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## Overview of a few General Relativistic Solitons

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### No Sigma-Model (Nonexistence: Bizon & Wasserman 2004) No Sine-Gordon ... Existence Proof: Bizon & Wasserman, Comm. Math. Phys. 215, 357-373 (2000) ... – PowerPoint PPT presentation

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Title: Overview of a few General Relativistic Solitons

1
Overview of a few General Relativistic Solitons
• Dr. Scott H. Hawley
• Center for Relativity
• Department of Physics
• University of Texas at Austin

4th IMACS International Conference on Nonlinear
Evolution Equations and Wave Phenomena
Computation and Theory April 11-14 , 2005
2
Extent of this Talk
• GR solitons is a huge topic.
• For further discussion, see Belinski Verdaguer,
Gravitational Soltions, Cambridge University
Press, 2001.
• My focus
• Particular interest in cases where no GR gt no
soliton!
• Numerical evolutions of scalar GR solitons in
asymptotically flat spacetimes.
• No Inverse Scattering Method
• No Sigma-Model (Nonexistence Bizon Wasserman
2004)
• No Sine-Gordon
• No gauge theories (Well, a little Yang-Mills...)
• Will skip regular, fluid stars (neutron stars,
WDs)

3
Introduction
• General Relativity (GR, Einsteins equations)
is a system of 10 coupled,nonlinear partial
differential equations Gmn 8 pTmn
m,n0...3 (1)
• Einstein tensor Gmn describes geometry of a
4-dimensional Riemannian manifold (spacetime)
with Lorenzian signature (-,,,), involves 2nd
derivatives of metric gmn.
• Stress-energy tensor Tmn describes matter (0
for vacuum). Examples fluid, E-M field, scalar
field.
• Solving Einsteins equations implies finding a
metric gmn which satisfies these equations (1).
• Exact solutions uncommon. Many solutions
obtained numerically, e.g. via 31
decomposition of 4D manifold into space
time gt Initial Value Problem.

4
GR Solitons DO...
• Well use sort of a physicists definition of
soliton, meaning a solution which is
• a solution to a nonlinear wave equation
• localized, i.e. compact
• very long lived stable
• exhibits particle-like behavior, moving at a
given speed (may be a speed of zero)

5
GR Solitons DONT...
• In general, they do not include the feature of
being able to pass through one another
unscathed such as KdV solitons posess,
because...
• When two GR solitons collide, they may form a
black hole (BH)
• Hoop Conjecture (Thorne 1972) Given enough
mass-energy in a given volume of space, a (BH)
horizon will form.
• If one or more of these solitons is already a BH,
then the result has always been a single BH
scattered waves (in countless numerical
evolutions).
• Theorem (Hawking) Event Horizon cannot
bifurcate. Unknown whether Apparent Horizon can
(inside EH)!

6
GR Solitons in Vacuum Black Holes
• The Schwarzschild solution itself is regarded as
a GR soliton. Describes a spherically-symmetric,
chargeless black hole (BH). E.g., in isotropic
coords
• Proofs of linear stability (Kay Wald, 1987)
• Also, Birchoffs Thm gt exterior to any
spherically-symmetric mass distribution is
Schwarzschild
• The Kerr solution describes an axisymmetric,
rotating BH
• Proof of linear stability (H. Beyer, 2001)

7
GR Solitons in Vacuum Plane Waves
8
Bartnik-McKinnon Solitons GR Y-M
• Yang-Mills matter field.... need to say more!

9
Boson Stars GR Complex, Massive SF
• Complex scalar field f(r,t) obeys massive
Klein-Gordon equation
• Let f(r,t) f0(r)eiwt, where f0(r) ? ?.
• Yields a static configuration (for metric), set
of ODEs to be solved via shooting (on
eigenvalue w2).
• Existence Proof Bizon Wasserman, Comm. Math.
Phys. 215, 357-373 (2000).
• Posess stable unstable branches (Kaup 1969,
Seidel Suen 1991, Li Peng 1989, Hawley
Choptuik 2000).
• Can add nonlinear potential to KG eq, to increase
mass of star (Colpi et al., 1986). (No
existence proof)

10
Boson Star Properties
• show graphs Mass vs. central density, mass vs.

11
Solitonic Collision of Boson Stars
• C.W. Lai, Ph.D. Dissertation, U. British
Columbia, 2004

12
Non-Solitonic Collision of Boson Stars
• Also C.W. Lai, Ph.D. Dissertation, U. British
Columbia, 2004?

13
Oscillatons GR Real, Massive SF
• Static solution exists, but is unstable, has
naked singularity -(
• Oscillating Soliton Stars (Seidel Suen,
1991) Ansatz of truncated Fourier series gt
yields system of ODEs for initial data.
Numerical evolution confirms ansatz.
• Macroscopically similar to boson stars (masses,
radii), except that all functions (metric, SF)
oscillate
• So similar to boson stars that BS f0(r) can be
used to nearly same effect (Hawley, 2002) Poor
Mans Soliton Star
• May not exist mathematically (Bizon, personal
comm.), but very long-lived
• FIX!! say some things about Mexico groups
work...
• Say something about dark matter candidates

14
Multi-Scalar Stars
• this one should be easy!

15
GR Solitons and Critical Collapse
• Critical phenomena in gravitational collapse
(Choptuik 1993)
• blah blah...
• blah
• blah?
• Critical solution for Y-M fields is (/can be)
Bartnik-McKinnon! (Choptuik, Bizon Chmaj 1996)
• Critical solution for complex, massive fields (at
long wavelengths) is boson star! (Hawley
Choptuik 2000) -------gt

16
GR Solitons and No-Hair Theorems

17
GR Ss Spontaneous Scalarization
• ?

18
Other Recent Work
• Again Belinski Verdaguer, Gravitational
Soltions, Cambridge University Press, 2001.
• Gonzalez Sudarsky, Scalar solitons in a
4-Dimensional curved space-time, 2001
• Oliynyk Kunzle, On all possible static
spherically symmetric EYM solitons and black
holes, 2001
• Rosu, Korteweg-de Vries adiabatic index solitons
in barotropic open FRW cosmologies, 2001
• Winstanley Sarbach, On the linear stability of
solitons and hairy black holes with a negative
cosmological constant the even-parity sector,
2001
• (FIX!! Will coveri earlier in talk)
• Nucamendi Salgado, Scalar hairy black holes
and solitons in asymptotically flat spacetimes