Solving Einstein s Equations: PDE Issues

Transcription

1 Solving Einstein s Equations: PDE Issues Lee Lindblom Theoretical Astrophysics, Caltech Mathematical and Numerical General Relativity Seminar University of California at San Diego 22 September 2011 Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 1 / 29

2 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 symmetric (ψ ab = ψ ba ) non-degenerate (ψ ab v b = 0 v a = 0) tensor field. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 2 / 29

3 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 symmetric (ψ ab = ψ ba ) non-degenerate (ψ ab v b = 0 v a = 0) tensor field. The metric ψ ab defines an inner product, e.g. ψ ab v a w b, which determines the physical angles between vectors for example. The spacetime metric determines the physical lengths of curves x a (λ) in spacetime, L 2 = ± ψ ab dx a dλ dx b dλ dλ. Coordinates 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. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 2 / 29

4 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 symmetric (ψ ab = ψ ba ) non-degenerate (ψ ab v b = 0 v a = 0) tensor field. The metric ψ ab defines an inner product, e.g. ψ ab v a w b, which determines the physical angles between vectors for example. The spacetime metric determines the physical lengths of curves x a (λ) in spacetime, L 2 = ± ψ ab dx a dλ dx b dλ dλ. Coordinates 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. The tensor ψ ab is the inverse metric, i.e. ψ ac ψ cb = δ a b. The metric and inverse metric are used to define the dual transformations between vector and co-vector fields, e.g. v a = ψ ab v b and w a = ψ ab w b. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 2 / 29

5 General Relativity Theory II The spacetime metric ψ ab is determined by Einstein s equation: R ab 1 2 Rψ ab = 8πT ab, where R ab is the Ricci curvature tensor associated with ψ ab, R = ψ ab R ab is the scalar curvature, and T ab is the stress-energy tensor of the matter present in spacetime. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 3 / 29

6 General Relativity Theory II The spacetime metric ψ ab is determined by Einstein s equation: R ab 1 2 Rψ ab = 8πT ab, where R ab is the Ricci curvature tensor associated with ψ ab, R = ψ ab R ab is the scalar curvature, and T ab is the stress-energy tensor of the matter present in spacetime. For vacuum spacetimes (like binary black hole systems) T ab = 0, so Einstein s equations can be reduced to R ab = 0. For spacetimes containing matter (like neutron-star binary systems) a suitable matter model must be used, e.g. the perfect fluid approximation T ab = (ɛ + p)u a u b + pψ ab. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 3 / 29

7 General Relativity Theory II The spacetime metric ψ ab is determined by Einstein s equation: R ab 1 2 Rψ ab = 8πT ab, where R ab is the Ricci curvature tensor associated with ψ ab, R = ψ ab R ab is the scalar curvature, and T ab is the stress-energy tensor of the matter present in spacetime. For vacuum spacetimes (like binary black hole systems) T ab = 0, so Einstein s equations can be reduced to R ab = 0. For spacetimes containing matter (like neutron-star binary systems) a suitable matter model must be used, e.g. the perfect fluid approximation T ab = (ɛ + p)u a u b + pψ ab. The Ricci curvature R ab is determined by derivatives of the metric: R ab = c Γ c ab a Γ c bc + Γ c cdγ d ab Γ c adγ d bc, where Γ c ab = 1 2 ψcd ( a ψ db + b ψ da d ψ ab ). Einstein s equations are second-order pde s that (should, hopefully) determine the spacetime metric, e.g. in vacuum R ab ( ψ, ψ, ψ) = 0. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 3 / 29

8 General Relativity Theory III Einstein s equations are second-order PDEs that (should, hopefully) determine the spacetime metric, e.g. in vacuum R ab ( ψ, ψ, ψ) = 0. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 4 / 29

9 General Relativity Theory III Einstein s equations are second-order PDEs that (should, hopefully) determine the spacetime metric, e.g. in vacuum R ab ( ψ, ψ, ψ) = 0. What are the properties of these PDEs? How do we go about solving them? What are the appropriate boundary and/or initial data needed to determine a unique solution to these equations? Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 4 / 29

10 General Relativity Theory III Einstein s equations are second-order PDEs that (should, hopefully) determine the spacetime metric, e.g. in vacuum R ab ( ψ, ψ, ψ) = 0. What are the properties of these PDEs? How do we go about solving them? What are the appropriate boundary and/or initial data needed to determine a unique solution to these equations? The important fundamental ideas needed to understand these questions are: gauge freedom, and constrints. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 4 / 29

