One of science’s most crucial yet underappreciated achievements is the description of the physical universe using mathematics—in particular, using continuous, smooth mathematical functions, like how a sine wave describes both light and sound. This is sometimes known as Newton’s zeroth law of motion in recognition of the fact that his famed three laws embody such functions.

In the early 20th century, Albert Einstein gave a profound jolt to the Newtonian universe, showing that space was both curved by mass and inherently linked to time. He called the new concept space-time. While this idea was shocking, its equations were smooth and continuous, like Newton’s.

But some recent findings from a small number of researchers suggest that randomness is actually inherent in space-time itself, and that Newton’s zeroth law also breaks down, on small scales.

Let’s explore what this means.

First, what is space-time? You probably recall from plane geometry that if you take two points on a plane and draw x and y axes through the first of those points (meaning that it is the *origin*), then the distance between the points is the square root of x^{2}+y^{2}, where x and y are the coordinates of the second point. In three dimensions, the analogous distance is the square root of x^{2}+y^{2}+z^{2}. And these distances are constant; their values don’t change if you draw your axes in some other way.

#### Ingenious: Robbert Dijkgraaf

This past week was the inauguration of Harvard University’s Black Hole Initiative. Stephen Hawking gave a lecture, media was gathered, and millions of dollars committed. A mural was also unveiled, full of fantastical dust swirls, particle jets, and an interstellar...**READ MORE**

Newton’s zeroth law is one of science’s most underappreciated achievements.

What about in four dimensions, where the fourth dimension is time? A point in a 4-dimensional coordinate system is called an *event*: a location specified by x, y, and z, at a particular time t. What then is the “distance” between two events? One might think, by analogy, it would be the square root of x^{2}+y^{2}+z^{2}+t^{2}. But it isn’t. If you draw the coordinates differently, that “distance” changes, so it can’t really be considered a distance. Einstein found that the constant distance was the square root of x^{2}+y^{2}+z^{2} – ct^{2}, where c is the speed of light. If you change the way you draw your coordinate axes, the values of x, y, z, and t will likely change, but the square root of x^{2}+y^{2}+z^{2} – ct^{2} won’t. For Einstein, the x, y, z, and t dimensions were really elements of a single concept, which he called space-time.

Einstein deduced, by a brilliant and highly complex chain of logic, that the explanation of gravity was the *geometry *of space-time itself—its curvature. And that curvature was the result of the presence of mass. According to Einstein, if there were no mass at all in the universe, space-time would be “flat,” meaning without curvature.

To imagine the curvature of space, think of a flat bug on the surface of a sphere. How would the bug know he was not on an infinite plane? If the bug walked in one direction for a while, he’d eventually come back to where he started. Or if, on the surface, the bug drew an x axis and y axis at right angles, he’d find that the distance from the origin to an arbitrary point would *not* be the square root of x^{2}+y^{2}. This clever bug might well deduce he was in a curved space.

So curvature influences the distance between two points, and mass determines the curvature.

That is essentially how Einstein thought about space-time. But his theories of relativity were just one of the two revolutionary triumphs of 20th-century physics; the other was quantum mechanics. It is natural, then, to ask: How does quantum mechanics affect the geometry of space-time? This is one of the biggest questions in physics today. And stochastic space-time seems as if it might well be part of the answer.

Quantum mechanics has at its core the Heisenberg uncertainty principle, which says (among other things) that every physical system must have some residual energy, even when its temperature is absolute zero. This residual energy is called the zero-point energy, and even an “empty” vacuum in space-time has it. In the vacuum, particles and antiparticles continually pop into existence, then collide together and annihilate each other. The sudden appearance and disappearance of particles causes the vacuum zero-point energy to fluctuate in time. Because energy is equivalent to mass (E=mc^{2}), and mass produces space-time curvature, vacuum energy fluctuations produce space-time curvature fluctuations. These in turn cause a fluctuation in the distance between points in space-time, which means that, at small scales, space-time is noisy and random, or “stochastic.” Distances and times become ill-defined.

