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The Sands of Spacetime: Researchers investigate two of physics’ greatest problems

Each year, nearly one million visitors are left breathless by the sand dunes of Death Valley in California, stunning structures that curve gracefully, rippling upwards to an impossibly crisp ridge winding its way down the length of each dune. To a distant observer, they could be a single solid mass that morphs and grows imperceptibly over the course of time. To a physicist, though, they could be a model of spacetime itself.

The sweeping dunes of Death Valley appear uniform but are actually made of countless sand particles. Image credit: Brocken Inaglory (via Wikimedia Creative Commons)

Ralph Bagnold was one of the first physicists to acknowledge what any family of tourists can see clearly: these massive structures are made of countless tiny grains of sand. He spent his life working to perfect a mathematical description of dunes based on their granular, not fluid, structure and correctly accounted for the rippling features that characterize many dune surfaces.

Spacetime is generally regarded in Einstein’s famous general relativity as perfectly smooth, if curved here and there. But some physicists think that it may actually be granular on the smallest of scales. Like Bagnold, these researchers look beyond the smooth big-scale structures and analyze the effect of each tiny grain. Although this idea is not yet mainstream in the physics community, a recent Physical Review Letter hints that granular spacetime could—just maybe—solve two of the most pressing problems in astronomy today.

The first is the inconsistency between two otherwise robust mathematical frameworks: general relativity and quantum mechanics. General relativity describes the behavior of mass and gravity through the introduction of a warped spacetime, while quantum mechanics focuses instead on the behavior of minuscule particles. Each works exceedingly well within the confines of its own regime, but the problem arises where a system has a very large mass in a very small space—for example, at the time of the Big Bang or at the center of a black hole. There, physicists find, the theories break down into mathematical gibberish, often flatly contradicting each other. The search is on for a so-called “Grand Unification Theory” to unite general relativity and quantum mechanics, but although theories abound, none has been satisfactorily proven.

The second problem is the expansion of the Universe. We have known for nearly a century that the space between galaxies is rapidly growing, but it was only a few decades ago that astronomers realized that this growth is actually accelerating, throwing scientists into great consternation. “A Universe that is expanding should be slowing down because gravity is attractive,” says Alejandro Perez of Aix-Marseille University. In the same way that an apple tossed in the air slows down before reversing its course, astronomers expected the expansion of the Universe to decelerate, not speed up. In response to this conundrum, physicists did the only logical thing they could think of: they added a mathematical term to counteract the effect of gravity and gave it a placeholder name, dark energy. Although it makes up 70% of the Universe, no one knows what dark energy is or why it exists, a challenge known as the dark energy problem.

So here we have two of the biggest outstanding problems in physics. Many brilliant minds have set themselves to the task of developing candidate theories to explain one or both of these mysteries, though none has been fully accepted in the scientific community. As Perez explains, however, among these theories certain concepts tend to pop up over and over again. “These theories are candidate theories, they are tentative theories, many questions remain open,” he cautions. “But there is a common idea from these theories, that is that spacetime might be discrete.”

According to some varieties of quantum gravity—one possibility for the Grand Unifying Theory—space is made up of a mind-boggling number of tiny particle-like entities, each on the order of 10-35 meters (technically speaking, the Planck length). As matter moves through spacetime, it hops from one of these particles to another—there is no such thing as “in-between”. We are so large in comparison with the granular structure that we only see the large-scale, apparently smooth curvature of spacetime, but that’s only part of the picture—it’s like studying a sand dune without considering the effect of each grain.

Spacetime is traditionally considered by physicists to be a continuous entity that distorts in the presence of matter; this image shows a two-dimensional visualization of this effect. However, it may actually be that spacetime is granular, not smooth. Image credit: Johnstone (via Wikimedia Creative Commons)

And these effects could be paradigm changing. Imagine riding a bicycle along the sandy base of a dune. If at any point you decide to stop pedaling, you will soon come to a stop as the kinetic energy of the bicycle is slowly lost, transformed into heat and sound energy and transferred to the surrounding air and sand. In a similar way, if spacetime is granular, the math suggests that small amounts of energy would be transformed away from matter—and that it would begin to behave exactly like dark energy.

