# Might gravity have mass?

The accelerating expansion of the universe could be explained by modifying general relativity so that gravity has mass – or so thinks a small group of physicists. Matthew Francis reports

When confronted with something unexplained in the data, scientists face several possibilities. Maybe there’s an error and the result is spurious. Maybe there’s a more mundane explanation they simply overlooked. Or perhaps the unexplained is a sign that a theory needs to be revised or supplanted. That last option is the rarest, at least when the theory in question is a successful one. After all, any new theory must explain all the same phenomena an old theory explained, and predict something new that can’t be handled with the old.

One unexplained result that’s been bugging physicists for more than 15 years is dark energy, which is the name we give to our ignorance. The universe is expanding at an accelerating rate, but we don’t know why. To make matters worse, dark energy comprises roughly three-quarters of the total energy content of the cosmos, so it’s not a minor thing we don’t get. For that reason, a small but dogged group of physicists thinks the existence of dark energy might be a clue that we need to revise one of the most successful theories we have: general relativity.

One way to revise general relativity is to modify the nature of the gravitational force so that it behaves as though it has mass. The alteration doesn’t have much effect on the motion of planets in the solar system. The most important consequence is instead at large distances, where the change would throttle the effects of gravity enough to account for the universe’s accelerating rate of expansion; dark energy would no longer be required.

### Holding it together

Gravity is one of the fundamental forces of nature. It literally holds the Earth and all planets together, keeps the solar system cycling predictably over billions of years, and dictates the structure of the universe itself. Isaac Newton and his fellow scientists established the connection between gravity and the motion of planets, providing a deep relationship between astronomy and physics, two fields many ancient thinkers thought were separate. Albert Einstein’s general theory of relativity, the modern theory that describes gravity using the structure of space and time, celebrated its 100th anniversary last year, and is still going strong.

Yet physicists know general relativity isn’t the last word on gravity. For one thing, nobody has yet found a complete and satisfactory quantum theory of gravity, a necessary step towards describing all the forces of nature within a single theory. Any changes to general relativity from quantum gravity, though, would take place on microscopic scales far smaller than anything we can probe in the foreseeable future. Those interested in explaining dark energy think general relativity might also break down on very large scales, bigger than galaxies. That hypothesis has led to a number of alternative theories of gravity, some of which are more radical reimaginings than others.

Physicists know general relativity isn’t the last word on gravity. For one thing, nobody has yet found a complete and satisfactory quantum theory of gravity

The “massive gravity” hypothesis is one such re-imagining, and an active area of research with many research articles published over the last few years. “Until the last five or so years, we didn’t even know that it was a possibility [for gravity to have mass],” says Rachel Rosen, a theoretical physicist at Columbia University in the US. But that possibility has now firmly emerged, spurred on by the desire to solve the looming problem of dark energy.

So what does it mean for gravity to have mass?

### A tangled mass

In the theory of gravity as laid out by Newton, mass is the reason for gravity. An attractive force exists between any two masses and causes both the motion of planets and the falling of objects near the Earth’s surface. The strength of that force decreases with the square of the distance between the masses, so doubling the distance results in a force four times weaker. This “inverse-square law” is what makes Newton’s theory valid for describing the planets and moons of the solar system, for plotting the trajectories of space probes, or for understanding the structure of galaxies.

In some situations, however, Newton’s theory of gravity is not sufficient. General relativity kicks in when gravity is strong – near black holes, neutron stars and other dense objects – or on large scales where the amount of mass in a volume of space reaches a significant point. That’s why general relativity turned cosmology from a branch of philosophy into a branch of science: it showed how gravity governs the cosmos on the largest scales.

But general relativity isn’t merely a slight modification of Newtonian gravity: it’s a fundamentally re-conceived notion of how gravity works. First, anything with energy can produce gravity or be affected by it, without the need for mass. That’s why paths of light are curved by gravity, producing gravitational lensing and other fun phenomena. (This is related to *E* = *mc*^{2}, but not identical to it.) Second, in Newtonian physics, a change in one mass produces an effect instantly through all of space, no matter how far away: if the Sun exploded (not that it will), Newton’s law says we’d feel the gravitational effect immediately, even though light from the Sun takes just over eight minutes to reach us. General relativity predicts that gravity propagates at the same speed as light.

From the quantum perspective, that means gravity behaves like a particle with no mass; this hypothetical particle is called a graviton. Gravitons are to gravity what photons are to light, and the two types of particles have a lot in common. Both are massless, both move at the speed of light and both have two basic types of polarization (though there are also differences that aren’t relevant in the present discussion). The first polarization type is labelled “+”, and it resembles squeezing a circle alternately horizontally and vertically if the wave is moving directly towards you; the second type is “×” polarization, and it’s the same deal, only squeezing the circle at a 45˚ angle. Unlike light, gravity is too weak for us to detect individual gravitons; instead, we see and feel the effects of countless numbers of them working, much as we usually only see huge amounts of photons at once.

