Lab Notes
The New Four-Dimensional World

Please welcome guest columnist Gregory Benford, who is filling in this month for Wil McCarthy, our regular Lab Notes columnist. Wil McCarthy will be returning next month with his regularly scheduled column.

Since Edwin Abbott's fantastic 1884 novel Flatland, a curious notion has drifted at the edge of scientific thinking—that our three-dimensional world might be embedded in a larger, four-dimensional universe. Now it has become the hottest idea in theoretical physics, with huge implications.

Physicists began envisioning higher dimensions because they got from it a simpler dynamic picture, at the price of apparent complication. More dimensions to deal with certainly strains the imagination, but can lead to beauties that theorists love, abstruse elegances. Einstein, in his 1916 theory of general relativity, invoked the simplicity that objects move in "geodesics"—undisturbed paths, the equivalent of a straight line in Euclidean, rectangular 3-D geometry, or a great circle on a sphere—in a four-dimensional (4-D) space-time. He gained the clarity of concept in return for the complication of a higher dimension.

Matter curved his 4-D space-time, an effect we know as gravity. This replaced a classical idea, force, with a modern geometrical view, curvature of a 4-D world. Now our hottest grand Theory of Everything replaces Einstein's picture of point particles moving in a geometric world with vibrating strings. Such tiny objects have tension and so can hum with characteristic tones like piano strings. These timbres correspond to the masses of elementary particles. Presto, string theory yields every particle we know, plus gravity (communicated by a particle called the "graviton") and even more.

The payoff of extra dimensions? Just maybe, an explanation of literally Everything.

The dimensions are limitless

The trick is that these strings are incredibly tiny and live in a 10-dimensional space—the four we are now used to, including time, plus six more. How come we don't see them? Because they rolled up into infinitesimal size at the very birth of the Big Bang, so we haven't noticed them.

In Flatland, Abbott made Flatlanders real by adding a tiny height to them, so they do exist in our 3-D world—though they don't know it. This was strikingly prescient. String theories use an analogy with a garden hose. From far away the hose looks like a 1-D object, a string—but up close, we see that it has another dimension, perpendicular to the long direction. This direction is one of the extra dimensions, unobservably small—perhaps a billion, billion, billion times more tiny than an atom.

Perhaps we are living in a universe only apparently spatially three-dimensional; infinitesimal but real dimensions lurk all about us. Sweep your arm and you move through the six invisible dimensions, wrapping around them many times, like an insect circling that garden hose. You take no notice.

Why are the extra dimensions tiny? So far, this occurs by what some dismiss as "wantum mechanics"—we want it, so it must happen. Current theories have hand-waving explanations that roll up dimensions by tying them like a roll of dollar bills, with strings like rubber bands. Some such processes must have happened. Without them we would end up with unworkable universes which could not support life. For example, in field theories with more than three persisting (not rolled-up) spatial dimensions there could be no stable atoms, and thus no matter more complex than single particles.

Why stop at only three spatial dimensions, then? Only in odd-numbered dimensions can waves propagate sharply, without weird reverberations that could make life impossible by confusing the physical world. So for smart life like us, 3-D is favored over 2-D. In this view, we live not only in the best of all possible worlds, but the only possible one. Maybe others exist, beyond our seeing—but they're boring.

But not all the items theorists exclude from the Menu of Universes are dull. You might innocently ask why the extra dimensions must be spatial at all. Why not an extra time dimension? We know the idea of repeating a pattern in space, returning to the same spot. The bug on the garden hose can circumnavigate its world and come home again—but what would moving through an extra dimension in time feel like? The bug would return to a prior moment—time travel. Again, we would not notice our arm sweeping through tiny slices of past time. But events could cycle in time, changing the way atoms or particles work together. This weird notion so violates our intuition that few string theorists have taken it up. There seems no fundamental reason why it should be excluded, but it's, well, disturbing. Even theory has its limits.

Dimensions are a matter of gravity

Can such small dimensions matter? They do lead to beautiful equations, but the larger picture could lead to major changes in the way we think.

Gravity is key—it may be the only force that senses these other dimensions, and so on such small scales gravity will change. We've never noticed it in the motions of planets, but on tiny scales we can think of the extra dimensions as added space to stuff the gravitational force into. Picture gravity as spreading over an expanding area, the way light dims far from a street lamp. Get closer and the attractive force rises.

Now add extra dimensions. The closer two small, gravitating masses are, the steeper their gravitational attraction gets, because it has extra room to spread. This means stronger attraction, so gravity can draw the masses tighter together. Near enough, and gravity wins over any repulsive force (say, if the particles had the same charge). Presto, the amplified gravity will slam the particles together, forming a black hole.

