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,

d*s*^{2} = –d*t*^{2} +
d*x*^{2} + d*y*^{2} +
d*z*^{2}
= –d*t*^{2} + d*r*^{2} + *r*^{2} (d*θ*^{2} +
sin^{2}*θ* d*φ*^{2})

=
–*ρ*^{2} d*τ*^{2} +
d*ρ*^{2} + (*ρ*^{2}–*τ*^{2})
(d*θ*^{2}
+ sin^{2}*θ* d*φ*^{2})
= –d*u* d*v* + \(1\over4\)(*u*–*v*)^{2} dΩ^{2} ,

where *ρ* =
(*r*^{2}–*t*^{2})^{1/2},
*τ* = arctan(*t*/*r*), or *r* =
*ρ* cosh *τ*, *t* =
*ρ* sinh *τ*; and *u* = *t *+ *r*, *v* = *t* – *r* 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 *r*^{2} – *t*^{2} =
constant hyperboloids, corresponding to a linear expansion, d*s*^{2} = –d*τ*^{2} + *τ*^{2} d*σ*^{2};
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,

d*s*^{2} = –d*t*^{2} + *a*^{2}*t*^{2} d*x*^{2}
= exp{2*a**η*}(d*η*^{2} +
d*x*^{2}) .

* __Relationship__: Can
be obtained as the future light cone in Minkowski spacetime
–d*T*^{ 2} + d*X*^{ 2},
with coordinates

*T* = *a*^{–1} exp{*aη*}
cosh *ax* , *X* = *a*^{–1}
exp{*aη*}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]; Ling a1706 [and cosmology].

**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]; Fajman et al a1707 [for the Einstein-Vlasov system].

@ __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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sep 2017