Physicists have a problem with time.
Whether through Newton’s gravitation, Maxwell’s electrodynamics, Einstein’s special and general relativity or quantum mechanics, all the equations that best describe our universe work perfectly if time flows forward or backward.
Of course the world we experience is entirely different. The universe is expanding, not contracting. Stars emit light rather than absorb it, and radioactive atoms decay rather than reassemble. Omelets don’t transform back to unbroken eggs and cigarettes never coalesce from smoke and ashes. We remember the past, not the future, and we grow old and decrepit, not young and rejuvenated. For us, time has a clear and irreversible direction. It flies forward like a missile, equations be damned.
For more than a century, the standard explanation for “time’s arrow,” as the astrophysicist Arthur Eddington first called it in 1927, has been that it is an emergent property of thermodynamics, as first laid out in the work of the 19th-century Austrian physicist Ludwig Boltzmann. In this view what we perceive as the arrow of time is really just the inexorable rearrangement of highly ordered states into random, useless configurations, a product of the universal tendency for all things to settle toward equilibrium with one another.
Informally speaking, the crux of this idea is that “things fall apart,” but more formally, it is a consequence of the second law of thermodynamics, which Boltzmann helped devise. The law states that in any closed system (like the universe itself), entropy—disorder—can only increase. Increasing entropy is a cosmic certainty because there are always a great many more disordered states than orderly ones for any given system, similar to how there are many more ways to scatter papers across a desk than to stack them neatly in a single pile.
The thermodynamic arrow of time suggests our observable universe began in an exceptionally special state of high order and low entropy, like a pristine cosmic egg materializing at the beginning of time to be broken and scrambled for all eternity. From Boltzmann’s era onward, scientists allergic to the notion of such an immaculate conception have been grappling with this conundrum.
Boltzmann, believing the universe to be eternal in accordance with Newton’s laws, thought that eternity could explain a low-entropy origin for time’s arrow. Given enough time—endless time, in fact—anything that can happen will happen, including the emergence of a large region of very low entropy as a statistical fluctuation from an ageless, high-entropy universe in a state of near-equilibrium. Boltzmann mused that we might live in such an improbable region, with an arrow of time set by the region’s long, slow entropic slide back into equilibrium.
Today’s cosmologists have a tougher task, because the universe as we now know it isn’t ageless and unmoving: They have to explain the emergence of time’s arrow within a dynamic, relativistic universe that apparently began some 14 billion years ago in the fiery conflagration of the big bang. More often than not the explanation involves ‘fine-tuning’—the careful and arbitrary tweaking of a theory’s parameters to accord with observations.
Many of the modern explanations for a low-entropy arrow of time involve a theory called inflation—the idea that a strange burst of antigravity ballooned the primordial universe to an astronomically larger size, smoothing it out into what corresponds to a very low-entropy state from which subsequent cosmic structures could emerge. But explaining inflation itself seems to require even more fine-tuning. One of the problems is that once begun, inflation tends to continue unstoppably. This “eternal inflation” would spawn infinitudes of baby universes about which predictions and observations are, at best, elusive. Whether this is an undesirable bug or a wonderful feature of the theory is a matter of fierce debate; for the time being it seems that inflation’s extreme flexibility and explanatory power are both its greatest strength and its greatest weakness.
For all these reasons, some scientists seeking a low-entropy origin for time’s arrow find explanations relying on inflation slightly unsatisfying. “There are many researchers now trying to show in some natural way why it’s reasonable to expect the initial entropy of the universe to be very low,” says David Albert, a philosopher and physicist at Columbia University. “There are even some who think that the entropy being low at the beginning of the universe should just be added as a new law of physics.”
That latter idea is tantamount to despairing cosmologists simply throwing in the towel. Fortunately, there may be another way.
Tentative new work from Julian Barbour of the University of Oxford, Tim Koslowski of the University of New Brunswick and Flavio Mercati of the Perimeter Institute for Theoretical Physics suggests that perhaps the arrow of time doesn’t really require a fine-tuned, low-entropy initial state at all but is instead the inevitable product of the fundamental laws of physics. Barbour and his colleagues argue that it is gravity, rather than thermodynamics, that draws the bowstring to let time’s arrow fly. Their findings were published in October in Physical Review Letters.
