Generalized Harmonic Gauge Drivers Lee Lindblom Caltech Collaborators: Keith Matthews, Oliver Rinne, and Mark Scheel GR18 Sydney, Australia 13 July 2007 Gauge conditions are specified in the GH Einstein system by the gauge source function H a c c x a. How do you choose H a corresponding to the familiar gauge conditions of numerical relativity without destroying the hyperbolicity of the system? Lee Lindblom (Caltech) Generalized Harmonic Gauge Drivers GR18 7/13/07 1 / 7
Gauge Conditions and Hyperbolicity The principal parts of the GH Einstein equations may be written as ψ cd c d ψ ab = a H b + b H a + Q ab (ψ, ψ), where ψ ab is the spacetime metric. These equations are manifestly hyperbolic when H a is specified as a function of x a and ψ ab : H a = H a (x, ψ). In this case the principal parts of the GH Einstein system are simple wave operators on each component of ψ ab : ψ cd c d ψ ab = ˆQ ab (x, ψ, ψ). Lee Lindblom (Caltech) Generalized Harmonic Gauge Drivers GR18 7/13/07 2 / 7
Imposing Useful Gauge Conditions Specifying the gauge source function H a places constraints on the derivatives of the spacetime metric: H a = c c x a = ψ bc Γ a bc Γ a. where Γ a bc is the Christoffel symbol. The quantity Γ a depends on the time derivatives of the lapse N and shift N i. For example, Γ t = N 3( ) t N N k k N + N 2 K, where K is the trace of the extrinsic curvature. One could impose the slicing condition t N N k k N = 2NK, for example, by setting H t = N 2 (N 2)K. Unfortunately this choice has the form H a = H a (x, ψ, ψ) which destroys the hyperbolicity of the GH Einstein system. Lee Lindblom (Caltech) Generalized Harmonic Gauge Drivers GR18 7/13/07 3 / 7
Solution: Gauge Driver Equations Elevate H a to the status of a dynamical field (Pretorius) and evolve it along with the spacetime metric ψ ab : ψ cd c d H a = Q a (x, H, H, ψ, ψ), ψ cd c d ψ ab = Q ab (x, H, H, ψ, ψ), Any gauge driver of this form produces a symmetric hyperbolic combined evolution system. Choose Q a so that H a evolves toward the desired gauge target F a as the system evolves: H a F a. For example, consider the simple gauge driver: ψ cd c d H a = Q a = µ 2 (H a F a ) + 2µN 1 t H a. For constant F a, this gauge driver causes H a F a + O(e µt ). Lee Lindblom (Caltech) Generalized Harmonic Gauge Drivers GR18 7/13/07 4 / 7
Improved Gauge Driver Equations In the time independent (but spatially inhomogeneous) limit, the simple gauge driver evolution equation reduces to, ψ jk j k H a = µ 2 (H a F a ). so this equation does not achieve H a F a unless k H a = 0. The system can be improved by adding a time averaging field, t θ a + µθ a = ψ jk j k H a, which is used to modify the gauge driver: ψ cd c d H a = Q a = µ 2 (H a F a ) + 2µN 1 t H a + µθ a. The resulting gauge driver system is symmetric hyperbolic for all F a = F a (x, ψ, ψ), has solutions H a that exponentially approach any time independent F a when the background geometry is fixed, and reduces to H a = F a in any time independent state. Lee Lindblom (Caltech) Generalized Harmonic Gauge Drivers GR18 7/13/07 5 / 7
Decoupled Gauge Driver Test: Test the gauge driver equation on a fixed (flat) background spacetime. Use a time and space dependent gauge target: F = [ 3 + e ] (t 10)2 /9 sin(10x). 10 0 10-2 H - F / F 10-4 10-6 10-8 10-10 µ = 1 µ = 2 µ = 4 µ = 10 F = [3 + e -(t-10)2 /9 ] sin(10x) 0 5 10 15 20 25 t Lee Lindblom (Caltech) Generalized Harmonic Gauge Drivers GR18 7/13/07 6 / 7
Fully Coupled Gauge Driver Test: Test the gauge driver equation (with a conformal gamma driver target F a ) coupled to the GH Einstein equations for a perturbed Schwarzschild spacetime: δn i = 0.01ˆr i Y 30 e (r 15)2 /9. 10-1 10-2 10-3 H - F / F 10-4 µ = 0.2 µ = 0.5 10-5 µ = 1.0 µ = 2.0 µ = 5.0 µ = 10.0 10-6 0 20 40 60 80 100 t Lee Lindblom (Caltech) Generalized Harmonic Gauge Drivers GR18 7/13/07 7 / 7