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 \(\mathbb R\)n, 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 (dθ2 + sin2θ dφ2)
                        = –ρ2 dτ2 + dρ2 + (ρ2τ2) (dθ2 + sin2θ dφ2) = –du dv + \(1\over4\)(uv)22 ,

where ρ = (r2t2)1/2, τ = arctan(t/r), or r = ρ cosh τ, t = ρ sinh τ; and u = t + r, v = tr are null coordinates.
* Uniqueness: A stationary, asymptotically flat spacetime satisfying the vacuum Einstein equation must be Minkowski spacetime; This was proved by Lichnerowicz, but the stronger conjecture by Einstein that the result should hold for any asymptotically flat complete vacuum Einstein spacetime was proved wrong in the 500-page work of Christodoulou and Klainerman of 1990; However, the conjecture has been proved to be true by Schoen and Yau for Einsteinian space times with zero ADM mass, that is tending fast enough to flatness at spatial infinity (faster than a Newtonian potential).
@ Properties: in Lichnerowicz 39, 55; Formiga & Romero AJP(06)nov-gq [differential geometry of curves and Serret-Frenet equations]; Giulini a0802-in [structure]; Catoni et al 08, Naber 11 [geometry]; > s.a. Hypersurfaces.
@ Causal structure: Thomas & Wichmann JMP(97) [and quantum field theory]; Kim CQG(10)-a1003 [group of causal automorphisms]; de Seguins Pazzis & Šemrl a1502 [continuous transformations preserving light cones]; > s.a. models of spacetime [axiomatic].
@ 3D: Kim & Yoon JGP(04) [ruled surfaces].
@ 2D: Mermin AdP(05)gq/04 [history and geometry].
@ 2D, causal isomorphisms: Kim CQG(10); García-Parrado & Minguzzi CQG(11)-a1105; Low CQG(11); Burgos CQG(13); Kim JMP(13)-a1304; Kim GRG(17)-a1609 [and the invariance of wave equations]

Milne Universe
* Idea: An unconventional coordinatization of Minkowski space; Spatial slices are r2t2 = constant hyperboloids, corresponding to a linear expansion, ds2 = –dτ2 + τ2 dσ2; Interesting for quantum field theory.
* Line element: Can be obtained setting a(t) = t in the FLRW form; In 2D, using a for a constant parameter,

ds2 = –dt2 + a2t2 dx2 = exp{2aη}(dη2 + dx2) .

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

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

@ References: Milne QJM(34), QJM(34); McCrea & Milne QJM(34); Gilbert QJM(38); in Milne 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]; Culetu IJMPD(10) [and Rindler spacetime].

Special Topics > s.a. types of spacetimes [other flat ones]; gravitational instantons and semiclassical general relativity [stability, semiclassical].
@ Stability, classical: Christodoulou & Klainerman 93; Lindblad & Rodnianski AM(10)m.AP/04; Bieri JDG(10)-a0904; Maliborski PRL(12) [instability of spherically symmetric self-gravitating massless scalar field in a timelike worldtube]; Okawa et al PRA(14)-a1311, LeFloch & Ma a1511 [self-interacting scalar fields].
@ Lower-dimensional: Marolf & Patiño PRD(06)ht [2+1, energy]; Boozer AJP(10)dec [1+1, symmetries of periodic lattices].
@ Related topics: Peters AJP(86)apr [periodic boundary conditions with moving walls, and Lorentz symmetry]; Beem CMP(80) [local extendibility]; Jadczyk a1105 [compactified]; Moretti & Di Criscienzo FiP(13) [determining if a spacetime is flat]; Fareghbal & Naseh CQG(15)-a1408 [flat/ccft correspondence]; Orel JCTA(17)-a1410 [maps preserving lightlike relations, and finite geometry]; Nadeem a1505 [secure positioning and position-based quantum cryptography are possible]; > s.a. conservation laws.
> Related topics: see decomposition [of tensor fields]; deformed minkowski space; extrinsic curvature [constant mean curvature surfaces]; Polygons; spacetime subsets; vectors [indefinite inner product spaces].


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