Wednesday 18 March 2020

Special Relativity, Entropy, and the Direction of Time



Assuming that, as far as perception and experience are concerned, space is a three-dimensional, and time a one-dimensional, projection of a four-dimensional space-time, the following question arises: why does the brain, a material organ, break down a four-dimensional, relativistic space-time into three spatial dimensions and one temporal dimension? There is really only one answer that suggests itself, and it is that as biological products of natural selection our survival depends on tracking entropy: we need to find pockets of low entropy in order to discover sources of work that we may employ in order to maintain our own relatively low entropy, and since the entropic gradient is aligned along the temporal dimension, it is to our benefit to evolve to single out that dimension from the other three in order to focus on it and thereby be able to track entropy. That is, it is important to our survival that we find low entropy systems in order to secure energy.

This seems straightforward enough but it does, however, raise a deeper question: why is entropy organized along the temporal dimension alone; why not along another dimension, or two or more dimensions? Why did evolution find it necessary to endow us with perceptual and cognitive faculties that see the world as a three-dimensional array of material and objects that are evolving in a fourth dimension rather than some other way?

We know that in Minkowski space-time, neither spatial distances nor temporal durations have geometric significance on their own: only their combination in terms of Einstein’s Interval Formula:

     I2 = dt2 – ds2

has physical significance. What this entails is that there is no preferred set of spatial axes for the universe, no sense in which one can be right or wrong in labelling one direction “east” rather than “north” or “up” rather than “sideways”, and so on. All such spatial directions are significant only in relation to an observer; the universe itself doesn’t single out a particular direction in space.

Accordingly, if entropy increase were to be correlated with a spatial axis, it would be impossible for evolution to track it because it is impossible for physically real correlations to exist in relation to one or more spatial axes. The pattern of entropic increase would simply disappear from all spatial perspectives except one, i.e. from those corresponding to the rotation of the particular choice of axes. If a pattern is deeply dependent on a directional convention in this way, then there is no way for the non-directed, probabilistic processes of evolution to latch onto it, for there is, in reality, no probabilistic pattern to be detected: the pattern would be an observer effect, just like some works of art are constructed so that only when looked at directly is an image visible; all other angles show no discernible pattern - there is no pattern inherent to the image itself in such a case. In essence, if entropy were to be aligned with one or more spatial axes, the pattern of the second law of thermodynamics would simply disappear with movement in space, rendering impossible to have both freedom of movement and the ability to track entropy.

Given the non-directionality of space, entropy increase could not be aligned along any spatial dimension, so there could be no fitness relevant pressure to track any such gradient spatially. The only remaining option is that what we evolved to detect, namely the entropy gradient, is aligned with the fourth, temporal, dimension. But how could this be the case? According to (I), temporal duration frame-dependent – how much time there is between any two events will vary, though in such a way as to preserve the interval, I, between them. Moreover, the direction of time does not make an appearance in (I) since whether we put a positive or negative sign in front of the (frame-dependent) measure of time will make no difference to the interval because the value for time is squared in (I). This is one of the senses in which it is thought that Relativity counts against the common-sense notion that time passes from past to future.

If we leave things here, however, then we are faced with a problem: we know that entropy is coordinated with the temporal dimension, but if that dimension lacks an inherent direction, then there is no reason for entropy to uniformly increase or decrease along it: any such pattern would only be apparent, like a spatially oriented second law of thermodynamics, that exists only from some points of view. So if there is a genuine entropy gradient along the temporal dimension, which would account for our having evolved to single out time from space, then we must conclude that as a matter of fact, if not mathematical necessity, the fourth, temporal of the universe is inherently directed or asymmetric.

However, precisely because the temporal duration is squared in (I), it makes no difference to the mathematics of the interval to suppose that there is an objective, physically significant direction to time. In other words, even if there is a real difference between +t and -t, the interval will wash that out by squaring the value. So, the mathematics of Special Relativity is not impacted by the assumption that time is directed in the following sense: at each four-dimensional space-time point, a unique direction is defined, from the past light cone to the future. We can think of it as an array of vectors originating at the point and directed toward each point on the interior of one half of the light cone only; alternatively, we can state that between any two time-like separated space-time points, there exists a vector that defines the past-future orientation. This way, local temporal asymmetry is relativistically invariant and from the frame of reference of the surface of the Earth, there is a genuine pattern for the non-directed mechanisms of evolution to stumble upon and register for future iterations of variation-selection-inheritance.

In other words, the very fact that the four-dimensional, invariant Interval is broken down by the brain into a three-plus-one dimensional cognitive-perceptual model provides excellent reason to believe that the direction of time is not an illusion but, rather, a relativistic invariant.

Why does this matter? Well, it matters because there is a strong current of argumentation in the philosophy of time/physics to the effect that both the passage and direction of time are illusory, mere mental projections onto a fully symmetric underlying physical reality. If the foregoing argument is correct, then this is unlikely to be true simply because it would be unable to account for the fact that evolution reached a wide-spread and relatively stable equilibrium in constructing human beings to experience and understand the world as 3+1-dimensional rather than 4+0-, 2+2, or 1+3-dimensional. More fundamentally, it cannot account for the fact that entropy is aligned with any of the dimensions in (I) at all: if there is no fact of the matter as to which direction in space-time is locally future-oriented, there would be no entropic gradient in the first place for evolution to eventually track. In other words, without an objective temporal direction, there could be no entropic gradient at all since from one temporal perspective the universe would consist of mostly improbable transitions and that perspective would have to be considered equally valid compared to the opposite perspective in which probable transitions are the rule. But any viewpoint according to which a highly probable and highly improbable set of transitions need to be considered equally valid must surely be incorrect.

So, it seems that the best way to view the implications of Relativity Theory for the debate over the nature of time is that it combines with the theory of evolution to suggest that Einstein-Minkowski space-time includes a temporally oriented vector field defined at every space-time point that determines the future light cone from that point. The vectors themselves are perhaps to be viewed as unit vectors with frame-dependent lengths determined by the time-dilation implied by Lorenz Transformations.

The upshot for the philosophy of time is that a naturalistic picture of time that combines Einstein-Minkowski space-time with Darwinian evolution is not only compatible with asymmetric, i.e. directed, time but in fact demands it. In short, the direction of time is not an illusion. Furthermore, so long as we can define temporal passage in terms of directed temporal ordering, then this view is compatible with objective temporal passage as well. Since, as I’ve argued elsewhere, there is no problem with a deflationary, relational conception of temporal passage, we can conclude that the Theory of Relativity is no threat to the idea that time genuinely passes from the past toward the future.

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