Dark Energy: Too Big To Fail

[1/31/13 update: Many thanks to David Wiltshire for clarifying some points in the comments, and for addressing critics of his ideas. -EC]

According to today’s mainstream science, about 71% of the Cosmos is made of something that scientists have never seen, detected, directly observed, or even indirectly observed. It’s “dark energy,” first described in the late 1990s as a repulsive or anti-gravitational property of empty space. By measuring supernovae in distant galaxies, astronomers inferred that not only is the universe getting larger, the expansion is getting faster with time. Physicists started speculating about what might be responsible for the acceleration, and the newly minted concept of dark energy quickly became the accepted explanation. In fact, today it’s built into the standard cosmological model of a universe that expands with an ever-accelerating rate.

Dark energy has also gone mainstream: As viewers of popular-science TV programs know well, in the very distant future, the repulsive effects of dark energy may become so dominant that they tear apart galaxies, stars, planets — eventually even the atoms that currently make up your body — in a highly dramatic “big rip” that marks the end of the universe.

From the outset, the concept of dark energy seemed to me like a bad idea. We had a set of observations we didn’t fully understand, and in response, we externalized the problem onto some unseen but supposedly real agent “out there” that must be to blame. It didn’t sound like the best approach. It seemed very possible that we misinterpreted the supernova measurements, so why the rush to invent a brand new form of energy (which would need to comprise three-quarters of the universe!) to explain this interpretation? Surely, no one can say that all other alternative explanations had been exhausted. Inventing dark energy was a bit like losing your homework assignment, and then promptly deciding that your family must have an invisible and undetectable dog who ate it.

It turns out that in 2007, David Wiltshire of New Zealand’s University of Canterbury explained how the apparently accelerating expansion of the universe could be an illusion due to our perspective in the Milky Way. According to Einstein’s general theory of relativity, a clock can run at different rates depending on the local strength of gravity; near a massive star, for example, a clock will run more slowly than an identical clock positioned far away. From the perspective of an observer close to the star, everything (including the nearby clock) will appear to be running normally, while the distant clock seems to run fast; to an observer near the clock in deep space, meanwhile, the local time will appear to pass normally as usual, but from that perspective, the gravitationally bound clock seems to run slow. Gravity’s effect on time is universally accepted. If your GPS receiver didn’t account for general relativity, your navigation directions would lose accuracy in minutes.

Wiltshire’s insight was that light from the supernovae traverses huge stretches of space where there is little to no matter, so clocks there would run fast as judged by observers near galaxies, such as ourselves. When astronomers study supernovae on the other side of these voids, the measurements are affected by the fact that the voids are expanding faster than the space closer to us (as judged here on Earth). According to Wiltshire’s calculations, if you lived in one of these voids, you’d judge the universe to be about 18 billion years old — several billion years older than how things appear here, where matter and gravity slow clocks as well as the expansion of space. The end result is that the Milky Way resides in a “Hubble bubble,” a local region beyond which all of the rest of the universe appears to be expanding away, at an accelerating rate. Similarly, if you lived in a galaxy on the other side of a large void, the Milky Way would appear to be accelerating away from you. You would be living in a Hubble bubble there, too.

The first time I read about Wiltshire’s insight, my reaction was disbelief: “Surely they must have thought of that!” But apparently, they hadn’t. The equations that cosmologists use to study the universe’s large-scale structure are extremely difficult to work with, so some simplifying assumptions have been made in order to construct workable models. One of these is to describe space everywhere as a uniformly expanding fluid. This assumption has been a part of the mathematics of cosmology (in the so-called Friedmann equations) since 1922, before we even knew there were other galaxies beyond our own. Given the brutally difficult equations, which could not otherwise be solved, the Friedmann solutions have been a part of cosmology bedrock ever since. However, we now know that the large-scale structure of the universe is not at all uniform; there are walls of galaxy clusters separated by enormous voids, so if Einstein is correct, the universe is not a uniformly expanding fluid. Time flows, and therefore expansion happens, faster in some places than in others.