If we look at the quantum fluctuations in a not-too-small region, the fluctuations within the region tend to average out. But if we look instead at an infinitesimally small region—a point—we find infinite energy. We might wonder: How small is small enough to capture the physics we are interested in, without being so small that energies become enormous—and what is an appropriate unit of measurement to use for that distance?

In the vacuum, particles and antiparticles continually pop into existence.

To answer that question, we follow the line of thinking of Max Planck, arguably the father of quantum mechanics, who wondered what a “natural unit” of distance might be—something not based on an arbitrary standard like meters or feet. He proposed a natural unit expressed using universal constants: the speed of light in a vacuum (*c*); the constant of gravitation, expressing the strength of the gravitational field (*G*); and what we now call Planck’s constant (*h*), expressing the relationship between a particle’s energy and its frequency. Planck found he could construct a distance, now known as the Planck length, *L _{P}*, with the formula

*L*=(

_{P}*hG*/ 2πc

^{3})

^{1/2}.

The Planck length turns out to be a very short distance: about 10^{-35} meters. It is a hundred million trillion times smaller than the diameter of a proton—too small to measure and, arguably, too small to *ever* be measured.

But the Planck length is significant. String theory has done away with points altogether and suggests that the Planck length is the shortest possible length. The newer quantum loop gravity theory suggests the same thing. The problem of infinite energies in very small volumes is neatly avoided because very small volumes are prohibited.

There is another important aspect of the Planck length. Relativity predicts that distances as measured by an observer in a fast-moving reference frame shrink—the so-called Lorentz contraction. But the Planck length is special—it’s the only length that can be derived from the constants *c*, *G*, and *h* without adding some arbitrary constant—so it may retain the same value in all reference frames, not subject to any Lorentz contraction. But the Planck length is derived from universal constants, so it must have the same value in all reference frames; it can’t change according to a Lorentz contraction. This implies that relativity theory does not apply at this size scale. We need some new scientific explanation for this phenomenon, and stochastic space-time might provide it. The idea that the Planck length cannot be shortened by the Lorentz contraction suggests that it is a fundamental quantum, or unit, of length. As a result, volumes with dimensions smaller than the Planck length arguably don’t exist. The Planck length then, is a highly likely candidate for the size of a space-time “grain,” the smallest possible piece of space-time.

Max Planck wondered what a natural unit of distance might be—something based on universal constants.

So now, finally, we can characterize our “stochastic space-time.” First, it is granular, at about the scale of the Planck length.

Second, the distances between these grains are not well-defined. Quantum mechanics says that the more massive an object is, the less pronounced its quantum properties will be. Therefore, we expect that as the mass in a region of space-time increases, the region will become less stochastic. (This is analogous to the case of relativity, where the more mass there is in a region, the more curvature the region exhibits.) We theorize that if there were no mass in the universe, space-time would not be flat, as in Einstein’s relativity, but completely stochastic: effectively undefined. Without mass, why would we need space?

Third, in stochastic space-time, unlike in string theory and quantum loop gravity theory, these grains are able to drift with respect to each other because of the randomness inherent in that size scale. Imagine the grains as a box of marbles. Stochasticity is like gently shaking the box so the marbles can move around. It is hoped that the drifting volume elements (marbles) might explain why relativity theory doesn’t seem to apply at the Planck length. That is because relativity is a theory requiring Newton’s zeroth law, which demands smooth and continuous mathematical functions—but near the Planck length, the smooth functions are thought to break down.

Isaac Newton would be surprised. He supposed that space and time were a featureless void, a mere framework onto which were imposed the equations of his three laws of motion. This is, after all, what each of us sees around us in everyday life. Instead, stochastic space-time theory posits a grainy, uncertain space-time beyond the reach of smooth, continuous functions.

The hope is that the equations of quantum mechanics will be derivable from the properties of space-time itself—not a roof thrown randomly on top of a building, but rather a beam built into the very foundation.

*After receiving his doctorate in theoretical physics, **Carl Frederick** first worked as a researcher at NASA and, after that, at Cornell. He now works at a high-tech start-up and is a pro science-fiction writer. *