Solving Einstein's Equations for Binary Black Hole Spacetimes

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Einstein’s Equation with the GH Method The spacetime Ricci tensor can be written as: R ab = − 1 2 ψcd ∂ c ∂ d ψ ab + ∇ (a Γ b) + Q ab (ψ, ∂ψ), where ψ ab is the 4-metric, and Γ a = ψ bc Γ abc . The Generalized Harmonic Einstein equation is obtained by replacing Γ a with −H a (x, ψ) = −ψ ab H b (x, ψ): R ab − ∇ (a [ Γb) + H b) ] = − 1 2 ψcd ∂ c ∂ d ψ ab − ∇ (a H b) + Q ab (ψ, ∂ψ). The vacuum GH Einstein equation, R ab = 0 with Γ a + H a = 0, is therefore manifestly hyperbolic, having the same principal part as Maxwell’s equation: ∇ c ∇ c A a = ψ cd ∂ c ∂ d A a + Q a (A, ∂A) = ∇ b H. Einstein’s equations can be solved by specifying initial values of ψ ab and ∂ t ψ ab (subject to the constraints C a = Γ a + H a = 0 and ∂ t C a = ∂ t Γ a + ∂ t H a = 0) at t = 0, and then integrating to find the solutions for t > 0. Lee Lindblom (Caltech) Binary Black Holes UW Milwaukee 10/14/2011 9 / 32
The Constraint Problem Fixing the gauge in an appropriate way makes the Einstein equations hyperbolic, so the initial value problem becomes well-posed mathematically. In a well-posed representation, the constraints, C = 0, remain satisfied for all time if they are satisfied initially. Lee Lindblom (Caltech) Binary Black Holes UW Milwaukee 10/14/2011 10 / 32
Page 1 and 2: Solving Einstein’s Equations for
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Page 9 and 10: General Relativity Theory Einstein
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High Spin Evolutions Lovelace, Sche
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Solving Einstein's Equations for Binary Black Hole Spacetimes

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