Chapter 4 Proper Curvature Collineations in Non- Static ...

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Oct 13, 2013 (3 years and 10 months ago)

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50
Chapter 4
Proper Curvature Collineations in Non-
Static Cylindrically Symmetric and
Special Non-Static Axially Symmetric
Spacetimes
4.1 Introduction
This chapter focuses on investigating proper curvature collineations in non-static
cylindrically symmetric and special non-static axially symmetric spacetimes. In this
chapter an approach, which is given in [8], is adopted to study proper CCS in the above
non-static spacetimes by using the rank of the
66
×
⁒楥= a湮a瑲楸⁡湤⁤楲散琠
楮瑥杲慴楯渠瑥捨湩煵敳⸠ih攠污祯畴e ⁴桩猠捨慰瑥爠楳 ⁡猠景汬潷猺†
卥捴楯渠㐮㈠桡猠扥敮牧慮楺敤⁴漠捬慳獩晹潮S 獴慴楣⁣祬楮摲楣慬汹s 獹sme瑲楣⁳灡捥瑩t敳e
慣捯牤楮朠瑯⁴桥楲⁃䍓⁷桥牥慳⁳散瑩潮‴⸳⁩= ⁤e癯瑥搠景爠晩湤楮朠灲潰敲⁃䍓⁩渠獰散楡氠
湯渭獴慴楣⁡硩慬汹⁳祭浥瑲楣⁳灡捥瑩ne献⁁⁢物s 映捯湣汵獩潮映瑨攠慢潶攠摩獣畳獩潮⁩猠
杩癥渠楮⁳散瑩潮‴⸴⸠g
㐮4 Proper Curvature Collineations in Non-Static
Cylindrically Symmetric Spacetimes
Consider the most general form of non-static cylindrically symmetric spacetime in the
usual coordinate system
),,,(
φ
θ
rt
(labeled by ),,,,(
3210
xxxx respectively) with line
element [32]

.
2),(2),(22),(2
φθ dededrdteds
rtWrtVrtU
+++−=
(4.2.1)

51
The above spacetime admits two linearly independent Killing vector fields which are

.,
φθ ∂




(4.2.2)
The non-zero independent components of the Riemann tensor are [35]
[ ]
,),(2),(
4
1
1
),(
2
0101
α=+−=
rtU
rrr
ertUrtUR

[ ]
,),(),(
4
1
),(),(),(2),(
4
1
2
),(),(),(
2
0202
α≡+−+−=
+
rtVrtUeertVrtUrtVrtVR
rr
rtVrtUrtV
ttttt

[ ]
,),(),(),(2),(),(
4
1
3
),(
0212
α≡−+−=
rtV
trtrrt
ertVrtUrtVrtVrtVR
[ ]
,),(),(
4
1
),(),(),(2),(
4
1
4
),(),(),(
2
0303
α≡+−+−=
+
rtWrtUeertWrtUrtWrtWR
rr
rtWrtUrtW
ttttt

[ ]
,),(),(),(2),(),(
4
1
5
),(
0313
α≡−+−=
e
rtWrtUrtWrtWrtWR
rtW
trtrrt

[ ]
,),(2),(
4
1
6
),(
2
1212
α=+−=
e
rtVrtVR
rtV
rrr

[ ]
,),(2),(
4
1
7
),(
2
1313
α=+−=
e
rtWrtWR
rtW
rrr

[ ]
.),(),(),(),(
4
1
8
),(),(
2323
α≡+−−=
+
e
rtWrtVrtWrtVR
rtWrtV
rrtt

Writing the curvature tensor with components
abcd
R at
p
as a
66
×
⁳祭me瑲楣a瑲楸†
=
.
00000
0000
0000
0000
0000
00000
8
75
63
54
32
1




















=
α
αα
αα
αα
αα
α
abcd
R

(4.2.3)
It is imperative to note that we will consider Riemann tensor components as
bcd
a
R
for
calculating CCS. Here, we are only interested in those cases when the rank of the
66
×
=
剩敭慮渠aa瑲楸⁩猠汥獳⁴桡渠ir⁥煵=氠瑯⁴=牥攮⁔h攠牥慳潮⁩a⁴桡t⁷攠歮=眠∂r潭⁴桥潲敭=
嬸Ⱐ㙝⁴桡琠睨敮⁴桥⁲慮ξ映瑨攠
66
×
⁒楥=慮渠aa瑲楸⁩猠杲敡瑥爠瑨慮⁴桲敥⁴桥牥⁥硩獴猠湯a
灲潰敲⁃䍓⸠周攠灲潣敤畲攠潦⁦楮摩湧⁴桥⁲慮欠潦 ⁒楥=慮渠浡瑲楸⁩猠 慬牥慤礠數灬慩湥搠楮a

52
section 3.2. Thus for better understanding the surviving possibilities when the rank of the
66×
Riemann matrix is three or less are classified as:
(A1) Rank=3,
,0),( =rtU
t

,0),( ≠rtW
t

,0),(
=
rtU
r
,0),( =
rtV
t

,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
rrr

,0),(2),(),(

+
rtWrtWrtW
trrt

,0),(2),(
2
≠+ rtWrtW
ttt

0),( ≠rtW
r
and
.0),(2),(
2
≠+ rtWrtW
rrr

(A2) Rank=3,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t
,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
rrr

,0),( ≠rtW
t

,0),(

rtW
r

,0),(2),(),(

+ rtWrtWrtW
trrt

0),(2),(
2
=+ rtWrtW
ttt
and .0),(2),(
2
≠+ rtWrtW
rrr

(A3) Rank=3,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( =rtV
r

,0),(2),(
2
=+ rtVrtV
ttt

,0),( ≠rtW
t

,0),(

rtW
r

,0),(2),(),(
=
+ rtWrtWrtW
trrt

0),(2),(
2
≠+ rtWrtW
ttt
and .0),(2),(
2
≠+ rtWrtW
rrr

(A4) Rank=3,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( =rtV
r

,0),(2),(
2
=+ rtVrtV
ttt

,0),( ≠rtW
t

,0),(

rtW
r

,0),(2),(),(

+ rtWrtWrtW
trrt

0),(2),(
2
=+ rtWrtW
ttt
and .0),(2),(
2
=+ rtWrtW
rrr

(A5) Rank=3,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
ttt
,0),(2),(
2
=+ rtVrtV
rrr
,0),(2),(),(
=
+ rtVrtVrtV
trrt

,0),( ≠rtW
t

,0),( ≠rtW
r

,0),(2),(),(

+
rtWrtWrtW
trrt

0),(2),(
2
=+ rtWrtW
ttt
and
.0),(2),(
2
≠+ rtWrtW
rrr

(A6) Rank=3,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t

,0),(

rtV
r

,0),(2),(
2
=+ rtVrtV
rrr

,0),( ≠rtW
t

,0),(

rtW
r

0),(2),(),( ≠
+
rtWrtWrtW
trrt
and
.0),(2),(
2
≠+ rtWrtW
rrr

(A7) Rank=3,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
ttt

,0),(2),(
2
=+ rtVrtV
rrr
,0),(2),(),(
=
+
rtVrtVrtV
trrt

,0),( ≠rtW
t

,0),(

rtW
r

,0),(2),(),( ≠+ rtWrtWrtW
trrt

0),(2),(
2
≠+ rtWrtW
ttt
and .0),(2),(
2
≠+ rtWrtW
rrr


53
(A8) Rank=3,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( =rtV
r

,0),(2),(
2
=+ rtVrtV
ttt

,0),( ≠rtW
t

,0),(

rtW
r

,0),(2),(),(

+ rtWrtWrtW
trrt

0),(2),(
2
≠+ rtWrtW
ttt
and .0),(2),(
2
≠+ rtWrtW
rrr

(A9) Rank=3,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t
,0),( ≠rtV
r

,0),(2),(
2
≠+ rtVrtV
rrr

,0),( ≠rtW
t

,0),(

rtW
r

,0),(2),(),(
=
+ rtWrtWrtW
trrt

0),(2),(
2
=+ rtWrtW
ttt
and .0),(2),(
2
=+ rtWrtW
rrr

(A10) Rank=3,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t
,0),( ≠rtV
r

,0),(2),(
2
≠+ rtVrtV
rrr

,0),( ≠rtW
t

,0),(

rtW
r

,0),(2),(),(
=
+ rtWrtWrtW
trrt

0),(2),(
2
≠+ rtWrtW
ttt
and .0),(2),(
2
=+ rtWrtW
rrr

(A11) Rank=3,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
ttt

,0),(2),(
2
≠+ rtVrtV
rrr
,0),(2),(),(
=
+
rtVrtVrtV
trrt

,0),( ≠rtW
t

,0),(

rtW
r

,0),(2),(),( =+ rtWrtWrtW
trrt

0),(2),(
2
≠+ rtWrtW
ttt
and .0),(2),(
2
=+ rtWrtW
rrr

(A12) Rank=3,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
ttt

,0),(2),(
2
≠+ rtVrtV
rrr
,0),(2),(),(
=
+
rtVrtVrtV
trrt

,0),( ≠rtW
t

0),( =rtW
r
and
.0),(2),(
2
≠+ rtWrtW
ttt

(A13) Rank=3,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( =rtV
r

,0),(2),(
2
≠+ rtVrtV
ttt

,0),( ≠rtW
t

,0),( ≠rtW
r

,0),(2),(),(
=
+
rtWrtWrtW
trrt

0),(2),(
2
=+ rtWrtW
ttt
and
.0),(2),(
2
≠+ rtWrtW
rrr

(A14) Rank=3,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
ttt
,0),(2),(
2
=+ rtVrtV
rrr
,0),(2),(),(
=
+ rtVrtVrtV
trrt

,0),( =rtW
t

0),( ≠rtW
r
and .0),(2),(
2
≠+ rtWrtW
rrr

(A15) Rank=3,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
≠+ rtVrtV
ttt

,0),(2),(
2
=+ rtVrtV
rrr
,0),(2),(),(
=
+
rtVrtVrtV
trrt

,0),( ≠rtW
t

,0),(

rtW
r

,0),(2),(),( =+ rtWrtWrtW
trrt

0),(2),(
2
=+ rtWrtW
ttt
and .0),(2),(
2
≠+ rtWrtW
rrr


54
(A16) Rank=3,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
≠+ rtVrtV
ttt

