Gen. Math. Notes, Vol. 30, No. 1, September 2015, pp.21-27
ISSN 2219-7184; Copyright c ICSRS Publication, 2015 www.i-csrs.org
Available free online at http://www.geman.in
Riemann Extension of Minkowski Line
Element in the Rindler Chart
H.G. Nagaraja 1 and D. Harish 2
1 , 2
Department of Mathematics, Bangalore University
Central College Campus, Bengaluru-560001, India
1 E-mail: hgnraj@yahoo.com
2
E-mail: itsme.harishd@gmail.com
(Received: 16-7-15 / Accepted: 29-8-15)
Abstract
In this paper, we discuss Riemannian extension of Minkowski metric in
Rindler coordinates and its geodesics.
Keywords: Riemannian curvature, Riemann extension, Flat metric, geodesic equations, constant positive curvature.
1 Introduction
Patterson and Walker[7] have defined Riemann extensions and showed how a
Riemannian structure can be given to the 2 n dimensional tangent bundle of an n − dimensional manifold with given non-Riemannian structure. This shows
Riemann extension provides a solution of the general problem of embedding a manifold M carrying a given structure in a manifold M
0 carrying another structure, the embedding being carried out in such a way that the structure on M
0 induces in a natural way the given structure on M. The Riemann extension of Riemannian or non-Riemannian spaces can be constructed with the help of the Christoffel coefficients Γ i jk connection coefficients Π i jk of corresponding Riemann space or with
in the case of the space of affine connection[5].
The theory of Riemann extensions has been extensively studied by Afifi[1] and
Dryuma[2],[3], [4] [5].

22 H.G. Nagaraja et al.
2 Preliminaries
Let ( M, g ) be a Riemannian manifold and let g ij tensor g .
The system of geodesic equations are be components of the metric d 2 x k ds 2
+ Γ k ij dx i ds dx j ds
= 0 , where Γ k ij are the Christoffel symbols constructed from g given by
(1)
Γ k ij
=
1
2 g km
( g mi,j
+ g mj,i
− g ij,m
) .
The geodesic deviation equations or the Jacobi equations are given by
(2)
D 2 η i ds 2
+ R i kjm dx k ds dx m
η j ds
= 0 , where R i kjm is Riemann curvature tensor of the manifold and
Dη i ds
= dη i ds
+ Γ i jk
η j dx k
.
ds
Using (1) and simplifying the equation (3) reduces to
(3)
(4) d 2 η i ds 2
+ 2Γ i lm dx m ds dη l ds
+
∂ Γ i kl
∂x j dx k ds dx l
η j ds
= 0 .
(5)
The Riemann extension[7] of the metric
g is given by ds
2
= − 2Γ k ij
( x l
) ψ k dx i dx j
+ 2 dψ k dx k
, (6) where ψ k are the coordinates of additional space. The geodesic equations of
(6) consists of two parts,
d 2 x k ds 2
+ Γ k ij dx i ds dx j ds
= 0 (7) and
δ 2 ψ k ds 2
+ R l kji dx j ds dx i
ψ l
= 0 , ds
(8) where
δ 2 ψ k ds 2
= dψ k ds
− Γ l jk
ψ l dx j
.
ds
(9)