This is a way of making microscopic black holes. Smack particles together at high energies and see if a new kind of "particle" emerges—infinitesimal black holes that last only an instant, allowing us a look at such exotic beasts—before they evaporate in a spray of radiation.

All the speculations about how to use black holes have focused on holes with the mass of stars. Tiny holes could be both disposal sites and energy sources. Throw garbage down one and it's gone forever—but in the last instant before vanishing down gravity's gullet, the heated mass radiates back out to us high-energy jets of light. Harnessing this might be possible, making a genuinely nonpolluting energy source.

Such practical ideas are fine, but the wonder of living in a many-dimensioned world is more fun.

Some recent theories believe there may be very large extra dimensions, not rolled up at all. These would be spaces above and beyond our own.

Seeing the world with 4-D eyes

Edwin Abbott taught us to think about a 2-D being suddenly moving through our 3-D world. That Flatlander would see only slices of our reality—cross sections of trees and rocks and moving cars. How could it stitch this into a coherent view? If it did, it could make a 2-D symbol or picture and we would understand it as a flat scene. But for the 2-D creature what we took as an image would be the whole object, not just a photograph.

Similarly, we (the dimensionally destitute trapped in our narrow 3-D) could not see dimensions above. To process light we use a basically 2-D retina fit at the back of a spherical eyeball. Then we reconstitute in our brains our 3-D world picture.

A 4-D creature must have a 3-D eye, then. A retinal sphere could provide the same service to image its 4-D world. In this way, maybe we can intuit 4-D, even if we can't see it. A Flatlander would see a human finger as a 2-D blob that changed size and shape as the finger thrust down through its world, punching through the 2-D sheet it lived on. We would see 3-D blobs appearing mysteriously, moving past in confusing ways because we could not get how they fitted together into something that extended away, in an unseen direction. If the 4-Ders spoke, we would hear a strange symphony of booms and clatters and screeches coming from the air all around, and even from inside us. These would be sounds as they are in 4-D, where the waves spread out in a different way, in packets and eddies.

Some distinctions in 3-space mean nothing when viewed in 4-D. Consider 2-D first—say, the sense of left and right. Picture a hand drawn on transparent glass. Viewed from one side of the glass, it looks right-handed. From the other side, it looks left-handed. Handedness is a 2-D distinction. Being able to look at a 2-D difference fully makes it trivial to the 3-D person viewing it.

A 4-D entity looking at 3-D objects has a wider understanding. It sees all aspects of our 3-D world, simultaneously, with nothing blocked by angle of view.

When we look at a 2-D painting, we see everything in it from one viewpoint. Suppose you go to an art exhibit to see a sculpture of, say, a woman. There are 10 copies of it, each one rotated a bit, standing against a wall. You stand in one place and can look at the entire woman without moving. This has been tried. People who see such a sculpture often do not recognize the ten different angles as showing the same object.

Maybe this is something like being a 4-D creature, which can then see everything in a 3-D scene, without moving viewpoint.

Surprised by strange perspectives

Perhaps the most intriguing aspect of all these new theoretical models is what they don't explain.

If there are large extra dimensions, why don't we see any signs? Blobs appearing from nowhere, for example.

We don't intervene in any 2-D worlds, because we haven't found any. Geometrically, this may mean that we are isolated on a membrane ("brane theory") and can't reach down to a 2-D world. We're tightly bound to our 'brane. Similarly, 4-D spaces may not be able to reach into ours. When we know more about the forces keeping us in our 3-D universe, maybe we'll discover a way to venture into both 2-D and 4-D universes.

Or maybe we don't recognize the signs of intrusion? In our 3-D world, two 2-D planes can intersect, meeting along a line. Similarly, two 3-D spaces can intersect, meeting only in a plane. Clear evidence that somebody is tinkering from 4-D would be the sudden appearance of a plane slanting across our space, extending to infinity, mottled with passing images. When you approach the plane you can feel the objects in it. Though you flatten yourself against the plane, you cannot fully enter the other realm, because you can only insert a slice of yourself at a time.

Nobody sees such things in everyday life, of course. But maybe we're not making allowances for how utterly different a 4-D world might be. Objects that appear only on a plane would be mysterious images without solid manifestations. We would see them moving, but not find any hard evidence.

This is oddly like the many UFO sightings that never yield solid proof. What if, for reasons we do not fathom, 4-D beings appear only in our skies, not on the ground? Then they might seem to be cruising forms, ships, glowing lights—then gone, vanished into other 4-D realms we do not know.

This is just a speculation, but it shows how little we truly know about the possibilities of 4-D, or more-D. As theories mount, we should keep in mind that as mind-wrenching a notion as higher dimensions will have ramifications far beyond the conventional.

They may be hard to recognize. The most valuable tool in the scientist's kit is the ability to be surprised.