Well-Posedness Constrained Evolution of 3+1 formulations of General Relativity

1. Well-Posedness Constrained Evolution of 3+1 formulations of General Relativity Vasileios Paschalidis (A. M. Khokhlov & I.D. Novikov) Dept. of Astronomy & Astrophysics The University of Chicago

2. Overview • What is the problem? • Approach to Well-Posedness of constrained evolution • Application to the standard ADM 3+1 formulation of GR • The well-posedness of a constrained evolution depends on the properties of the gauge • Results for several types of gauges • Conslusions V. Paschalidis 18/11/2006

3. Understanding the problem • The Einstein equations in a 3+1 split approach consist of a set of evolution equations and a set of constraint equations which must be satisfied on every time slice. Physical solutions must satisfy the constraint equations. • Well-posed formulations of GR have been used in free evolution 3D simulations, but after some time the solution turns unphysical. This is termed as Error blow-up. V. Paschalidis 18/11/2006

4. Understanding the problem • The community has turned to enforcing (some of) the constraint equations after each time-step in a free evolution. This is not a unique procedure and there is no theory describing its well-posedness. • Recent success in simulating BBH by • 1) Frans Pretorius using generalized harmonic coordinates and constraint damping • 2) Goddard Space center relativity group using BSSN with sophisticated gauge condition, enforcing some of the constraints. • However, we still lack a general gauge free approach V. Paschalidis 18/11/2006

5. Sources of Instabilities • Physical Instabilities – Singularities • Gauge Instabilities – Coordinate perturbations – Coordinate singularities • Constraint violating modes V. Paschalidis 18/11/2006

6. Well-Posedness • Loosely speaking well-posedness means continuous dependence of the solution on initial conditions • Quasi-linear PDEs • Well posedness does not guarantee global solutions in time, only short time existence but is a necessary condition for stability. V. Paschalidis 18/11/2006

7. Well-posedness ofConstrained evolution • Stability of a quasi-linear PDE system with constraints • , n unknown variables , m constraints • Against high-frequency and small amplitude harmonic perturbations V. Paschalidis 18/11/2006

8. Well-posedness ofConstrained evolution • The n evolution equations yield: • The m constraints yield: • Substitution of the former in the latter results in an eigenvalue problem for the independent perturbation amplitudes, given by theminimal set • The minimal set controls the well-posedness of the constrained evolution. The m remaining perturbation amplitudes are determined by the solutions of the minimal set. V. Paschalidis 18/11/2006

9. Hypebolicity and Well-Posednessof a minimal set • From the characteristic matrix Aq • Weakly hyperbolic: if all eigenvalues λ of Aq real • Strongly hyperbolic: if Aq has complete set of eigenvectors and λ real for all directions k • Strongly hyperbolic systems have a well-posed Cauchy problem • Weakly hyperbolic sets are ill-posed. V. Paschalidis 18/11/2006

10. Applications to 3+1 formulations of GR • Applying the preceding approach to GR gives us the minimal set of the Einstein equations • Analysis of the minimal set shows that it consists of two subsets: a) A subset corresponding to gravitational waves. Waves are described by strongly hyperbolic equations which are well posed b) A subset corresponding to the gauge. This subset is not necessarily well posed. The gauge has to be chosen carefully so that this subset is strongly hyperbolic, too. • The well-posedness of the Constrained Evolution depends entirely on the properties of the gauge. V. Paschalidis 18/11/2006

11. The Standard ADM formulationwith several gauge conditions • Algebraic Gauges Well-posed constrained evolution requires that • Examples • Densitized lapse Well-posed if and only if • 1+log slicing Well-posed V. Paschalidis 18/11/2006

12. The ADM formulationGauges & Well-posedness conditions • Geodesic Slicing Ill-posed • Maximal Slicing (MS) Ill-Posed • Parabolic Extension of MS Ill-Posed • K-driver Well-Posed V. Paschalidis 18/11/2006

13. Resolution of the fact that maximal slicing is coordinate singularity-free • Maximal slicing means • If we impose this condition on the evolution equation of the trace of the extrinsic curvature we obtain • The differential maximal slicing is ill-posed because the perturbations of the extrinsic curvature are not necessarily 0. These satisfy • If one however imposes the algebraic condition of maximal slicing at all times then the perturbations of the trace of K are identically 0 and the constrained evolution is well-posed. V. Paschalidis 18/11/2006

14. Conclusions • We have developed a new approach to study well-posedness of constrained evolution of quasi-linear sets of PDEs. • This approach when applied to GR • Tells us that the well-posedness of a constrained evolution depends on the properties of the gauge • It provides us with conditions of well-posedness that a gauge has to satisfy, in order for the constrained evolution to be well-posed. • It provides us with a consistent way of finding new well-behaved gauges. • A well-behaved gauge does not imply well posed free evolution. But, a gauge leading to ill-posed constrained evolution will result in an ill-posed free evolution, too • The most desirable approach is the one which eliminates the constraint violating modes. However, to have successful free evolution we might as well damp or at least control the growth of the constraint violating modes. V. Paschalidis 18/11/2006

15. Other formulations • The Kidder-Scheel-Teukolsky (KST) formulation • If the ADM constrained evolution is well-posed for a specific gauge the constrained evolution of the aforementioned class of formulations will-be well posed and vice versa. • The Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation introduces 5 additional variables and evolves a total of 17 variables. It is derived by a non-linear invertible transformation of the ADM variables. • If ADM with a given gauge has well-posed constrained evolution then any other formulation derived from ADM via a general (non-linear) invertible transformation will also be well posed. V. Paschalidis 17/11/2006