Rindler Space |

**In General** > minkowski space.

* __Idea__: Minkowski spacetime
with coordinates adapted to a boost Killing vector field, i.e., to a uniformly
accelerated observer.

* __Coordinates__: If (*X*, *T*)
are the Minkowski coordinates, Rindler coordinates (*x*, *t*)
are defined on the right wedge (*X* > 0, |*T*| < |*X*|) by

*X* = *g*^{–1 }e^{gx} cosh *gt* , *T* =
*g*^{–1} e^{gx} sinh *gt* ,

and coordinates (*x*', *t*') on the left wedge (*X* < 0,
|*T*| < |*X*|)
are defined by

*X* = –*g*^{–1} e^{gx'} cosh *gt*'
, *T* =
–*g*^{–1} e^{gx'} sinh *gt*'
;

In either case, the inverse transformation is given by

*t*, *t*' = *g*^{–1} tanh^{–1}(*T*/*X*)
, *x*, *x*'
= (2*g*)^{–1} ln[*g*^{2}(*X*^{ 2}–*T*^{ 2})]
;

The lines *t* = constant are straight half-lines, while *x* =
constant are hyperbolae of acceleration *g* e^{–gx}.

* __Line element__: Given
by

d*s*^{2} = e^{2gx} (–d*t*^{2} +
d*x*^{2})
,

so proper time is related to coordinate time by *τ* =
e^{gx} *t*.

@ __General references__: Born AdP(09) [precursor];
Rindler AJP(66)dec;
Felix da Silva & Dahia IJMPA(07) [non-Euclidean geometry of spatial sections].

@ __Related topics__: Kowalski-Glikman PRD(09)-a0907 [deformed, *κ*-Rindler space];
Daszkiewicz MPLA(10)-a1004 [twisted];
Chung PRD(10) [asymptotic symmetries];
Bianchi & Satz PRD(13)-a1305 [mechanical laws of the Rindler horizon];
Araya & Bars PRD(18)-a1712
[infinite stack of identical Minkowski geometries as a multiverse model];
> s.a. black-hole geometry [interior];
modified theories of gravity [Rindler force];
tests of general relativity [Rindler-type acceleration].

> __Online resources__:
see Wikipedia page;
't Hooft page
with animated gif on Rindler coordinates.

**And Classical Field Theory** > see dirac fields.

**And Quantum Theory** > s.a. gravitational thermodynamics.

* __Idea__: The Minkowski
vacuum looks like a thermal state in Rindler space, for an observer moving
along *x* = constant, with temperature depending on its acceleration;
This makes it useful for mimicking black-hole radiation.

@ __Thermal properties__: Fulling PRD(73);
Unruh PRD(76);
Lapedes JMP(78);
Dray & Manogue pr(87);
Laflamme PLB(87);
Nikolić MPLA(01)gq [criticism of use];
Xiang & Zheng IJTP(01) [horizon entropy];
Socolovsky a1304 [application to the Unruh effect];
Kolekar & Padmanabhan PRD(14)-a1309 [Rindler-Rindler spacetime];
> s.a. radiation; quantum field theory in curved backgrounds.

@ __Quantum mechanics__: Dai PLA(16)-a1609 [hydrogen atom energy eigenvalues and wave functions].

@ __Quantum field theory__: Michel a1612 [quantization of scalar and gauge fields]; > s.a. mirrors.

@ __Related topics__: Balasubramanian et al JHEP(13) [entropy of a "spherical Rindler space" hole in spacetime].

> __Related topics__:
see quantum technology [communication].

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send feedback and suggestions to bombelli at olemiss.edu – modified 1 jun 2018