Of course, there is an obvious problem with this theory: if energy were being “lost” into spacetime, wouldn’t we have noticed? Instead, all of physics is built upon the notion that energy cannot disappear, an idea known as conservation of energy. Technically speaking, this theory doesn’t do away with the conservation of energy, since the energy is merely transformed, but the fact remains that it would disappear from our measurements. In their recent paper, Perez and his colleague, Daniel Sudarsky of the Universidad Nacional Autónoma de México, tried to shed some light on this question with a series of order-of-magnitude calculations.

To begin with, they reasoned, spacetime would be granular only on the very smallest of scales, far smaller than we could hope to measure. The energy transferred away from matter as a result of this granular structure must be correspondingly minuscule. They also calculated that the amount of energy lost is proportional to density squared; since the modern universe is relatively rarified, current energy losses would be tiny. In fact, the entire planet Earth would take 10 million years to lose the energetic equivalent of a single electron mass through this process! Current technology is nowhere near the capabilities that would be required to measure such tiny effects, and so it would be impossible for researchers to measure in the lab.

But, tiny effects can accumulate into larger ones. Starting at a 10-11 seconds after the Big Bang*, Perez and Sudarsky added together all of the energy expected to have been lost in the Universe to date. Very little is known about the theoretically granular nature of spacetime; “We have only hints about it,” Perez comments. Nevertheless, using these hints they were able to produce an order of magnitude estimate of the result by introducing only a single parameter encoding the theoretical uncertainties.

The exciting thing about their result is that the energy lost through this mechanism corresponds to the dark energy observed in the Universe today for this free constant of order unity! “If this is correct, it would be the first observable manifestation of quantum gravity,” Perez says. Not only that, it would solve the mystery of dark energy—effectively killing two very elusive birds with one beautiful theory.

Of course, this is all still speculation. Perez is the first to admit that the theory needs more work, which could lead to testable predictions. He is particularly interested in what the implications are for black holes, which formally have infinite density at their singularities. Does this mean that dark energy is being produced at infinitely high rates in these regions? He shakes his head. “I don’t know the answers to these questions,” he says. However, if astronomers discover higher concentrations of dark energy surrounding black holes, that could be a point in favor of this theory.

While this idea is still a long way from becoming an accepted part of the cosmological model, it is awfully intriguing. Perez thinks of it in terms of Planck’s original hypothesis of energy quantization, the implications of which were not fully understood until years later. “We don’t really understand the physics at the Planck scale,” he says “but I would say that nobody understands the physics at the Planck scale.” Maybe it’s just a matter of time.

–Eleanor Hook

*The Big Bang presents an interesting nuance to this theory. Recall that the energy lost is proportional to matter density squared, so when the Universe began with an infinite density one would expect an infinite amount of energy to be transferred into spacetime—not a very helpful (or accurate) prediction. Fortunately, according to their ideas, the equations only involve the density of massive particles with spin, which did not emerge until after the density of the Universe had become finite.


  1. my 1987 work The Fundamental Quanta covered this: motion is the fundamental quanta, not an attendant property of forces or matter; it has an indivisible unit size; it has a shape (generally spheroid); and gives rise to all forces and matter - it is the stuff of which they are made ... further, there is no space and there is no time, hence no 'fabric' or space-time and gravitation is obviously a two-fold force and centripetal and centrifugal forces are direct manifestations of 'falling in' we know from Newton's apple and 'spatial distention' which spread things out

  2. Polite cough: spacetimes models space at all times. It’s a static abstract thing. Space isn’t. Talking of which Einstein said a gravitational field was a place where space is “neither homogeneous nor isotropic”, and where the speed of light is “spatially variable”. That’s why light curves. Like any other wave curves when there’s an orthogonal gradient in wave speed. At the black hole event horizon the coordinate speed of light is zero, and it can’t go lower than that. So there’s no more gravity, and no central singularity. We have no evidence whatsoever that space is in any way granular. It isn’t going to solve the problem that arises when a system has a very large mass in a small space. Understanding general relativity solves that. And understanding that Einstein considered space to be some kind of gin-clear ghostly elastic continuum tells you an awful lot about dark energy. It seems to be both Schrödinger’s cosmic pressure, and the reducing tensile “strength of space” mentioned on page 5 of Milgrom's MOND paper: . Meanwhile quantum gravity is a wild goose chase based on a misunderstanding of both general relativity and electrodynamics. See my physics detective articles for details:


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