11 General Relativity Theory III Einstein s equations are second-order PDEs that (should, hopefully) determine the spacetime metric, e.g. in vacuum R ab ( ψ, ψ, ψ) = 0. What are the properties of these PDEs? How do we go about solving them? What are the appropriate boundary and/or initial data needed to determine a unique solution to these equations? The important fundamental ideas needed to understand these questions are: gauge freedom, and constrints. Maxwell s equations are a simpler system in which these same fundamental issues play analogous roles. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 4 / 29

12 Gauge and Hyperbolicity in Electromagnetism The usual representation of the vacuum Maxwell equations split into evolution equations and constraints: t E = B, E = 0, t B = E, B = 0. These equations are often written in the more compact 4-dimensional form a F ab = 0 and [a F bc] = 0, where F ab has components E and B. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 5 / 29

13 Gauge and Hyperbolicity in Electromagnetism The usual representation of the vacuum Maxwell equations split into evolution equations and constraints: t E = B, E = 0, t B = E, B = 0. These equations are often written in the more compact 4-dimensional form a F ab = 0 and [a F bc] = 0, where F ab has components E and B. 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. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 5 / 29

14 Gauge and Hyperbolicity in Electromagnetism The usual representation of the vacuum Maxwell equations split into evolution equations and constraints: t E = B, E = 0, t B = E, B = 0. These equations are often written in the more compact 4-dimensional form a F ab = 0 and [a F bc] = 0, where F ab has components E and B. 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 manifestly hyperbolic by choosing the gauge correctly, e.g., let a A a = H(x, t, A), giving: a a A b = b H. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 5 / 29

15 Gauge and Hyperbolicity in Electromagnetism The usual representation of the vacuum Maxwell equations split into evolution equations and constraints: t E = B, E = 0, t B = E, B = 0. These equations are often written in the more compact 4-dimensional form a F ab = 0 and [a F bc] = 0, where F ab has components E and B. 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 manifestly 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. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 5 / 29

16 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 equation can not be solved without specifying suitable gauge conditions. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 6 / 29

17 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 equation can not be solved without specifying suitable gauge conditions. The gauge freedom in general relativity theory is the freedom to represent the equations using any coordinates x a on spacetime. Solving the equations requires some specific choice of coordinates be made. Gauge conditions are used to impose the desired choice. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 6 / 29

18 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 equation can not be solved without specifying suitable gauge conditions. The gauge freedom in general relativity theory is the freedom to represent the equations using any coordinates x a on spacetime. Solving the equations requires some specific choice of coordinates be made. Gauge conditions are used to impose the desired choice. One way to impose the needed gauge conditions is to specify H a, the source term for a wave equation for each coordinate x a : H a = c c x a = ψ bc ( b c x a Γ e bc e x a ) = Γ a, where Γ a = ψ bc Γ a bc and ψ ab is the 4-metric. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 6 / 29

19 Gauge Conditions in General Relativity Specifying coordinates by the generalized harmonic (GH) method is accomplished by choosing a gauge-source function H a (x, ψ), e.g. H a = ψ ab H b (x), and requiring that H a (x, ψ) = Γ a = 1 2 ψad ψ bc ( b ψ dc + c ψ db d ψ bc ). Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 7 / 29

20 Gauge Conditions in General Relativity Specifying coordinates by the generalized harmonic (GH) method is accomplished by choosing a gauge-source function H a (x, ψ), e.g. H a = ψ ab H b (x), and requiring that H a (x, ψ) = Γ a = 1 2 ψad ψ bc ( b ψ dc + c ψ db d ψ bc ). Recall that the spacetime Ricci tensor is given by R ab = 1 2 ψcd c d ψ ab + (a Γ b) + Q ab (ψ, ψ). The Generalized Harmonic Einstein equation is obtained by replacing Γ a = ψ ab Γ b 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). Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 7 / 29

21 Gauge Conditions in General Relativity Specifying coordinates by the generalized harmonic (GH) method is accomplished by choosing a gauge-source function H a (x, ψ), e.g. H a = ψ ab H b (x), and requiring that H a (x, ψ) = Γ a = 1 2 ψad ψ bc ( b ψ dc + c ψ db d ψ bc ). Recall that the spacetime Ricci tensor is given by R ab = 1 2 ψcd c d ψ ab + (a Γ b) + Q ab (ψ, ψ). The Generalized Harmonic Einstein equation is obtained by replacing Γ a = ψ ab Γ b 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 the scalar wave equation: 0 = a a Φ = ψ ab a b Φ + F( Φ). Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 7 / 29

22 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) Einstein s Equations: PDE Issues UCSD 9/22/11 8 / 29

23 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. There is no guarantee, however, that constraints that are small initially will remain small. Constraint violating instabilities were one of the major problems that made progress on solving the binary black hole problem so slow. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 8 / 29