,0),(2),(
2
≠+ rtVrtV
rrr
,0),(2),(),(

+
rtVrtVrtV
trrt

,0),( ≠rtW
t

0),( =rtW
r
and
.0),(2),(
2
=+ rtWrtW
ttt

(A17) Rank=3,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
≠+ rtVrtV
ttt

,0),(2),(
2
≠+ rtVrtV
rrr
,0),(2),(),(

+
rtVrtVrtV
trrt

,0),( =rtW
t

0),( ≠rtW
r
and
.0),(2),(
2
=+ rtWrtW
rrr

(A18) Rank=3,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
ttt

,0),(2),(
2
=+ rtVrtV
rrr
,0),(2),(),(

+
rtVrtVrtV
trrt

,0),( =rtW
t

0),( ≠rtW
r
and
.0),(2),(
2
=+ rtWrtW
rrr

(A19) Rank=3,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
≠+ rtVrtV
rrr
,0),(2),(),(
=
+
rtVrtVrtV
trrt

,0),( =rtW
t

0),( ≠rtW
r
and
.0),(2),(
2
=+ rtWrtW
rrr

(A20) Rank=3,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
ttt

,0),(2),(
2
≠+ rtVrtV
rrr
,0),(2),(),(
=
+ rtVrtVrtV
trrt

,0),( =rtW
t

0),( ≠rtW
r
and .0),(2),(
2
=+ rtWrtW
rrr

(A21) Rank=3,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
≠+ rtVrtV
ttt

,0),(2),(
2
≠+ rtVrtV
rrr
,0),(2),(),(
=
+
rtVrtVrtV
trrt

,0),( ≠rtW
t

0),( =rtW
r
and
.0),(2),(
2
=+ rtWrtW
ttt


(A22) Rank=3,
,0),( =rtU
t

,0),(

rtU
r
,0),(2),(
2
=+ rtUrtU
rrr
,0),(
=
rtV
t

,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
rrr

,0),( ≠rtW
t

0),( =rtW
r
and
.0),(2),(
2
=+ rtWrtW
ttt


(A23) Rank=3,
,0),( ≠rtU
t

,0),(

rtU
r
,0),(2),(
2
=+ rtUrtU
rrr
,0),(
=
rtV
t

,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
rrr

,0),( ≠rtW
t

0),( =rtW
r
and
.0),(2),(
2
=+ rtWrtW
ttt



55
(A24) Rank=3,
,0),( ≠rtU
t
,0),(2),(
2
=+ rtUrtU
rrr
,0),( =rtV
t
,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
rrr

0),( ≠rtW
t
and
.0),(
=
rtW
r


(A25) Rank=3,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
ttt

,0),(2),(
2
=+ rtVrtV
rrr
,0),(2),(),(

+
rtVrtVrtV
trrt

,0),( ≠rtW
t

0),( =rtW
r
and
.0),(2),(
2
=+ rtWrtW
ttt


(A26) Rank=3,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
ttt

,0),(2),(
2
=+ rtVrtV
rrr
,0),(2),(),(

+
rtVrtVrtV
trrt

,0),( =rtW
t

0),( ≠rtW
r
and
.0),(2),(
2
=+ rtWrtW
rrr

(A27) Rank=3,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
ttt

,0),(2),(
2
=+ rtVrtV
rrr
,0),(2),(),(

+
rtVrtVrtV
trrt

,0),( ≠rtW
t

,0),(

rtW
r

,0),(2),(),( =+ rtWrtWrtW
trrt

0),(2),(
2
=+ rtWrtW
ttt
and .0),(2),(
2
=+ rtWrtW
rrr

(A28) Rank=3,
,0),( ≠rtU
t

,0),(

rtU
r
,0),(2),(
2
=+ rtUrtU
rrr
,0),(

rtV
t

,0),( =rtV
r

,0),(2),(
2
=+ rtVrtV
ttt

,0),( =rtW
t

0),( ≠rtW
r
and
.0),(2),(
2
=+ rtWrtW
rrr

(A29) Rank=3,
,0),( =rtU
t

,0),(

rtU
r

,0),(2),(
2
=+ rtUrtU
rrr
,0),(

rtV
t

,0),( =rtV
r

,0),(2),(
2
≠+ rtVrtV
ttt

,0),( =rtW
t

0),( ≠rtW
r
and
.0),(2),(
2
=+ rtWrtW
rrr

(A30) Rank=3,
,0),( =rtU
t

,0),(

rtU
r

,0),(2),(
2
=+ rtUrtU
rrr

,0),( =rtV
t

,0),(

rtV
r

,0),(2),(
2
=+ rtVrtV
rrr

,0),( ≠rtW
t

0),(
=
rtW
r
and
.0),(2),(
2
≠+ rtWrtW
ttt


(B1)

Rank=2, ,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(

rtV
t

,0),(
=
rtV
r

,0),(2),(
2
=+ rtVrtV
ttt
,0),(
=
rtW
t

0),(

rtW
r
and .0),(2),(
2
≠+ rtWrtW
rrr

(B2)

Rank=2, ,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(

rtV
t

,0),( =rtV
r
,0),(
=
rtW
t

0),( ≠rtW
r
and .0),(2),(
2
≠+ rtWrtW
rrr


56
(B3)

Rank=2, ,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t

,0),(
=
rtV
r

,0),(2),(
2
≠+ rtVrtV
ttt
,0),(
=
rtW
t

0),(

rtW
r
and .0),(2),(
2
≠+ rtWrtW
rrr


(C1)

Rank=3, ,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t

,0),( ≠rtV
r
,0),(
=
rtW
t

,0),( ≠rtW
r

0),(2),(
2
≠+ rtVrtV
rrr
and
.0),(2),(
2
≠+ rtWrtW
rrr

(C2) Rank=3, ,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(

rtV
t

,0),( =rtV
r
,0),(

rtW
t

,0),( =rtW
r

0),(2),(
2
=+ rtVrtV
ttt
and
.0),(2),(
2
≠+ rtWrtW
ttt

(C3) Rank=3, ,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(

rtV
t

,0),( =rtV
r
,0),(

rtW
t

0),( =rtW
r
and
.0),(2),(
2
=+ rtWrtW
ttt

(C4) Rank=3, ,0),( =rtU
t

,0),(

rtU
r
,0),(2),(
2
≠+ rtUrtU
rrr
,0),(
=
rtV
t

,0),( =rtV
r
,0),( ≠rtW
t

0),(
=
rtW
r
and
.0),(2),(
2
=+ rtWrtW
ttt

(C5) Rank=3, ,0),( =rtU
t

,0),(

rtU
r

,0),(2),(
2
≠+ rtUrtU
rrr
,0),(
=
rtV
t

,0),( ≠rtV
r
,0),(2),(
2
≠+ rtVrtV
rrr
0),(
=
rtW
t
and
,0),(
=
rtW
r

(C6) Rank=3, ,0),( ≠rtU
t

,0),(

rtU
r
,0),(2),(
2
≠+ rtUrtU
rrr
,0),(
=
rtV
t

,0),( ≠rtV
r

,0),(2),(
2
≠+ rtVrtV
rrr
0),(
=
rtW
t
and
,0),(
=
rtW
r

(C7) Rank=3, ,0),( ≠rtU
t

,0),(

rtU
r
,0),(2),(
2
≠+ rtUrtU
rrr
,0),(

rtV
t

0),( ≠rtV
r
,0),(2),(
2
≠+ rtVrtV
rrr
0),(
=
rtW
t
and
.0),(
=
rtW
r

(C8) Rank=3, ,0),( =rtU
t

,0),(

rtU
r

,0),(2),(
2
≠+ rtUrtU
rrr
,0),(

rtV
t

0),( ≠rtV
r
,0),(2),(
2
≠+ rtVrtV
rrr
0),(
=
rtW
t
and
.0),(
=
rtW
r

(C9) Rank=3, ,0),( =rtU
t

,0),(

rtU
r
,0),(2),(
2
≠+ rtUrtU
rrr
,0),(

rtV
t

,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
rrr
0),(
=
rtW
t
and
.0),(
=
rtW
r

(C10) Rank=3, ,0),( =rtU
t

,0),(

rtU
r
,0),(2),(
2
≠+ rtUrtU
rrr
,0),(

rtV
t

,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
ttt
,0),(2),(
2
=+ rtVrtV
rrr
0),( =rtW
t
and
.0),( =rtW
r


57
(C11) Rank=3, ,0),( ≠rtU
t

,0),(

rtU
r
,0),(2),(
2
≠+ rtUrtU
rrr
,0),(

rtV
t

,0),( =rtV
r
0),( =rtW
t
and
.0),(
=
rtW
r

(C12) Rank=3, ,0),( ≠rtU
t

,0),(

rtU
r

,0),(2),(
2
≠+ rtUrtU
rrr
,0),(

rtV
t

,0),( =rtV
r

,0),(2),(
2
=+ rtVrtV
ttt
0),(
=
rtW
t
and
.0),(
=
rtW
r

(C13) Rank=2,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t
,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
rrr

