On the constraints problem for the Einstein-Yang-Mills-Higgs system. (English) Zbl 1304.35682
Author’s abstract: We provide proofs of some key propositions that were used in previous work by M. Dossa and the author [AMRX, Appl. Math. Res. Express 2010, 154–231 (2010; Zbl 1205.35162)] and by F. Hirosawa [in: Further progress in analysis. Proceedings of the 6th international ISAAC congress, Ankara, Turkey, August 13–18, 2007. Hackensack, NJ: World Scientific. 444–453 (2009; Zbl 1184.35086)] dealing with the characteristic initial value problem for the Einstein-Yang-Mills-Higgs (EYMH) system. The aforesaid proofs were missing, making the considered work difficult to understand. This work is presented with a view to have an almost self-contained paper. With this respect we completely recall the process of constructing initial data for the EYMH system on two intersecting smooth null hypersurfaces as done in the work of Dossa and Tadmon mentioned above. This is achieved by successfully adapting the hierarchical method set up by A. D. Rendall [Pitman Res. Notes Math. Ser. 253, 154–163 (1992; Zbl 0795.35127)] to solve the same problem for the Einstein equations in vacuum and with perfect fluid source. Many delicate calculations and expressions are given in details so as to address, in a forthcoming work, the issue of global resolution of the characteristic initial value problem for the EYMH system. The method obviously applies to the Einstein-Maxwell and the Einstein-scalar field models as well.
Reviewer: Anthony D. Osborne (Keele)
MSC:
| 35Q75 | PDEs in connection with relativity and gravitational theory |
| 35L15 | Initial value problems for second-order hyperbolic equations |
| 35L70 | Second-order nonlinear hyperbolic equations |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 46J10 | Banach algebras of continuous functions, function algebras |
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
| 83C10 | Equations of motion in general relativity and gravitational theory |
| 83C20 | Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory |
Keywords:
Einstein-Yang-Mills-Higgs system; null hypersurfaces; characteristic initial value problem; constraints problem; harmonic gauge; Lorentz gaugeReferences:
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