24 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. There is no guarantee, however, that constraints that are small initially will remain small. Constraint violating instabilities were one of the major problems that made progress on solving the binary black hole problem so slow. Special representations of the Einstein equations are needed that control the growth of any constraint violations. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 8 / 29

25 Constraint Damping in Electromagnetism Electromagnetism is described by the hyperbolic evolution equation a a A b = b H. Are there any constraints? Where have the usual E = B = 0 constraints gone? Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 9 / 29

26 Constraint Damping in Electromagnetism Electromagnetism is described by the hyperbolic evolution equation a a A b = b H. Are there any constraints? Where have the usual E = B = 0 constraints gone? Gauge condition becomes a constraint: 0 = C b A b H. Maxwell s equations imply that this constraint is preserved: a a ( b A b H) = a a C = 0. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 9 / 29

27 Constraint Damping in Electromagnetism Electromagnetism is described by the hyperbolic evolution equation a a A b = b H. Are there any constraints? Where have the usual E = B = 0 constraints gone? Gauge condition becomes a constraint: 0 = C b A b H. Maxwell s equations imply that this constraint is preserved: a a ( b A b H) = a a C = 0. Modify evolution equations by adding multiples of the constraints: a a A b = b H+γ 0 t b C = b H+γ 0 t b ( a A a H). These changes effect the constraint evolution equation, a a C γ 0 t b b C = 0, so constraint violations are damped when γ 0 > 0. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 9 / 29

28 Constraints in the GH Evolution System The GH evolution system has the form, 0 = R ab (a Γ b) (a H b), = R ab (a C b), where C a = H a + Γ a plays the role of a constraint. Without constraint damping, these equations are very unstable to constraint violating instabilities. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 10 / 29

29 Constraints in the GH Evolution System The GH evolution system has the form, 0 = R ab (a Γ b) (a H b), = R ab (a C b), where C a = H a + Γ a plays the role of a constraint. Without constraint damping, these equations are very unstable to constraint violating instabilities. Imposing coordinates using a GH gauge function profoundly changes the constraints. The GH constraint, C a = 0, where C a = H a + Γ a, depends only on first derivatives of the metric. The standard Hamiltonian and momentum constraints, M a = 0, are determined by derivatives of the gauge constraint C a : M a [ R ab 1 2 ψ abr ] t b = [ (a C b) 12 ψ ab c C c ] t b. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 10 / 29

30 Constraint Damping Generalized Harmonic System Pretorius (based on a suggestion from Gundlach, et al.) modified the GH system by adding terms proportional to the gauge constraints: 0 = R ab (a C b) + γ 0 [ t(a C b) 1 2 ψ ab t c C c ], where t a is a unit timelike vector field. Since C a = H a + Γ a depends only on first derivatives of the metric, these additional terms do not change the hyperbolic structure of the system. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 11 / 29

31 Constraint Damping Generalized Harmonic System Pretorius (based on a suggestion from Gundlach, et al.) modified the GH system by adding terms proportional to the gauge constraints: 0 = R ab (a C b) + γ 0 [ t(a C b) 1 2 ψ ab t c C c ], where t a is a unit timelike vector field. Since C a = H a + Γ a depends only on first derivatives of the metric, these additional terms do not change the hyperbolic structure of the system. Evolution of the constraints C a follow from the Bianchi identities: 0 = c c C a 2γ 0 c[ t (c C a) ] + C c (c C a) 1 2 γ 0 t a C c C c. This is a damped wave equation for C a, that drives all small short-wavelength constraint violations toward zero as the system evolves (for γ 0 > 0). Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 11 / 29

32 Numerical Tests of the GH Evolution System 3D numerical evolutions of static black-hole spacetimes illustrate the constraint damping properties of the GH evolution system. These evolutions are stable and convergent when γ 0 = 1. The boundary conditions used for this simple test problem freeze the incoming characteristic fields to their initial values. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues UCSD 9/22/11 12 / 29

33 Solving Einstein s Equations: PDE Issues II Lee Lindblom Theoretical Astrophysics, Caltech Mathematical and Numerical General Relativity Seminar University of California at San Diego 29 September 2011 Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 13 / 29

34 Summary of the GH Einstein System Choose coordinates by fixing a gauge-source function H a (x, ψ), e.g. H a = ψ ab H b (x), and requiring that H a (x, ψ) = c c x a = Γ a = 1 2 ψad ψ bc ( b ψ dc + c ψ db d ψ bc ). Gauge condition H a = Γ a is a constraint: C a = H a + Γ a = 0. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 14 / 29

