Conformal geometry and elementary particles

Abstract

Then, since J has properties (P and Q, sets G, Ti, and Di may be constructed so that the sets G u u Ti u u Di and ( u u Zi u u Ti are homeomorphic.The extension of this homeomorphism to a self-homeomorphism of R3 which maps J onto a is then made. In obtaining this extension strong use is made of the result due to Alexander7 that one of the complementary domains of a polyhedral torus in spherical 3-space is homeomorphic to a complementary domain of a standard torus.It is known8 that if Si, S1' and S2, S2' are two pairs of polyhedral spheres in RI, with Si contained in the interior of Si' (Int Si'), then there is a homeomorphism of the closure of the set (Int Si') n (Ext Si) onto the closure of (Int S2') n (Ext S2).It is evident that quite a diverse collection of types of regions are bounded by pairs of disjoint polyhedral tori, but an extension of the above-mentioned result to suitably restricted pairs of "concentric" polyhedral tori is made. This provides a chief tool for the proof of the above theorem.III. If the 1-manifold J has properties (P and Q at each point, then J is locally tame and conversely.* This work was begun jointly during the tenure of a Naitional Science Foundation Fellowship by Griffith. I L. Antoine, "Sur la possibility d'entendre l'hom6omorphisme de deux figures a leur voisinage,"

Relativistic mechanics in multiple time dimensions

“…(40), (41), (44), (48), (51), and (53).] If P k h¼1 k 00 hl 6 ¼ P k h¼1 k hl (that is, K 00 l 6 ¼ 1, l ¼ 1, 2, .…”

“…Let us now consider generalized five-dimensional space-time instead of the four-dimensional of space-time of Minkowski, where the fifth dimension corresponds to proper time and consequently is connected with the rest mass. 11,[35][36][37][38][39][40][41][42][43][44][45][46] It turns out that from the geometric point of view, chirality P 5 in fact means inversion of the fifth axis (i.e., inversion of the proper time or accordingly of the rest mass). Taking into account Eq.…”

Conformal Klein-Gordon Equations and Quasinormal Modes

“…On the other hand (our second, biprojective case) we can interpret the extra coordinate, x 5 , as a second projective coordinate. We then obtain the so-called conformal projective relativity, proposed by Arcidiacono [20], which extends in cosmic scale the theory proposed by Ingraham [21], but with a different physical interpreta-tion. In this theory we have another universal constant, r 0 , which can be taken as r/r 0 = N , where r is the radius of the hypersphere and N is the cosmological number appearing in the Eddington-Dirac theory [22].…”

Pioneer Anomaly and Accelerating Universe as Effects of the Minkowski Space Conformal Symmetry

“…If it turns out that possible values of the parameter |b| = aη are bounded below by b 0 = a 0 η 0 , the permissible time interval is bounded above by t max = (2a 0 η 0 c) −1 . A nonlinear character of the transformations (10) suggests nonuniformity of time. Geometrically, these transformations represent a deformation (respectively compression or extension) of the light cone generating lines.…”

“…Because of this, the approach requires no specific dynamic substantiation. In fact, the question is about the exhibited time inhomogeneity induced by the cosmological expansion and described in the general case by equation (10). In the approximation of H 0 t ≪ 1, valid in a very wide time interval, this inhomogeneity is exhibited as a term quadratic in t, that appears in the expression determining the nonlinear dependence of t ′ on t.…”