But the properties of gravitons are inferred from general relativity, so alternative theories can predict different behaviours. Case Western Reserve University physicists Claudia de Rham and Andrew Tolley, along with colleagues, have worked on various models examining how a graviton with a non-zero mass could solve the dark-energy problem in cosmology. If the graviton has mass, gravity will no longer obey the inverse-square law precisely. Instead, the force will decrease *faster* with distance, depending on the mass of the graviton. A relatively large graviton mass means a sharp cut-off for gravitational attraction at short distances; this is precisely the case for the strong force binding the nucleus of an atom together. A sufficiently small graviton mass, however, will produce a force nearly identical to the predictions of general relativity.

“We wouldn’t want the graviton mass to be much larger than 10^{–32} electron volts or something like that,” says de Rham. For comparison, the electron mass is about 500,000 electron volts, so gravitons would have “the smallest mass you can ever imagine”, she says. That tiny mass is what could produce deviations from general relativity on cosmological distances and time scales.

### Constant solution

Ironically, de Rham, Tolley and their collaborators actually use the graviton to explain why cosmic acceleration is so *small*. According to particle physics, empty space is actually a stew of “virtual particles”, which may be better thought of as *potential *particles: a froth that could produce real particles under the right circumstances. Add up all the contributions from all these virtual particles, and you find that empty space contributes something called a cosmological constant. The cosmological constant looks like dark energy, but if calculations are correct, the universe should have a factor of 10^{100} more dark energy than we see. Since the universe isn’t accelerating that much, this is known as the cosmological constant problem.

“The cosmological constant problem is perhaps one of the most compelling current problems in theoretical physics,” says Rosen. “[To solve it,] we’re looking at every possible approach, but it basically comes down to: either our understanding of quantum theory needs to be modified, or our understanding of gravity needs to be modified.” A popular possibility for the first option is string theory, a modification of quantum theory that allows for a huge number of different cosmological constants. The second option includes the massive graviton hypothesis. If gravity has a cut-off, it would dampen that acceleration down from the amount expected from the cosmological constant to the relatively small amount we observe today. “The cosmological constant could actually be large, as large as particle physics would like it to be, but we observe just a fraction of it,” de Rham says.

There is a catch, however, which lies in how the graviton’s behaviour comes about. In the massive gravity theory that de Rham and Tolley propound, the observable universe is like the surface of a bigger reality, with two or three extra dimensions lying “beneath” what we see. This concept is known as a “braneworld”, where “brane” is short for “membrane”. The mass of the graviton comes from the particular way gravity acts when it is trapped on the surface of the brane that is our universe. In physics terms, we say gravitons “acquire” mass, in somewhat the same way that electrons and quarks acquire mass through interaction with the Higgs field.

As the extra dimensions are the reason for the graviton mass, they are also only indirectly knowable through the effect they have on the force of gravity in our observable universe: there’s no way to independently confirm their existence. But the extra dimensions also bring a cost in terms of complication. General relativity is well known for being difficult to work with, but the massive gravity theory is far more so. The mathematical description of the structure of the universe requires a long chain of reasoning in the massive graviton theory, whereas the general relativity version – one of the huge early successes of the theory – is remarkably straightforward.

However, the real test of any theory isn’t its mathematical simplicity, but how well it matches real-world data. Tolley and de Rham point out that a massive graviton would have an effect on the same gravitational-wave spectrum that the BICEP2 telescope at the South Pole and other experiments strive to measure. The next generation of experiments, then, could conceivably provide a good test if they can overcome BICEP2’s particular difficulties of observing an unwanted foreground of cosmic dust that obscures a possible gravitational-wave signature. The modified gravitational force law would also affect the number and distribution in space of the earliest galaxies, so various astronomers are looking at whether large galaxy surveys are consistent with the massive graviton hypothesis.

However, adding mass to the graviton changes more than just the force law. “The usual massless graviton only has two degrees of freedom, similar to the two polarizations of a photon,” says Rosen. A massive graviton, by contrast, has several more polarizations, and those lead to a number of subtle – but possibly detectable – effects. In addition to the “+” and “×” modes, the circular cross-section of a wave coming towards you could rock back and forth, or shrink and expand. “Adding a small mass to the graviton is not just a large-distance modification. You can see effects at shorter distances potentially as well.”

The proliferation of theories shows both how little we understand dark energy and cosmic acceleration, but also how creative minds are working to resolve it

Gravitational-wave detectors such as the upgraded Laser Interferometer Gravitational-wave Observatory (LIGO) may be able to see some of the differences in these ripples of space–time from these extra polarizations. Surprisingly, though, other experiments within the solar system could provide the fastest answers, even though the effect of massive gravity is very small. Measurements of the Moon’s position using the reflectors placed by the Apollo astronauts would be almost precise enough to see the tiny difference in the Moon’s orbit produced by massive gravity’s different behaviour.

Of course, the massive graviton is one hypothesis among many for explaining dark energy; even the version de Rham and Tolley describe isn’t the only version of massive gravity out there. The proliferation of theories shows both how little we understand dark energy and cosmic acceleration, but also how creative minds are working to resolve it. “Should we believe that massive gravity is the true theory of nature?” Rosen asks rhetorically. “The big deal is that [a massive graviton] is suddenly a possibility, where before we weren’t sure it was a possibility or not. The truth is we don’t know.”

When the data are signposts pointing us into the unknown, it’s hard to know exactly what guide is best to choose. One possibility may be to change – again – the way we think about gravity.

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