Riemann Extension of Minkowski Line...
23
The system of equations (7) is the system of geodesic equations for geodesics
of basic space with local coordinates x i and it does not contain the coordinates
ψ k
.
The system of equations (8) is a
n × n linear matrix system of second order in the form d 2 ψ ds 2
+ A ( s ) dψ ds
+ B ( s ) ψ = 0 , (10) where A ( s ) and B ( s ) are matrices.
3 Minkowski Metric in Rindler Coordinates
In relativistic physics, the Rindler coordinate chart is an important and useful coordinate chart representing part of flat spacetime, also called the Minkowski vacuum. The Rindler coordinate system or frame describes a uniformly accelerating frame of reference in Minkowski space. In special relativity, a uniformly accelerating particle undergoes hyperbolic motion. For each such particle a
Rindler frame can be chosen in which it is at rest.
The Rindler chart is named after Wolfgang Rindler who popularised its
use, although it was already well known in 1935[6].
In Rindler chart the Minkowski line element is given by ds
2
= dx
2
+ dy
2
+ dz
2 − a
2 x
2 dt
2
, (11) where x, y and z are space coordinates, t is time coordinate and ’a’ is a constant called proper acceleration. The non vanishing Christoffel symbols of the metric
(11) are
Γ 1
44
= a 2 x, Γ 4
14
= 1 x
.
(12)
The spatial metric of any four dimensional metric ds
2
= g
αβ dx
α dx
β
+ 2 g
0 α dx
0 dx
α
+ g
00 dx
0 dx
0
(13) has the form dl
2
= γ
αβ dx
α dx
β
, where
γ
αβ
= − g
αβ
+ g
0 α g
0 β g
00 is a three dimensional tensor determining the properties of the space.
In the case of the Minkowski metric in Rindler coordinates,
− dl
2
= dx
2
+ dy
2
+ dz
2
,
(14)
(15)
(16) which is a flat metric.

24 H.G. Nagaraja et al.
3.1
Geodesic Deviation Equations
The connection coefficients are clearly very simple. Using the Christoffel sym-
bols (12) in equation (5) and simplifying we get a system of Jacobi equations.
d 2 η 1 ds 2 d 2 η 2
+ 2 a
2 x dt ds dη ds
4
+ a
2 dt ds
2
η
1
= 0 , ds 2 d
2
η
3 ds 2 d 2 η 4 ds 2
= 0 ,
= 0 ,
2 dt dη 1
+ x ds ds
1
− x 2 dx ds dt ds
η
1
= 0 ,
(17)
(18)
(19)
(20) where d
2 x
+ a
2 x ds 2 dt ds
2
= 0 (21) and d 2 t ds 2
+
2 dx dt x ds ds
= 0 .
(22)
Equations (21) and (22) are the geodesic equations obtained from (1). Simpli-
fying and substituting for ’t’ the system reduces to d
2
η
1
2 a
2 k
2 ds 2 d 2 η 2
+ x
= 0 ds 2 d 2 η 3
= 0 ds 2 d
2
η
4 d ds
2 ds x
2
2
2 k
2
+
+ a
2 x 3 k 4 x 3 dη ds dη
4
1 ds
= 0 .
−
+ k x
2
4 a
2 k
4
η
1 x 4 dx ds
η
1
= 0
= 0
Solving we get
η
1
= x ( c
1
(
Z
+ c
2
(
Z (
( ak
2 ak
2
+
+
√ a 2 k 4 + c 2 x 2 )
√
5
√ a 2 k 4 + c 2 x 2
√ a 2 k 4 + c 2 x 2 )
−
√
5
√ a 2 k 4 + c 2 x 2 x
−
√
5 x
√
5 dx ) dx )) + c
3
,
(23)
(24)
(25)
(26)
(27)
(28)

Riemann Extension of Minkowski Line...
25
η
η
η
2
3
4 and x
2
= a
1 s + a
2
, (29)
= a
3 s + a
4
,
=
Z c
2
+ x
1
1+
√
5
1
√ a 2 k 4 + c 2 x 2
+ 2( ak
2
( x ((
+
√
Z a 2 k x 3
4
√ a 2 k 4 + c 2 x 2
+ c 2 x
1
2 )
√
5 c
2 x
(2(
1 −
√
5 ak
√ a 2 k 4 + c 2 x 2 dx ) k
2
+ c
4
))) dx + c
5
, x
η
1
2
+
(30)
√ a 2 k 4 + c 2 x 2 )
−
√
5
(31)
(32)
=
1 c 2
[ c
4
( s + d )
2 − a
2 k
4
] , (33) where a
1
, a
2
, a
3
, a
4
, c
1
, c
2
, c
3
, c
4
, c
5
, d are arbitrary constants.
3.2
Riemann Extension
The metric in the extended space is given by ds
2
= − 2 xa
2
P dt
2
−
4 x
V dxdt + 2 dxdP + 2 dydQ + 2 dzdU + 2 dtdV.
(34)
The Christoffel symbols in the extended space are
Γ 5
14
Γ 8
14
=
2 V x
= a 2
2
P
Γ 5
44
Γ 8
18
= a 2 P Γ 5
48
1
= − x
Γ 8
44
1
= − x
= 2 V a 2
2 V
Γ 8
11
= x 2
Γ 8
45
= − a 2 x.
The geodesic equations are given by d 2 x
+ a
2 x ds 2 d
2 y
= 0 ds 2 d 2 z
= 0 ds 2 dt ds
2
= 0
(35)
(36)
(37)
(38)

