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A novel solution to Einstein’s gravitational equations, discovered in 1963, turned out to describe the curvature of space around every astrophysical black hole.

General relativity describes how mass and energy induce curvature in spacetime. The math is intricate, so exact solutions to Einstein’s equations are rare. One important solution appeared in Physical Review Letters in 1963. Analysis over the following years showed it to be the unique description of curved spacetime around a spinning black hole. Since all black holes spin, this solution has been essential to astrophysicists studying the behavior of black holes and of matter in their vicinity.

Solutions to Einstein’s general relativity equations describe the curvature of space with a mathematical function called the metric tensor, or “metric.” Given the coordinates of two points in space, the metric tells you how to compute the distance between them, since the usual Pythagorean theorem doesn’t apply in curved space. On the surface of a sphere, for example, the Pythagorean answer is always too small.

Einstein published his theory of gravitation in 1916. That same year, the German mathematician Karl Schwarzschild found a spherically symmetric solution for empty space, with no time variation but with curvature everywhere. Mathematicians eventually showed that Schwarzschild’s metric had a central singularity—corresponding to a finite mass packed into a point—and a surrounding spherical surface called the “event horizon” at what became known as the Schwarzschild radius. This metric is now known as the description of a static black hole (see 2004 Focus Landmark).

In 1918 theorists used approximate methods to show that a rotating mass also distorts spacetime via an effect called frame dragging [1]. An example is that bodies traveling around the Earth on identical orbits, but in opposite directions, will measure slightly different times for one circuit. A complete solution to the Einstein equations for a rotating body would have the symmetry of a cylinder, but even this modest departure from spherical symmetry made solving the equations fearsomely difficult.

Roy Kerr, a New Zealand relativist at the University of Texas in Austin and Wright-Patterson Air Force Base in Ohio, came at the problem from a different angle. Exploring a certain class of metrics, he found an exact solution with two free parameters. One corresponded to the mass parameter in the Schwarzschild metric, but the significance of the other was not obvious. By examining the form of the new solution at large distances from the origin and by comparing it to known, approximate solutions for a rotating object, Kerr showed that the second parameter represented angular momentum—essentially, the amount of spin [2].

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