35 Summary of the GH Einstein System Choose coordinates by fixing a gauge-source function H a (x, ψ), e.g. H a = ψ ab H b (x), and requiring that H a (x, ψ) = c c x a = Γ a = 1 2 ψad ψ bc ( b ψ dc + c ψ db d ψ bc ). Gauge condition H a = Γ a is a constraint: C a = H a + Γ a = 0. Principal part of evolution system becomes manifestly hyperbolic: R ab (a C b) = 1 2 ψcd c d ψ ab (a H b) + Q ab (ψ, ψ). Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 14 / 29

36 Summary of the GH Einstein System Choose coordinates by fixing a gauge-source function H a (x, ψ), e.g. H a = ψ ab H b (x), and requiring that H a (x, ψ) = c c x a = Γ a = 1 2 ψad ψ bc ( b ψ dc + c ψ db d ψ bc ). Gauge condition H a = Γ a is a constraint: C a = H a + Γ a = 0. Principal part of evolution system becomes manifestly hyperbolic: R ab (a C b) = 1 2 ψcd c d ψ ab (a H b) + Q ab (ψ, ψ). Add constraint damping terms for stability: 0 = R ab (a C b) + γ 0 [ t(a C b) 1 2 ψ ab t c C c ], where t a is a unit timelike vector field. Since C a = H a + Γ a depends only on first derivatives of the metric, these additional terms do not change the hyperbolic structure of the system. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 14 / 29

37 Numerical Tests of the GH Evolution System 3D numerical evolutions of static black-hole spacetimes illustrate the constraint damping properties of the GH evolution system. These evolutions are stable and convergent when γ 0 = 1. The boundary conditions used for this simple test problem freeze the incoming characteristic fields to their initial values. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 15 / 29

38 ADM 3+1 Approach to Fixing Coordinates Coordinates must be chosen to label points in spacetime before the Einstein equations can be solved. For some purposes it is convenient to split the spacetime coordinates x a into separate time and space components: x a = {t, x i }. (t + δt, x k ) Construct spacetime foliation by spacelike slices. Choose time function with t = const. on these slices. Choose spatial coordinates, x k, on each slice. t = τ (t, x k ) k t Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 16 / 29

39 ADM 3+1 Approach to Fixing Coordinates Coordinates must be chosen to label points in spacetime before the Einstein equations can be solved. For some purposes it is convenient to split the spacetime coordinates x a into separate time and space components: x a = {t, x i }. (t + δt, x k ) Construct spacetime foliation by spacelike slices. Choose time function with t = const. on these slices. t = τ Choose spatial coordinates, x k, on each slice. (t, x k ) Decompose the 4-metric ψ ab into its 3+1 parts: ds 2 = ψ ab dx a dx b = N 2 dt 2 + g ij (dx i + N i dt)(dx j + N j dt). k t Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 16 / 29

40 ADM 3+1 Approach to Fixing Coordinates Coordinates must be chosen to label points in spacetime before the Einstein equations can be solved. For some purposes it is convenient to split the spacetime coordinates x a into separate time and space components: x a = {t, x i }. (t + δt, x k ) Construct spacetime foliation by spacelike slices. Choose time function with t = const. on these slices. t = τ Choose spatial coordinates, x k, on each slice. (t, x k ) Decompose the 4-metric ψ ab into its 3+1 parts: ds 2 = ψ ab dx a dx b = N 2 dt 2 + g ij (dx i + N i dt)(dx j + N j dt). The unit vector t a normal to the t =constant slices depends only on the lapse N and shift N i : t = τ = x a τ a = 1 N t Nk N k. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 16 / 29 k t

41 ADM Approach to the Einstein Evolution System Decompose the Einstein equations R ab = 0 using the ADM 3+1 coordinate splitting. The resulting system includes evolution equations for the spatial metric g ij and extrinsic curvature K ij : t g ij N k k g ij = 2NK ij + g jk i N k + g ik j N k, t K ij N k k K ij = NR (3) ij + K jk i N k + K ik j N k i j N 2NK ik K k j + NK k kk ij. The resulting system also includes constraints: 0 = R (3) K ij K ij + (K k k) 2, 0 = k K ki i K k k. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 17 / 29

42 ADM Approach to the Einstein Evolution System Decompose the Einstein equations R ab = 0 using the ADM 3+1 coordinate splitting. The resulting system includes evolution equations for the spatial metric g ij and extrinsic curvature K ij : t g ij N k k g ij = 2NK ij + g jk i N k + g ik j N k, t K ij N k k K ij = NR (3) ij + K jk i N k + K ik j N k i j N 2NK ik K k j + NK k kk ij. The resulting system also includes constraints: 0 = R (3) K ij K ij + (K k k) 2, 0 = k K ki i K k k. System includes no evolution equations for lapse N or shift N i. These quanties can be specified freely to fix the gauge. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 17 / 29

