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
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

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.
V. Paschalidis
18/11/2006
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
Sources of Instabilities

Physical Instabilities – Singularities

Gauge Instabilities – Coordinate perturbations
– Coordinate singularities

Constraint violating modes
V. Paschalidis
18/11/2006
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
Well-posedness of
Constrained 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
Well-posedness of
Constrained 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.
V. Paschalidis
18/11/2006
Hypebolicity and Well-Posedness
of 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
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
The Standard ADM formulation
with several gauge conditions
Algebraic Gauges
Well-posed constrained evolution requires that


Examples

Densitized lapse
Well-posed if and only if

V. Paschalidis
1+log slicing
18/11/2006
Well-posed
The ADM formulation
Gauges & Well-posedness conditions

Geodesic Slicing

Maximal Slicing (MS)

Parabolic Extension of MS

K-driver
V. Paschalidis
18/11/2006
Ill-posed
Ill-Posed
Ill-Posed
Well-Posed
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
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
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