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.
This is way out of my wheelhouse...
I'm confused about what a "timelike geodesic" is. Objects can move in space and geodesics are the lines or shortest distances of travel.
I guess relativity explains how, for light, a geodesic is different than one in space.
But what is going on with the "timelike geodesic" ?
When you say 2+1-dimensional is that a 2D plane plus time? so the model of our universe would need to be (3+1)-dimensional? Is this flatland stuff?
@futurebird @gregeganSF And if something were traveling in spacetime along a timelike geodesic, it would be going faster than the speed of light.
And perhaps traveling backwards in time, from a relativity point of view. The symmetry transformations of the space don't necessarily preserve forward vs backward in time, for timeline geodesics. That's another way of saying that two different observers looking at the two endpoints of a timelike geodesic might disagree about which end "happened first."
No, that’s exactly wrong. What you’re saying here applies to spacelike geodesics. Every ordinary object’s world line is a timelike geodesic; no object’s world line is a spacelike geodesic, as that would mean travelling faster than light.
Greg Egan
myrmepropagandist
Michael Kleber