Loop Special Relativity: Kaluza-Klein area metric as a line element for stringy events

Abstract

Let a physical event constitute a simple loop in spacetime. This in turn calls for a generalized loop line element (= distance$^2$ between two neighboring loops) capable of restoring, at the shrinking loop limit, the special relativistic line element (= distance$^2$ between the two neighboring centers-of-mass, respectively). Sticking at first stage to a flat Euclidean/Minkowski background, one is led to such a preliminary loop line element, where the role of coordinates is played by the oriented cross-sections projected by the loop event. Such cross-sections are generically center-of-mass independent, unless (owing to a topological term) the loop events are intrinsically wrapped around a Kaluza-Klein like compact fifth dimension. Serendipitously, it is the Kaluza-Klein ingredient which, on top of its traditional assignments, is shown to govern the extension of Pythagoras theorem to loop space. Associated with $M_4 \otimes S_1$ is then a 10-dim loop spacetime metric, whose 4-dim center-of-mass core term is supplemented by a 6-dim Maxwell-style fine structure. The imperative inclusion of a positive (say Nambu-Goto) string tension within the framework of Loop Special Relativity is fingerprinted by a low periodicity breathing mode. Nash global isometric embedding is conjectured to play a major role in the construction of Loop General Relativity.