According to Einstein’s theory of General Relativity, gravity is curvature in the fabric of spacetime. Shockwaves can distort spacetime, causing singularities where the laws of physics appear to break down.

Now two mathematicians at UC Davis have come up with equations that remove these singularities. In doing so, they also extend a theorem called Uhlenbeck Compactness to the setting of General Relativity.

“We prove that such singularities can always be removed by coordinate transformation,” said Blake Temple, professor of mathematics at UC Davis. “The proof is based on the discovery of a new elliptic system of partial differential equations and the construction of an existence theory for these equations.”

A coordinate transformation is the set of equations needed to locate the same object in two different systems of coordinates.

The work by Temple and Moritz Reintjes, a former UC Davis graduate student now at the University of Konstanz, Germany, was published Sept. 16 in Proceedings of the Royal Society A.

“The result is so general that the equations apply to the curvature of any metric, (the fundamental mathematical entities which define a geometry), including the dynamical gravitational metrics of General Relativity,’’ Temple said. “A consequence of this is that our equations extend Uhlenbeck Compactness to the dynamical setting of physics.’’

Compactness is a technique for proving that a solution to an equation exists by analyzing a sequence of approximate solutions. Uhlenbeck Compactness shows that one need only analyze the curvature of the approximations. It says that curvature alone sets the bounds on all the derivatives (rates of change) needed to establish that the approximation strategy is valid.

Karen Uhlenbeck discovered and proved this fundamental theorem for static geometries in Yang-Mills theory, Temple said, but it wasn’t previously known whether the principle could be applied to the dynamical spacetime geometries of physics. The new Reintjes-Temple equations show that it can.

“Turns out our method for regularizing singularities at shock waves in General Relativity gives you Uhlenbeck Compactness for free!” Temple said.

— Andy Fell, UC Davis News and Media Relations, wrote this article for the Egghead blog.