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

抄録

We define an observer-centered drawing map for the causal future of an origin event in 2+1-dimensional Minkowski spacetime. Time is assigned to a Euclidean drawing radius r = ct, spatial direction to an angle θ, and distance from the central observer to a cylinder height h. At each time, the reachable disk is represented by the lateral surface 0 ≤ h ≤ r, and successive surfaces form concentric cylinders. The map is a diffeomorphism onto its image away from the observer’s worldline; the bottom circle carries redundant direction labels for that worldline, not distinct physical positions. The physical Lorentzian metric is retained and is not the Euclidean metric of the drawing.

The construction supplies an inverse, a physical metric, and explicit formulas for event-to-event reachability, re-centering, and Lorentz boosts. A finite causal-diamond example organizes two-way communication by time shell. Cosmological applications distinguish the selected reachable region from an entire spatial slice and classify particle and event horizons by conformal-time integrals with explicit topology restrictions. An imposed initial cutoff produces a central hole under a fixed drawing origin. The Milne example compares inertial and proper-time slicings, without identifying its regular Minkowski origin with a singular FLRW boundary.

The construction retains direction and distance in a radial-time picture and complements conventional and conformal diagrams. Its auxiliary plane-to-cylinder logarithmic map is distinguished from physical spacetime conformal maps. Black-hole figures and quantum-theory connections remain sketches or analogies; neither entropy nor singularity resolution follows from drawing area or a cutoff. Three-spatial-dimensional figures require projection or additional encoding, and educational effectiveness remains to be tested.