43 ADM Approach to the Einstein Evolution System Decompose the Einstein equations R ab = 0 using the ADM 3+1 coordinate splitting. The resulting system includes evolution equations for the spatial metric g ij and extrinsic curvature K ij : t g ij N k k g ij = 2NK ij + g jk i N k + g ik j N k, t K ij N k k K ij = NR (3) ij + K jk i N k + K ik j N k i j N 2NK ik K k j + NK k kk ij. The resulting system also includes constraints: 0 = R (3) K ij K ij + (K k k) 2, 0 = k K ki i K k k. System includes no evolution equations for lapse N or shift N i. These quanties can be specified freely to fix the gauge. Resolving the issues of hyperbolicity (i.e. well posedness of the initial value problem) and constraint stability are much more complicated in this approach. The most successful version is the BSSN evolution system used by many (most) codes. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 17 / 29

44 Dynamical GH Gauge Conditions The spacetime coordinates x b are fixed in the generalized harmonic Einstein equations by specifying H b : a a x b H b. The generalized harmonic Einstein equations remain hyperbolic as long as the gauge source functions H b are taken to be functions of the coordinates x b and the spacetime metric ψ ab. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 18 / 29

45 Dynamical GH Gauge Conditions The spacetime coordinates x b are fixed in the generalized harmonic Einstein equations by specifying H b : a a x b H b. The generalized harmonic Einstein equations remain hyperbolic as long as the gauge source functions H b are taken to be functions of the coordinates x b and the spacetime metric ψ ab. The simplest choice H b = 0 (harmonic gauge) fails for very dynamical spacetimes, like binary black hole mergers. This failure seems to occur because the coordinates themselves become very dynamical solutions of the wave equation a a x b = 0 in these situations. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 18 / 29

46 Dynamical GH Gauge Conditions The spacetime coordinates x b are fixed in the generalized harmonic Einstein equations by specifying H b : a a x b H b. The generalized harmonic Einstein equations remain hyperbolic as long as the gauge source functions H b are taken to be functions of the coordinates x b and the spacetime metric ψ ab. The simplest choice H b = 0 (harmonic gauge) fails for very dynamical spacetimes, like binary black hole mergers. This failure seems to occur because the coordinates themselves become very dynamical solutions of the wave equation a a x b = 0 in these situations. Another simple choice keeping H b fixed in the co-moving frame of the black holes works well during the long inspiral phase, but fails when the black holes begin to merge. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 18 / 29

47 Dynamical GH Gauge Conditions II Some of the extraneous gauge dynamics could be removed by adding a damping term to the harmonic gauge condition: a a x b = H b = µt a a x b = µt b = µnψ tb. This works well for the spatial coordinates x i, driving them toward solutions of the spatial Laplace equation on the timescale 1/µ. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 19 / 29

48 Dynamical GH Gauge Conditions II Some of the extraneous gauge dynamics could be removed by adding a damping term to the harmonic gauge condition: a a x b = H b = µt a a x b = µt b = µnψ tb. This works well for the spatial coordinates x i, driving them toward solutions of the spatial Laplace equation on the timescale 1/µ. For the time coordinate t, this damped wave condition drives t to a time independent constant, which is not a good coordinate. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 19 / 29

49 Dynamical GH Gauge Conditions II Some of the extraneous gauge dynamics could be removed by adding a damping term to the harmonic gauge condition: a a x b = H b = µt a a x b = µt b = µnψ tb. This works well for the spatial coordinates x i, driving them toward solutions of the spatial Laplace equation on the timescale 1/µ. For the time coordinate t, this damped wave condition drives t to a time independent constant, which is not a good coordinate. A better choice sets t a H a = µ log g/n 2. The gauge condition in this case becomes t a a log g/n 2 = µ log g/n 2 + N 1 k N k This coordinate condition keeps g/n 2 close to unity, even during binary black hole mergers (where it became of order 100 using simpler gauge conditions). Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 19 / 29

50 First Order Generalized Harmonic Evolution System For some purposes, like constructing appropriate boundary conditions, it is useful to transform second-order hyperbolic equations into first-order systems. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 20 / 29

51 First Order Generalized Harmonic Evolution System For some purposes, like constructing appropriate boundary conditions, it is useful to transform second-order hyperbolic equations into first-order systems. GH evolution system can be written as a symmetric-hyperbolic first-order system (Fischer and Marsden 1972, Alvi 2002): t ψ ab N k k ψ ab = N Π ab, t Π ab N k k Π ab + Ng ki k Φ iab 0, t Φ iab N k k Φ iab + N i Π ab 0, where Φ kab = k ψ ab. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 20 / 29

