Since my first question was answered in GQ (about FTL and time travel)…I thought I’d throw this one out.
I either read or saw an article about higher-dimensional universes. The question here relates to the ‘fact’ that there are no stable orbits for hyper-sphere planets around hypersphere stars.
Is this true? If so, why? I would assume that 4-D space would have an inverse cubed law of the strength of gravitation…does that have something to do with it?
That’s it in a nutshell. Circular orbits are possible for any sort of gravity at all, but in most cases, they’re not stable. In our normal inverse-square gravity, if you slightly perturb an object in a circular orbit (say, slow it down slightly), it’ll move closer to the star for half an orbit, then come back out for the other half, ending up back where it started after tracing out a complete ellipse. But in inverse-cube (or higher powers) gravity, if you nudge an object in a circular orbit, it’ll either spiral into the star, or out off into infinity, and never return to its original distance.
This is not an insurmountable obstacle for higher-dimensional theories. First of all, this only applies when the central object is a point source or the equivalent (a hypersphere that fills all of the available dimensions is equivalent to a point source). If, instead, your central object is a very long line extending through one of the dimensions, then its gravity will fall off as the inverse square of the distance from that line, and objects restricted to the space perpendicular to that line will still follow familiar Keplerian orbits.
In A.K. Dewdney’s The Planiverse, IIRC, he claims that there are stable orvits for two-dimensional “infra-spheres” (circles), however.
That depends on how your gravity works. Surprisingly, 2+1 dimensional general relativity doesn’t even result in gravity as an attractive force at all. There are still gravitational effects, but they take the form of a deficit angle, rather than any attraction.
If you contrived by some other means to produce a 1/r force, then I think that you can get stable orbits, but I’ve never actually done the math on that one. They might not be closed, though (so you’d get a Spirograph sort of pattern).
What does “deficit angle” mean?
Roughly speaking, it means that there are points in the 2-D space where the ratio of a circle’s circumference to its radius is not 2*pi, no matter how small you make the circle. You can view this as a “deficit” in the angles you measure at this point; if you stood at the point and started turning around, you’d end up facing the same way before you had turned a full 360 degrees. In more practical terms, it means that any triangles enclosing this special point would have interior angles summing to something other than 180 degrees, or that light rays that passed on opposite sides of the singular point would meet, or other such strange effects.
Thanks, MikeS. So Chronos’s remark was that, as in 3-D space, 2-D space in the vicinity of masses is non-euclidean, but this condition doesn’t manifest itself as an attractive force. But then I’m confused by your remark that “light rays that passed on opposite sides of the singular point would meet”. Wouldn’t that amount to saying that the paths of the light rays were bent by a gravitational force? My layperson’s understanding of GR was that it re-interpreted gravitational attraction as the result of objects traveling along geodesics in a manifold curved by the presence of mass. If that is still happening in 2-D, then in what sense is there no gravitational attraction?
Not quite. Such a space is locally Euclidean almost everywhere. The simplest model of a space with a deficit angle is a cone: Take a piece of paper, cut a wedge out of it (the angle of this wedge is the deficit angle), and then glue the cut edges together. Since you made the cone out of a flat piece of paper, and didn’t have to squeeze or stretch it, it’s still flat (in the geometric sense), except right at the vertex. Any triangle which does not have the vertex inside of it will still total 180 degrees, and two parallel lines which pass on the same side of the vertex will stay parallel. But a triangle which contains the vertex will have angles adding up to more than 180 degrees (in fact, to 180 degrees plus the deficit angle), and two parallel lines which pass on opposite sides of the vertex will end up converging.
Man…there is so much I don’t know. I wouldn’t even no where to begin to see that a 2-d universe, gravity isn’t attractive…or even have thought of it.
In a 4-D universe, orbiting a line would give inverse-square gravity and, therefore, be stable. However, I don’t see how a ‘star’ would be in shape of a line in 4-D space. Is this true or my limited 3-D thinking affecting me?
I ask because I also read an article saying 4-D life was impossible…because you cannot tie knots in 4-D space. This confused me…because couldn’t you tie planes into knots? Why wouldn’t 4-D creatures use knotted planes…why do they need line? These people were supposed experts so my thinking might be flawed…Therefore, I know I know nothing. 
Do masses in 2-D only produce singularities like that? I had thought that in our space there is a smooth but (in general) non-euclidean geometry except at special points where something bad happens, like black holes. Is that not the case? Or is it that way in 3-D, while in 2-D masses would induce a euclidean geometry with singularities?
Where to begin is to learn all of the mathematical techniques of GR in 3+1 dimensions, and then figure out how to generalize that to n+1 dimensions, and then let n = 2. It’s unfortunately not something that can really be explained in a message board post.
The simplest way is if the extra dimension is identified (looped around) on a length scale significantly smaller than the size of a star (or other familiar objects). Effectively, then, what you would have would be an identical copy of every object on each “layer”, which for most cases could be approximated as a line.
As for the life question, I agree that you could tie 4-d knots in planes: If nothing else, you could just take a 3-d knot and extend it infinitely along an extra dimension. Plus, of course, you could probably come up with some novel knots without 3-d analogues (which my brain is insufficient to visualize). Further, I don’t even see why knots would even be necessary for life.
Didn’t see this before:
I was a bit sloppy in my statements before. In a 2-d universe, there is (nonzero, finite) curvature wherever there’s mass. Everywhere outside of the masses, though, the space is conically flat. So an embedding diagram of the space around a star would be a cone with a rounded tip. A pointy cone would correspond (approximately) to a black hole (but only approximately: In 2+1 dimensions, mass is unitless, so there’s no Schwartzschild radius or horizon).
Thanks for the reply, Chronos. I’m still trying to make sense of this in terms of my layperson’s understanding. If there is a “rounded tip” where the mass is, wouldn’t that mean that a trajectory that passed through the “rounded tip” would appear to be deflected as though the mass had attracted it? I’m thinking of all those bowling-ball-on-a-trampoline depictions of gravity that one sees in popularizations of GR.
This is, technically speaking, correct. As a practical matter, however, any path through the “rounded” region would also have to contend with the matter present in that region. The fundamental difference between 2D and 3D gravity is that in 3D gravity, the region outside the matter is also curved and thus “deflects trajectories”, while in 2D gravity the region outside the matter doesn’t deflect trajectories at all.
Of course, we’re playing a little fast & loose with what it means to “deflect” a trajectory in 2D. In 3D, the metric is going to be “pretty close to flat” everywhere, and so we can talk about comparing the trajectory a particle actually takes to the path it would have taken if spacetime had been flat. However, it’s not so clear to me that you can do this easily in 2D, since the metric has this deficit angle for any circle enclosing the central matter. (In fancy words, it’s not clear to me whether one can legitimately view a 2D metric with a deficit angle as a small perturbation to an asymptotically flat metric.) The best way to think about whether spacetime is flat or not is to do tidal measurements, as explained by Cecil towards the end of this column. The gedankenwrenches he mentions there will diverge for a spaceship in orbit about a 3D mass, but no such divergence will happen for a 2D spaceship.
Fascinating. Thanks for the replies, MikeS and Chronos.