Visualization of flat and curved spacetimes with simple cartography tools

Abstract

Abstract Well-developed tools from cartography are adapted and used to visualize curved and flat spacetime to further understanding of relativistic effects, while interpretation of these tools is aided through their application to the familiar world map. First spacetime is cut into two-dimensional subspaces that are mapped to the Minkowski or Euclidean plane using map projections. Then, exploiting the cartographic technique of map interruptions, the two-dimensional subspaces are further cut into ‘wedges’ to minimize the distortion. The mental exercise of sewing the wedges back together aids in visualizing and interpreting the geometry of the spacetime slice. Map distortion indicators alert the reader to the inevitable effects of distortion. This wedge diagram provides a simple yet powerful tool to visualize both flat and curved spacetime geometry. It is more general than the familiar embedding diagram and avoids the pitfall of confusing the space in question with the embedding space. The wedge diagram is applied first to Minkowski spacetime with the cartographers’ azimuthal equidistant map projection and, when combined with the wedge technique, provides a simple visualization of the twin paradox. The wedge diagram is then applied to the exterior Schwarzschild spacetime to visualize the physical significance of spacetime curvature. In particular it is shown how gravitational time dilation results from intrinsic curvature of a radial-time slice of spacetime. Then this diagram is applied to the Friedmann–Robertson–Walker (FRW) spacetime revealing how expansion of the Universe results from a qualitatively similar curvature of a radial-time slice, but with the roles of space and time reversed. The analogy between the two spacetimes suggests novel interpretations of the two spacetimes. We interpret the Schwarzschild spacetime as time expanding with increasing distance from the horizon. And we interpret the FRW spacetime as the Universe expanding as the influence of the big bang peters out.