52 First Order Generalized Harmonic Evolution System For some purposes, like constructing appropriate boundary conditions, it is useful to transform second-order hyperbolic equations into first-order systems. GH evolution system can be written as a symmetric-hyperbolic first-order system (Fischer and Marsden 1972, Alvi 2002): t ψ ab N k k ψ ab = N Π ab, t Π ab N k k Π ab + Ng ki k Φ iab 0, t Φ iab N k k Φ iab + N i Π ab 0, where Φ kab = k ψ ab. This system has two immediate problems: This system has new constraints, C kab = k ψ ab Φ kab, that tend to grow exponentially during numerical evolutions. This system is not linearly degenerate, so it is possible (likely?) that shocks will develop (e.g. the components that determine shift evolution have the form t N i N k k N i 0). Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 20 / 29

53 A New Generalized Harmonic Evolution System We can correct these problems by adding additional multiples of the constraints to the evolution system: t ψ ab (1 + γ 1 )N k k ψ ab = NΠ ab γ 1 N k Φ kab, t Π ab N k k Π ab + Ng ki k Φ iab γ 1 γ 2 N k k ψ ab γ 1 γ 2 N k Φ kab, t Φ iab N k k Φ iab + N i Π ab γ 2 N i ψ ab γ 2 NΦ iab. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 21 / 29

54 A New Generalized Harmonic Evolution System We can correct these problems by adding additional multiples of the constraints to the evolution system: t ψ ab (1 + γ 1 )N k k ψ ab = NΠ ab γ 1 N k Φ kab, t Π ab N k k Π ab + Ng ki k Φ iab γ 1 γ 2 N k k ψ ab γ 1 γ 2 N k Φ kab, t Φ iab N k k Φ iab + N i Π ab γ 2 N i ψ ab γ 2 NΦ iab. This new generalized-harmonic evolution system has several nice properties: This system is linearly degenerate for γ 1 = 1 (and so shocks should not form from smooth initial data). The Φ iab evolution equation can be written in the form, t C iab N k k C iab γ 2 NC iab, so the new constraints are damped when γ 2 > 0. This system is symmetric hyperbolic for all values of γ 1 and γ 2. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 21 / 29

55 Constraint Evolution for the New GH System The evolution of the constraints, c A = {C a, C kab, F a t c c C a, C ka k C a, C klab = [k C l]ab } are determined by the evolution of the fields u α = {ψ ab, Π ab, Φ kab }: t c A + A k A B(u) k c B = F A B(u, u) c B. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 22 / 29

56 Constraint Evolution for the New GH System The evolution of the constraints, c A = {C a, C kab, F a t c c C a, C ka k C a, C klab = [k C l]ab } are determined by the evolution of the fields u α = {ψ ab, Π ab, Φ kab }: t c A + A k A B(u) k c B = F A B(u, u) c B. This constraint evolution system is symmetric hyperbolic with principal part: t C a 0, t F a N k k F a Ng ij i C ja 0, t C ia N k k C ia N i F a 0, t C iab (1 + γ 1 )N k k C iab 0, t C ijab N k k C ijab 0. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 22 / 29

57 Constraint Evolution for the New GH System The evolution of the constraints, c A = {C a, C kab, F a t c c C a, C ka k C a, C klab = [k C l]ab } are determined by the evolution of the fields u α = {ψ ab, Π ab, Φ kab }: t c A + A k A B(u) k c B = F A B(u, u) c B. This constraint evolution system is symmetric hyperbolic with principal part: t C a 0, t F a N k k F a Ng ij i C ja 0, t C ia N k k C ia N i F a 0, t C iab (1 + γ 1 )N k k C iab 0, t C ijab N k k C ijab 0. An analysis of this system shows that all of the constraints are damped in the WKB limit when γ 0 > 0 and γ 2 > 0. So, this system has constraint suppression properties that are similar to those of the Pretorius (and Gundlach, et al.) system. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 22 / 29

58 Numerical Tests of the New GH System 3D numerical evolutions of static black-hole spacetimes illustrate the constraint damping properties of our GH evolution system. These evolutions are stable and convergent when γ 0 = γ 2 = 1. C γ 0 = γ 2 = 0.0 γ 0 = 1.0, γ 2 = 0.0 γ 0 = 0.0, γ 2 = 1.0 C {N r, L max } = {9, 7} {11, 7} 10-8 γ 0 = γ 2 = {13, 7} t/m t/m The boundary conditions used for this simple test problem freeze the incoming characteristic fields to their initial values. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 23 / 29

