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. geodesics; 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__: It 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*aη*}(d*η*^{2}
+ d*x*^{2}) .

* __Relationship__:
It 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,
a1803 [and cosmology, symmetries].

**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];
Briscese & Modesto a1811 [in non-local gravity];
Graf a2010 [global nonlinear stability].

@ __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];
linearized quantum gravity [stability, quantum]; lorentz group [origin of Lorentz symmetry]; Polygons;
spacetime subsets; spacetime topology;
vectors [indefinite inner product spaces].

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