Wiltshire argues that a necessarily uneven expansion of the universe previously hadn’t been considered because (1) physicists are used to considering relativistic effects on time only in the extreme conditions of black holes and particle-accelerator experiments, and (2) they have assumed that such effects in intergalactic space would be extremely weak. They are indeed weak, but Wiltshire has done the math to show how the cumulative effects become huge over billions of years and across vast distances of space.

According to Google Scholar, Wiltshire’s original paper has been cited 123 times to date; it’s certainly not being ignored. There appear to be few or no papers that refute the concept of non-uniform expansion due to relativistic effects. In fact, most of the papers build upon the idea.

So why, in 2013, is “dark energy” still a thing? Beats me! In his excellent book The Trouble With Physics, Lee Smolin describes how modern physics can unwittingly construct opaque, monstrous theories that become intellectual dead-ends with no practical uses, but which nevertheless persist for sociological and financial reasons. First, a few innocent working assumptions are laid down, and then models are built on those assumptions, and then others build upon those models, and so on — until there’s a mini-industry of researchers working in the field and collecting funding, with the original assumptions rarely re-examined. This is what has happened with dark energy: Even though we now have a viable explanation for the appearance of cosmic acceleration, a bad alternative has become entrenched. Careers have been forged based on the cosmological model that includes dark energy. And in a potential embarrassment for the ages, the 2011 Nobel Prize in Physics was awarded for what the prize committee called “the discovery of the accelerating expansion of the universe.” Dark energy is too big to fail!

Beyond academia, the public never should have been asked to have faith in a mysterious, undetectable force that’s out there but at the same time all around us. Invoking such ideas makes it impossible to maintain the clear distinctions between religion and science, at least in the public eye. But even today, the History Channel tells us “we now know” that most of the universe is made of mysterious dark energy. NASA does the same, on a public-education page called “Universe 101.” Houston, we have a problem.

Perhaps there are physicists who secretly hope the dark-energy concept will gradually die out. But in popular science, which is usually well behind the curve, the myth of dark energy will undoubtedly linger for years on the websites and the TV programs. We deserve better.

Gravity Is Not A Rubber Sheet

Every physics demonstration of gravity uses the familiar “rubber sheet” model: We are shown a stretched piece of rubber, or perhaps the surface of a trampoline. A heavy ball is placed in the middle, distorting the sheet. Now a smaller ball, pushed in the general direction of the heavy ball, will follow a curved path, as if “attracted” by the mass. If given a particular kind of shove, it will circle around the heavy ball for a while, “orbiting” like a planet around a star. Thus the model demonstrates how an object with mass warps the fabric of space, causing the paths of other objects to curve in the direction of the larger object. Objects follow straight paths through space, but if that space happens to be curved by a massive object nearby, their paths will curve. Since Einstein, we’ve known that this is what causes gravitational attraction.

When I was first getting interested in physics, the rubber-sheet model of gravity bothered me. For one thing, it only works in gravity! It seemed that the rolling ball was just curving downhill. Tilt the sheet without warping it, and its path will curve the same way. In the weightlessness of the International Space Station, I figured, the model wouldn’t do anything. I didn’t like that gravity was required in order to demonstrate how gravity works. It was like a model that shows where wind comes from, but which only works when it’s windy.

Something else disturbed me. When the rubber-sheet model is presented in diagram form (in books, for example), the diagrams are often inconsistent. Empty space is depicted as a flat grid of straight lines, but when a massive object is added, some of the lines suddenly form circles. The graph-paper grid turns into a pushed-in dartboard or spider-web pattern, with circular elements representing potential orbits around the mass. Thinking that maybe I had discovered something, I wondered: At what point do the open-ended straight lines of empty space start joining together to form closed circles? If we took an empty region of space and gradually started adding mass to it, when would the circles appear? I was perplexed — the diagrams never show that transition, just the before and after!

