Dynamical evolution of unstable selfgravitating scalar solitons'

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Dynamical evolution of unstable self-gravitating
scalar solitons.
José A. González
In collaboration with M.Alcubierre, M.Salgado
and D. Sudarsky
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Overview
I. Spherically symmetric code

• Introduction
• Regularity conditions
• Regularization algorithm
• Examples

II. Scalar solitons

• Theories with scalar hair
• Numerical evolutions

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Introduction
When developing spherically symmetric codes in
numerical relativity, the coordinate singularity
at the origin can be a source of serious
instabilities caused by the lack of regularity of
the geometric variables there. The problem
arises because of the presence of terms in the
evolution equations that go as 1/r near the
origin. Regularity of the metric guarantees the
exact cancellations of such terms at the origin,
thus ensuring well-behaved solutions.
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This exact cancellation however, though certainly
true for analytical solutions, usually fails to
hold for numerical solutions. One the finds that
the 1/r terms do not cancel and the numerical
solution becomes ill-behaved near r0. The usual
way to deal with this problem is to use the areal
gauge, where the radial coordinate r is chosen in
such a way that the proper area of spheres of
constant r is always . If, moreover, one
also choses a vanishing shift vector one ends up
in the standar polar/areal gauge.
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In this gauge the exact cancellation of the 1/r
terms is now reduced to imposing the boundary
condition at ,which can be
easily done if one solves for from the
hamiltonian constraint and ignores its evolution
equation. The main drawback of the standard
approach is that the gauge choice has been
completely exhausted. In particular, the
polar/areal gauge can not penetrate apparent
horizons, since inside an apparent horizon it is
impossible to keep the areas of spheres
fixed without a non-trivial shift vector.
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31 formalism
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Constraints
Evolution equations
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Regularity Conditions
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Evolution equations
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(No Transcript)
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Constraints
Hamiltonian
Momentum
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with
perhaps functions of time
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Regulatization Algorithm
Arbona and Bona (1999) develop a
regularization technique for the spherically
symmetric version of the Bona-Masso evolution
system. Their technique is based on redefining
the auxiliary Dynamical varible that is part
of the standard BM formulation. Our
regularization method is similar in spirit, if
not in detail, to this one. The main difference
being that the approach of AB was tied to the use
of the BM evolution system, while our algorithm
is much more general.
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We define an auxiliary variable
The behavoir at the origin can easily be
imposed numerically using a grid that staggers
the origin, and asking for to be odd
across We need the evolution equation for .
This can be obtained directly from its definition.
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The last evolution equation clearly has
the dangerous
term, but this can be removed with the help of
the momentum constraint to find
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Consider some arbitrary first order formulation
of the Einstein evolution equations in spherical
symmetry that has the generic form
where
The formulation might be hyperbolic or not,
depending on the characteristic structure of the
matrix M. We will assume that one has arrived at
such a formulation by adding multiples of the
constraints
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Using the evolution equation of
If we define
then
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The final step is to substitute the spatial
derivative of
for that of
with
Using that the spatial derivative of is
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we finally find
This last equation is now regular, and has
precisely the same characteristic structure as
the original system.
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Example
Using the Hamiltonian and Momentum constraints to
remove the terms
and
in the evolution equations of and
respectively. We introduced the variables
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Bona-Masso slicing condition (harmonic)
Initial data
with
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Lapse not regularized
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Metric not regularized
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Lapse regularized
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Metric regularized
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Lapse regularized
Lapse not regularized
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No hair conjecture

• Stationary black holes are completely specified
  by the mass, charge and angular momentum.
• Einstein-Yang-Mills
• Einstein-Skyrme
• Einstein-Yang-Mills-dilaton
• Einstein-Yang-Mills-Higgs
• Einstein-non-Abelian-Procca

Counterexamples
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• One place where it seemed for a while that there
  was hope for a restricted form of the conjecture
  was the scalar field arena.
• No hair theorems
• Bekenstein ? Convex potentials
• Sudarsky ? Arbitrary positive potentials

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Torii, Maeda y Narita. PRD 64 044007 (2001)
Scalar fields minimally coupled to gravity in
spherically simmetric asymptotically anti-de
Sitter spacetimes
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U. Nucamendi y M. Salgado. PRD 68 044026 (2003)
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Asymptotically flat scalar hairy black holes
and solitons
Unstable under radial perturbations
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Numerical evolution
M.Alcubierre, J.A. González y M. Salgado (2004)
Gauge conditions
Collapse
Explosion
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Initial data (soliton)
Solve the constraint and the Klein-Gordon
equation in the static case a perturbation
in
Collapse
Explosion
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Evolution of the scalar field (Collapse)
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Evolution of the lapse function (Collapse)
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Evolution of the metric function A (Collapse)
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Aparent horizon (Collapse)
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Convergence of the hamiltonian constraint
(Collapse)
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Evolution of the scalar field (Explosion)
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Evolution of the lapse function (Explosion)
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Evolution of the metric function A (Explosion)
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Convergence of the Hamiltonian constraint
(Explosion)
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Summary

• Spherically symmetric code
• Regularization
• Can handle with aparent horizons
• Numerical evolution of the scalar soliton
  Collapse and Explosion