well-posedness constrained evolution of 3+1 formulations of general relativity

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

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

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.


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


Sources of Instabilities

Physical Instabilities – Singularities

Gauge Instabilities – Coordinate perturbations – Coordinate singularities

Constraint violating modes


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.


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


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 the minimal 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.


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.


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.


Algebraic Gauges

Well-posed constrained evolution requires that

Examples Densitized lapse

Well-posed if and only if

1+log slicing Well-posed

The Standard ADM formulationwith several gauge conditions


Geodesic Slicing Ill-posed

Maximal Slicing (MS) Ill-Posed

Parabolic Extension of MS Ill-Posed

K-driver Well-Posed

The ADM formulationGauges & Well-posedness conditions


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.


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.


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.


well-posedness constrained evolution of 3+1 formulations of general relativity

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