Astronomy study proves mathematics theorem

The gravitational lens G2237 + 0305, dubbed the "Einstein Cross", shows four images of a very distant bright galaxy called a quasar whose light has been bent by a relatively nearby galaxy acting as a gravitational lens (Image: NASA)

A gravitational lens can do more than reveal details of the distant universe. In an unexpected collision of astrophysics and algebra, it seems that this cosmic mirage can also be used to peer into the heart of pure mathematics.

In a gravitational lens, the gravity of stars and other matter can bend the light of a much more distant star or galaxy, often fracturing it into several separate images (see image at right). Several years ago, Sun Hong Rhie, then at the University of Notre Dame in Indiana, US, was trying to calculate just how many images there can be.

It depends on the shape of the lens – that is, how the intervening matter is scattered. Rhie was looking at a lens consisting of a cluster of small, dense objects such as stars or planets. If the light from a distant galaxy reaches us having passed through a cluster of say, four stars, she wondered, then how many images might we see?

She managed to construct a case where just four stars could split the galaxy into 15 separate images, by arranging three stars in an equilateral triangle and putting a fourth in the middle.

Later, she found that a similar shape works in general for a lens made of n stars (as long as there are more than one), producing 5n - 5 images. She suspected that was the maximum number possible, but she couldn't prove it.

At about the same time, two mathematicians were working on a seemingly unrelated problem. They were trying to extend one of the foundation stones of mathematics, called the fundamental theorem of algebra.

Algebraic roots

It governs polynomial equations, which involve a variable raised to powers. For example, the equation x³ + 4x - 3 = 0 is a polynomial of degree three – the highest power of x is x³.

The fundamental theorem of algebra, proved back in the 18^th century, says that a polynomial of degree n has exactly n solutions. (Though in general, the variable x has to be a complex number, involving the square root of -1.)

"The fundamental theorem of algebra has been a true beacon, where modern algebra started," says Dmitry Khavinson of the University of South Florida in Tampa, US.

Khavinson and Genevra Neumann of the University of Northern Iowa in Cedar Falls, US, wanted to take this further, by looking at more complicated mathematical objects called rational harmonic functions. These involve one polynomial divided by another.

Upper limit

In 2004, they proved that for one simple class of rational harmonic functions, there could never be more than 5n - 5 solutions. But they couldn't prove that this was the tightest possible limit; the true limit could have been lower.

It turned out that Khavinson and Neumann were working on the same problem as Rhie. To calculate the position of images in a gravitational lens, you must solve an equation containing a rational harmonic function.

When mathematician Jeff Rabin of the University of California, San Diego, US, pointed out a preprint describing Rhie's work, the two pieces fell into place. Rhie's lens completes the mathematicians' proof, and their work confirms her conjecture. So 5n - 5 is the true upper limit for lensed images.

"This kind of exchange of ideas between math and physics is important to both fields," Rabin told New Scientist.

Rhie no longer works in academia, having run out of funding. "I didn't even bother to submit my papers to journals because I had been so much harassed by the referees [of earlier papers]," she told New Scientist. "I was new to gravitational lensing at that time. What I said and the way I said it must have been unfamiliar to the gravitational lensing experts."

Spread out

Theoretically, the work is valid for any type of gravitational lens, but its practical applications are not yet clear. That's because the objects in Rhie's sample lens all lie in the same plane and are simple point sources, with nothing between them.

Actual gravitational lenses tend to be much more complicated, and can be made up of clusters of hundreds of galaxies. These are spread out over large regions of space and contain a lot of gas between and within individual galaxies.

And although there are gravitational lenses comprised of just a few stars or planets, they produce images that are too close together for present-day telescopes to resolve.

But such "microlensing" can reveal the existence of planets around other stars. And in the future, a technique called optical interferometry, which links together the observations of more than one telescope, might make it possible to see the multiple lensed images produced by the planets of another star system.

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