# The Arrow of Time The laws of nature, _except the second law of thermodynamics_, are symmetric in time. Reversing the time in the dynamical equations of motion simply describes everything going backwards. The second law is different. Entropy must never decrease in time, except statistically and briefly. Many natural processes are apparently _irreversible_. [Irreversibility](https://www.informationphilosopher.com/problems/reversibility/) is intimately connected to the direction of time. Identifying the physical reasons for the observed irreversibility, the [origin of irreversibility](https://www.informationphilosopher.com/articles/Origin_Irreversibility.pdf), would contribute greatly to understanding the apparent asymmetry of nature in time, despite nature's apparently perfect symmetry in space. In 1927, [Arthur Stanley Eddington](https://www.informationphilosopher.com/solutions/scientists/eddington/) coined the term "Arrow of Time" in his book _The Nature of the Physical World_. He connected "Time's Arrow" to the one-way direction of increasing entropy required by the second law of thermodynamics. This is now known as the _"thermodynamic arrow_." > > (_Nature of the Physical World_, 1927, p.328-9) In his later work, Eddington identified a _"cosmological arrow_," the direction in which the universe is expanding, as shown by Edwin Hubble about the time Eddington first defined the thermodynamic arrow. > > _New Pathways in Science_, 1937, p.328-9) There are now at least five other proposed arrows of time (discussed below). We can ask whether one arrow is a "master arrow" that all the others are following, or perhaps time itself is just a given property of nature that is otherwise irreducible to something more basic, as is space. Given the four-dimensional space-time picture of special relativity, and given that the laws of nature are symmetric in space, we may expect the laws to be _invariant_ under a change in time direction. The laws do not depend on position in space or direction, they are _invariant_ under translations and rotations, space is assumed uniform and isotropic. But time is not just another spatial dimension. It enters into calculations of event separations as an imaginary term (multiplied by the square root of minus 1). Nevertheless, all the dynamical laws of motion are symmetric under time reversal. So the basic problem is - how can macroscopic [irreversibility](https://www.informationphilosopher.com/problems/reversibility/) result from microscopic processes that are fundamentally reversible? Long before Eddington, scientists asked deep questions about the direction of time. Perhaps the first to explore the connection with physics was [Ludwig Boltzmann](https://www.informationphilosopher.com/solutions/scientists/boltzmann/), who with [James Clerk Maxwell](https://www.informationphilosopher.com/solutions/scientists/maxwell/) investigated the statistical motions of the atoms and molecules of gases. If the laws of nature are time symmetric, perhaps the "arrow of time" is to be found in the "initial" conditions, although this may be a circular concept, since "initial,"current," and "final" states are all defined with respect to time. Since the dynamical laws are time reversible, scientists as early as [Isaac Newton](https://www.informationphilosopher.com/solutions/scientists/newton/) understood that one could calculate all the motions of a system by assuming "final conditions" and working backwards in time. Nevertheless, most physicists have assumed the universe must have begun in a highly ordered (low entropy) state and it has been "running down" (entropy or disorder increasing) ever since. In the nineteenth century, this was called the "heat death" of the universe. This view has the unfortunate implication that all the information in the current universe was present at the beginning, which is friendly to some theological ideas like [pre-destination](https://www.informationphilosopher.com/freedom/pre-determinism.html), but distinctly unfriendly to ideas of human [free will](https://www.informationphilosopher.com/freedom/). Boltzmann assumed that the universe was infinitely old and that our current state is the consequence of a massive statistical fluctuation away from equilibrium and maximum entropy, a condition to which we must ultimately return. Would time itself be reversed if we could make the entropy decrease? That is unlikely, since entropy decrease anywhere (creating negative entropy or negentropy, a term coined by [Leon Brillouin](https://www.informationphilosopher.com/solutions/scientists/brillouin/)) must be accompanied by an increase elsewhere, to satisfy the second law. Otherwise we could use the local reduction in the entropy to build a perpetual motion machine. Put another way, if we could reverse the time, would entropy decrease? What can time reversal really mean? A thought experiment suggests not. Consider a closed perfume bottle inside a large empty container. Remove the bottle top and what would happen assuming that time is flowing backwards? It seems likely that the perfume molecules would leave the bottle whatever time is doing. For Aristotle, time was a measure of motion and change and for practical purposes, many scientists have thought