Those who followed Gottfried Leibniz, who lived at approximately the same time, saw time as a relative measurement between events. If there were no events, there would be no time, because time is simply the separation between events.
Both views found ways to conceptualize the directionality of time. The Newtonians saw time as a one-way street: events happen at points on the timeline, and an observer is moving along the timeline in one direction only, from one event to the next.
The Leibnizians denied an independent reality to the timeline, saying rather that a later event and a prior event have a relationship to each other which is not symmetrical or reciprocal: an analogy to the parent-child relationship reveals that the parent and the child are real, while the concept of parenthood is a merely relative abstraction from the two real things. So it is also, Leibniz would suggest, with time.
When the question about the possibility of bidirectional time is raised — when one asks about time moving backwards — a challenge arises both for Newton and for Leibniz.
Attempts to explain the directionality of time often incorporate the concept of causality or the second law of thermodynamics or both.
In an intuitive and naive sense, it seems obvious that later events cannot cause prior events. This is an instinctive argument for the directionality of time.
The second law of thermodynamics is subject to many different phrasings, but a simplistic version says that entropy never decreases and that systems always tend toward maximum entropy. The directionality of time, then, is marked out as entropy increases, or at least fails to decrease.
Ludwig Boltzmann was a physicist and philosopher in Vienna. He did his work in the late 1800s and early 1900s. Much of his work dealt with the physical chemistry of gasses. In particular, he refined the mathematical formulation of Brownian motion and how gasses move toward entropy, equilibrium, and homogeneity.
Along the way, Boltzmann obtained some results which are perhaps counterintuitive and which challenge the common understanding of the directionality of time.
Boltzmann discovered that a gas, if contained in a finite space which changes neither in shape nor in total volume, and if in a state of equilibrium, will spontaneously develop local regions of disequilibrium. This seems like a violation of the intuitive understanding of the second law of thermodynamics.
Further, Boltzmann came to reject a simplistic version of Newtonian time, in which it would be said that systems move toward entropy over time. He came instead to view the movement toward entropy as time. On Boltzmann’s view, then, it would be said that the movement of systems toward entropy is time: time is the increase of entropy.
Combining these two ideas, Boltzmann concluded that there are instances in which time runs backward: times when pockets of disequilibrium develop in a system which has already obtained maximum entropy. As author Martin Gardner writes:
The most popular way to give an operational meaning to “backward time” was by imagining a world in which shuffling processes went backward, from disorder to order. Ludwig Boltzmann, the 19th-century Austrian physicist who was one of the founders of statistical thermodynamics, realized that after the molecules of a gas in a closed, isolated container have reached a state of thermal equilibrium — that is, are moving in complete disorder with maximum entropy — there will always be little pockets forming here and there where entropy is momentarily decreasing. These would be balanced by other regions where entropy is increasing; the overall entropy remains relatively stable, with only minor up-and-down fluctuations.
As counterintuitive as these results are, Boltzmann went even further. If that these principles hold for a gas in an unchanging container — imagine a corked test tube in a chemistry laboratory — then these principles will also hold true for the universe as a whole.
On the grand cosmic scale, Boltzmann hypothesizes, there might be regions within the universe in which time is running backward.
If one is to speak of time running backward, then it must be decided whether this will be explained in Newtonian terms, Leibnizian terms, or Boltzmann’s terms. In intuitive Newtonian terms, some sense can be made of time as an existing framework in which it might happen that entropy would decrease instead of increase: but if time is independent of the events which happen in it, then this decrease in entropy would not qualify as a reversal of time’s direction, even if it is a violation of the laws of thermodynamics.
In Leibnizian terms, one event succeeding another, or one state succeeding another, shapes the direction of time, and so likewise this decrease in entropy would not be a reversal of time.
On Boltzmann’s own terms, in which time is the movement toward entropy, this can be seen as a reversal of time. Yet it should be asked: does Boltzmann need to assume a larger Newtonian framework of independent time, in order to determine that time in the smaller region is running backward?
To complicate matters further, Boltzmann implies that it would be possible to have in the universe regions, some of which are moving toward equilibrium, and some of which are moving away from it. Here one wants to add the phrase: “at the same time.” But if those competing regions within the universe define their time as Boltzmann suggests, i.e., by the movement toward equilibrium, how then could it be said that these regions have time moving in opposite directions, unless there were a larger framework, a meta-time, of Newtonian nature, against which the direction of time in the smaller regions could be measured?
The question is: Does Boltzmann need a Newtonian meta-time to make his view of time succeed?
Martin Gardner continues:
Boltzmann imagined a cosmos of vast size, perhaps infinite in space and time, the overall entropy of which is at a maximum but which contains pockets where for the moment entropy is decreasing. (A “pocket” could include billions of galaxies and the “moment” could be billions of years.) Perhaps our flyspeck portion of the infinite sea of space-time is one in which such a fluctuation has occurred. At some time in the past, perhaps at the time of the “big bang,” entropy happened to decrease; now it is increasing. In the eternal and infinite flux a bit of order happened to put in its appearance; now that order is disappearing again, and so our arrow of time runs in the familiar direction of increasing entropy. Are there other regions of space-time, Boltzmann asked, in which the arrow of entropy points the other way? If so, would it be correct to say that time in such a region was moving backward, or should one simply say that entropy was decreasing as the region continued to move forward in time?
Boltzmann further concludes that in regions, or in the universe at large, in which equilibrium as been achieved, i.e., in which entropy is at its maximum, there is no time, or in Boltzmann’s own words, it is “dead.”
Can it make sense to speak of time running backwards, or in the case of an achieved equilibrium, of time stopping, unless there is some meta-time, some perspective from a higher level, from which it could be observed that time was so behaving? Martin Gardner asks:
If things come to a standstill in time and “then” reverse, what does the word “then” mean? It has meaning only if we assume a more fundamental kind of time that continues to move forward, altogether independent of how things in the universe move. Relative to this meta-time — the time of the hypothetical observer who has slipped unnoticed into the picture — the cosmos is indeed running backward. But if there is no meta-time — no observer who can stand outside the entire cosmos and watch it reverse — it is hard to understand what sense can be given to the statement that the cosmos “stops” and “then” starts moving backward.
There is no doubt that Boltzmann was an exceptionally brilliant thinker. Yet there are some difficult questions for him to answer.
Did he over-rely on the analogy to gasses? What might be provable or observable about a corked test tube filled with air might not apply to the universe as a whole. What justifies the transference? If principles have been understood from gasses in a finite container of unchanging shape and size, why would these principles apply to the universe as a whole?
Is Boltzmann justified in asserting that the universe as a whole is in a state of equilibrium? He makes the assertion that “the universe” is “everywhere in thermal equilibrium and therefore dead,” with the exception of small regions which “depart from thermal equilibrium” for a “relatively short time.”
Yet the universe as it is known demonstrates sharp distinctions between vacuums and dense astronomical bodies. It displays, not chaotic Brownian motion, but predictable Newtonian and Keplerian orbits. Observations, whether by optical telescope or by radar telescope or by space travel, do not reveal a homogenized universe.
The reader will want to consult Boltzmann’s Vorlesungen über Gastheorie, Band II, Kapitel 90.
Boltzmann made remarkable discoveries and had brilliant insights. Yet many questions about the direction of time remain to be answered.
