Self-similar spherically symmetric wave maps coupled to gravity

@article{Bizon2000SelfsimilarSS,
  title={Self-similar spherically symmetric wave maps coupled to gravity},
  author={Piotr Bizo'n and Arthur G. Wasserman},
  journal={Physical Review D},
  year={2000},
  volume={62},
  pages={084031}
}
We investigate spherically symmetric continuously self-similar (CSS) solutions in the SU(2) sigma model coupled to gravity. Using mixed numerical and analytical methods, we provide evidence for the existence (for small coupling) of a countable family of regular CSS solutions. This fact is argued to have important implications for the ongoing studies of black hole formation in the model. 

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References

SHOWING 1-10 OF 28 REFERENCES
On the existence of self-similar spherically symmetric wave maps coupled to gravity
We present a detailed analytical study of spherically symmetric self-similar solutions in the SU(2) sigma model coupled to gravity. Using a shooting argument, we prove that there is a countable
Static spherically symmetric solutions of the Einstein-Yang-Mills equations
We study the global behaviour of static, spherically symmetric solutions of the Einstein-Yang-Mills equations with gauge groupSU(2). Our analysis results in three disjoint classes of solutions with a
Equivariant Self-Similar Wave Maps from Minkowski Spacetime into 3-Sphere
Abstract: We prove existence of a countable family of spherically symmetric self-similar wave maps from 3+1 Minkowski spacetime into the 3-sphere. These maps can be viewed as excitations of the
Critical Phenomena in Nonlinear Sigma Models
We consider solutions to the nonlinear sigma model (wave maps) with target space S3 and base space 3+1 Minkowski space, and we find critical behavior separating singular solutions from nonsingular
Self-similarity in general relativity
The different kinds of self-similarity in general relativity are discussed, with special emphasis on similarity of the `first' kind, corresponding to spacetimes admitting a homothetic vector. We then
Type II Critical Collapse of a Self-Gravitating Nonlinear Sigma Model
We report on the existence and phenomenology of type II critical collapse within the one-parameter family of SU(2) σ models coupled to gravity. Numerical investigations in spherical symmetry show
Type II Critical Collapse of a Self-Gravitating Nonlinear σ-Model
We report on the existence and phenomenology of type II critical collapse within the one-parameter family of SU(2) σ-models coupled to gravity. Numerical investigations in spherical symmetry show
S duality at the black hole threshold in gravitational collapse.
TLDR
A new critical solution is derived that is spherically symmetric and continuously self-similar in classical low energy string theory, at the threshold for black hole formation.
Understanding critical collapse of a scalar field
I construct a spherically symmetric solution for a massless real scalar field minimally coupled to general relativity which is discretely self-similar (DSS) and regular. This solution coincides with
A New Tradition between Discrete and Contiuous Self-Similarity in Critical Gravitational Collapse
Gravitational collapse at the threshold of black hole formation ties together many of the fundamental issues of general relativity, such as the global aspects of solutions, the structure of
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