The team’s conclusions come from studying an exceedingly simple proxy for our universe, a computer simulation of 1,000 pointlike particles interacting under the influence of Newtonian gravity. They investigated the dynamic behavior of the system using a measure of its "complexity," which corresponds to the ratio of the distance between the system’s closest pair of particles and the distance between the most widely separated particle pair. The system’s complexity is at its lowest when all the particles come together in a densely packed cloud, a state of minimum size and maximum uniformity roughly analogous to the big bang. The team’s analysis showed that essentially every configuration of particles, regardless of their number and scale, would evolve into this low-complexity state. Thus, the sheer force of gravity sets the stage for the system’s expansion and the origin of time’s arrow, all without any delicate fine-tuning to first establish a low-entropy initial condition.
From that low-complexity state, the system of particles then expands outward in both temporal directions, creating two distinct, symmetric and opposite arrows of time. Along each of the two temporal paths, gravity then pulls the particles into larger, more ordered and complex structures—the model’s equivalent of galaxy clusters, stars and planetary systems. From there, the standard thermodynamic passage of time can manifest and unfold on each of the two divergent paths. In other words, the model has one past but two futures. As hinted by the time-indifferent laws of physics, time’s arrow may in a sense move in two directions, although any observer can only see and experience one. “It is the nature of gravity to pull the universe out of its primordial chaos and create structure, order and complexity,” Mercati says. “All the solutions break into two epochs, which go on forever in the two time directions, divided by this central state which has very characteristic properties.”
Although the model is crude, and does not incorporate either quantum mechanics or general relativity, its potential implications are vast. If it holds true for our actual universe, then the big bang could no longer be considered a cosmic beginning but rather only a phase in an effectively timeless and eternal universe. More prosaically, a two-branched arrow of time would lead to curious incongruities for observers on opposite sides. “This two-futures situation would exhibit a single, chaotic past in both directions, meaning that there would be essentially two universes, one on either side of this central state,” Barbour says. “If they were complicated enough, both sides could sustain observers who would perceive time going in opposite directions. Any intelligent beings there would define their arrow of time as moving away from this central state. They would think we now live in their deepest past.”
What’s more, Barbour says, if gravitation does prove to be fundamental to the arrow of time, this could sooner or later generate testable predictions and potentially lead to a less “ad hoc” explanation than inflation for the history and structure of our observable universe.
This is not the first rigorous two-futures solution for time’s arrow. Most notably, California Institute of Technology cosmologist Sean Carroll and a graduate student, Jennifer Chen, produced their own branching model in 2004, one that sought to explain the low-entropy origin of time’s arrow in the context of cosmic inflation and the creation of baby universes. They attribute the arrow of time’s emergence in their model not so much to entropy being very low in the past but rather to entropy being so much higher in both futures, increased by the inflation-driven creation of baby universes.
A decade on, Carroll is just as bullish about the prospect that increasing entropy alone is the source for time’s arrow, rather than other influences such as gravity. “Everything that happens in the universe to distinguish the past from the future is ultimately because the entropy is lower in one direction and higher in the other,” Carroll says. “This paper by Barbour, Koslowski and Mercati is good because they roll up their sleeves and do the calculations for their specific model of particles interacting via gravity, but I don’t think it’s the model that is interesting—it’s the model’s behavior being analyzed carefully…. I think basically any time you have a finite collection of particles in a really big space you’ll get this kind of generic behavior they describe. The real question is, is our universe like that? That’s the hard part.”
Together with Alan Guth, the Massachusetts Institute of Technology cosmologist who pioneered the theory of inflation, Carroll is now working on a thermodynamic response of sorts to the new claims for a gravitational arrow of time: Another exceedingly simple particle-based model universe that also naturally gives rise to time’s arrow, but without the addition of gravity or any other forces. The thermodynamic secret to the model’s success, they say, is assuming that the universe has an unlimited capacity for entropy.
