The human view of the arrow of time

On why we need to think beyond entropy to understand our experience of the arrow of time.

In Kurt Vonnegut’s classic war novel, Slaughterhouse-Five, we follow the journey of Billy Pilgrim, a poorly trained, fatalistic soldier in the United States army in World War II. On the night of his daughter’s wedding, Billy is abducted by aliens in a flying saucer, and taken to a distant planet named Tralfamadore, where he is held captive. Now, these are not your garden-variety aliens — Tralfamadorians do not experience time like we do. As described by Vonnegut -

“The Tralfamadorians can look at all the different moments just the way we can look at a stretch of the Rocky Mountains, for instance. They can see how permanent all the moments are, and they can look at any moment that interests them. It is just an illusion we have here on Earth that one moment follows another one, like beads on a string, and that once a moment is gone it is gone forever.”

That is undeniably lovely writing, but as far as scientific accuracy goes, it is seemingly nothing more than spectacular science fiction. It is interesting though that back here on Earth, our own comprehension of time itself has a rather tenuous relationship with the fundamental laws of physics. For a start, we unambiguously experience an arrow of time; raindrops fall, trees grow, people age, matter decays, stars explode; one moment follows another one, like beads on a string. On the contrary, even a cursory look at the fundamental laws of physics reveals that they are completely agnostic to the direction of time. This is as true of Newton’s laws of motion as it is of quantum theory and Einstein’s theory of relativity.

Time-reversal invariance

The fact that the fundamental laws don’t pick a special direction of time is usually referred to as ‘time-reversal invariance’, i. e. the laws of nature remain unchanged when the direction of time is reversed. An intuitive approach to this is to think of time-reversal as a ‘reversal in the direction of motion’, as said by physicist Eugene Wigner. In other words, according to time-reversal invariance, if you were to film something occurring and then played the film backwards, this ‘reversed’ movie would still be completely compatible with the fundamental laws. An audience with knowledge of just the fundamental laws would be incapable of identifying which movie was played ‘forward’ and which one ‘backward’.

It is easiest to see this in the trajectory of an isolated object, for example a lone gas molecule bouncing around in a box. If I reverse the direction of motion of this molecule at an instant of time, the particle, following Newton’s laws of motion, would exactly retrace its path. There is nothing special about one direction of time versus the other. This is the essence of time-reversal invariance.

The very ideas of ‘forward’ and ‘backward’ time, or for that matter, past and future, are absent in the fundamental laws.

Entropy and the arrow of time

Why are the fundamental laws seemingly at odds with our experience of the macroscopic world? Movies played backwards look absurd, even a toddler would be able to say something was terribly wrong. The textbook explanation of this is of course, the second law of thermodynamics. The entropy of any closed system increases with time, establishing a macroscopic arrow of time. Even as every individual process happens in a manner consistent with time-reversal invariance, the system as a collective moves in the direction of increasing disorder. It is this direction of increasing disorder that establishes an emergent arrow of time in the macroscopic world.

The fact that entropy increases with time is itself thought to be a consequence of something much more profound — the highly ordered initial state of the universe at the time of the Big Bang. Physicist and author Sean Carroll illustrates this in an intuitive manner, by drawing parallels between the asymmetry of time and the asymmetry of space. Space, left to itself is completely uniform. In outer space, there isn’t any distinction between forward and backward, or left and right, or up and down. However, when you are near a planet or a star (or for that matter any object with a mass), the gravity of the object creates a force field that unambiguously distinguishes motion towards and away from it. Notions of ‘up’ and ‘down’ are meaningless in outer space but have a definitive meaning and consequence on the surface of a planet. In the same way, when you are proximate (in time) to a symmetry-breaking event like the Big Bang, the ‘forward’ and ‘backward’ directions of time are no longer the same. There is a definite evolution in time from the initial highly ordered state towards an increasingly disordered state, establishing the asymmetry between the forward and backward directions of time.

The conversation usually ends here — special conditions during the Big Bang gave rise to the second law of thermodynamics, which in turn establishes the arrow of time that we observe. However, there is a missing link in this chain of reasoning. It is understood that the Big Bang established the second law of thermodynamics for the universe as a whole. We learn and observe in everyday life though, that the second law of thermodynamics holds not just for the universe in its entirety, but for any closed system. Think of water evaporating, or coffee and milk mixing in a cup, or a dead plant decaying. Our experience of the arrow of time is due to entropy increasing not at a universal scale, but much more locally. Explaining this requires the consideration of an additional detail — one that is conceptually almost trivially simple, but in my opinion, important to connect the fundamental laws to our experience of time.

A missing link: The everyday breaking of the symmetry of time

It helps to revisit the ubiquitous example of gas molecules in a box. The second law of thermodynamics for isolated systems is often illustrated by means of model systems such as this, where an initially ordered state (i. e. low entropy) is shown to evolve over time into a disordered state (i. e. high entropy). The argument is that when you have a large number of molecules, applying Newton’s laws of motion (and the machinery of statistical mechanics), it is astronomically more probable for the system to evolve into a state of higher disorder than it is for the reverse to occur, thus establishing the asymmetry of time.

