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Penrose, R. and Rindler, W. 1986, "Spinors and space-time. Vol. 2: Spin or and twistor methods in space-time geometry," Cambridge University Press, Cambridge. S. 1977, Phys. , 61A, 81-2. S. N. J. Baston), LMS Lecture Note Series, 156, Cambridge University Press, Cambridge, pp246-259. Ward, R. S. and Wells, R. O. (1990). Twistor geometry and field theory. Cambridge University Press, Cambridge. SOLITON EQUATIONS AND CONNECTIONS WITH SELF-DUAL CURVATURE. I. K ABSTRACT. This article describes how soliton equations of 'zero curvature' type can be used to satisfy the self-dual Yang-Mills equations; this produces connections which need not possess translationally invariant self-dual curvature.
The equations can be explicitly solved, and the solution, the so-called Schwarz schild metric, has a singularity at r = 0, the centre of the spherical symmetry . Physically, this occurs because there is nothing to balance out the gravitational attraction. If one attempts to obtain a balance by introducing electro-magnetic forces, coupling Einstein's equations with Maxwell's equations, there is still no smooth, radially symmetric, static solution. The only solution, the Reissner-Nordstrom metric, is again singular at the origin .
L is the Lorentz index running from 0 to 2 for respectively t, x and y. The finite energy static configurations of the model correspond to the mappings of the two dimensional sphere into itself and so these field configurations are characterised by an integer-valued topological charge describing such mappings. As the time evolution corresponds to a continuous transformation, the topological charge remains constant and so is conserved. Thus we can think of our extended structures as providing us with (2 + 1) dimensional analogues of, say, monopoles and antimonopolcs or protons and antiprotons.