,0),(
=
rtW
t

0),(

rtW
r
and .0),(2),(
2
≠+ rtWrtW
rrr

(C14) Rank=2,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t

,0),(

rtV
r

,0),(2),(
2
≠+ rtVrtV
rrr

,0),( =rtW
t

0),(

rtW
r
and .0),(2),(
2
=+ rtWrtW
rrr

(C15) Rank=2,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( =rtV
r

,0),(2),(
2
=+ rtVrtV
ttt

,0),( ≠rtW
t

0),( =rtW
r
and
.0),(2),(
2
≠+ rtWrtW
ttt

(C16) Rank=2,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( =rtV
r

,0),(2),(
2
≠+ rtVrtV
ttt

,0),( ≠rtW
t

0),( =rtW
r
and
.0),(2),(
2
=+ rtWrtW
ttt

(C17) Rank=2,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t
,0),( =rtV
r
,0),(

rtW
t

,0),( ≠rtW
r

0),(2),(),(

+ rtWrtWrtW
trrt
and .0),(2),(
2
=+ rtWrtW
rrr

(C18) Rank=2,
,0),( ≠rtU
t

,0),(

rtU
r
,0),(2),(
2
=+ rtUrtU
rrr
,0),(
=
rtV
t

,0),( =rtV
r
,0),( ≠rtW
t

,0),(

rtW
r

0),(2),(
2
=+ rtWrtW
ttt
and
.0),(2),(
2
=+ rtWrtW
rrr

(C19) Rank=2,
,0),( =rtU
t

,0),(

rtU
r
,0),(2),(
2
=+ rtUrtU
rrr
,0),(
=
rtV
t

,0),( =rtV
r
,0),( ≠rtW
t

,0),(

rtW
r

0),(2),(
2
=+ rtWrtW
ttt
and
.0),(2),(
2
=+ rtWrtW
rrr

(C20) Rank=2,
,0),( ≠rtU
t

,0),(

rtU
r
,0),(2),(
2
=+ rtUrtU
rrr
,0),(
=
rtV
t

,0),( =rtV
r
,0),( ≠rtW
t

0),(

rtW
r
and .0),(2),(
2
=+ rtWrtW
rrr

(C21) Rank=2,
,0),( =rtU
t

,0),(

rtU
r
,0),(2),(
2
=+ rtUrtU
rrr
,0),(
=
rtV
t

,0),( =rtV
r
,0),( ≠rtW
t

0),(

rtW
r
and .0),(2),(
2
=+ rtWrtW
rrr


58
(C22) Rank=2,
,0),( =rtU
t

,0),(

rtU
r
,0),(2),(
2
=+ rtUrtU
rrr
,0),(
=
rtV
t

,0),( =rtV
r
,0),( ≠rtW
t

0),(

rtW
r
and
.0),(2),(
2
≠+ rtWrtW
rrr

(C23) Rank=2,
,0),( ≠rtU
t

,0),(

rtU
r
,0),(2),(
2
=+ rtUrtU
rrr
,0),(
=
rtV
t

,0),( =rtV
r
,0),( ≠rtW
t

0),(

rtW
r
and .0),(2),(
2
≠+ rtWrtW
rrr

(C24) Rank=2,
,0),( ≠rtU
t

,0),(

rtU
r

,0),(2),(
2
=+ rtUrtU
rrr
,0),(
=
rtV
t

,0),( =rtV
r
,0),( ≠rtW
t

,0),(

rtW
r

0),(2),(
2
=+ rtWrtW
ttt
and
.0),(2),(
2
≠+ rtWrtW
rrr

(C25) Rank=2,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t
,0),( =rtV
r
,0),(

rtW
t

,0),( ≠rtW
r

,0),(2),(),(

+
rtWrtWrtW
trrt

0),(2),(
2
≠+ rtWrtW
ttt
and
.0),(2),(
2
≠+ rtWrtW
rrr

(C26) Rank=2,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t
,0),( =rtV
r
,0),(

rtW
t

,0),( ≠rtW
r

0),(2),(),(

+ rtWrtWrtW
trrt
and
.0),(2),(
2
≠+ rtWrtW
rrr

(C27) Rank=2,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t
,0),( =rtV
r
.0),(

rtW
t

,0),( ≠rtW
r

,0),(2),(),(
=
+
rtWrtWrtW
trrt

0),(2),(
2
=+ rtWrtW
ttt
and
.0),(2),(
2
≠+ rtWrtW
rrr

(C28) Rank=2,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t
,0),( =rtV
r
,0),(

rtW
t

,0),( ≠rtW
r

,0),(2),(),(

+
rtWrtWrtW
trrt

0),(2),(
2
=+ rtWrtW
ttt
and
.0),(2),(
2
≠+ rtWrtW
rrr

(C29) Rank=2,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t
,0),( =rtV
r
,0),(

rtW
t

,0),( ≠rtW
r

,0),(2),(),(
=
+
rtWrtWrtW
trrt

0),(2),(
2
≠+ rtWrtW
ttt
and
.0),(2),(
2
≠+ rtWrtW
rrr

(C30) Rank=2,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t

,0),(
=
rtV
r
,0),(

rtW
t

,0),( ≠rtW
r

,0),(2),(),(

+
rtWrtWrtW
trrt

0),(2),(
2
=+ rtWrtW
ttt
and
.0),(2),(
2
=+ rtWrtW
rrr


59
(C31) Rank=2,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t
,0),( =rtV
r
,0),(

rtW
t

,0),( ≠rtW
r

,0),(2),(),(
=
+
rtWrtWrtW
trrt

0),(),(
2
=+ rtWrtW
ttt
and
.0),(2),(
2
≠+ rtWrtW
rrr

(C32) Rank=2,
,0),( ≠rtU
t

,0),(

rtU
r
,0),(2),(
2
≠+ rtUrtU
rrr
,0),(
=
rtV
t

,0),( =rtV
r
,0),( =rtW
t

0),(

rtW
r
and .0),(2),(
2
=+ rtWrtW
rrr

(C33) Rank=2,
,0),( =rtU
t

,0),(

rtU
r
,0),(2),(
2
≠+ rtUrtU
rrr
,0),(
=
rtV
t

,0),( =rtV
r
,0),( =rtW
t

0),(

rtW
r
and .0),(2),(
2
=+ rtWrtW
rrr

(C34) Rank=2,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(

rtV
t

,0),(

rtV
r

,0),(2),(),( ≠+ rtVrtVrtV
trrt

,0),(2),(
2
≠+ rtVrtV
rrr

0),( =rtW
t
and
.0),( =rtW
r

(C35) Rank=2,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
≠+ rtVrtV
ttt

,0),(2),(),( ≠+ rtVrtVrtV
trrt

,0),(2),(
2
≠+ rtVrtV
rrr

0),( =rtW
t
and
.0),( =rtW
r

(C36) Rank=2,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
ttt
,0),(2),(),(

+
rtVrtVrtV
trrt
,0),(2),(
2
≠+ rtVrtV
rrr

0),( =rtW
t
and
.0),( =rtW
r

(C37) Rank=2,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(

rtV
t

,0),(

rtV
r

,0),(2),(),( ≠+ rtVrtVrtV
trrt
,0),(2),(
2
=+ rtVrtV
rrr

0),( =rtW
t
and
.0),( =rtW
r

(C38) Rank=2,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
ttt
,0),(2),(),(

+
rtVrtVrtV
trrt
,0),(2),(
2
=+ rtVrtV
rrr

0),( =rtW
t
and
.0),( =rtW
r

(C39) Rank=2,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
≠+ rtVrtV
ttt
,0),(2),(),(
=
+
rtVrtVrtV
trrt
,0),(2),(
2
≠+ rtVrtV
rrr

0),( =rtW
t
and
.0),( =rtW
r

(C40) Rank=2,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
ttt
,0),(2),(),(
=
+
rtVrtVrtV
trrt
,0),(2),(
2
≠+ rtVrtV
rrr

0),( =rtW
t
and
.0),( =rtW
r


60
(C41) Rank=2,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
ttt
,0),(2),(),(

+
rtVrtVrtV
trrt
,0),(2),(
2
=+ rtVrtV
rrr

0),( =rtW
t
and
.0),( =rtW
r

(C42) Rank=2,
,0),( ≠rtU
t

,0),(

rtU
r

,0),(2),(
2
≠+ rtUrtU
rrr
,0),(
=
rtV
t

,0),( ≠rtV
r
,0),(2),(
2
=+ rtVrtV
rrr

0),( =rtW
t
and
.0),(
=
rtW
r


(C43) Rank=3,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
ttt

,0),(2),(),( =+ rtVrtVrtV
trrt
,0),(2),(
2
≠+ rtVrtV
rrr

,0),( ≠rtW
t

,0),(

rtW
r

,0),(2),(),( =+ rtWrtWrtW
trrt

0),(2),(
2
=+ rtWrtW
ttt
and .0),(2),(
2
≠+ rtWrtW
rrr

(C44) Rank=3,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( =rtV
r

,0),(2),(
2
≠+ rtVrtV
ttt

,0),( ≠rtW
t

,0),( ≠rtW
r

,0),(2),(),(
=
+
rtWrtWrtW
trrt
0),(2),(
2
=+ rtWrtW
rrr
and
.0),(2),(
2
≠+ rtWrtW
ttt

(C45) Rank=3,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( =rtV
r

,0),(2),(
2
=+ rtVrtV
ttt

,0),( ≠rtW
t

,0),(

rtW
r

0),(2),(),( =
+
rtWrtWrtW
trrt
and
.0),(2),(
2
=+ rtWrtW
rrr

(C46) Rank=3,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( =rtV
r

,0),( ≠rtW
t

,0),(

rtW
r

,0),(2),(),( =+ rtWrtWrtW
trrt
0),(2),(
2
=+ rtWrtW
rrr
and
.0),(2),(
2
=+ rtWrtW
ttt

(C47) Rank=3,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( =rtV
r

,0),(2),(
2
=+ rtVrtV
ttt

,0),( ≠rtW
t

,0),(

rtW
r

,0),(2),(),(
=
+ rtWrtWrtW
trrt

0),(2),(
2
=+ rtWrtW
ttt
and .0),(2),(
2
=+ rtWrtW
rrr

(C48) Rank=3,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
≠+ rtVrtV
ttt

,0),(2),(),( =+ rtVrtVrtV
trrt
0),(2),(
2
=+ rtVrtV
rrr

,0),( ≠rtW
t

0),( =rtW
r
and
.0),(2),(
2
≠+ rtWrtW
ttt


61
(C49) Rank=3,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
ttt
,0),(2),(),(
=
+
rtVrtVrtV
trrt
,0),(2),(
2
=+ rtVrtV
rrr

0),( ≠rtW
t
and
.0),( =rtW
r

(C50) Rank=3,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(),( =+ rtVrtVrtV
trrt