What’s wrong with this picture?

The problem of course lies not in Einstein’s theory, but in the rubber-sheet model. It isn’t a perfect analogy for gravity.

It’s a coincidence that real gravity on Earth causes a rubber sheet to warp in a manner that suggests the warping of space. You could just as easily turn the model upside down, and push the ball up against the rubber sheet, and the sheet would be warped in the same way (just in the opposite direction). The rubber-sheet model of gravity is intended to demonstrate how a massive object causes space to curve, so it’s the warping of the sheet that’s important, not the direction.

When a two-dimensional surface is curved into a third dimension, its geometry changes. No longer do the laws of Euclid, which most of us learned in 9th grade, apply: The angles of a triangle do not add up to 90°, for example. In ordinary geometry, two parallel lines never meet; in the non-Euclidean geometry of a curved surface, parallel lines can meet. Imagine that you and a friend began walking from the equator to the north pole. Initially, your paths would be exactly parallel, but since the Earth’s surface curves, the paths would intersect at your destination. Similarly, if two objects were moving in parallel from empty space toward a star, their paths would eventually converge — even with no sideways forces acting upon them.

As it happens, the rubber-sheet model would work in zero gravity, if you warped the sheet with some other force (say, by pushing the end of a broomstick against it), and if you got the rolling ball to remain on the surface somehow (perhaps with a bit of static electricity). In that case, the ball’s path would appear to curve as it attempted to follow a straight line on this non-flat surface. And two balls, nudged along parallel paths toward the depression, would approach each other as the surface under them began to curve.

As for the grid that’s often laid over the rubber sheet, it’s only there to help you see the shape of the surface. The straight or circular lines are a human invention; there is no such grid in space. The actual paths that objects trace through warped space are, well, the actual paths that they trace. These can be circles, ellipses, parabolas, or hyperbolas, depending on the trajectory of the object.

Planets and comets go their own way — they have no use for grid lines.

The rubber-sheet model does give a general idea of how gravity deflects the path of an object. But it’s a crude demonstration, as the Earth’s gravity fouls the geometric effect that the model is intended to demonstrate.* When you see the rolling ball get “attracted” to the larger ball, much of that deflection is just the ball rolling downhill, as it would on a tilted, flat surface. A true tabletop demonstration of gravity, where objects follow stable orbits along a surface due to geometry alone — would be interesting to watch. Until then, don’t take the conventional version too seriously.



* Consider what would happen if you rolled a ball inside the surface of a vertical tube in a frictionless vacuum. Under Earth’s gravity, the ball would inevitably spiral down to the floor. But in zero G, it would follow a circular path forever. This circular orbit, not the spiral, is the accurate representation of the “straight-line path” that would be followed on the surface due only to its geometry.

Resolving the “Twin Paradox”

Fans of science-fiction space travel know that if someone goes on a rocket at close to the speed of light, he will age more slowly than someone back home. Returning to Earth and reuniting with a twin, he would find that the twin had aged more, perhaps by years. If the trip is long enough and gets really close to the speed of light, the returning traveler could find an Earth that’s millions of years in the future, maybe even devoid of human life. For many people, that’s the “twin paradox” — how could something this strange possibly happen?

Actually, twins aging differently isn’t the issue; no paradox there. The “paradox” lies in the fact that according to relativity (the very effect that causes the twins to age differently), the ideas of motion and rest are relative. Suppose rather than leaving from Earth, the experiment is done in deep space. Twin A takes off and leaves Twin B behind. But, once the twins are separated, who’s to say which twin is moving and which is at rest? After all, if you consider the picture from the perspective of either twin, it’s the other one that’s moving. Twin A sees Twin B receding rapidly in the rear-view mirror, just as if Twin B was the one who had taken off. In relativity, the question of moving vs. stationary depends upon the perspective of the observer. Shouldn’t this mean that the aging will be equal for both twins when they finally reunite? That’s the “twin paradox.”