59 Boundary Condition Basics We impose boundary conditions on first-order hyperbolic evolution systems, t u α + A k α β(u) k u β = F α (u) in the following way (where in our case u α = {ψ ab, Π ab, Φ kab }): Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 24 / 29

60 Boundary Condition Basics We impose boundary conditions on first-order hyperbolic evolution systems, t u α + A k α β(u) k u β = F α (u) in the following way (where in our case u α = {ψ ab, Π ab, Φ kab }): We first find the eigenvectors of the characteristic matrix n k A k α β at each boundary point: e ˆα α n k A k α β = v (ˆα) e ˆα β, where n k is the (spacelike) outward directed unit normal; and then define the characteristic fields: u ˆα = e ˆα αu α. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 24 / 29

61 Boundary Condition Basics We impose boundary conditions on first-order hyperbolic evolution systems, t u α + A k α β(u) k u β = F α (u) in the following way (where in our case u α = {ψ ab, Π ab, Φ kab }): We first find the eigenvectors of the characteristic matrix n k A k α β at each boundary point: e ˆα α n k A k α β = v (ˆα) e ˆα β, where n k is the (spacelike) outward directed unit normal; and then define the characteristic fields: u ˆα = e ˆα αu α. Finally we impose a boundary condition on each incoming characteristic field (i.e. every field with v (ˆα) < 0), and impose no condition on any outgoing field (i.e. any field with v (ˆα) 0). Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 24 / 29

62 Boundary Condition Basics We impose boundary conditions on first-order hyperbolic evolution systems, t u α + A k α β(u) k u β = F α (u) in the following way (where in our case u α = {ψ ab, Π ab, Φ kab }): We first find the eigenvectors of the characteristic matrix n k A k α β at each boundary point: e ˆα α n k A k α β = v (ˆα) e ˆα β, where n k is the (spacelike) outward directed unit normal; and then define the characteristic fields: u ˆα = e ˆα αu α. Finally we impose a boundary condition on each incoming characteristic field (i.e. every field with v (ˆα) < 0), and impose no condition on any outgoing field (i.e. any field with v (ˆα) 0). At internal boundaries (i.e. interfaces between computational subdomains) use outgoing characteristics of one subdomain to fix data for incoming characteristics of neighboring subdomain. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 24 / 29

63 Evolutions of a Perturbed Schwarzschild Black Hole The simplest boundary conditions that correspond (roughly) to no incoming waves set u ˆα = 0 for each incoming field, or d t u ˆα e ˆα β t u β = 0 for fields that include static Coulomb parts. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 25 / 29

64 Evolutions of a Perturbed Schwarzschild Black Hole The simplest boundary conditions that correspond (roughly) to no incoming waves set u ˆα = 0 for each incoming field, or d t u ˆα e ˆα β t u β = 0 for fields that include static Coulomb parts. A black-hole spacetime is perturbed by an incoming gravitational wave that excites quasi-normal oscillations. Use boundary conditions that Freeze the remaining incoming characteristic fields: d t u ˆα = 0. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 25 / 29

65 Evolutions of a Perturbed Schwarzschild Black Hole The simplest boundary conditions that correspond (roughly) to no incoming waves set u ˆα = 0 for each incoming field, or d t u ˆα e ˆα β t u β = 0 for fields that include static Coulomb parts. A black-hole spacetime is perturbed by an incoming gravitational wave that excites quasi-normal oscillations. Use boundary conditions that Freeze the remaining incoming characteristic fields: d t u ˆα = 0. The resulting outgoing waves interact with the boundary of the computational domain and produce constraint violations. C {N r, L max } = {13, 11} {17, 15} {11, 9} {21, 19} t/m Play Constraint Movie Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 25 / 29

66 Constraint Preserving Boundary Conditions Construct the characteristic fields, ĉâ = eâac A, associated with the constraint evolution system, t c A + A k A B k c B = F A Bc B. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 26 / 29

67 Constraint Preserving Boundary Conditions Construct the characteristic fields, ĉâ = eâac A, associated with the constraint evolution system, t c A + A k A B k c B = F A Bc B. Split the constraints into incoming and outgoing characteristics: ĉ = {ĉ, ĉ + }. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 26 / 29

68 Constraint Preserving Boundary Conditions Construct the characteristic fields, ĉâ = eâac A, associated with the constraint evolution system, t c A + A k A B k c B = F A Bc B. Split the constraints into incoming and outgoing characteristics: ĉ = {ĉ, ĉ + }. The incoming characteristic fields mush vanish on the boundaries, ĉ = 0, if the influx of constraint violations is to be prevented. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 26 / 29