,0),(2),(
2
=+ rtVrtV
rrr

,0),( ≠rtW
t

0),( =rtW
r
and
.0),(2),(
2
=+ rtWrtW
ttt

(C51) Rank=3,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
ttt
,0),(2),(),(
=
+
rtVrtVrtV
trrt
,0),(2),(
2
=+ rtVrtV
rrr

,0),( ≠rtW
t

0),( =rtW
r
and
.0),(2),(
2
=+ rtWrtW
ttt

(C52) Rank=3,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(),( =+ rtVrtVrtV
trrt
,0),(2),(
2
=+ rtVrtV
rrr

,0),( ≠rtW
t

,0),(

rtW
r

0),(2),(),( =+ rtWrtWrtW
trrt
and
.0),(2),(
2
=+ rtWrtW
rrr

(C53)

Rank=3, ,0),( =rtU
t

,0),(

rtU
r
,0),(2),(
2
=+ rtUrtU
rrr
,0),(
=
rtV
t

,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
rrr
,0),(
=
rtW
t

0),( ≠rtW
r
and
.0),(2),(
2
=+ rtWrtW
rrr

(C54) Rank=2,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( =rtV
r

,0),(2),(
2
=+ rtVrtV
ttt

,0),( ≠rtW
t

,0),( ≠rtW
r

,0),(2),(),(
=
+
rtWrtWrtW
trrt

0),(2),(
2
=+ rtWrtW
ttt
and
.0),(2),(
2
≠+ rtWrtW
rrr

(C55) Rank=2,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t

,0),(

rtV
r

,0),(2),(
2
=+ rtVrtV
ttt

,0),(2),(),( =+ rtVrtVrtV
trrt
,0),(2),(
2
=+ rtVrtV
rrr
,0),(
=
rtW
t

0),( ≠rtW
r
and
.0),(2),(
2
≠+ rtWrtW
rrr

(C56) Rank=2,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
ttt

,0),(2),(),( =+ rtVrtVrtV
trrt

,0),(2),(
2
=+ rtVrtV
rrr
,0),(

rtW
t

,0),(

rtW
r

,0),(2),(),( =+ rtWrtWrtW
trrt

0),(2),(
2
=+ rtWrtW
ttt
and .0),(2),(
2
≠+ rtWrtW
rrr


62
(C57) Rank=2,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
ttt

,0),(2),(),( =+ rtVrtVrtV
trrt

,0),(2),(
2
≠+ rtVrtV
rrr
,0),(

rtW
t

0),( =rtW
r
and
.0),(2),(
2
=+ rtWrtW
ttt

(C58) Rank=2,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
ttt

,0),(2),(),( =+ rtVrtVrtV
trrt

,0),(2),(
2
≠+ rtVrtV
rrr
,0),(
=
rtW
t

0),( ≠rtW
r
and
.0),(2),(
2
=+ rtWrtW
rrr

(C59) Rank=2,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t
,0),( ≠rtV
r

,0),(2),(
2
≠+ rtVrtV
rrr
,0),(

rtW
t

,0),(

rtW
r

0),(2),(),( =
+
rtWrtWrtW
trrt
and
.0),(2),(
2
=+ rtWrtW
rrr

(C60) Rank=2,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
≠+ rtVrtV
ttt

,0),(2),(),( =+ rtVrtVrtV
trrt
,0),(2),(
2
=+ rtVrtV
rrr
,0),(

rtW
t

0),( =rtW
r
and
.0),(2),(
2
≠+ rtWrtW
ttt

(C61) Rank=2,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
≠+ rtVrtV
ttt

,0),(2),(),( =+ rtVrtVrtV
trrt
,0),(2),(
2
=+ rtVrtV
rrr
,0),(

rtW
t

0),( =rtW
r
and
.0),(2),(
2
=+ rtWrtW
ttt

(C62) Rank=2,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
=+ rtVrtV
ttt

,0),(2),(),( =+ rtVrtVrtV
trrt
,0),(2),(
2
=+ rtVrtV
rrr
,0),(
=
rtW
t

0),( ≠rtW
r
and
.0),(2),(
2
=+ rtWrtW
rrr

(C63) Rank=2,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r

,0),(2),(
2
≠+ rtVrtV
ttt

,0),(2),(),( =+ rtVrtVrtV
trrt
,0),(2),(
2
=+ rtVrtV
rrr
,0),(
=
rtW
t

0),( ≠rtW
r
and
.0),(2),(
2
=+ rtWrtW
rrr

(C64) Rank=2,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t

,0),(
=
rtV
r

,0),(2),(
2
≠+ rtVrtV
ttt

,0),( ≠rtW
t

,0),( ≠rtW
r

,0),(2),(),(
=
+
rtWrtWrtW
trrt

0),(2),(
2
=+ rtWrtW
ttt
and
.0),(2),(
2
=+ rtWrtW
rrr


63
(D1) Rank=1,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t
,0),( =rtV
r
,0),(
=
rtW
t

0),( ≠rtW
r
and .0),(2),(
2
≠+ rtWrtW
rrr

(D2) Rank=1,
,0),( ≠rtU
t

,0),(

rtU
r
,0),(2),(
2
≠+ rtUrtU
rrr
,0),(
=
rtV
t

,0),( =rtV
r
0),( =rtW
t
and
.0),(
=
rtW
r

(D3) Rank=1,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( =rtV
r
,0),(
=
rtW
t

0),( =rtW
r
and
.0),(2),(
2
≠+ rtVrtV
ttt

(D4) Rank=1,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t
,0),( =rtV
r
,0),(
=
rtW
t

0),( =rtW
r
and
.0),(2),(
2
=+ rtVrtV
ttt

(D5) Rank=1,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t
,0),( =rtV
r
,0),(

rtW
t

0),( =rtW
r
and
.0),(2),(
2
=+ rtWrtW
ttt

(D6) Rank=1,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t

,0),(
=
rtV
r
,0),(

rtW
t

0),( =rtW
r
and
.0),(2),(
2
≠+ rtWrtW
ttt

(D7) Rank=1,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t
,0),( ≠rtV
r
,0),(
=
rtW
t

0),( =rtW
r
and .0),(2),(
2
≠+ rtVrtV
rrr

(D8) Rank=1,
,0),( =rtU
t

,0),(

rtU
r
,0),(2),(
2
≠+ rtUrtU
rrr
,0),(
=
rtV
t

,0),( =rtV
r
0),( =rtW
t
and
.0),(
=
rtW
r


(D9) Rank=1,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r
,0),(
=
rtW
t

,0),( =rtW
r
0),(2),(),(
=
+ rtVrtVrtV
trrt
and .0),(2),(
2
=+ rtVrtV
rrr

(D10) Rank=1,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t

,0),(

rtV
r
,0),(
=
rtW
t

,0),( =rtW
r
0),(2),(),(
=
+
rtVrtVrtV
trrt
0),(2),(
2
=+ rtVrtV
rrr
and
.0),(2),(
2
≠+ rtVrtV
ttt

(D11) Rank=1,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t
,0),( =rtV
r
,0),(

rtW
t

,0),( ≠rtW
r

,0),(2),(),(
=
+
rtWrtWrtW
trrt
0),(2),(
2
=+ rtWrtW
rrr
and
.0),(2),(
2
≠+ rtWrtW
ttt


64
(D12) Rank=1,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t
,0),( =rtV
r
,0),(

rtW
t

,0),( ≠rtW
r

,0),(2),(),(
=
+
rtWrtWrtW
trrt
0),(2),(
2
=+ rtWrtW
rrr
and
.0),(2),(
2
=+ rtWrtW
ttt

(D13) Rank=1,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t
,0),( =rtV
r
,0),(

rtW
t

,0),( ≠rtW
r

0),(2),(),(
=
+ rtWrtWrtW
trrt
and
.0),(2),(
2
=+ rtWrtW
rrr

(D14) Rank=1,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r
,0),(
=
rtW
t

,0),( =rtW
r
,0),(2),(),(
=
+
rtVrtVrtV
trrt

0),(2),(
2
=+ rtVrtV
ttt
and
.0),(2),(
2
≠+ rtVrtV
rrr

(D15) Rank=1,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t
,0),( =rtV
r
,0),(

rtW
t

,0),( ≠rtW
r

,0),(2),(),(
=
+
rtWrtWrtW
trrt

0),(2),(
2
=+ rtWrtW
ttt
and
.0),(2),(
2
≠+ rtWrtW
rrr

(D16) Rank=1,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( =rtV
r
,0),(

rtW
t

,0),( =rtW
r

0),(2),(
2
=+ rtWrtW
ttt
and
.0),(2),(
2
=+ rtVrtV
ttt

(D17) Rank=1,
,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( =rtV
r
,0),(
=
rtW
t

,0),( ≠rtW
r

0),(2),(
2
=+ rtWrtW
rrr
and
.0),(2),(
2
=+ rtVrtV
ttt

(D18) Rank=1,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( =rtV
r
,0),(
=
rtW
t

,0),( ≠rtW
r
0),(2),(
2
≠+ rtWrtW
rrr
and
.0),(2),(
2
=+ rtVrtV
ttt


(D19) Rank=1,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r
,0),(
=
rtW
t

,0),( ≠rtW
r
,0),(2),(),(
=
+
rtVrtVrtV
trrt

,0),(2),(
2
=+ rtVrtV
ttt

0),(2),(
2
=+ rtVrtV
rrr
and .0),(2),(
2
=+ rtWrtW
rrr

(D20) Rank=1,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( ≠rtV
r
,0),(

rtW
t

,0),( =rtW
r
,0),(2),(),(
=
+
rtVrtVrtV
trrt

,0),(2),(
2
=+ rtVrtV
ttt

0),(2),(
2
=+ rtVrtV
rrr
and
.0),(2),(
2
=+ rtWrtW
ttt


65
(D21) Rank=1,
,0),( =rtU
t

,0),(
=
rtU
r
,0),(

rtV
t
,0),( =rtV
r
,0),(
=
rtW
t

,0),( ≠rtW
r

0),(2),(
2
≠+ rtVrtV
ttt
and .0),(2),(
2
=+ rtWrtW
rrr

Some illustrative cases are discussed below. The remaining possibilities yield somewhat
similar results and there detailed discussion has been omitted.
Case A1
In this case one has 0,0,0,0
7546321



