A Cylindrical Spacetime Diagram with Time as the Radial Coordinate: An Observer-Centered Visualization of the Light-Cone Interior

抄録

The light cone of special relativity is conventionally drawn as a cone with 45° slopes, and it serves as a standard representation of causality. This paper takes time as a radius r and maps the region reachable from the origin at each time onto the surface of a cylinder of radius r with coordinates (θ, h), the observer's distance becoming the height h. The reachable region is then the single condition h ≤ r; the regions of successive times are nested concentrically rather than stacked along an axis, and direction and distance are carried by separate coordinates. The construction is a drawing embedding fixed to one central observer, mapping the causal future of the origin (minus the observer's worldline) one-to-one into a cylindrical drawing space; it is related to the Milne universe, to the standard cylindrical spacetime diagram recovered under a logarithmic map, and to conformal compactification.

As applications, the resolution known from standard cosmology for the conflict about the Big Bang—"a single point, or everywhere at once?"—becomes one picture of an expanding observer-centered reachable region; a cutoff r₀ assumed at an initial time appears as a hole at the centre, provided the drawing origin is kept at the extrapolated start; and the particle horizon, inflation, and the outer wall r_max of accelerated expansion fall within the same scheme, models being classified by whether the nest has a centre and an outer wall. These are the claims of the paper. The future of a surface point further admits two readings—keeping the origin, or re-centering on it—congruent in the conventional diagram by translation invariance but different figures here. Black holes, and correspondences with radial quantization, dS/CFT, and cyclic cosmologies, are kept separate as sketch and outlook.

What this paper changes is not the physics but the drawing. As a supplementary figure depicting observer-centered causal structure in a way consistent with intuition, we expect it to be of educational value and to help generate new physical ideas.