Solving Einstein's Equations for Binary Black Hole Spacetimes

General Relativity Theory
Einstein’s theory of gravitation, general relativity theory, is a
geometrical theory in which gravitational effects are described as
geometrical structures on spacetime.
The fundamental “gravitational” field is the spacetime metric ψ ab
— a non-degenerate symmetric tensor field.
Coordinates x a are used to label points in spacetime.
The spacetime metric determines the physical lengths of curves
x a (λ) in spacetime, L 2 = ± ∫ dx
ψ a dx b
ab dλ dλ dλ.
Coordinates x a can be chosen in any convenient way.
For example x a can be chosen at any point in spacetime so that
ds 2 = ψ ab dx a dx b = −dt 2 + dx 2 + dy 2 + dz 2 at that point.
In these special coordinates the second-order differential operator
ψ ab ∂ a ∂ b reduces to the standard wave operator:
.
ψ ab ∂ a ∂ b = −∂ 2 t + ∂ 2 x + ∂ 2 y + ∂ 2 z
Lee Lindblom (Caltech) Binary Black Holes UW Milwaukee 10/14/2011 4 / 32