedness in these approaches is not given by the basic properties of cosmology. In this section, I focus on approaches that define the arrow of time in cosmology via timeasymmetric behaviour of entropy. More precisely, the future direction in such approaches is given by the time direction in which entropy increases according to the second law of thermodynamics. Of course, statistical mechanics show that most types of entropy also would (theoretically) increase in the time-mirrored direction. Thus, it is necessary to set some boundary conditions for the past, specifically, that the universe has low entropy in the past (see, for example, Albert [2000]). However, approaches that focus on the temporal behaviour of entropy can be subsumed under (at least) two different topics, as analysed by Price [(2002)]. To make this point clear, in this subsection I only focused on attempts to base the cosmological time asymmetry on the entropy behaviour during a time evolution of the three dimensional universe. My claim is, in full agreement with Price [(2002)], to underline the point that such approaches are unable to provide an explanation for the occurring of a cosmological time arrow. Moreover, I consider an third kind of account, which is not captured by the analysis from Price [(2002)] and which, I think, is advocated in a very adequate way by Ćirković and Miloševic-Zdjelar [(2004)]. I will argue against this ‘Acausal-Anthropic’ attempt to the entropy behaviour and defending the result from Price [(2002)] that, given the actual theories in physics, the entropy behaviour is not a valid feature to yield the directedness of time. Regarding the three mentioned types of entropy based attempts: 1. Causal–general approaches: This class of approaches seeks an explanation of the lowentropy state near the Big Bang by fundamental physical laws and an associated dynamic explanation for the specialness of the ‘early’ state. But, no current physical theory can explain the specialness of the early universe only by invoking dynamic laws. All approaches that I know refer to boundary conditions of some type (such as the past hypothesis of Albert [2000]). It seems that with our current knowledge of physics, we cannot formulate approaches that deduce an arrow of time from dynamic laws (see also Wald [2006]). Thus, causal-general approaches seem unsuccessful in deducing a time direction or even the directedness of time (in a fundamental way) in classical cosmology as long as boundary conditions are not understood as fundamental properties. Given modern physical theories, it seems in fact that boundary conditions are not understood as fundamental properties. I also believe that many philosophers and physicists are aware of this situation and agree well with this view, and hence, this point is not discussed in more detail. 2. Acausal–particular approaches: This class of approaches describes the specialness (or the low entropy) of states near the Big Bang by the existence of boundary conditions. As mentioned above, boundary conditions should not be understood as fundamental properties of physics, at least in modern physical theories. Thus, neither of these two approaches explains the occurrence of a fundamental (in some reasonable sense) arrow of time in classical cosmology. However, as mentioned earlier, some authors have argued for another possibility (for example, Ćirković and Miloševic-Zdjelar [2004]). They argued that some type of cosmological theory describing a multiverse (more precisely, a type that includes the existence of many cosmic domains) could provide another possibility for defining the directedness of time, because some multiverse theories (see, for example, Linde [1990]) could explain the fact that our universe has a very smooth matter distribution near the Big Bang. In the classes of multiverse theories, we have more than one universe, where each universe can be called a cosmic domain. One of these domains is our particular universe. Each of these domains could have different initial conditions. Thus, if the occurrence of smooth states near the Big Bang has a probability of, for example 123 10 1:10 (Penrose [1979]), the prediction of the existence of cosmic domains that include such smooth early states seems very plausible, as long as the number of cosmic domains is assumed to be much larger than the reciprocal of the probability of the occurrence of such smooth early states. This prediction is as plausible as the prediction of getting a ‘6’ at least once if a fair six-sided dice is rolled n times, where n is much larger than six. Authors, who support this view, occasionally call it an anthropic approach (Ćirković and Miloševic-Zdjelar [2004]) because of the following reasons. At first glance, it may appear very surprising in such a theory that the observable universe belongs to this minority of domains, which are only as likely as 123 10 1:10 (for example). If other domains are more likely, why do we not observe such a domain in our cosmic environment? At this point, it becomes possible to consider the anthropic principle. The answer to the question would be that the domains that are more likely cannot be observed unlike other domains because no human being can survive in such a universe in which even the existence of stars or atoms is unlikely. Thus, the anthropic principle is not used to explain a cosmological fact. The explanation of