In 3D Euclidean space, the surface equidistant from a point is a sphere, with constant curvature.
In (2+1)-dimensional Minkowski spacetime, the equidistant surface is a (1+1)-dimensional de Sitter spacetime, which also has constant curvature.
The image shows the spacelike (blue), timelike (black) and lightlike (white) geodesics through a point.
What about lightlike geodesics? Do they converge, diverge, or remain equally spaced? That’s tricky, because in 2D spacetime the only direction perpendicular to a lightlike geodesic is along the geodesic itself, so you can’t measure the separation along a perpendicular vector.
Here lightlike geodesics grow further apart, as measured along the grey circles. This is how the cosmological red shift works: wavefronts in an expanding universe move apart. But that’s a measurement by *particular observers*, at rest in galaxies that are themselves moving apart.
A more objective answer is that the lightlike geodesics remain “equally spaced” … but the spacing itself has no observer-independent value!
What does that mean? The cyan curves in this image join neighbouring geodesics with equal-length segments, while always meeting the geodesics at the same angle. But we could always choose a *different* set of curves that made that equal length take a different value.
So in a sense the separation is unchanging, while having no specific value.
Greg Egan