=
=
==
α
α
α
α
α
α
α
and .0
8

α
From these
constraints, we have
,0),(
=
rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t

,0),(

rtV
r

,0),(2),(
2
=+ rtVrtV
rrr

,0),( ≠rtW
t

,0),(

rtW
r

,0),(2),(),(

+ rtWrtWrtW
trrt

0),(2),(
2
≠+ rtWrtW
ttt
and
.0),(2),(
2
≠+ rtWrtW
rrr
Here, the rank of the
66
×
=
剩敭慮渠aa瑲楸⁩猠瑨牥t⁡湤⁴桥=攠數楳瑳漠湯i ⁴物癩慬⁳=汵瑩ln=⁥煵慴楯渠⠲⸶⸱⤮⁕n楮朠
瑨攠慢潶攠楮景tm慴楯測⁷攠来琠,),(
artU
=
,)ln(),(
2
crbrtV
+= ),,(
rtWW
= where
).0(,,≠∈
bRcba
Substituting the values of ),(
rtU
and ),(
rtV
in (4.2.1) and after a
rescaling of
,t
the line element can be written in the form

.)(
2),(22222
φθ
dedcrbdrdtds
rtW
++++−=
(4.2.4)
The above spacetime (4.2.4) belongs to curvature class A. In class A the rank of the 66
×
=
剩敭慮渠浡瑲楸⁩猠㘬‵Ⱐ㐬″爠㈠⡥硣畬摩 湧⁴桥⁣污獳⁂⤠慮搠瑨敲攠數楳瑳漠湯渠瑲楶楡氠
獯汵瑩潮=⁥煵慴楯q
㈮=⸱⤠.8].⁔桥⁣污獳⁁⁩== 獡楤⁴i⁧敮=物r⁩渠瑨攠獥湳=⁴桡t⁥癥r礠䍃y
楳散敳獡物汹⁡⁨潭潴桥瑩挠癥捴潲⁦楥汤⁛㠬o 㙝⸠䡥湣攠楮⁴6楳⁣慳攠湯⁰牯灥爠䍃⁥硩獴献†
Cases (A2-A30)

All the cases from A2 to A30 also belong to curvature class A. In these cases rank of the
66× Riemann matrix is three and there exists no non trivial solution of equation (2.6.1).
CCS here are precisely the homothetic vector fields. Therefore, no proper CCS exist in all
these cases.
Case B1
In this case 0,0,0
72865431


=
=
=
===
α
α
α
α
α
α
α
α
and the rank of
66
×
=
剩敭慮渠浡瑲楸⁩猠瑷漮⁈敲攬⁷e⁨慶攠瑨攠捯湤楴楯湳=,0),( ≠
rtU
t

,0),(
=
rtU
r

,0),( ≠
rtV
t

,0),( =rtV
r

,0),(2),(
2
=+ rtVrtV
ttt
,0),(
=
rtW
t

0),( ≠rtW
r
and
0),(2),(
2
≠+
rtWrtW
rrr
which imply that ),(
tUU
=

2
)ln()(
btatV
+= and ),(
rWW
=

66
where ).0(,≠∈
aRba
In this case there exists no solution of equation (2.6.1).
Substituting the information of ),,(
rtU
),(
rtV
and ),(
rtW
in equation (4.2.1), the line
element can be written in the form


(
)
.
2)(22
2
2)(2
φθ
dedrdbtadteds
rWtU
++++−=
(4.2.5)
The above spacetime is clearly 2+2 decomposable and belongs to curvature class B. CCS
in this case are [8] of the form

,
21
XXX
+
=

(4.2.6)
where
1
X
are CCS in the induced geometry on each of the two dimensional submanifolds
of constant
r
and ,
φ
⁡湤=
2
X
are CCS in the induced geometry on each of the two
dimensional submanifolds of constant
t
and
.
θ
⁔heext⁳tep⁩s⁴o⁷ork×t⁴he⁃CS⁩n=
瑨攠楮摵捥搠来潭整特映瑨攠獵扭慮楦潬摳映捯湳瑡湴a
r
and
.
φ
⁔桥e瑨潤⁦潲⁦楮摩湧t
䍃匠楮′ⵤ業e湳楯湡氠獵扭慮楦潬摳⁩猠杩癥渠楮 ⁳散瑩潮′⸹⸠䥦湥⁰牯捥敤猠晵牴桥爠瑨攠
杩癥渠湯湺敲漠捯g灯湥湴猠潦⁴桥⁩湤畣敤p 浥瑲楣渠敡捨映瑨攠瑷漠摩me湳楯湡氠
獵扭慮楦潬摳映捯湳瑡湴a
r
and
φ
⁡牥⁧楶敮⁢礠=
=
,

U
eg
−=
2
22
)(
btag
+=
(4.2.7)
Calculation shows that the nonzero components of the Ricci tensor turn out to be

,
)(2
00
U
bta
a
R
&
+
=
U
eUbtaaR

+−=
&
)(
2
1
22

(4.2.8)
and the Ricci scalar is
.
2
1
U
eaUR

−=
&
Accordingly, the tensor
2,0,;
2
=≡
βα
εβαβ
g
R
G has nonzero components

,
)(2
00
U
bta
a
G
&
+
=
U
eUbtaaG

+−=
&
)(
2
1
22

(4.2.9)
A method of to find CCS in two dimensional space is given in section 2.9. If one
proceeds further can find that

,0)(2)(
0
0,
0
=
+
+
+
X
bta
U
X
bta
U
&
&
&

(4.2.10)

,0)(
0
2,
2
0,
2
=−+ XeXbta
U

(4.2.11)

67

.0)(2))((
2
2,
0
=+++
−−
XeUbtaXeUbta
UU
&
&
&

(4.2.12)
From equations (4.2.10) and (4.2.11), we get

),()()(,))((
2
2
1
2
3
12
2
1
10
θθθ
θ
AdteUbtaAX
bta
U
AX
U
++=
+
=

−−−
&
&

(4.2.13)
where )(
1
θA and )(
2
θA are functions of integration. Proceeding further with equation
(4.2.12), we get the condition

.
)(
)(
)()
)(2
))((
(
1
1
2
1
2
3
2
3
2
1
η
θ
θ
θθ
=−=+
+
+


A
A
eUbta
Ubta
eeUbta
U
UU
&
&
&
&
&

(4.2.14)
There exist following two possibilities:
(a)
,)()
)(2
))((
(
2
1
2
3
2
3
2
1
η=+
+
+


U
UU
eUbta
Ubta
eeUbta
&
&
&
&
&

(b)
,)()
)(2
))((
(
2
1
2
3
2
3
2
1
η≠+
+
+


U
UU
eUbta
Ubta
eeUbta
&
&
&
&
&
where .R

η

First considering the subcase (a). In this subcase there exist three possibilities which are
(i) ,0>
η
(ii) ,0
=
η
† ⡩楩⤠.0
<
η

We⁷楬氠捯=s楤敲⁥慣栠ao獳楢楬楴礠楮⁴畲渮†
Case B1 (ai)
In this case 0>
η
and equation (4.2.14) implies that .0)()(
11
=+ θηθ
θθθ
AA The solution
of this equation is
,)sinhcosh()(
32
1
θηθηθ
ccA +=
where .,
32
Rcc ∈ Using the
above information, one can find that CCS in this case are

,)()sinhcosh(
,)()sinhcosh(
4
2
1
2
3
23
2
2
1
32
0
cdteUbtaccX
bta
U
ccX
U
+++=
+
+=

−−

&
&
θηθηη
θηθη

(4.2.15)
provided that
,)()
)(2
))((
(
2
1
2
3
2
3
2
1
U
UU
eUbta
Ubta
eeUbta
−−

+−=
+
+
&
&
&
&
&
η where .,,
432
Rccc




68
Case B1 (aii)
In this case
0=
η
and equation (4.2.14) implies that ,)(0)(
32
11
ccAA θθθ
θθ
+=⇒=
where .,
32
Rcc ∈ After a straight forward calculation it can be found that the CCS in this
case are


,)
2
()(
,)()(
42
2
3
2
1
2
3
3
2
2
1
32
0
cc
c
NdteUbtacX
bta
U
ccX
U
++−+=
+
+=

−−

θθ
θ
&
&

(4.2.16)
provided that
,
)(2
))((
2
3
2
1
N
Ubta
eeUbta
UU
=
+
+

&
&
&
where .,,,
432
RNccc


Case B1 (aiii)
Here .0<
η
Substituting ,
ξ
η
−= where R

ξ

(
)
0>
ξ
in equation (4.2.14), we get
,)sincos()(0)()(
32
111
θξθξθθξθ
θθθ
ccAAA +=⇒=−
where .,
32
Rcc ∈ Using the
above information, one can easily find that CCS in this case are

,)()sincos(
,)()sincos(
4
2
1
2
3
23
2
2
1
32
0
cdteUbtaccX
bta
U
ccX
U
++−=
+
+=

−−

&
&
θξθξξ
θξθξ

(4.2.17)
provided that
,)()
)(2
))((
(
2
1
2
3
2
3
2
1
U
UU
eUbta
Ubta
eeUbta
−−

+=
+
+
&
&
&
&
&
ξ where .,,
432
Rccc


Case B1b
In this subcase (b) the CCS are given by

,,0
5
20
cXX ==
(4.2.18)
where .
5
Rc ∈ This completes the calculation of
.
1
X

We now proceed with the calculation of the CCS in the induced geometries on each of
the two-dimensional submanifolds of constant t and
θ
⁩⹥⸠
.
2
X
The nonzero components

69
of the induced metric on each of the two-dimensional submanifolds of constant t and
θ
=
慲攠杩癥渠批†
=
,1

=g

.
33
W
eg =

(4.2.19)
Calculation shows that the nonzero components of the Ricci tensor turn out to be