this fact comes from the cosmological theory, independent of any anthropic considerations. The anthropic principle is only used to clarify that the fact that our particular universe belonging to the small minority is not surprising because otherwise it would not have been our particular universe. However, what is interesting is that we find a cosmological theory that explains the occurrence of certain initial conditions in a particular cosmic domain. Thus, according to such types of multiverse (and often inflation) theory, an acausal–particular approach could be understood as fundamental in the sense that the basic properties of the laws of the universe, in such theories, show that some particular boundary conditions occur and provide an time asymmetry. Thus, in addition to causal–general and acausal–particular approaches, which were also analysed by Price ([2002]), we could consider this acausal–anthropic approach in this study. I use this name as it is used in Ćirković and Miloševic-Zdjelar ([2004]). The approach based on the possibility of time-asymmetric behaviour of entropy in our particular cosmic domain is called, from now on in this investigation, the entropic–anthropic approach. However, upon closer examination, this time asymmetry cannot be understood as being based on the laws of fundamental physics, for the following reason. The multiverse theories predict a very large number of cosmic domains. In addition, the time parameter, which is fundamental in this context, is a quantum parameter independent of particular cosmic times in some cosmic domains. The laws that give rise to the fundamental processes of creating different cosmic domains (which could have different cosmic times) are processes in quantum physics, which can be time symmetric according to the quantum physical time parameter. The fundamental laws and mechanisms of those theories also allow many cosmic domains, which could be time symmetric in terms of their cosmic times. Thus, in such domains, the behaviour of entropy is symmetric or the value of entropy is constant apart from fluctuations; obviously, this value is also symmetric. 4 This is true if we assume the standard interpretation of quantum physics in which, for example, complex conjugation will not change the physical meaning of a quantum dynamic equation such as the Schrödinger equation. Thus, according to the entropic–anthropic approach, we find an explanation for the occurrence of time-asymmetric behaviour in our particular cosmic domain, but this asymmetry occurs by accident. The entropic–anthropic approach explains that it is not surprising that we find ourselves in a cosmic environment such as the observable universe; nevertheless, the temporal direction is not based on basic properties of the theory, which is treaded as the fundamental one (here quantum field theory). Thus, we have seen that all the approaches described in this section fail to describe the existence of a time asymmetry that could reasonably be called fundamental (for some reasonable understanding of fundamentality in this context). This holds for: a) all the causalgeneral accounts, b) the acausal-particular accounts and c) the acausal-anthropic considerations. Thus, I will discuss another prominent approach to define an arrow of time in cosmology. This approach is independent of the behaviour of entropy. 1.3 Hyperbolic curved spacetimes Many cosmological models of the evolution of our particular universe have a property that we have not yet discussed in this paper. Observations and theoretical work support the idea that the three-dimensional universe exhibits accelerated expansion in cosmic time (e.g. Riess et al. [1998] on supernovae observations; Barlett and Blanchard [1996] on the cosmic virial theorem; Fan, Bahcall and Cen [1997] on mass indicators in galaxy clusters; Bertschinger [1998] on large-scale velocity maps and Kochanek [1995] or Coles and Ellis [1994]). This could be described in general relativity as a large positive value of the cosmological constant. Moreover, this seems to indicate that it is plausible that the universe has an open geometry, not only because the matter and energy density are too low to overcome expansion but also because the expansion is accelerated by a ‘force’ described by the cosmological constant. Now, the question arises whether this large value of the cosmological constant occurs accidentally or owing to some law of physics. If the accelerated expansion of the universe were the result of a fundamental law, time asymmetry could be defined regarding the direction pointing to the open end of spacetime and this time asymmetry could be based on the properties of the fundamental law. However, I must argue that this is not convincing given our current knowledge of cosmology. Note first that the value and origin of the cosmological constant still remains unsolved if we consider a one universe picture (no many cosmic domains). Moreover, even if multiverse pictures are assumed a particular value of a cosmological constant in one cosmic domain would occur accidently in those pictures and not for fundamental reasons (see above). Nevertheless, I briefly examine one possible account because some authors (Ćirković and Miloševic-Zdjelar [2004]) had attempted to argue that a particular account of the origin of the cosmological constant yields a special understanding of the arrow of time in