Abstract
The definition of a sphere depends on the definition of distance in the embedding three-dimensional space; the classification is straightforward, but should be clearly understood to appreciate the richness and the variety of this concept in geometry and mathematical physics. There are essentially three loci of points with the same "distance" from a centre O: the ordinary sphere S2, the single-sheet hyperboloid AdS2 and the two-sheet hyperboloid script H sign. The extraordinary fertility of AdS 2 and script H sign began in the XIX century, when the Italian mathematician E. Beltrami discovered an intrinsic metric on a disk on the plane which is a canonical realization of two-dimensional hyperbolic geometry with constant curvature, but did not recognize that it is just a stereographic projection of double-struck H sign. Subsequently, AdS2 became an important geometrical building block in special relativity, in cosmology and, in 1959, in a solution of Einstein-Maxwell's field equations corresponding to a uniform electromagnetic field. This space-time, here called BR after Bertotti and Robinson's papers of 1959, consists in the combination of two (generalized) spheres and can be obtained in a purely geometric way. The BR geometry has played a relevant role in the search for new unifying fundamental laws, in particular in very high-energy physics, and has provided examples to test and exemplify new physical principles. In sect. 4 we briefly outline three general areas. The first area is the extension of a classical field theory to the complex domain (in particular, in relation to quantum gravity). Ideally, the most interesting complexification of a Riemannian manifold consists in endowing it with a Kählerian structure; it turns out that, while this is possible for a definite signature in many cases, in space-time a Kählerian manifold is obtained only if it is just the BR solution (or, trivially, if it is flat). The other two areas are: the exploration of the quantum properties of the horizon of a black hole and the holographic principle; string theory, and the fundamental role of a scalar field, the dilaton. We give, and clarify, examples in which the BR solution has been applied.
Lingua originale | Inglese |
---|---|
pagine (da-a) | 1-23 |
Numero di pagine | 23 |
Rivista | La Rivista del Nuovo Cimento |
Volume | 29 |
Numero di pubblicazione | 1 |
DOI | |
Stato di pubblicazione | Pubblicato - 2006 |
Pubblicato esternamente | Sì |