),2(
4
1
2
11
WWR
′′
+

−=
.)2(
4
1
2
33
W
eWWR
′′
+

−=
(4.2.20)
and the Ricci scalar is given by
).2(
2
1
2
WWR
′′
+

−= Accordingly, the tensor
3,1,;
2
=≡ βα
αβαβ
g
R
G
has nonzero components

),2(
4
1
2
11
WWG
′′
+

−=
.)2(
4
1
2
33
W
eWWG
′′
+

−=
(4.2.21)
It follows from [8] that the Killing fields for this tensor, which are precisely the CCS in
the induced geometry, are the solutions to the equations

,0)2(2)2(
1
1,
212
=
′′
+

+
′′′
+

XWWXWW

(4.2.22)

,0
3
1,
1
3,
=+ XeX
W

(4.2.23)

.0)2(2))2(
3
3,
212
=
′′
+

+
′′′
+

XeWWXeWW
WW

(4.2.24)
Solving equations (4.2.22) and (4.2.23) we get

),()2()(,)2()(
2
2
1
213
2
1
211
φφφ
φ
AdreWWAXWWAX
W
+
′′
+

−=
′′
+

=


−−

(4.2.25)
where )(
1
φA and )(
2
φA are functions of integration. Proceeding further with equation
(4.2.24), we get the condition

.
)(
)(
)2()
)2(2
))2((
(
1
1
2
1
2
2
3
2
2
Ω==
′′
+
′′
′′
+

′′′
+
′′

φ
φ
θθ
A
A
eWWe
WW
eWW
WW
W

(4.2.26)
There exist following two possibilities:
(c)
,)2()
)2(2
))2((
(
2
1
2
2
3
2
2
Ω=
′′
+
′′
′′
+

′′′
+
′′
− WW
W
eWWe
WW
eWW

(d)
,)2()
)2(2
))2((
(
2
1
2
2
3
2
2
Ω≠
′′
+
′′
′′
+

′′′
+
′′
− WW
W
eWWe
WW
eWW
where
.R∈Ω


70
First consider the subcase (c). In this subcase there exist three possibilities which are
(i) ,0>Ω (ii) ,0
=
Ω
† ⡩楩⤠
.0
<
Ω

We⁷楬氠捯湳楤敲⁥慣栠灯獳楢楬n 瑹⁩渠瑵牮⁡摯t瑩湧⁴桥⁳a浥⁰牯捥摵re⁡猠摩獣=獳s搠楮≤
捡獥映捡汣畬慴楮朠
.
1
X

Case B1 (ci)
In this case CCS are given by

,)2()sinhcosh(
,)2()sinhcosh(
8
2
1
2
76
3
2
1
2
67
1
cdreWWccX
WWccX
W
+
′′
+

Ω+ΩΩ=
′′
+

Ω+Ω=




φφ
φφ

(4.2.27)
provided that
,)2()
)2(2
))2((
(
2
1
2
2
3
2
2
Ω=
′′
+
′′
′′
+

′′′
+
′′
− WW
W
eWWe
WW
eWW
where .,,
876
Rccc


Case B1 (cii)
CCS in this case can be expressed as

,)
2
()2(
,)2()(
87
2
6
2
1
2
6
3
2
1
2
76
1
cc
c
MdreWWcX
WWccX
W
++−
′′
+

=
′′
+

+=




φφ
φ

(4.2.28)
provided that
,
)2(2
))2((
2
3
2
2
Me
WW
eWW
W
W
=
′′
+

′′′
+


where .,,,
876
RMccc


Case B1 (ciii)
Here
.0<Ω
Substituting
,
σ
−=Ω
where
R

σ

(
)
.0>
σ
CCS in this case are given
below

,)2()sincos(
,)2()sincos(
8
2
1
2
76
3
2
1
2
67
1
cdreWWccX
WWccX
W
+
′′
+

−=
′′
+

+=




φσφσσ
φσφσ

(4.2.29)
provided that
,)2()
)2(2
))2((
(
2
1
2
2
3
2
2
WW
W
eWWe
WW
eWW



′′
+

−=

′′
+

′′′
+

σ where .,,
876
Rccc ∈

71
Case B1d
In subcase (d) CCS are found to be

,,0
9
31
cXX ==
(4.2.30)
where .
9
Rc ∈ This completes the calculation of CCS in Case B1.
Cases (B2-B3)
.

In both cases B2 and B3 the rank of the
66
×
⁒楥=慮渠aa瑲楸⁩t⁡汳漠瑷漮⁍o牥潶敲r⁩==
瑨敳攠捡獥猠瑨攠獰慣整業e
㐮㈮ㄩ⁢散潭e 猠㈫㈠摥捯浰潳慢汥⁡湤⁢敬潮杳⁴漠瑨s=
捵牶慴′r攠捬慳猠䈮⁃慬捵l慴楯渠景爠䍃匠楳⁰牥捩ae汹⁳慭e⁡猠瑨慴映捡獥⁂ㄮ†
Case C1
In this case one has 0,0,0,0
87654321



=
=
=
==
α
α
α
α
α
α
α
α
and the rank of the
66×
Riemann matrix is three and there exists a unique (up to multiple) no where zero
vector field
aa
tt
,
=
such that
.0
;
=
ba
t
From Ricci identity .0=
d
bcd
a
tR Using the above
constraints we have ,0),( ≠rtU
t

,0),(
=
rtU
r
,0),(
=
rtV
t

,0),( ≠rtV
r
,0),(
=
rtW
t

,0),(2),(
2
≠+ rtVrtV
rrr

0),(

rtW
r
and
0),(2),(
2
≠+ rtWrtW
rrr
which imply that
),(tUU = )(rVV = and ).(rWW = After a suitable rescaling of t the line element can
be written in the form


.
2)(2)(222
φθ dededrdtds
rWrV
+++−=
(4.2.31)
The above spacetime is clearly 1+3 decomposable. CCS in this case [8] are of the form

,)( X
t
tfX

+


=
(4.2.32)
where )(tf is an arbitrary function of t and
X

⁩猠愠桯=o瑨整楣⁶散瑯爠tiel≤⁩渠瑨==
楮摵捥搠ieom整特渠敡捨映瑨攠瑨牥攭ei me湳楯湡氠獵扭an楦潬摳映捯湳瑡湴i.t The
induced metric
αβ
g (where 3,2,1,
=
β
α
⤠睩瑨潮⁺敲漠捯)灯湥湴猠楳⁧楶敮⁢礠n
=
,1

=g
,
22
V
eg
=
.
22
W
eg
=

(4.2.33)
A vector field
X

is called homothetic vector field if it satisfies ,2
αβαβ
gcgL
X
=

where
.
Rc

One can expand homothetic equation by using (4.2.33) to get

,
1
1,
cX
=

(4.2.34)

,0
2
1,
1
2,
=+
XeX
V

(4.2.35)

72

,0
3
1,
1
3,
=+
XeX
W

(4.2.36)

,22
2
2,
1
cXXV
=+


(4.2.37)

,0
3
2,
2
3,
=+
XeXe
WV

(4.2.38)

.22
3
3,
1
cXXW
=+


(4.2.39)
From equations (4.2.34), (4.2.35) and (4.2.36), we get

(
)
( )
,,),(
,,),(),,(
313
21211
φθφθ
φθφθφθ
φ
θ
AdreAX
AdreAXAcrX
W
V
+−=
+−=+=





(4.2.40)
where
( )
φθ,
1
A
( )
φθ,
2
A and
(
)
φθ,
3
A are functions of integration. Proceeding further
with some calculation there arise two conditions
(i)
( )
0≠

−WV (ii)
( )
.0=

−WV
Considering the condition (i) and after some straight forward calculation one can easily
find that the solution of the equations from (4.2.34) to (4.2.39) is

,,,
54
3
32
2
1
1
ccXccXccrX +=+=+=
φθ

(4.2.41)
where
,)ln()(
)1(2
1
2
c
c
ccrrV

+=
,)ln()(
)1(2
1
4
c
c
ccrrW

+= Rcccccc ∈
54321
,,,,, and
).,,0,0,0,(
424242
ccccccccc

≠≠≠≠≠
It follows from the above calculation that the
induced metric in the induced geometry on each of the three-dimensional submanifolds of
constant t admits proper homothetic vector field. CCS in this case are given by the use of
equations (4.2.32) and (4.2.41) as

,,,),(
54
3
32
2
1
10
ccXccXccrXtfX +=+=+==
φθ

(4.2.42)
where )(tf is an arbitrary function of .t One can write the above equation (4.2.42) after
subtracting homothetic vector fields as

).0,0,0),((
tfX
=

(4.2.43)
CCS clearly form an infinite dimensional vector space
In subcase (ii) we have
( )
,0 λ+=⇒=

− WVWV where
R

λ
and
0



V

(and so
0≠
′′
W
). Solution of the above equations from (4.2.34) to (4.2.39) is

,,,
532
3
432
2
1
1
ccecXcccXccrX +−=++=+=
λ
θφφθ
(4.2.44)

73
where
λ++=
− )1(2
1
2
)ln()(
c
c
ccrrV and ).0,0,(,,,,,
2254321
≠≠


ccccRcccccc It
follows from the above calculation that the induced metric in the induced geometry on
each of the three-dimensional submanifolds of constant t admits proper homothetic
vector field. CCS in this case are given by the use of equations (4.2.32) and (4.2.44) as

.,,),(
532
3
432
2
1
10
ccecXcccXccrXtfX +−=++=+==
λ
θφφθ

(4.2.45)
One can write the above equation (4.2.45) after subtracting homothetic vector fields as in
equation (4.2.43).
Cases (C2-C12)


Here, the rank of the
66×
Riemann matrix in cases C2 to C12 is three and there exists a
unique (up to multiple) no where zero vector

field which is the solution of equation
(2.6.1). By using the constraints given by these cases the spacetime (4.2.1) becomes 1+3
decomposable and belong to curvature C. CCS can be found by the method described in
case.C1.
Cases (C13)
In this case 0,0,0
87654321


