Einstein, Lie, and Black Holes

Date: Friday, Sept. 22
Time: 3:30 pm
Location: Room 206 Arts Building, 9 Campus Dr., Saskatoon

Free and open to the public

This talk is supported by funding from the Pacific Institute for the Mathematical Sciences (PIMS).

About this event

Sophus Lie invented what came to be known as Lie groups to study differential equations systematically. However, while Lie groups gained prominence in mathematics and physics, their original application to differential equations fell into obscurity for about half a century. General Relativity is considered the most nonlinear theory of nature that describes the gravitational field by a set of ten nonlinear second-order partial differential equations, the Einstein equations. Although Lie groups were used by relativists in connection with idealized spacetimes from the beginning --- which made Einstein's equations more tractable --- applying Lie groups to study the "hidden" symmetries of Einstein's equations was not considered until Lie’s original ideas were revived and expanded in the 60s. This led to the search for solution generation techniques in general relativity in which one obtains a solution, or a family of solutions, from a "seed" solution that would otherwise be impossible to obtain. This talk will review static Einstein equations with axial symmetry in vacuum, i.e., Ricci-flat metrics with two commuting hypersurface-orthogonal Killing vector fields. It will present a recently found unexpected symmetry of the system that can be effected algebraically to obtain new solutions and will show how such symmetries are useful in studying black holes and other spacetimes of current physical interest as well as Ricci solitons.

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