“If we assume there is no maximum possible entropy for the universe, then any state can be a state of low entropy,” Guth says. “That may sound dumb, but I think it really works, and I also think it’s the secret of the Barbour et al construction. If there’s no limit to how big the entropy can get, then you can start anywhere, and from that starting point you’d expect entropy to rise as the system moves to explore larger and larger regions of phase space. Eternal inflation is a natural context in which to invoke this idea, since it looks like the maximum possible entropy is unlimited in an eternally inflating universe.”
The controversy over time’s arrow has come far since the 19th-century ideas of Boltzmann and the 20th-century notions of Eddington, but in many ways, Barbour says, the debate at its core remains appropriately timeless. “This is opening up a completely new way to think about a fundamental problem, the nature of the arrow of time and the origin of the second law of thermodynamics,” Barbour says. “But really we’re just investigating a new aspect of Newton’s gravitation, which hadn’t been noticed before. Who knows what might flow from this with further work and elaboration?”
“Arthur Eddington coined the term ‘arrow of time,’ and famously said the shuffling of material and energy is the only thing which nature cannot undo,” Barbour adds. “And here we are, showing beyond any doubt really that this is in fact exactly what gravity does. It takes [entropic] systems that look extraordinarily disordered and makes them wonderfully ordered. And this is what has happened in our universe. We are realizing the ancient Greek dream of order out of chaos.”
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Additional References
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FROM THE BLOG OF SEAN CARROLL
Why Does Dark Energy
Make the Universe Accelerate?
http://www.preposterousuniverse.com/blog/2013/11/16/why-does-dark-energy-make-the-universe-accelerate/
November 16, 2013
Peter Coles has issued a challenge: explain why dark energy makes the universe accelerate in terms that are understandable to non-scientists. This is a pet peeve of mine — any number of fellow cosmologists will recall me haranguing them about it over coffee at conferences — but I’m not sure I’ve ever blogged about it directly, so here goes. In three parts: the wrong way, the right way, and the math.
The Wrong Way
Ordinary matter acts to slow down the expansion of the universe. That makes intuitive sense, because the matter is exerting a gravitational force, acting to pull things together. So why does dark energy seem to push things apart?
The usual (wrong) way to explain this is to point out that dark energy has “negative pressure.” The kind of pressure we are most familiar with, in a balloon or an inflated tire, pushing out on the membrane enclosing it. But negative pressure — tension — is more like a stretched string or rubber band, pulling in rather than pushing out. And dark energy has negative pressure, so that makes the universe accelerate.
If the kindly cosmologist is both lazy and fortunate, that little bit of word salad will suffice. But it makes no sense at all, as Peter points out. Why do we go through all the conceptual effort of explaining that negative pressure corresponds to a pull, and then quickly mumble that this accounts for why galaxies are pushed apart?
So the slightly more careful cosmologist has to explain that the direct action of this negative pressure is completely impotent, because it’s equal in all directions and cancels out. (That’s a bit of a lie as well, of course; it’s really because you don’t interact directly with the dark energy, so you don’t feel pressure of any sort, but admitting that runs the risk of making it all seem even more confusing.) What matters, according to this line of fast talk, is the gravitational effect of the negative pressure. And in Einstein’s general relativity, unlike Newtonian gravity, both the pressure and the energy contribute to the force of gravity. The negative pressure associated with dark energy is so large that it overcomes the positive (attractive) impulse of the energy itself, so the net effect is a push rather than a pull.
This explanation isn’t wrong; it does track the actual equations. But it’s not the slightest bit of help in bringing people to any real understanding. It simply replaces one question (why does dark energy cause acceleration?) with two facts that need to be taken on faith (dark energy has negative pressure, and gravity is sourced by a sum of energy and pressure). The listener goes away with, at best, the impression that something profound has just happened rather than any actual understanding.
The Right Way
The right way is to not mention pressure at all, positive or negative. For cosmological dynamics, the relevant fact about dark energy isn’t its pressure, it’s that it’s persistent. It doesn’t dilute away as the universe expands. And this is even a fact that can be explained, by saying that dark energy isn’t a collection of particles growing less dense as space expands, but instead is (according to our simplest and best models) a feature of space itself. The amount of dark energy is constant throughout both space and time: about one hundred-millionth of an erg per cubic centimeter. It doesn’t dilute away, even as space expands.