=
=
=
===
α
α
α
α
α
α
α
α
and the rank of
66
×
=
剩敭慮渠ma瑲楸⁩猠瑷漠慮搠瑨ire⁥=楳瑳⁡⁵i 楱ie
異⁴漠=×汴楰汥⤠湯⁷桥牥⁺敲漠癥捴潲′
晩敬搠
aa
tt
,
= such that 0=
d
abcd
tR and .0
;
=
ba
t From the above constraints, we have
,0),( =rtU
t

,0),( =rtU
r
,0),(
=
rtV
t
,0),( ≠rtV
r
,0),(2),(
2
=+ rtVrtV
rrr

,0),(
=
rtW
t

0),( ≠rtW
r
and
.0),(2),(
2
≠+ rtWrtW
rrr
It is deduced that
,
kU
=

2
)ln()( barrV +=
and ),(rWW = where ).0(,,

∈ aRkba After a suitable rescaling of t the line element
can be written in the form

.)(
2)(22222
φθ
dedbardrdtds
rW
++++−=
(4.2.46)
The above spacetime is clearly 1+3 decomposable. CCS in this case are of the form

,)( X
t
tNX

+


=
(4.2.47)
where )(tN is an arbitrary function and
X

⁩猠愠桯=潴桥瑩挠癥捴潲⁦i敬搠楮⁴桥⁩湤畣敤=
来潭整特渠敡捨映瑨攠瑨牥攭= ime湳楯湡氠n×扭慮楦潬摳映捯湳瑡at=.t The induced
metric
αβ
g (where 3,2,1,=
β
α
) with non zero components is given by

74

,1
11
=
g
,)(
2
22
barg += .
)(
22
rW
eg =
(4.2.48)
A vector field
X

is called homothetic vector field if it satisfies ,2
αβαβ
gcgL
X
=

where
.
Rc

Expanding homothetic equation by using (4.2.48) to get

,
1
1,
cX
=

(4.2.49)

,0)(
2
1,
21
2,
=++
XbarX

(4.2.50)

,0
3
1,
1
3,
=+
XeX
W

(4.2.51)

),()(
2
2,
1
barcXbarXa
+=++

(4.2.52)

,0)(
3
2,
2
3,
2
=++
XeXbar
W

(4.2.53)

.22
3
3,
1
cXXW
=+


(4.2.54)
If one proceeds further, after some calculation one can find that the solution of the above
equations from (4.2.49) to (4.2.54) is

,,,
43
3
2
2
1
1
ccXcXccrX +==+=
φ

(4.2.55)
where
,)ln()(
)(2
1
3
c
cc
ccrrW

+= )0(,,,,
4321


cRccccc and
.0
1
=

bcac
It follows from
the above calculation that the induced metric in the induced geometry on each of the
three-dimensional submanifolds of constant t admits proper homothetic vector field.
CCS in this case are given by the use of equations (4.2.47) and (4.2.55) as

.,,),(
43
3
2
2
1
10
ccXcXccrXtNX +==+==
φ

(4.2.56)
One can write the above equation (4.2.56) after subtracting homothetic vector fields as in
equation (4.2.43).
Cases (C14-C42)

The rank of
66×
Riemann matrix in all these cases is two and there exists a unique (up
to multiple) no where zero vector field which is the solution of equation (2.6.1) and is
covariantly constant. If we use constraints given by cases C14 to C42 in equation (4.2.1)
then it becomes 1+3 decomposable. CCS can be calculated by the similar way as
discussed in case C13.




75
Case C43
In this case one has 0,0,0,0
87654321



=
=
=
==
α
α
α
α
α
α
α
α
and the rank of the
66×
Riemann matrix is three and there exists a unique (up to multiple) no where zero
vector field
aa
tt
,
= such that
0=
d
abcd
tR
and .0
;

ba
t From the above constraints, we
have ,0),( =rtU
t

,0),( =
rtU
r
,0),(

rtV
t

,0),(

rtV
r
,0),( ≠rtW
t

0),(

rtW
r

,0),(2),(),( =+ rtVrtVrtV
trrt

,0),(2),(
2
=+
rtVrtV
ttt
,0),(2),(
2
≠+ rtVrtV
rrr

,0),(2),(
2
=+
rtWrtW
ttt
0),(2),(),(
=
+
rtWrtWrtW
trrt
and
.0),(2),(
2
≠+ rtWrtW
rrr

Which imply that ,kU =
2
))(ln(),( rbatrtV
α
++= and ,))(ln(),(
2
rdctrtW
β
++=
where )0,(,,,,≠∈ caRkdcba and )(r
α
and )(r
β
are no where zero and no where
equal functions of integration. After a suitable rescaling of t the line element can be
written in the form

.))(())((
2222222
φβθα
drdctdrbatdrdtds +++++++−=
(4.2.57)
Substituting the above information into CC equations (2.7.5) to (2.7.19) and after
simplifying we have ,0,,,
3
0
2
0
1
0
=== XXX ,0
1
=X ,0,,,,
3
2
2
2
1
2
0
2
==== XXXX
.0,,,,
3
3
2
3
1
3
0
3
==== XXXX From these equations one has ),(
0
tfX = ,0
1
=X
1
2
cX = and ,
2
3
cX = where
.,
21
Rcc

CCS in this case are

,,,0),(
2
3
1
210
cXcXXtfX ====
(4.2.58)
where )(tf is an arbitrary function. One can write the above equation (4.2.58) after
subtracting Killing vector fields as given in equation (4.2.43).

Cases (C44-C53)

In all these cases the rank of
66×
Riemann matrix is three and there exists a unique (up
to multiple) no where zero vector field which is the solution of equation (2.6.1) and is not
covariantly constant. CCS from cases C44 to C53 can be determined by the use of CC
equations from (2.7.5) to (2.7.19).
Case C54
In this case one has 0,0,0
7654321

=
=
=
=
==
α
α
α
α
α
α
α
and .0
8

α
Here, the rank
of the
66×
Riemann matrix is two and there exists a unique (up to multiple) no where

76
zero timelike vector field
aa
tt
,
= which is the solution of equation (2.6.1) and is not
covariantly constant. From the above constraints, we have
,0),( =rtU
t

,0),(
=
rtU
r

,0),( ≠rtV
t
,0),( =rtV
r

,0),(2),(
2
=+
rtVrtV
ttt
,0),(

rtW
t

,0),(

rtW
r

,0),(2),(),( =+ rtWrtWrtW
trrt

0),(2),(
2
=+
rtWrtW
ttt
and .0),(2),(
2
≠+ rtWrtW
rrr

From the above information, we get
,
mU
=

2
)ln()( battV += and
,))(ln(),(
2
rdctrtW
β
++= where
)0,(,,,,


caRmdcba
and
)(
r
β
no where zero
function of integration. After rescaling of t the line element takes the form


.))(()(
2222222
φβθ
drdctdbatdrdtds ++++++−=
(4.2.59)
Substituting the above information into CC equations (2.7.5) to (2.7.19) and after
simplifying them, one has ,0,,,
3
0
2
0
1
0
=== XXX ,0
1
=X
,0,,,,
3
2
2
2
1
2
0
2
==== XXXX ,0,,,,
3
3
2
3
1
3
0
3
==== XXXX which imply ),(
0
tfX =
,0
1
=X
1
2
cX = and ,
2
3
cX = where
.,
21
Rcc

CCS in this case after subtracting
Killing vector fields are given in equation (4.2.43).
Cases (C55-C64)

In the following cases rank of the
66
×
⁒楥=慮渠aa瑲楸⁩猠瑷i⁡湤⁴桥r攠數楳瑳⁡⁵=楱略=
⡵瀠瑯×汴楰汥⤠湯⁷桥牥⁺敲漠癥捴lr⁦楥汤⁷桩= 栠楳⁴桥⁳潬畴楯渠潦⁥煵慴楯渠⠲⸶⸱⤠慮搠楳=
湯琠捯癡ni慮瑬礠捯湳瑡湴t⁏湥⁣慮⁥=獩汹⁦i湤⁴ne⁃䍓⁢礠畳楮朠瑨攠楮=orm慴楯渠杩癥渠批a
瑨敳攠捡獥猠楮⁃䌠敱畡瑩潮猠晲潭
㈮㜮㔩⁴漠⠲‷⸱㤩⸠=
Case D1
In this case 0,0
78654321

=
=
=
=
===
α
α
α
α
α
α
α
α
and the rank of the
66
×
=
剩敭慮渠aa瑲楸⁩猠潮攠en搠睥⁨慶攠瑨攠捯湤楴楯湳n,0),(
=
rtU
t

,0),( =
rtU
r
,0),(
=
rtV
t

,0),( =
rtV
r
0),( =rtW
t

0),(

rtW
r
and .0),(2),(
2
≠+ rtWrtW
rrr
From the above
constraints, we have
,
1
kU
=

2
kV
=
and ),(rWW
=
where
.,
21
Rkk

Here, there exist
two linearly independent solutions
a
a
tt
,
=
⁡湤=
a
a,
θ
θ
=
映敱畡瑩潮
㈮㘮ㄩ⁡湤⁳慴楳晹楮==
0
;
=
ba
t and .0
;
=
ba
θ
The line element after rescaling of t and
θ
⁣慮⁢攠睲楴瑥渠=s=


.
2)(2222
φθ
deddrdtds
rW
+++−=
(4.2.60)

77
The above spacetime is clearly 1+1+2 decomposable and belongs to the curvature class
D. CCS in this case are [8] of the form

,),(),( XtK
t
tJX

+


+


=
θ
θθ

(4.2.61)
where ),(
θ
tJ and ),(
θ
tK are arbitrary functions of t and
θ
⁡湤=
X

⁩猠愠䍃渠敡捨映
瑷漠摩te湳楯湡氠獵戠na湩景汤猠潦⁣潮獴慮琠 t and
θ
⁷桩捨⁡牥⁧i癥渠楮⁴桥⁃a獥䈱⸠

Cases (D2-D8)
In cases D2 to D8 the rank of the
66
×
⁒楥= 慮渠aa瑲楸⁩猠tn攠慮搠瑨敲攠數楳琠瑷漠
汩湥慲汹⁩湤数敮摥湴⁳潬畴楯湳l ⁥煵慴楯渠⠲⸶⸱⤠慮搠扯瑨⁡牥⁣潶慲楡湴汹⁣潮獴慮琮⁉渠慬氠
瑨敳攠捡獥猠瑨攠獰慣整業攠⠴⸲⸱⤠扥捯)e猠ㄫ ㄫ㈠摥捯1灯獡扬攠慮搠扥汯湧⁴漠捵牶慴畲攠
捬慳猠䐮⁏湥⁣慮⁦楮搠䍃匠批⁡摯灴楮朠瑨攠獡ge⁰牯捥摵牥⁷桩捨⁩猠=i癥渠楮⁣慳攠䐱⸠v
Case D9
In this case one has 0
8765431
=
=
=
=
=
==
α
α
α
α
α
α
α
and
.0
2

