Minkowski Spacetime  

In General > s.a. history of relativity; lorentz invariance; rindler space; special relativity.
* Idea: The only stationary, asymptotically flat, smooth and topologically trivial solution in 4D spacetime; The only zero of the energy for asymptotically flat spacetimes.
$ Def: An affine space over Rn, equipped with a flat Lorentzian metric.
* Line element: For n = 4, in a few of the usual sets of coordinates,

ds2 = –dt2 + dx2 + dy2 + dz2 = –dt2 + dr2 + r2 (d2 + sin2 d2)
                        = –2 d2 + d2 + (22) (d2 + sin2 d2) = –du dv + (uv)2 d2 ,

where = (r2t2)1/2, = arctan(t/r), or r = cosh , t = sinh ; u = t + r, v = tr.
@ Properties: in Lichnerowicz 55; Formiga & Romero gq/06 [differential geometry of curves and Serret-Frenet equations]; Giulini a0802 [structure]; > s.a. Hypersurfaces.
@ Causal structure: Thomas & Wichmann JMP(97) [and quantum field theory]; > s.a. models of spacetime [axiomatic].
@ 3D: Kim & Yoon JGP(04) [ruled surfaces].
@ 2D: Mermin AdP(05)gq/04 [history and geometry].

Milne Universe
* Idea: An unconventional coordinatization of Minkowski space; Spatial slices are r2t2 = constant hyperboloids, corresponding to a linear expansion, ds2 = –d2 + 2 d2; Interesting for quantum field theory.
* Line element: Can be obtained setting a(t) = t in the FRW form; In 2D, using a for a constant parameter,

ds2 = –dt2 + a2t2 dx2 = exp{2a}(d2 + dx2) .

* Relationship: Can be obtained as the future light cone in Minkowski spacetime –dT 2+dX 2, with coordinates

T = a–1 exp{a} cosh ax ,   X = a–1 exp{a}sinh ax .

@ References: Milne QJM(34), QJM(34); McCrea & Milne QJM(34); Gilbert QJM(38); Milne in 51; Robertson ZfAp(38); Sandage ApJ(61); in Birrell & Davies 82, 5.3; Dunning-Davies ap/04 [and Newtonian cosmology]; Chodorowski PASA(05)ap [and supernova magnitude-redshift relation]; Macleod phy/05 [and observational cosmology].

Special Topics > s.a. types of spacetimes [other flat ones]; gravitational instantons and semiclassical general relativity [stability, semiclassical].
@ Stability, classical: Christodoulou & Klainerman AMS; Lindblad & Rodnianski m.AP/04.
@ Deformations: Majid ht/94; de Azcárraga & Rodenas JPA(96), qa/96-in [h-deformed, calculus]; Bauer & Wachter EPJC(03)mp/02 [q-deformed]; D'Andrea JMP(06)ht/05 [and Snyder's non-commutative geometry, coordinate algebra as operators on Hilbert space]; Ghosh ht/06/PLB [-Minkowski and DSR]; Meljanac et al a0705 [star product realizations]; Amelino-Camelia et al a0707 [boosts and space-rotations in -Minkowski]; Freidel & Kowalski-Glikman a0710 [symmetries and field theory]; > s.a. non-commutative geometry.
@ Related topics: Peters AJP(86) [periodic boundary conditions with moving walls, and Lorentz symmetry]; Beem CMP(80) [local extendibility]; Marolf & Patiño PRD(06)ht [2+1, energy]; > s.a. conservation laws.
> Related topics: see extrinsic curvature [constant mean curvature surfaces]; Polygons.


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