Given that, all you need to accept is that Einstein’s formulation of gravity says “the curvature of spacetime is proportional to the amount of stuff within it.” (The technical version of “curvature of spacetime” is the Einstein tensor, and the technical version of “stuff” is the energy-momentum tensor.) In the case of an expanding universe, the manifestation of spacetime curvature is simply the fact that space is expanding. (There can also be spatial curvature, but that seems negligible in the real world, so why complicate things.)
So: the density of dark energy is constant, which means the curvature of spacetime is constant, which means that the universe expands at a fixed rate.
The tricky part is explaining why “expanding at a fixed rate” means “accelerating.” But this is a subtlety worth clarifying, as it helps distinguish between the expansion of the universe and the speed of a physical object like a moving car, and perhaps will help someone down the road not get confused about the universe “expanding faster than light.” (A confusion which many trained cosmologists who really should know better continue to fall into.)
The point is that the expansion rate of the universe is not a speed. It’s a timescale — the time it takes the universe to double in size (or expand by one percent, or whatever, depending on your conventions). It couldn’t possibly be a speed, because the apparent velocity of distant galaxies is not a constant number, it’s proportional to their distance. When we say “the expansion rate of the universe is a constant,” we mean it takes a fixed amount of time for the universe to double in size. So if we look at any one particular galaxy, in roughly ten billion years it will be twice as far away; in twenty billion years (twice that time) it will be four times as far away; in thirty billion years it will be eight times that far away, and so on. It’s accelerating away from us, exponentially. “Constant expansion rate” implies “accelerated motion away from us” for individual objects.
There’s absolutely no reason why a non-scientist shouldn’t be able to follow why dark energy makes the universe accelerate, given just a bit of willingness to think about it. Dark energy is persistent, which imparts a constant impulse to the expansion of the universe, which makes galaxies accelerate away. No negative pressures, no double-talk.
The Math
So why are people tempted to talk about negative pressure? As Peter says, there is an equation for the second derivative (roughly, the acceleration) of the universe, which looks like this:
(I use a for the scale factor rather than R, and sensibly set c=1.) Here, ρ is the energy density and p is the pressure. To get acceleration, you want the second derivative to be positive, and there’s a minus sign outside the right-hand side, so we want (ρ + 3p) to be negative. The data say the dark energy density is positive, so a negative pressure is just the trick.
But, while that’s a perfectly good equation — the “second Friedmann equation” — it’s not the one anyone actually uses to solve for the evolution of the universe. It’s much nicer to use the first Friedmann equation, which involves the first derivative of the scale factor rather than its second derivative (spatial curvature set to zero for convenience):
Here H is the Hubble parameter, which is what we mean when we say “the expansion rate.” You notice a couple of nice things about this equation. First, the pressure doesn’t appear. The expansion rate is simply driven by the energy density ρ. It’s completely consistent with the first equation, as they are related to each other by an equation that encodes energy-momentum conservation, and the pressure does make an appearance there. Second, a constant energy density straightforwardly implies a constant expansion rate H. So no problem at all: a persistent source of energy causes the universe to accelerate.
Banning “negative pressure” from popular expositions of cosmology would be a great step forward. It’s a legitimate scientific concept, but is more often employed to give the illusion of understanding rather than any actual insight.
Scott Barry Kaufman is Scientific Director of The Imagination Institute and a researcher in the Positive Psychology Center at the University of Pennsylvania, where he investigates the measurement and development of imagination. Kaufman is also co-founder of The Creativity Post and author of Ungifted: Intelligence Redefined, in which he presents a holistic approach to achievement that takes into account each person’s personal goals, individual psychology, and developmental trajectory. Kaufman regularly gives keynotes and workshops for educators and business leaders on the development of intelligence, creativity, and human potential (see appearances). To book a speaking engagement, contact The Lavin Agency Speakers Bureau atinfo@thelavinagency.com. To reprint one of his articles, contactsales@featurewell.com.
