Global dynamics of a Yang-Mills field on an asymptotically hyperbolic space
HTML articles powered by AMS MathViewer
- by Piotr Bizoń and Patryk Mach PDF
- Trans. Amer. Math. Soc. 369 (2017), 2029-2048 Request permission
Erratum: Trans. Amer. Math. Soc. 369 (2017), 3013-3013.
Supplementary material: Mathematica notebook demo
Abstract:
We consider a spherically symmetric (purely magnetic) $SU(2)$ Yang-Mills field propagating on an ultrastatic spacetime with two asymptotically hyperbolic regions connected by a throat of radius $\alpha$. Static solutions in this model are shown to exhibit an interesting bifurcation pattern in the parameter $\alpha$. We relate this pattern to the Morse index of the static solution with maximal energy. Using a hyperboloidal approach to the initial value problem, we describe the relaxation to the ground state solution for generic initial data and unstable static solutions for initial data of codimension one, two, and three.References
- Douglas M. Eardley and Vincent Moncrief, The global existence of Yang-Mills-Higgs fields in $4$-dimensional Minkowski space. I. Local existence and smoothness properties, Comm. Math. Phys. 83 (1982), no. 2, 171–191. MR 649158, DOI 10.1007/BF01976040
- Demetrios Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math. 39 (1986), no. 2, 267–282. MR 820070, DOI 10.1002/cpa.3160390205
- Piotr Bizoń, Tadeusz Chmaj, and Andrzej Rostworowski, Late-time tails of a Yang-Mills field on Minkowski and Schwarzschild backgrounds, Classical Quantum Gravity 24 (2007), no. 13, F55–F63. MR 2334886, DOI 10.1088/0264-9381/24/13/F01
- Piotr T. Chruściel and Jalal Shatah, Global existence of solutions of the Yang-Mills equations on globally hyperbolic four-dimensional Lorentzian manifolds, Asian J. Math. 1 (1997), no. 3, 530–548. MR 1604914, DOI 10.4310/AJM.1997.v1.n3.a4
- Piotr Bizoń, Andrzej Rostworowski, and Anıl Zenginoğlu, Saddle-point dynamics of a Yang-Mills field on the exterior Schwarzschild spacetime, Classical Quantum Gravity 27 (2010), no. 17, 175003, 11. MR 2671555, DOI 10.1088/0264-9381/27/17/175003
- J. A. Wheeler, Geometrodynamics, Academic Press, New York, 1962.
- Piotr Bizoń, Formation of singularities in Yang-Mills equations, Acta Phys. Polon. B 33 (2002), no. 7, 1893–1922. MR 1923684
- Kevin Corlette and Robert M. Wald, Morse theory and infinite families of harmonic maps between spheres, Comm. Math. Phys. 215 (2001), no. 3, 591–608. MR 1810946, DOI 10.1007/PL00005545
- Jan Dereziński and MichałWrochna, Exactly solvable Schrödinger operators, Ann. Henri Poincaré 12 (2011), no. 2, 397–418. MR 2774864, DOI 10.1007/s00023-011-0077-4
- Hidekazu Nariai, On a new cosmological solution of Einstein’s field equations of gravitation [ MR0055837 (14,1133f)], Gen. Relativity Gravitation 31 (1999), no. 6, 963–971. MR 1693455, DOI 10.1023/A:1026602724948
- Helmut Friedrich, Cauchy problems for the conformal vacuum field equations in general relativity, Comm. Math. Phys. 91 (1983), no. 4, 445–472. MR 727195, DOI 10.1007/BF01206015
- Anıl Zenginoğlu, Hyperboloidal foliations and scri-fixing, Classical Quantum Gravity 25 (2008), no. 14, 145002, 19. MR 2430521, DOI 10.1088/0264-9381/25/14/145002
- Anıl Zenginoğlu, A hyperboloidal study of tail decay rates for scalar and Yang-Mills fields, Classical Quantum Gravity 25 (2008), no. 17, 175013, 13. MR 2430682, DOI 10.1088/0264-9381/25/17/175013
- Oliver Rinne and Vincent Moncrief, Hyperboloidal Einstein-matter evolution and tails for scalar and Yang-Mills fields, Classical Quantum Gravity 30 (2013), no. 9, 095009, 27. MR 3046474, DOI 10.1088/0264-9381/30/9/095009
- B. G. Schmidt, On relativistic stellar oscillations, Gravity Research Foundation essay (1993).
- Claude M. Warnick, On quasinormal modes of asymptotically anti-de Sitter black holes, Comm. Math. Phys. 333 (2015), no. 2, 959–1035. MR 3296168, DOI 10.1007/s00220-014-2171-1
- Piotr Bizoń, Tadeusz Chmaj, and Andrzej Rostworowski, Asymptotic stability of the skyrmion, Phys. Rev. D 75 (2007), no. 12, 121702, 5. MR 2326835, DOI 10.1103/PhysRevD.75.121702
- F. E. Neumann, Beiträge zur Theorie der Kugelfunctionen, II, Leipzig, 1878.
- J. C. Adams, On the expression of the product of any two Legendre’s coefficients by means of a series of Legendre’s coefficients, Proc. Roy. Soc. London 27 (1878), 63-71.
- P. Bizoń and M. Kahl, Yang-Mills field on the extremal Reissner-Nordström black hole, arXiv:1603.0475.
Additional Information
- Piotr Bizoń
- Affiliation: Institute of Physics, Jagiellonian University, Kraków, Poland – and – Max Planck Institute for Gravitational Physics (A. Einstein Institute), Golm, Germany
- MR Author ID: 37460
- Email: piotr.bizon@aei.mpg.de
- Patryk Mach
- Affiliation: Institute of Physics, Jagiellonian University, Kraków, Poland
- Email: patryk.mach@uj.edu.pl
- Received by editor(s): March 17, 2015
- Published electronically: May 17, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 2029-2048
- MSC (2010): Primary 35Q75
- DOI: https://doi.org/10.1090/tran/6807
- MathSciNet review: 3581226