We rely on maps to make sense of the world around us. This isn’t just true for sailors navigating the oceans or families taking to the highways on vacations; it’s true for scientists who use maps of spacetime to study how empty space behaves around celestial objects such as black holes.
But for most maps, points of extreme irregularity pop up. Those points are called singularities.
What are singularities?
Singularities are geometric points where maps break down into distortions.
On our standard two-dimensional map of Earth, singularities occur at the North and South Poles. Divorced from the three-dimensional, spherical shape of the Earth, the latitude lines on a two-dimensional map stretch infinitely, leaving the polar regions inscrutable by the map’s standards.
But the poles aren’t actually distorted, they’re perfectly regular out in the real world.
“We say the poles are removable singularities in the map,” said Blake Temple, a distinguished professor in the Department of Mathematics at the University of California, Davis. “Similarly, the event horizon of a black hole in a map of spacetime is a celebrated removable singularity in Albert Einstein’s theory of general relativity.”
But shockwaves, and the center of a black hole (into which everything eventually crashes after passing through an event horizon), are non-removable geometric singularities in the real world. For physicists studying such phenomena, these essential singularities are mathematically difficult to model. It’s also difficult to distinguish them from the removable kind.
“Arguably, the oldest problem in differential geometry is to determine whether singularities in a map, like the North Pole in the Mercator map, or a black hole or shockwave in a map of spacetime, are removable distortions in the map or true defects in the object mapped,” said Temple.
New equations distinguish real singularities from mapping errors
In a new study appearing in Proceedings of the Royal Society A, Temple and Moritz Reintjes of the City University of Honk Kong, report the discovery of a set of equations capable of untangling the distortions caused by singularities. The equations can determine if a singularity in a map is a removable distortion, such as those that exist at the North and South Poles in the two-dimensional Mercator map, or real.
In mathematical terms, it applies to any map that represents the geodesics of a connection, Blake said.
“We prove that any singularity can be lifted by coordinate transformation, explicitly by our equations,” Temple said. “Our theory tells you whether the problem is with your map or a real singularity in the object being mapped. If it’s a coordinate problem, then our equations provide coordinate transformations which convert your map over to one in which the singularity is as regular as possible.”
In essence, this process smooths out removable singularities (problems with the map) while leaving true singularities, such as shockwaves, intact and discernible.
How this discovery could improve physics research
The equations, Temple said, will help mathematicians and physicists overcome long-standing barriers in accurately mapping true singularities. When trying to model things such as merging black holes in general relativity, the equations can identify and help computer simulations work through non-essential coordinate singularities.
“But the most amazing intellectual consequence of this new theory, not known before, is that every singularity or irregularity in a map can be lifted to a quantifiable highest possible numerical level of smoothness, its essential regularity,” Temple said. “We prove you get the same number starting from any two maps of the same thing.”
“The essential regularity thus represents the discovery of a new geometric property of any object that can be mapped,” Temple said, “and you can compute it using our equations starting from any map at all.’’
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