The same branch of mathematics that helped Einstein to formulate his theory of general relativity could now allow scientists to peer with unprecedented accuracy into impenetrable objects—such as the Earth itself.
In a talk last week, András Vasy of Stanford University presented the work that he and two collaborators produced. The proof hasn’t been thoroughly checked by the mathematics community, but if it holds up, it could solve a decades long problem known as the boundary-rigidity conjecture.
Gunther Uhlmann of the University of Washington, one of Vasy’s coauthors, has been working on the problem for since the late 1990s. But now—for the first time—they’ve solved it for spaces that have three or more dimensions.
Here’s Davide Castelvecchi, reporting for Nature News:
Mathematicians already knew that the way in which curvature varies from place to place inside a ‘Riemannian manifold’ — the mathematical jargon for curved space — determines the shortest paths between any two points.
The conjecture flips things around: it says that knowing the lengths of the shortest paths between points on a boundary essentially determines the curvature throughout. (The geometry is therefore said to be ‘rigid’.) Thus, by measuring how fast waves travel inside a space, one could work out the shortest paths, and theoretically, the overall structure.
As is the case with light rays, sound waves travel differently depending on the substance in which they’re moving. Depending on the density of the object at various points, the shortest path a wave takes won’t always be a straight line. Those deviations reflect the internal structure of the object.
Scientists already use sound waves to map the Earth’s internal structure—geophysicists use them to help map the mantle, for example, and geologists do when prospecting for oil and gas—but until now, we didn’t know the limits of using seismic waves to understand our planet. Could sound waves be enough to fully describe its innards? This proof, if correct, suggests that they will suffice.
There are still some bumps in the road for Vasy and his colleagues proof, which he says he plans on posting to the arXiv preprint server in the next few weeks. The first is the scrutiny it’ll undergo in the hands of fellow mathematicians. The others will be in its application—a seismologist Castelvecchi interviewed said there are some assumptions in the proof that will be hard to implement, especially when using it to try to describe the interior of the entire planet.
Once mathematicians are able to work out all the kinks, though, this discovery appears poised to help refine ultrasound procedures and advance seismological research. And if we could just get the right equipment to promising exoplanets, these mathematics could even give astrobiologists new ways of finding life beyond Earth.