69 Constraint Preserving Boundary Conditions Construct the characteristic fields, ĉâ = eâac A, associated with the constraint evolution system, t c A + A k A B k c B = F A Bc B. Split the constraints into incoming and outgoing characteristics: ĉ = {ĉ, ĉ + }. The incoming characteristic fields mush vanish on the boundaries, ĉ = 0, if the influx of constraint violations is to be prevented. The constraints depend on the primary evolution fields (and their derivatives). We find that ĉ for the GH system can be expressed: ĉ = d û + ˆF(u, d u). Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 26 / 29

70 Constraint Preserving Boundary Conditions Construct the characteristic fields, ĉâ = eâac A, associated with the constraint evolution system, t c A + A k A B k c B = F A Bc B. Split the constraints into incoming and outgoing characteristics: ĉ = {ĉ, ĉ + }. The incoming characteristic fields mush vanish on the boundaries, ĉ = 0, if the influx of constraint violations is to be prevented. The constraints depend on the primary evolution fields (and their derivatives). We find that ĉ for the GH system can be expressed: ĉ = d û + ˆF(u, d u). Set boundary conditions on the fields û by requiring d û = ˆF(u, d u). Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 26 / 29

71 Physical Boundary Conditions The Weyl curvature tensor C abcd satisfies a system of evolution equations from the Bianchi identities: [a C bc]de = 0. The characteristic fields of this system corresponding to physical gravitational waves are the quantities: ŵ ± ab = (P a c P b d 1 2 P abp cd )(t e n e )(t f n f )C cedf, where t a is a unit timelike vector, n a a unit spacelike vector (with t a n a = 0), and P ab = ψ ab + t a t b n a n b. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 27 / 29

72 Physical Boundary Conditions The Weyl curvature tensor C abcd satisfies a system of evolution equations from the Bianchi identities: [a C bc]de = 0. The characteristic fields of this system corresponding to physical gravitational waves are the quantities: ŵ ± ab = (P a c P b d 1 2 P abp cd )(t e n e )(t f n f )C cedf, where t a is a unit timelike vector, n a a unit spacelike vector (with t a n a = 0), and P ab = ψ ab + t a t b n a n b. The incoming field w ab can be expressed in terms of the characteristic fields of the primary evolution system: ŵ ab = d û ab + ˆF ab (u, d u). We impose boundary conditions on the physical graviational wave degrees of freedom then by setting: d û ab = ˆF ab (u, d u) + ŵ ab t=0. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 27 / 29

73 Imposing Neumann-like Boundary Conditions Consider Neumann-like boundary conditions of the form e ˆα βn k k u β d u ˆα = d u ˆα BC. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 28 / 29

74 Imposing Neumann-like Boundary Conditions Consider Neumann-like boundary conditions of the form e ˆα βn k k u β d u ˆα = d u ˆα BC. The characteristic field projections of the evolution equations are: d t u ˆα e ˆα β t u β = e ˆα β( A kβ γ k u γ + F β) D t u ˆα. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 28 / 29

75 Imposing Neumann-like Boundary Conditions Consider Neumann-like boundary conditions of the form e ˆα βn k k u β d u ˆα = d u ˆα BC. The characteristic field projections of the evolution equations are: d t u ˆα e ˆα β t u β = e ˆα β( A kβ γ k u γ + F β) D t u ˆα. We impose these Neumann-like boundary conditions by changing the appropriate components of the evolution equations at the boundary to: d t u ˆα = D t u ˆα + v (ˆα) ( d u ˆα d u ˆα BC ). Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 28 / 29

76 Tests of Constraint Preserving and Physical BC Evolve the perturbed black-hole spacetime using the resulting constraint preserving boundary conditions for the generalized harmonic evolution systems. Play Movies RΨ t/m Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 29 / 29

77 Tests of Constraint Preserving and Physical BC Evolve the perturbed black-hole spacetime using the resulting constraint preserving boundary conditions for the generalized harmonic evolution systems. Play Movies RΨ t/m Evolutions using these new constraint-preserving boundary conditions are still stable and convergent. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 29 / 29

78 Tests of Constraint Preserving and Physical BC Evolve the perturbed black-hole spacetime using the resulting constraint preserving boundary conditions for the generalized harmonic evolution systems. Play Movies RΨ t/m Evolutions using these new constraint-preserving boundary conditions are still stable and convergent. The Weyl curvature component Ψ 4 shows clear quasi-normal mode oscillations in the outgoing gravitational wave flux when constraint-preserving boundary conditions are used. Lee Lindblom (Caltech) Einstein s Equations: PDE Issues II UCSD 9/29/11 29 / 29