However, it is important to remember that such an asymmetry cannot be established purely by the laws of motion (which we note, are completely time-reversal invariant). After all by this logic, entropy should increase in both the forward and backward directions of time.

It requires one additional, singularly important ingredient — the artificially created, highly ordered initial state of the system. Critically, the creation of this initial state necessarily requires an external hand that breaks the time-reversal invariance of the otherwise ‘isolated’ box of gas molecules. This is in spirit similar to the argument about the Big Bang establishing the second law of thermodynamics for the universe as a whole! Again, it is necessary to be proximate (in time) to this artificial, highly ordered initial state to observe the asymmetry of time. For example, if we allow the box of gas molecules to develop into a highly disordered state (over a sufficiently long period of time) and then film it, there will no longer be a noticeable contrast between the movies of the gas molecules played forwards and backwards. Indeed, once maximum disorder is achieved, it would be impossible to identify which was forward and which was backward. The symmetry of time encoded in the fundamental laws is recovered!

Our everyday observations of the arrow of time are simply dressed-up versions of the story of the box of gas molecules. The coffee and milk in a cup start off in a highly ordered state, created by an external hand, with the coffee and milk neatly separated. As the coffee and milk particles randomly bump into each other, the combined coffee-milk system evolves over time into a fully disordered state where the milk is completely randomly distributed in the coffee (or vice-versa, depending on how you like your coffee). While the arrow of time is striking and obvious at early times, it is seemingly invisible after enough time passes and the disordered state is achieved. Again, it wasn’t merely the interactions between particles that caused entropy to increase and create an asymmetry in time, it was the (symmetry-breaking) creation of a highly ordered initial state that did.

The human experience of time

Time and chance happeneth to them all. — Ecclesiastes 9:11

Physical arguments about the inherent symmetry of time in the fundamental laws can only go so far when faced with the formidable bias of our own cognition. The very ideas of ‘past’ and ‘future’ and the construction of human languages in such terms are a reflection of the chasm between the inherent symmetry in the physical description of time and our perception of an emergent arrow of time. However, we can begin to bridge this gap by deconstructing our perception of time and using some of the ideas I just described.

For example, the fundamental laws don’t make a distinction between the past and the future. On the other hand, in our human experience, we perceive a glaring contrast between the two — the past is determined, while the future is open, unknown. The resolution lies in that this contrast is only true in an emergent way, at the macroscopic scale.

Consider a simple system made of a few objects (say, a few gas molecules, a pair of billiards balls, or even planets in orbit). Here, future is as determined as the past. Trajectories can be mapped out with equal precision, both forwards and backwards in time — it’s as if time is a knob that you can turn back and forth. You could project into the future in the same way that you remember the past.

It is only in the macroscopic world, with a large number of objects (molecules, cells, dust particles) that one encounters, both physically and cognitively, the asymmetry and uncertainty imposed by the forward march of entropy. The past is determined. In contrast, at an emergent level, the future, with increasing entropy, may take one of several paths, each with a different probability, and is thus ‘open’. As macroscopic beings living in a macroscopic world, we perceive this asymmetry — we remember events that occurred, make a prediction of the immediate future, and based on what occurs, update our memory. This is in part what gives us the subjective experience of ‘flowing’ through time. It is even argued that this experience is the result of a form of evolutionary adaptation, a way for us to adapt and survive in a macroscopic world with a clear, unambiguous, emergent arrow of time. For an eloquently written, lucid exposition of some of these ideas, I recommend reading philosopher Jenann Ismael’s paper on this subject here.

From the classical into the unknown

All the examples we have explicitly talked about so far involve classical mechanics and Newton’s laws of motion. At its most elementary level though, nature is described instead by quantum mechanics (for the really small) and Einstein’s theory of relativity (for the really fast, and for the really massive). The governing equations of these theories are indeed time-reversal invariant, just like Newton’s laws. However, some details in these theories add fascinating new wrinkles to this discussion.

We have already seen how the past and future are apparently no different from each other in the fundamental laws of physics. With the theory of relativity, the idea of an absolute ‘present’ is not well-defined either. In relativity, space and time are intertwined such that different frames of reference, traveling at different velocities, experience different times. The notion of absolute space and absolute time are no longer valid. In quantum mechanics, once again, the equations of motion are clearly time-reversal invariant. However, the ‘measurement’ of a quantum state and the subsequent collapse of the quantum wavefunction is still a mystery, with unclear implications on the arrow of time.

The answers to these questions are intertwined with questions surrounding the origin of the universe itself — a topic that is decidedly beyond my paygrade. I will leave you instead, in the able hands of physicist Lee Smolin, here.

For some more earth-bound contemplations on time and our interaction with it, check out this timeless (see what I did there?) piece of standup comedy by George Carlin.

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Hari Padmanabhan

Postdoctoral researcher at Harvard, studying the physics of materials.