Solving Einstein's Equations for Binary Black Hole Spacetimes

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Gauge and Hyperbolicity in Electromagnetism Maxwell’s equations split into evolution equations and constraints: ∂ t ⃗ E = ⃗ ∇ × ⃗ B, ∇ · ⃗ E = 0, ∂ t ⃗ B = − ⃗ ∇ × ⃗ E, ∇ · ⃗ B = 0. Introduce Maxwell tensor, F ab with components are ⃗ E and ⃗ B. The Maxwell equations then reduce to ∇ a F ab = 0 and ∇ [a F bc] = 0. Maxwell’s equations can be solved in part by introducing a vector potential F ab = ∇ a A b − ∇ b A a . This reduces the system to the single equation: ∇ a ∇ a A b − ∇ b ∇ a A a = 0. This form of the equations can be made hyperbolic by choosing the gauge correctly, e.g., let ∇ a A a = H(x, t, A), giving: ∇ a ∇ a A b = ( −∂ 2 t + ∂ 2 x + ∂ 2 y + ∂ 2 z ) Ab = ∇ b H. Hyperbolic equations (like Maxwell’s) can be solved by giving values of the fields, A a and ∂ t A a , at an initial time t = 0, and integrating to determine the fields A a for future times t > 0. Lee Lindblom (Caltech) Binary Black Holes UW Milwaukee 10/14/2011 7 / 32
Gauge and Hyperbolicity in General Relativity The spacetime Ricci curvature 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 ad ψ bc Γ d bc . Like Maxwell’s equations, these equations can not be solved without specifying suitable gauge conditions. Lee Lindblom (Caltech) Binary Black Holes UW Milwaukee 10/14/2011 8 / 32
Page 1 and 2: Solving Einstein’s Equations for
Page 3 and 4: History of Numerical Solution of th
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Page 9 and 10: General Relativity Theory Einstein
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Page 13 and 14: General Relativity Theory II The sp
Page 15 and 16: General Relativity Theory III Einst
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Page 19 and 20: Gauge and Hyperbolicity in Electrom
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Page 27 and 28: The Constraint Problem Fixing the g
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Page 31 and 32: Constraint Damping in Electromagnet
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Page 35 and 36: Constraint Damping Generalized Harm
Page 37 and 38: Outline of Talk: Brief History of t
Page 39 and 40: Numerical Solution of Evolution Equ
Page 41 and 42: Numerical Solution of Evolution Equ
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Page 51 and 52: Comparing Different Numerical Metho
Page 53 and 54: Moving Black Holes in a Spectral Co
Page 65 and 66: Horizon Tracking Coordinates Coordi
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Page 71 and 72: Caltech/Cornell Spectral Einstein C
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Accurate Long Waveform Simulations
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Post-Merger Recoils with Spin Merge
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High Spin Evolutions Lovelace, Sche
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Summary The NR community has made g
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Solving Einstein's Equations for Binary Black Hole Spacetimes

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