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 Anti-deSitter Space (AdS)
• 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)

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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.

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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)

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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)!

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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)

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GR Solitons in Vacuum Plane Waves
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Bartnik-McKinnon Solitons GR Y-M

• Yang-Mills matter field.... need to say more!

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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)

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Boson Star Properties

• show graphs Mass vs. central density, mass vs.
  radius...

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Solitonic Collision of Boson Stars

• C.W. Lai, Ph.D. Dissertation, U. British
  Columbia, 2004

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Non-Solitonic Collision of Boson Stars

• Also C.W. Lai, Ph.D. Dissertation, U. British
  Columbia, 2004?

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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

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Multi-Scalar Stars

• this one should be easy!

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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

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GR Solitons and No-Hair Theorems

• Salgado, Alcubierre et al...

17
GR Ss Spontaneous Scalarization

• ?

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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
• Alcubierre, Gonzalez Salgado, Dynamical
  evolution of unstable self-gravitating scalar
  solitons, 2004

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Conclusions
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Bonus Nonlinear Waves Gausszilla