Applications of symmetry to general relativity

Abstract:
The equations of general relativity are highly nonlinear partial differential equations and require special techniques to solve exactly.  Symmetry considerations have sometimes helped find solutions.  Three examples will be considered here.  The first considers a scaling symmetry in the equations for spherical symmetry with a perfect fluid, which is of historical importance in understanding neutron stars and black holes.  The second involves symmetries in the Ernst equation for stationary axially symmetric fields, which are related to the methods worked out in the 1970s for finding solutions to that equation.  The third example is an ongoing investigation into critical gravitational collapse, a topic of current interest, using a symmetry of the equations and other analytic techniques.