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.
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?
A timelike geodesic is the world line for a free-falling object. In the absence of gravity, spacetime is flat (that is Minkowski spacetime, the spacetime equivalent of flat Euclidean space) and the world line of a free-falling object is just a straight line through spacetime, which we would see as the object moving along a straight line in space at a uniform velocity.
But where there is gravity, spacetime is curved, so just as the geodesics on a curved surface are not straight lines, the geodesics through spacetime are no longer straight lines.
For example, the timelike geodesic of the Earth as it orbits the Sun is a kind of elliptical helix through spacetime, as it follows an ellipse in space while its time coordinate increases.
“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?”
Yes, I’m starting with “flatland” (2D space plus time), and then carving out a hyperboloid within it that is even lower-dimensional, just 1D space plus time.
So this (1+1)-dimensional spacetime is the simplest spacetime with constant curvature, a bit like the 2D surface of a sphere is the simplest space with constant curvature.
But what’s cool is that, while the geodesics on a sphere behave the same way whichever direction they point — a bunch of parallel geodesics will always converge — the geodesics in (1+1)-dimensional de Sitter spacetime behave in three different ways! If they point in a “spacelike” direction [more spacey than timey] they converge. If they point in a “timelike” direction [more timey than spacey] they diverge.
And if they point in a “lightlike” direction [equal parts spacey and timey, at least in units where c=1] they neither grow closer nor further apart.
@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.
happily wading throughs acres of shitposts to get to this content.
Greg Egan
myrmepropagandist
Michael Kleber
Я⊛ḶḶ model ☭