Results already obtained involving similar but not equal tetrads will be useful in our constructions and demonstrations. These fields will constitute the structure needed to build local tetrads of a new kind that allow for the proof to be carried out with simplicity. In order to prove this claim we will introduce a system of differential equations involving several kinds of fields-Abelian, non-Abelian, spinor, gravitational. In order to accomplish this goal and since the group of translations is four-dimensional we will prove first that it is isomorphic to (⊗ LB1) 4. A special case of the latter is the Bondi–Metzner–Sachs subgroup of supertranslations. By using the geometric structures built to prove this elementary result we will generalize it to the case of what we might call local translations. Translations have been extensively studied under the scope of gauge theories.
WORKING WITH INFINITESIMALS PLUS
LB1 is the group composed of SO(1, 1) plus two kinds of discrete transformations. We will prove that there is a direct relationship between the Poincaré subgroup of translations, and the group of tetrad transformations LB1 introduced in a previous paper. We thank Ryszard Kostecki for a careful perusal of the manuscript and insightful comments. The category of functorial differential spaces can indeed serve as a machine Acknowledgements This is well resonant with the present tendency of geometrization and algebraization in contemporary theoretical physics. The scheme has a strong, both differential and algebraic, aspects. The first result of the present work is the formulation of a very general scheme - the category of functorial differential spaces - unifying many, known so far, conceptions of differential spaces, and enabling to produce new ones. There are two projections: π 1: Δ × Σ → Δ and π Concluding remarks It is a ringed space (M, C), where C is a set of real-valued functions on M satisfying the following conditions: Our starting point is a Sikorski differential space. Later on, many works and several generalizations of this concept were proposed (for a review of literature till 1992 see ). One of the first to introduce differential spaces were Postnikov, Sikorski, Spallek and Aronszajn (but see also ). Differential spaces encode the structure of space in a ring of functions rather than in atlases and maps, and rings of functions are a natural place for introducing infinitesimals, as it is done in synthetic differential geometry.
These two objectives are not independent. This could prove interesting when dealing with the classical singularity problem, when the space shrinks to infinitesimal dimensions. The present paper has two objectives: first, to generalize and then to explore the concept of differential spaces as a tool for developing differential geometry of space-time second, to enrich this geometry with infinitesimals that would allow us to penetrate into the superfine structure of space-time which is inaccessible to the standard geometric methods.
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