26 H.G. Nagaraja et al.
d 2 t ds 2 d
2
P
+
+ ds 2
2 dx x ds
4 x
V
2 dt ds dx dt ds
= 0 ds
+ a
2
P dt ds
2
2 dt
− x ds dV ds d 2 Q
= 0 ds 2 d
2
U
= 0 ds 2 d 2 V ds 2
+
2 V x 2
( dx ds
)
2
+ 2 a
2
P dx dt ds ds
2 dx dV
− x ds ds
= 0
+ 2 a
2
V
− 2 a
2 x dt ds dP ds
= 0 .
Solving the above equations, we obtain dt ds
2
(39)
(40)
(41)
(42)
(43)
(44) y = a
1 s + b
1
, z = a
2 s + b
2
, Q = a
3 s + b
3
, U = a
4 s + b
4
, t = c 2
2 a log c c
2
2
( s + d ) − ak
( s + d ) + ak
2
2 and x 2 =
1 c 2
[ c 4 ( s + d ) 2 − a 2
Transforming the remaining equations from s to x , we get k 4 ].
x
3
( c
2 x
2
+ a
2 k
4
) d
2
P dx 2
− a
2
= 2 k k
2
4 x
2 dP
+ a
2 k
4 xP dx
√ a 2 k 4 + c 2 x 2 x dV dx
− 2 V
(45) and x
2
( c
2 x
2
+ a
2 k
4
) d 2 V dx 2
− (3 a
2
= 2 a
2 k k
2
4 x + 2 c
2 x
3
) dV
+ (4 a
2 dx x
√ c 2 x 2 + a 2 k 4 x dP dx k
4
−
+ 2
P c
.
2 x
2
) V
(46)
Solving (45) and (46) we get,
P ( x ) = b
1
( a
2 k
4
+ c
2 x
2
) + b
2
√ a 2 k 4 + c 2 x 2 + b
3 x
(47) and
V ( x ) = x
2 b
4
+
Z 2 k 4 a 2 ( k 4 b
1
+ b
3
) + c 2 x 2 ( b k 2 x 3
1 a
√
2 k 4 + b
3
) + 2 a 2 k 4 + c 2 x 2
√ a 2 k 4 + c 2 x 2 b
2 a 2 k 4 dx
!
,
(48)

Riemann Extension of Minkowski Line...
27 where b
1
, b
2
, b
3
, b
4 are constants determined by the given initial conditions.
Thus we have obtained Riemann extension of Minkowski metric in Rindler coordinates and have obtained solution of its geodesic equations.
Acknowledgements: This work is supported by CSIR:09/039(0106)2012-
EMR-I.
References
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,
Oxford Ser., 2(5) (1954), 312-320.
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Mat. Fiz.
, 146(1) (2006), 42-54.
[3] V.S. Dryuma, On geodesical extension of the Schwarzshild space-time,
Proceedings of Institute of Mathematics of NAS of Ukraine , 50(3) (2004),
1294-1300.
[4] V.S. Dryuma, On Riemann extension of the Schwarzschild metric, Bul.
Acad. S , 3(2003), 92-103.
[5] V.S. Dryuma, Four dimensional Einstein spaces on six dimensional Ricci flat base spaces, arXiv:gr-qc/0601051, January (2006), v1 12.1.
[6] A. Einstein and N. Rosen, A particle problem in the general theory of relativity, Physical Review , 48(1935), 73-77.
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,
Oxford Ser., 2(3) (1952), 19-28.