Friday, June 11, 2010

Deriving Part of the Kerr Geometry - Greg Egan

"In Chapter 20 of Incandescence, the Splinterites derive a spacetime geometry that is symmetrical under rotations around a single axis. This is a much harder feat than the case of a geometry that has spherical symmetry. In our own history, the spherically symmetrical case was solved by Schwarzschild in 1915, but the Kerr spacetime, with axial symmetry, was not found until 1963! The Kerr solution describes a black hole formed by the collapse of a rotating star, and the imprint of the star's angular momentum can be seen in the geometry surrounding the hole.

The spherically symmetrical problem is essentially one-dimensional; everything depends only upon r, the radial coordinate, and so it boils down to solving equations for functions of one variable, known as ordinary differential equations. In the axially symmetrical version, the geometry depends not only upon r, but also upon the “latitude” of each point, which measures its relationship to the axis of symmetry. So we're forced to solve equations for functions of two variables: partial differential equations. These are not so simple to deal with."

5 out of 5

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