摘要
In this paper, we present a new form of “special relativity” (BSR), which is isomorphic to Einstein’s “special relativity” (ESR). This in turn proves the non-uniqueness of Einstein’s “special relativity” and implies the inconclusiveness of so-called “relativistic physics”. This work presents new results of principal significance for the foundations of physics and practical results for high energy physics, deep space astrophysics, and cosmology as well. The entire exposition is done within the formalism of the Lorentz <em>SL</em>(2<em>C</em>) group acting via isometries on <strong>real 3-dimensional Lobachevskian (hyperbolic) spaces</strong> <em>L</em><sup>3</sup> regarded as quotients <span style="white-space:nowrap;"><em>SL</em>(2<em>C</em>)/<em>SU</em>(2)</span>. We show via direct calculations that both ESR and BSR are parametric maps from Lobachevskian into Euclidean space, namely a <strong>gnomonic</strong> (central) map in the case of ESR, and a<strong> stereographic </strong>map in the case of BSR. Such an identification allows us to link these maps to relevant models of Lobachevskian geometry. Thus, we identify ESR as the physical realization of the Beltrami-Klein (non-conformal) model, and BSR as the physical realization of the Poincare (conformal) model of Lobachevskian geometry. Although we focus our discussion on ball models of Lobachevskian geometry, our method is quite general, and for instance, may be applied to the half-space model of Lobachevskian geometry with appropriate “Lorentz group” acting via isometries on (positive) half space, resulting yet in another “special relativity” isomorphic with ESR and BSR. By using the notion of a<strong> homotopy</strong> of maps, the identification of “special relativities” as maps from Lobachevskian into Euclidean space allows us to justify the existence of an uncountable infinity of hybrid “special relativities” and consequently an uncountable infinity of “relativistic physics” built upon them. This is another new result in physics and it states that so called “relativistic physics” is unique only up to a homotopy. Finally, we show that “paradoxes” of “special relativities” in either ESR or BSR are simply common distortions of maps between non-isometric spaces. The entire exposition is kept at elementary level accessible to majority of students in physics and/or engineering.
In this paper, we present a new form of “special relativity” (BSR), which is isomorphic to Einstein’s “special relativity” (ESR). This in turn proves the non-uniqueness of Einstein’s “special relativity” and implies the inconclusiveness of so-called “relativistic physics”. This work presents new results of principal significance for the foundations of physics and practical results for high energy physics, deep space astrophysics, and cosmology as well. The entire exposition is done within the formalism of the Lorentz <em>SL</em>(2<em>C</em>) group acting via isometries on <strong>real 3-dimensional Lobachevskian (hyperbolic) spaces</strong> <em>L</em><sup>3</sup> regarded as quotients <span style="white-space:nowrap;"><em>SL</em>(2<em>C</em>)/<em>SU</em>(2)</span>. We show via direct calculations that both ESR and BSR are parametric maps from Lobachevskian into Euclidean space, namely a <strong>gnomonic</strong> (central) map in the case of ESR, and a<strong> stereographic </strong>map in the case of BSR. Such an identification allows us to link these maps to relevant models of Lobachevskian geometry. Thus, we identify ESR as the physical realization of the Beltrami-Klein (non-conformal) model, and BSR as the physical realization of the Poincare (conformal) model of Lobachevskian geometry. Although we focus our discussion on ball models of Lobachevskian geometry, our method is quite general, and for instance, may be applied to the half-space model of Lobachevskian geometry with appropriate “Lorentz group” acting via isometries on (positive) half space, resulting yet in another “special relativity” isomorphic with ESR and BSR. By using the notion of a<strong> homotopy</strong> of maps, the identification of “special relativities” as maps from Lobachevskian into Euclidean space allows us to justify the existence of an uncountable infinity of hybrid “special relativities” and consequently an uncountable infinity of “relativistic physics” built upon them. This is another new result in physics and it states that so called “relativistic physics” is unique only up to a homotopy. Finally, we show that “paradoxes” of “special relativities” in either ESR or BSR are simply common distortions of maps between non-isometric spaces. The entire exposition is kept at elementary level accessible to majority of students in physics and/or engineering.
