Cavatappi 2.0: More of the same but better

TL;DR

This paper explores a family of geometrically interesting surfaces inspired by pasta shapes, extending to Lorentzian geometry to model relativistic phenomena like the classical electron orbit, providing visual tools for relativity concepts.

Contribution

It introduces a 4-parameter family of surfaces, including relativistic models of electrons, useful for visualizing spacetime splittings in general relativity.

Findings

Derived a continuous family of invariant surfaces including spheres, tori, and cylinders.

Modeled the Fermi Born rigid model of the classical electron in relativistic spacetime.

Provided visualizations of spacetime splittings and coordinate systems in stationary spacetimes.

Abstract

Innocent musing on geodesics on the surface of helical pasta shapes leads to a single continuous 4-parameter family of surfaces invariant under at least a 1-parameter symmetry group and which contains as various limits spheres, tori, helical tubes, and cylinders, all useful for illustrating various aspects of geometry in a visualizable setting that are important in special and general relativity. In this family the most aesthetically pleasing surfaces come from screw-rotating a plane cross-sectional curve perpendicular to itself, i.e., orthogonal to the tangent vector to a helix. If we impose instead this orthogonality in the Lorentzian geometry of 3-dimensional Minkowski spacetime with a timelike helical "central" world line representing a circular orbit, we can model the Fermi Born rigid model of the classical electron in such an orbit around the nucleus, and visualize the Fermi coordinate grid and its intersection with the world tube of the equatorial circle of the spherical surface of the electron (suppressing one spatial dimension). This leads what we might playfully term "relativistic pasta". This is a useful 2-dimensional stationary spacetime with closed spatial slices on which to illustrate the slicing and threading splittings of general relativity relative to a Killing congruence, like stationary axisymmetric spacetimes including the rotating black hole family.