Minkowski
Spacetime| 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 (d
2 +
sin2 d
2)
=
–
2 d
2 +
d2 + (2–2)
(d2
+ sin2 d2)
= –du dv + 
(u–v)2 d
2 ,
where =
(r2–t2)1/2,
=
arctan(t/r),
or r =
cosh , t =
sinh ; u = t + r, v = t – r.
@ 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 r2 – t2 =
constant hyperboloids, corresponding to a linear expansion, ds2 = –d2 + 2 d
2;
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;
Bieri a0904.
@ Related topics: Peters AJP(86)apr
[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
decomposition [of tensor fields]; extrinsic
curvature [constant mean curvature surfaces]; Polygons.
Deformations, Non-Commutative Versions
* Idea: 
-Minkowski
space has a commutative spatial structure, but t does not commute with spatial
coordinates.
@ General references: 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]; Meljanac et al EPJC(08)-a0705 [star
product realizations]; Freidel & Kowalski-Glikman a0710-in
[symmetries and field theory]; Bentalha & Tahiri PRD(08); > s.a. non-commutative
geometry, quantum group.
@ Kappa-Minkowski spacetime: Ghosh PLB(07)ht/06 [and
DSR]; Amelino-Camelia et al PLB(09)-a0707 [boosts
and space-rotations];
Freidel et al IJMPA(08)-a0706 [free
scalar field]; Agostini IJMPA(09)-a0711 [covariant
formulation of Noether's theorem]; Arzano et al a0908 [Lorentz-invariant
field
theory]; Harikumar & Sivakumar a0910 [and hydrogen atom spectrum].
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nov 2009