cosmology. This account is that the cosmological constant is a result of vacuum polarisation. The effect of vacuum polarisation is surely very fundamental; moreover, it arises from the laws of quantum field theory (QFT). I shall show that this assumption does not allow us to conclude that the open geometry of spacetime yields a time asymmetry based on the properties of a fundamental law. This is because the force of gravity, which opposes the cosmological constant, depends on the matter and energy density of the universe. Thus, the critical value that the cosmological constant must exceed to yield an accelerated universe in such a semi-classical model depends on the matter and energy density. This density does not seem to be determined by QFT in general. Moreover, some types of cosmological theory suggest that this density could vary among cosmic domains in the multiverse picture. If such variation is possible, it follows that the effect of the accelerated expansion of the universe could only be used to define an time asymmetry in some particular cosmic domains; However, I think this asymmetry should not be understood as fundamental, because we notice the same situation in standard classical cosmology in which a spacetime could be open or closed, given the mass and energy density. Thus, an account based on the large value of our particular cosmological constant yields the same problems as the accounts mentioned above: a closed universe is possible even if the value of the cosmological constant is assumed to be ‘large’ for fundamental rezones. As long as we prefer to call a time asymmetry a fundamental one only if the directedness of time is based on fundamental properties of the theory used in the model, Ćirković and MiloševicZdjelar ([2004]) cannot provide new insight on this question by considering the origin of the large value of the cosmological constant in vacuum polarisation. As mentioned, this is because a closed spacetime geometry is not ruled out (in principle) by a large value of the cosmological constant. Thus, the fact that our particular universe or cosmic domain seems to present an accelerated expansion cannot be used to conclude that we can define a cosmological time asymmetry based on the fundamental properties of a used physical description. Hence, the attempts to define a time asymmetry in cosmology, which we discussed till this point, do not yield a fundamental understanding of this asymmetry. Thus, in the following section, I present a more precise understanding of fundamentality, which allows an understanding of the time asymmetry as some kind of ‘fundamental’ property of nature according to classical cosmology. I will show more precise what I mean by ‘some kind of’ in section 3, where I try to apply my definition to classical cosmology. 2 A new understanding of fundamentality The first step in building my argument is well known and has been analysed by many authors in both philosophy and physics. It arises from the distinction between the property of time reversal invariance and the property of time asymmetry in general. In this paper, ‘time reversal invariance’ will be understood as a property of dynamical equations and ‘time asymmetry’ will be considered in particular as a property of the solutions of those equations. Here ‘time symmetry’ is given by a solution f(t) if f(t0+t)=f(t0-t) holds for one t0, hence in particular if a solution describes closed curves in phase space. Moreover we can identify dynamical equations with physical laws and solutions to those equations with physical models that satisfy such laws. Therefore, we can combine these properties in four different ways: a) time reversal invariance and only time-symmetric solutions, b) time reversal invariance and some time-asymmetric solutions, c) no time reversal invariance and only time-symmetric solutions and d) no time reversal invariance and some time-asymmetric solutions. For combination a), it is easy to find physical examples. However, this combination is not useful if we are interested in time asymmetries. Combination b) looks interesting because it shows that time-reversal-invariant laws (TRIL) could have time-asymmetric solutions. Simple examples are given in classical electrodynamics; nevertheless, such traditional asymmetries are not understood as fundamental time asymmetries because they occur only in some special models of the TRIL and the occurrence of such asymmetries is explained mostly by boundary conditions. The applicability of combination c) or d) in fundamental physics is at least problematic. It seems that non-TRIL’s cannot be found within the laws of fundamental physics in the standard interpretation. Hence those combinations seem applicable only in some special formulation of quantum laws, for example in some formulations of the rigged Hilbert space approach (see, for example, Bohm, Gadella and Wickramasekara (1999); Bishop (2004), Castagnino, Gadella and Lombardi (2005) or Castagnino, Gadella and Lombardi (2006)). However, it will be shown below that combination c) or d) need not be used to understand the time-asymmetries in a fundamental manner. In the following, I shall show that combination b) indicates another plausible way to define ‘fundamentality’ for time-asymmetries, based only on the structure of the solution space of physical equations.

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تاریخ انتشار 2012