I would like to understand basic concepts of the general relativity in 2+1 spacetime. As far as I know, GR predicts that such a spacetime is flat everywhere except for the point masses which create angular deficit proportional to their mass. Flatland with one point mass is like surface of cone. I imagine that when one adds other point masses the Flatland can be folded to a (convex) polyhedron (then there is the constraint on total masses, since total angular deficit is 720 degrees) (see note #1). I assume that a 2d Flatlander would not (at least locally) notice crossing the edges when moving from one face of polyhedron to another.

The problem I have with this model is that when one heavy body which defines the Flatland is set to motion, its mass must change and - more surprisingly from a local point of view - also masses of the neighboring bodies to keep the total of 720 degrees. The image shows cube with a vertex moving along edge to its middle with corresponding angular deficits.

On the other hand, I know that 2+1 gravity and motion of point masses have been considered seriously by Gott (in his two strings time machine), Caroll, Guth, t' Hooft and others. Where is error in my naive model?

Edited: Given the first answer and comments I should maybe be more precise:

Is a motion which requires change of angular deficit (and hence mass) of the surrounding point masses possible, or is possible only motion when all angular deficits are kept constant? Anyway, for a Flatender living on the polyhedron surface the situation looks like there is an interaction between the point masses, despite the fact the spacetime is flate in between them. Or is such a configuration (initial condition) simply impossible?

Edited: I have overlooked the fact that a point mass cannot be just "set to motion" by a miracle - total momentum must be conserved. I will think it over and prepare a better example.

Edited: This papers by 't Hooft may contain answer:

The evolution of gravitating point particles in 2+1 dimensions (pdf)

Three-dimensional Einstein gravity: dynamics of flat space (pdf)

enter image description here

Notes (added in later edits):

1) Gott & Alpert: General Relativity in a (2+1)-Dimensional Space-Time (Gen. Relat. Gravit. 16:243-247, 1984):

"Consider a convex polyhedron with a finite number of faces. The faces and edges have no intrinsic curvature and represent solutions to the vacuum field equations. The vertices each have an angle deficit (like the vertex of a cone) and represent point masses. For example, a universe shaped like the surface of a cube represents a vacuum with 8 point masses of $M = \pi/2$ each (three squares meet at each vertex giving each an angle deficit of $\pi/2$). The Einstein static universe of equation (6) may be approximated by a polyhedron of many faces containing many vertices each with small angle deficits. The total mass in such a closed universe is always $M_u = 4\pi$."

In my opinion, there are also some nonconvex polyhedra which work well.

  • $\begingroup$ Polyhedron is kind of a special case (because it implies finite volume). Why not consider first two point particles with small angular deficit. This could be mapped onto 2d plane (times time) with two slices cut out (and glued along the line of cut) $\endgroup$ – user23660 Sep 22 '13 at 14:11
  • $\begingroup$ @user23660 Because in such a case there is only upper limit of the total angular deficit and hence mass (in open Flatland): 360 degrees (otherwise you cannot unfold the Flatland without overlaping). Then I understand that the two particles can move freely. $\endgroup$ – Leos Ondra Sep 22 '13 at 14:22
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    $\begingroup$ So what you are asking is, if compactness of multiconic space-time (i.e. polyhedron) would imply additional constraints? $\endgroup$ – user23660 Sep 22 '13 at 14:45
  • $\begingroup$ @user23660 I guess I do :-) $\endgroup$ – Leos Ondra Sep 22 '13 at 14:49
  • $\begingroup$ For minimal case of 4 particles (tetrahedron) the angular deficits of vertices define the tetrahedron up to rescaling. The only allowable motion is enlargement (or contraction) of the tetrahedron (cosmological expansion of sorts). $\endgroup$ – user23660 Sep 22 '13 at 16:19

Differential geometry predicts that the Weyl tensor vanishes in 2+1 dimensions. General relativity predicts that the Ricci curvature vanishes in a vacuum -- i.e. no forces at a distance. So acceleration due to gravity can still exist in 3 dimensional spacetime, but only in a region with a non-zero stress-energy tensor.

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    $\begingroup$ It looks like the OP's question is phrased in terms of Regge-style discrete spacetime, so in this context his statement that the spacetime is flat everywhere except at vertices carrying mass points would be right? $\endgroup$ – twistor59 Sep 22 '13 at 11:43
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    $\begingroup$ I don't quite understand the answer, I'm afraid, but it seems to me that your reasoning is not restricted to the closed Flatland (where total sum of angular deficits equals 720 degrees). Does it mean that Gott time machine using two cosmic strings is impossible simply because the string (which are analog to the point mass or vertices here) can't move relative to each other? $\endgroup$ – Leos Ondra Sep 22 '13 at 13:29
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    $\begingroup$ I don't see the reasoning behind the phrase "So to answer your question, the masses would not be able to move unless they're somewhere like inside the Earth". Why in order to move mass have to be in some curved space-time? Point particle can move in flat space-time independent of presence of gravity at all and peculiar type of such gravity should not exclude such motion (just change the interactions between such particles). $\endgroup$ – user23660 Sep 22 '13 at 13:51
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    $\begingroup$ @AlexeyBobrick: But in 2+1 $R_{\mu\nu}=0$ implies $R_{\mu\nu\lambda\rho}=0$. So the geometry is flat. $\endgroup$ – user23660 Sep 22 '13 at 16:55
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    $\begingroup$ @LeosOndra: Such motion is possible even for convex polyhedron as long as there are enough vertices, just the number of degrees freedom will be smaller then in open (noncompact) space. $\endgroup$ – user23660 Sep 22 '13 at 17:18

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