Study of AdS Soliton Instability

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Study of AdS Soliton Instability

• The 18th Workshop on Numerical Relativity
• Changheon Oh
• 2007. 3. 31

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• 0. Motivation
• In 1998, Horowitz and Myers proposed a new
  positive energy
• theorem conjecture.
• -gt It states that the AdS soliton spacetime has
  the least energy among
• all spacetime configurations which are
  asymptotically AdS.
• It states that the AdS soliton spacetime has the
  least energy among
• all spacetime configurations which are
  asymptotically AdS.
• It is belived thatr this spacetime must be stable
  under small perturbations
• at least in order to serve as a ground state.

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• What is the AdS Soliton

(1) How to gain the AdS Soliton solution in the 5
dimension

• The 5-dim. AdS Black Hole metric

where
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(2) Why we call it AdS Soliton
- We can easily find singular points at r0 and
Vs(r) 0. ( )- To check
whether r is physical or coordinate singularity.
Taylor rxpansion around the
Therefore metric becomes
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To avoid a cononical singularity
Now we can determine parameters ranges as below
Therefore r is not singular point, and this
metric is regular all allowed region.
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In the r is infinity
This is a AdS5 spacetime metric in Poincare coord.
So we call AdS Soliton
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• 2. Numerically reproduce the AdS Soliton solution
• Coordinate change from Spherical coordinates
• to Kruskal coordinates

Why??
Null infinity is time-like, and any observer
living in AdS spacetime can send and receive
light-like signals to and from null infinity in
finite proper time.
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How??
We set the metric ansatz as below
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(2) Solve the inverse function r(r) by
numerically
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(3) Equations set
Constraints equations!!!
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Evolution equations!!!
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(4) How to impose the initial data
First of all, we are interested in static
solution. So we impose the time Symmetric initial
data.
For the satisfying constrains equations, assume
the c(r) is constant and A(r) has gaussian
profile.
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