that time reversal is approximated by the reversal of all the velocities or momenta of material particles at an instant, starting from their current positions. If we could reverse the motions of every material body (a practical impossibility, and perhaps a violation of Heisenberg's [uncertainty principle](https://www.informationphilosopher.com/freedom/uncertainty.html)), would that make the entropy decrease? [Ludwig Boltzmann](https://www.informationphilosopher.com/solutions/scientists/boltzmann/) agreed that it might, but only for a while. His intuition was that a system could not return to a highly ordered original state, such as every molecule back in the perfume bottle. J. Willard Gibbs thought otherwise, if the detailed path information in all the macroscopic motions is still available as microscopic information (if information is a conserved quantity), then reversal of all the motions should be exactly like a movie played backwards. The fundamental question of [information philosophy](https://www.informationphilosopher.com/) is cosmological and ultimately metaphysical. What is the process that creates [information](https://www.informationphilosopher.com/introduction/information/) structures in the universe? > Given the second law of thermodynamics, which says that any system will over time approach a thermodynamic equilibrium of maximum disorder or entropy, in which all information is lost, and given the best current model for the origin of the universe, which says everything began in a state of equilibrium some 13.75 billion years ago, how can it be that living beings are creating and communicating new information every day? Why are we not still in that state of thermal equilibrium? It is perhaps easier for us to see the increasing complexity and order of information structures on the earth than it is to notice the increase in chaos that comes with increasing entropy, since the entropy is radiated away from the earth into the night sky, then away to the cosmic microwave background sink of deep space. [David Layzer](https://www.informationphilosopher.com/solutions/scientists/layzer/) is a Harvard cosmologist who in the early 1970's made it clear that in an expanding universe the entropy would increase, as required by the second law of thermodynamics, but that the maximum possible entropy of the universe might increase faster than the actual entropy increase. This would leave room for an increase of order or information at the same time the entropy is increasing! Layzer pointed out that if the equilibration rate of the matter (the speed with which matter redistributes itself randomly among all the possible states) was slower than the rate of expansion, then the "negative entropy" or "order" (defined as the difference between the maximum possible entropy and the actual entropy) would also increase. [Claude Shannon](https://www.informationphilosopher.com/solutions/scientists/shannon/) identified this negative entropy with [information](https://www.informationphilosopher.com/introduction/information/), though visible structural information in the universe may be less than this "potential" for information.  Layzer called the direction of information increase the "_historical arrow_." In a 1975 article for _Scientific American_ called [_The Arrow of Time_](https://www.informationphilosopher.com/problems/arrow_of_time/Arrow_of_Time.pdf), Layzer wrote: > the complexity of the astronomical universe seems puzzling.  > > This is the fundamental question of [information philosophy](https://www.informationphilosopher.com/) > > Isolated systems inevitably evolve toward the featureless state of thermodynamic equilibrium. Since the universe is in some sense an isolated system, why has it not settled into equilibrium? One answer, favored by many cosmologists, is that the cosmological trend is in fact toward equilibrium but that too little time has elapsed for the process to have reached completion... I shall argue that this view is fundamentally incorrect. The universe is not running down, and it need not have started with a marked degree of disequilibrium; the initial state may indeed have been wholly lacking in macroscopic as well as microscopic information. > > Suppose that at some early moment local thermodynamic equilibrium prevailed in the universe. The entropy of any region would then be as large as possible for the prevailing values of the mean temperature and density. As the universe expanded from that hypothetical state the local values of the mean density and temperature would change, and so would the entropy of the region. For the entropy to remain at its maximum value (and thus for equilibrium to be maintained) the distribution of energies allotted to matter and to radiation must change, and so must the concentrations of the various kinds of particles. The physical processes that mediate these changes proceed at finite rates; if these "equilibration" rates are all much greater than the rate of cosmic expansion, approximate local thermodynamic equilibrium will be maintained; if they are not, the expansion will give rise to significant local departures from equilibrium. > > This is the Layzer's seminal theory of the [growth of order in the