4,008 research outputs found
Lattice Universes in 2+1-dimensional gravity
Lattice universes are spatially closed space-times of spherical topology in
the large, containing masses or black holes arranged in the symmetry of a
regular polygon or polytope. Exact solutions for such spacetimes are found in
2+1 dimensions for Einstein gravity with a non-positive cosmological constant.
By means of a mapping that preserves the essential nature of geodesics we
establish analogies between the flat and the negative curvature cases. This map
also allows treatment of point particles and black holes on a similar footing.Comment: 14 pages 7 figures, to appear in Festschrift for Vince Moncrief (CQG
Fuchsian polyhedra in Lorentzian space-forms
Let S be a compact surface of genus >1, and g be a metric on S of constant
curvature K\in\{-1,0,1\} with conical singularities of negative singular
curvature. When K=1 we add the condition that the lengths of the contractible
geodesics are >2\pi. We prove that there exists a convex polyhedral surface P
in the Lorentzian space-form of curvature K and a group G of isometries of this
space such that the induced metric on the quotient P/G is isometric to (S,g).
Moreover, the pair (P,G) is unique (up to global isometries) among a particular
class of convex polyhedra, namely Fuchsian polyhedra. This extends theorems of
A.D. Alexandrov and Rivin--Hodgson concerning the sphere to the higher genus
cases, and it is also the polyhedral version of a theorem of
Labourie--Schlenker
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