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Overview of a few General Relativistic Solitons


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

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
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
  • 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
  • No Sine-Gordon
  • No gauge theories (Well, a little Yang-Mills...)
  • Will skip regular, fluid stars (neutron stars,

  • 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
  • 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.

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)

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,
  • 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
  • Theorem (Hawking) Event Horizon cannot
    bifurcate. Unknown whether Apparent Horizon can
    (inside EH)!

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
  • Proofs of linear stability (Kay Wald, 1987)
  • Also, Birchoffs Thm gt exterior to any
    spherically-symmetric mass distribution is
  • The Kerr solution describes an axisymmetric,
    rotating BH
  • Proof of linear stability (H. Beyer, 2001)

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

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)

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

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

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

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)
  • 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
  • Say something about dark matter candidates

Multi-Scalar Stars
  • this one should be easy!

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

GR Solitons and No-Hair Theorems
  • Salgado, Alcubierre et al...

GR Ss Spontaneous Scalarization
  • ?

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,
  • (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

Bonus Nonlinear Waves Gausszilla
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