universe](https://www.informationphilosopher.com/problems/arrow_of_time/growth_of_order) > > These departures represent macroscopic information; the quantity of macroscopic information generated by the expansion is the difference between the actual value of the entropy and the theoretical maximum entropy at the mean temperature and density. > > > (_Scientific American_, December, 1975, p.68) In his 1989 book _The Emperor's New Mind_, [Roger Penrose](https://www.informationphilosopher.com/solutions/scientists/penrose/) speculated on the connection between information, entropy, and the [arrow of time](https://www.informationphilosopher.com/problems/arrow_of_time/). > Recall that the primordial fireball was a thermal state — a hot gas in expanding thermal equilibrium. Recall, also, that the term 'thermal equilibrium' refers to a state of maximum entropy. (This was how we referred to the maximum entropy state of a gas in a box.) However, the second law demands that in its initial state, the entropy of our universe was at some sort of minimum, not a maximum! > > What has gone wrong? One 'standard' answer would run roughly as follows: > > > True, the fireball was effectively in thermal equilibrium at the beginning, but the universe at that time was very tiny. The fireball represented the state of maximum entropy that could be permitted for a universe of that tiny size, but the entropy so permitted would have been minute by comparison with that which is allowed for a universe of the size that we find it to be today. As the universe expanded, the permitted maximum entropy increased with the universe's size, but the actual entropy in the universe lagged well behind this permitted maximum. The second law arises because the actual entropy is always striving to catch up with this permitted maximum. > > (_The Emperor's New Mind_, p.328-9) Penrose's "standard" answer is a clear reference to the pioneering work of Harvard cosmologist [David Layzer](https://www.informationphilosopher.com/solutions/scientists/layzer/), especially his 1975 _Scientific American_ article "[The Arrow of Time](https://www.informationphilosopher.com/solutions/scientists/layzer/Arrow_of_Time.pdf)." Layzer explained the [the growth of order in the universe](https://www.informationphilosopher.com/solutions/scientists/layzer/growth_of_order/) as the maximum possible entropy of the expanding universe increasing faster than the actual entropy, because the equilibration rates for matter and radiation are slower than the expansion rate. ## Other Arrows of Time ### The Radiation Arrow Whether they be electromagnetic waves or waves in water, we only observe wavelike disturbances that propagate outwards in space away from the disturbance. These waves are described by what is called the _retarded_ potential. In his 1909 discussion of waves and particles, Albert Einstein described the possibility of _incoming_ spherical waves: > According to our prevailing theory, an oscillating ion generates a spherical wave that propagates outwards. The inverse process does not exist as an elementary process. A converging spherical wave is mathematically possible, to be sure; but to approach its realization requires a vast number of emitting entities. The elementary process of emission is not invertible. In this, I believe, our oscillation theory does not hit the mark. Newton's emission theory of light seems to contain more truth with respect to this point than the oscillation theory since, first of all, the energy given to a light particle is not scattered over infinite space, but remains available for an elementary process of absorption. > > > ("[On the Development of Our Views Concerning the Nature and Constitution of Radiation](https://www.informationphilosopher.com/solutions/scientists/einstein/1909.html)," , orig. German, _Physikalische Zeitschrift_, 10, 817-825, 1909) In 1945, [John Wheeler](https://www.informationphilosopher.com/solutions/scientists/wheeler/) and his student [Richard Feynman](https://www.informationphilosopher.com/solutions/scientists/feynman/) attempted to symmetrize Maxwell's equations for electromagnetic fields with an "Absorber Theory of Radiation," that combined retarded (outgoing waves) and _advanced potentials_ (incoming spherical waves) for radiation. They later described the theory as a mistake. ### The Cosmological Arrow (expansion of the universe) We can define a cosmological direction of time as the direction in which the universe is expanding. There are excellent reasons for seeing this as the most fundamental of all arrows, even the one driving some of the others. Without expansion, a static universe would settle into thermal equilibrium and there would be no changes. There would be no entropy increase to show Eddington's thermodynamic arrow. There would be no information increase, as seen in Layzer's historical arrow. Without the cosmological arrow, all the other arrows could not be realized. ## **References** - Davies, P.C.W 1977 _The Physics of Time Asymmetry_, University of California Press. - Gold, T. 1967 _The Nature of Time_, Cornell University Press. - Reichenbach, H, 1956 _The Direction of Time_, University of California Press. - Zeh, H.D. 2010 _The Physical Basis of the Direction of Time_ 5th ed., Springer-Verlag Berlin.