The twin paradox isn’t really a paradox, because it can be resolved in several ways. For one thing, the situation isn’t symmetrical. One of the twins, Twin A, has to turn around at some point, whereas Twin B can just cool his heels. Physicists say that Twin B remains in an inertial reference frame — a state of constant motion (or rest), without any change in speed. Twin A, though, spends portions of his trip in two different inertial reference frames, one on the way out, and one on the way back, and in between he has to slow down, stop, and accelerate in the other direction. (Or, he could make a circular U-turn, but that still counts as an acceleration.) It’s clear how changing velocity affects the local passage of time, from the formula known as the Lorentz transformation: Higher velocities mean slower clocks, so as Twin A slows down, his onboard clock begins speeding up. Twin B’s velocity never changes, though. Therefore, it’s wrong to say the twins are moving in identical ways relative to the other.

Thinking this through, the situation is a bit peculiar. How does Twin A and his mechanical onboard clock “know” that they had turned around? Suppose Twin B, rather than staying put, secretly takes off in the other direction, goes even faster, turns around, and races back to the starting point, just in time to meet the returning Twin A. In that case, Twin B would age slower than Twin A — who, having just gone on a fast trip that included turning around, expects to meet an older Twin B. Instead he finds a younger Twin B. How does Twin B (and his clock) know that he had gone farther and faster than Twin A?

The solution all comes down to the turning-around part: acceleration. When you’re in an inertial reference frame, you can’t tell whether you’re moving or not. You could be sitting in your living room, or you could be on a rocket ship going 99% the speed of light. Close your eyes and the situations are indistinguishable. However, when you’re slowing down or speeding up, or making a turn, you can definitely tell that something’s going on. You feel a pull toward one direction, just as you do in a car with the brakes applied. This acceleration* is what causes a clock (biological or otherwise) to slow down. For Twin B who secretly went on a really fast trip, his larger accelerations slowed down his biological clock even more than Twin A’s, who now needs a facelift to look as young as his twin again.

When you add gravity into the mix, it gets even stranger. Twin A — rather than turning on the reverse-thrust engine to slow down and turn around — could instead have a close call with a massive star, and like Halley’s comet, take a tight orbit around and be fired back toward his starting point. In such a situation, the ship is in free fall with respect to the star, and counter-intuitively, doesn’t feel the acceleration as it gets slung back toward Twin B. Our traveler could continue his game of three-dimensional billiards in the weightless environment of his ship, even as it hooks sharply around the star. If there were no windows, he might not even know when he was passing behind and starting to head back. Yet incredibly, the curvature of space from the star’s gravity would slow down time aboard the ship. While the traveling twin works on his weightless billiards game, the stay-at-home twin would have the time to master not only billiards but also croquet and miniature golf, much to Twin A’s later envy.

The “twin paradox” is one of those cases where the universe just works out perfectly right, out of sheer mathematical consistency. Relative velocity determines the rate at which clocks tick; Einstein showed that this malleability of time is necessary in a universe where the speed of light is measured the same by all observers. Therefore, changing velocity (accelerating) changes the local rate at which time passes. All of this can be calculated from the Lorentz transformation. But a star’s gravity will also slow down time for Twin A and his ship, by exactly the same amount as if he had used his engines to slow down and turn around. The curvature of space due to gravity — by having the ability to sling a spaceship around and back in the other direction, on momentum alone — simply has to make an adjustment to the ship’s onboard clocks. Otherwise the math wouldn’t work out. And then we’d have a real paradox.

Anything in our universe with mass slows down clocks in its wake. For the Earth, the effect is not only measurable, it needs to be built into your GPS to avoid large, cumulative errors that would render it useless. Thanks to Einstein’s discovery, you can reliably arrive at your destination. Isn’t it nice when things work out?



* The term “acceleration” refers to both slowing down and speeding up. Slowing down is simply accelerating in the opposite direction.