It is known that there exists critical behavior in the SU(2) EYM-system.It occured at the boundary in phase space between initial data which eventually forms a black hole and data which do not. These non-abelian black holes (and cosmic strings) turns out to be very fragile objects. I conjecture that the conventional linear stability analysis is inadequate to be applied to the situation where a singularity is formed. I applied the multiple-scale method in order to handle the non-linear effects (i.e. the 'back-reaction', dispersion of high-frequency gravitational waves, interaction between the HF YM-components and the HF gravitational waves).
Although it is claimed that it will be exceedingly unlikely to obtain closed timelike curves in the spacetime of a spinning EYM-string, the problem turns out to be more deep seated. The topic is related to the mathematics of the (2+1)-dimensional spacetimes and dynamically topology changes in some EYMH-systems. The increase in interest in these models originates not only from the fact that causality violation could occur, but also from the conjecture that the solution of these controversies could be related to a possible quantum version of such systems. Further, the helical structure of time, introduced by the matching conditions of the interior spacetime on the exterior, remains intriguing. Although itis known that general relativity don't exclude a priori causality violation, it is commonly believed that in nature there will be an universal chronology protection mechanism. The proof of this conjecture will eventually be enforced within the framework of quantum gravity. Glimpses of the framework will be seen by considering quantum gauge fields in general relativity. This is one of the motivations of considering Yang-Mills fields in the axially symmetric spacetimes.
Topological defects such as cosmic strings, are remnants of the symmetry-breaking process in grand unified theories in the early beginning of the universe. These defects could have played a substantial role in the structure formation, as is now seen on large cosmological scales. The observed anisotropy from COBE of the cosmic microwave sky could indicate that cosmic string-induced structures are more likely than the other defects. One also assumes that cosmological inflation, necessary to solve major problems in standard cosmological models, can be induced by cosmic defects. The accelerated expansion of the universe is driven by the energy of the false vacuum of the Higgs field. Further, there is a profound similarity between condensed matter physics and comic defects in the early universe. In particular, the Aharonov-Bohm effect induced by spinning cosmic strings shows a striking analogue of the Iordanskii force acting on the vortex in superfluids. In cosmic string models one usually starts with an Abelian Higgs field as matter part for the equations of Einstein. The spactime around (spinning) gauge strings could have, besides the usual angle deficit feature, exotic properties, such as the violation of causality, time-delay effects, helical structure of time and frame dragging.
A natural step is then to consider the non-Abelian Yang-Mills model and parametrize the YM potential by a 2D Abelian one-form, a 3D Higgs field and a matrix valued scalar. Taking the YM model as the matter part of the equations of Einstein, one finds a rich spectrum of stationary regular and non-regular solutions. This is in contrast to the vacuum solution and the Einstein-Maxwell solution, where the Schwarzschild and Reissner-Nordstrom solution are the only static black hole solutions. In the Einstein-YM theory one finds not only the embedding of the Abelian RN black hole, but in addition there are genuine non-Abelian coloured black holes. Their coexistence gives rise to the violation of the no-hair conjecture since they carry the same (magnetic) charge. For the cylindrical symmetric situation, it is found that time-dependent rotating solutions are related to cosmic string solutions. The emitted gravitational waves have an impact on the properties of the string. When a matter field is incorporated, one expects that there will be interaction between the matter waves and the gravitational waves.Without a cosmological constant, it was found numerically that there are wave-like solutions which possess both electric and magnetic charge, and with an interaction between the gravitational waves and the YM waves. When a cosmological constant is introduced, it turns out that a new class of regular spatially compact solutions is found. The solutions with negative cosmological constants are quite surprising.
Physical models, in general, should not possess to many free parameters. In the Abelian string model, the are a lot of parameters, such as the shape of the Higgs potential. It is of interest to construct string-like solutions in the Einstein-Yang-Mills theory which resemble the Abelian Higgs model. Further, higher dimensional counterparts of compact objects like a cosmic string could possess unexpected properties
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