α
⁈敲攬⁴桥⁲慮欠潦=
瑨攠
66×
Riemann matrix is one. We have the conditions ,0),( ≠rtU
t

,0),(
=
rtU
r

,0),( ≠rtV
t

,0),( ≠
rtV
r
,0),(2),(),(
=
+
rtVrtVrtV
trrt
,0),(2),(
2
=+ rtVrtV
rrr

0),( =rtW
t
and
.0),( =
rtW
r
From the above constraints we have ),(tUU =
2
))(ln( tbarV
σ
++= and
,
1
kW
=
where
)0(,,
1


aRkba
and )(t
σ
is no where zero
function of integration and there exist two linearly independent solutions
aa
rr,= and
aa
,
φ
φ
= of equation (2.6.1). Here, the vector field
a
r is not covariantly constant where as
a
φ
is covariantly constant. The line element after rescaling of
φ
⁣慮⁢攠睲楴瑥渠慳†
=
.⤩((
22222)(2
φθσ
ddtbardrdteds
tU
+++++−=
(4.2.62)
The above spacetime (4.2.62) is clearly 1+3 decomposable and belongs to the curvature
class D. Substitution of the above information in CC equations gives

,0,,
3
0
1
0
== XX ,0,,
2
1
0
1
== XX ,0,,
3
2
1
2
== XX ,0,,
2
3
0
3
== XX
(4.2.63)

,0,,
2
0)(
0
2),(
=− XeXe
tUrtV

(4.2.64)

,0,)2(2)2(
0
0212
=−++

−+ XVUVVXVUVV
&&&&&&&&&&

(4.2.65)

.0,)2(2))2((
2
2212
=−++

−+ XeVUVVXeVUVV
VV
&&&&&&&&&&

(4.2.66)

78
It follows by calculation that the above system of equations has the following solutions,
which are precisely the CCS.

),,(,,0
3
1
210
φ
rfXcXXX ====
(4.2.67)
where
),(
φ
rf
is an arbitrary function and
.
1
Rc

One can write the above equation
(4.2.67) after subtracting Killing vector fields as

)),,(,0,0,0(
φ
rfX =
(4.2.68)
CCS clearly form an infinite dimensional vector space.
Cases (D10-D18)

In the following cases rank of the
66
×
⁒楥=慮渠aa瑲楸⁩猠潮攠慮搠瑨敲攠數楳瑳⁴睯e
癥捴潲⁦iel≤s⁷桩捨⁡牥⁳潬畴楯= sf⁥煵慴楯 渠⠲⸶⸱⤠慮搠潮攠潦n 瑨敭⁩猠捯癡物慮瑬= =
捯湳瑡′t⸠䥮⁴桥獥⁣慳敳⁴桥⁳灡捥瑩浥
㐮㈮.⤠扥捯)e猠ㄫ㌠摥捯sp潳慢汥⁡湤⁢敬潮杳⁴漠
捵牶慴′r攠䐮⁗e⁣慮⁦楮搠䍃匠楮⁴=敳攠捡獥猠批⁵獩湧⁴桥⁃C⁥煵慴楯=s⁦牯=
㈮㜮㔩⁴===
⠲‷⸱㤩⁰牥捩獥汹⁩渠瑨攠獡 me⁷慹⁡猠瑨慴映捡獥⁄㤮=
Case D19
In this case 0,0
87654321

=
=
=
=
===
α
α
α
α
α
α
α
α
and the rank of the
66
×
=
剩敭慮渠浡瑲楸⁩猠潮攮⁁汳漬⁷攠桡癥⁴桥⁣潮摩瑩潮s=,0),( =rtU
t

,0),(
=
rtU
r

,0),( ≠rtV
t

,0),( ≠
rtV
r
,0),(
=
rtW
t
,0),(2),(),(
=
+ rtVrtVrtV
trrt

,0),(2),(
2
=+ rtVrtV
rrr

,0),(2),(
2
=+
rtVrtV
ttt

0),( ≠
rtW
r
and
.0),(2),(
2
=+ rtWrtW
rrr
From the above constraints we have
,
1
kU
=

2
)ln( cbratV ++= and ,)ln(
2
qprW += where
).0,,(,,,,,
1


pbaRkqpcba
In this
case there exist two linearly independent solutions
a
a
tt
,
=
⁡湤=
a
a
rr
,
= of equation
(2.6.1) and both vector fields
a
t and
a
r are not covariantly constant. The line element
after rescaling of t can be written as

.)()(
2222222
φθ
dqprdcbratdrdtds ++++++−=
(4.2.69)
The above spacetime (4.2.69) belongs to the curvature class D. Substituting the above
information in CC equations (2.7.5) to (2.7.19) then one can easily find

,0,,
3
0
2
0
== XX ,0,,
3
1
2
1
== XX ,0,,
1
2
0
2
== XX ,0,,
1
3
0
3
== XX
(4.2.70)

,0,)(,)(
2
32
3
22
=++++ XqprXcbrat
(4.2.71)

79

,0,))((2))((
2
21
=+++++− XcbratqprXcatpbq
(4.2.72)

.0,))((2))((
3
31
=++++++− XcbratqprXcatpbq
(4.2.73)
It follows by calculation that the above system of equations has the following solutions,
which are precisely the CCS.

,,,0),,(
2
3
1
210
cXcXXrtMX ====
(4.2.74)
where ),( rtM is an arbitrary function and
.,
21
Rcc

One can write the above equation
(4.2.74) after subtracting Killing vector fields as

.)0,0,0),,(( rtMX =
(4.2.75)
CCS clearly form an infinite dimensional vector space.
Cases (D20-D21)

In these cases rank of the
66×
Riemann matrix is one and there exists two vector fields
which are solutions of equation (2.6.1) and both are not covariantly constant. We can find
CCS in these cases by using the CC equations from (2.7.5) to (2 7.19) exactly on the
same lines as described in case D19.

4.3 Proper Curvature Collineations in Special Non-Static
Axially Symmetric Spacetimes
Consider a non-static axially symmetric spacetimes in the usual coordinate system
),,,(
φ
θ
rt (labeled by ),,,,(
3210
xxxx respectively) with line element [33]

)(
222),,(2),,(2
φθ
θθ
dddredteds
rtBrtA
+++−=
(4.3.1)
The above spacetime admits only one Killing vector field which is
.
φ


The non-zero
independent components of the Riemann tensor are [36]
,
)),,(),,(),,(2),,((
)),,(),,(),,(),,(),,(2),,((
4
1
1
2),,(
2),,(
0101
α
θθθθ
θθθθθθ
θ
θθ
θ









−+−
+−+
=
rtBrtArtBrtBe
rtBrtArtBrtArtArtAe
R
ttttt
rtB
rrrrr
rtA
,
),,(),,(),,(),,(
),,(2),,(),,(
4
1
2
),,(
0102
α
θθθθ
θθθ
θθ
θθ
θ







−−
+
=
rtBrtArtBrtA
rtArtArtA
eR
rr
rr
rtA


80
[ ]
,),,(),,(),,(2
4
1
3
),,(
03230112
αθθθ
θθ
θ
≡−=−= rtBrtArtBeRR
tt
rtB

,
)),,(),,(),,(2),,((
)),,(),,(),,(),,(),,(2),,((
4
1
4
2),,(
2),,(
0202
α
θθθθ
θθθθθθ
θ
θθθθθ
θ









−+−
+−+
=
rtBrtArtBrtBe
rtBrtArtBrtArtArtAe
R
ttttt
rtB
rr
rtA
[ ]
,),,(),,(),,(2
4
1
5
),,(
03130212
αθθθ
θ
≡−−== rtBrtArtBeRR
trtr
rtB

,
)),,(),,(),,(2),,((
)),,(),,(),,(),,((
4
1
6
2),,(
),,(
0303
α
θθθθ
θθθθ
θ
θθ
θ









−+−
+
=
rtBrtArtBrtBe
rtBrtArtBrtAe
R
ttttt
rtB
rr
rtA

[ ]
,),,()),,(),,((2
4
1
7
2
),,(),,(),,(),,(
1212
αθθθ
θθ
θθ
θθ
≡−+−=

rtBeertBrtBeR
t
rtBrtA
rr
rtArtB

[ ]
,),,()),,(),,((
4
1
8
2
),,(),,(
2
),,(),,(
1313
αθθθ
θθ
θ
θθ
≡−+−=

rtBeertBrtBeR
t
rtBrtA
rr
rtArtB

[ ]
,),,(),,(),,(2
4
1
9
),,(
1323
αθθθ
θθ
θ
≡−−= rtBrtBrtBeR
rr
rtB

[ ]
.),,()),,(2),,((
4
1
10
2
),,(),,(
2
),,(),,(
2323
αθθθ
θθ
θθ
θθ
≡−+−=

rtBeertBrtBeR
t
rtBrtA
r
rtArtB

Writing the curvature tensor with components
abcd
R at point
p
of the manifold as a
66×
symmetric matrix

.
000
000
000
000
000
000
1093
985
753
356
542
321






















=
ααα
ααα
ααα
ααα
ααα
ααα
abcd
R

(4.3.2)
It is noticeable that we will consider Riemann tensor components as
bcd
a
R
for calculating
CCS. We know from theorem [8, 6] that when the rank of the
66
×
⁒楥浡湮a瑲楸⁩猠
杲敡瑥爠瑨慮⁴桲敥⁴桥牥⁥硩獴=漠灲潰敲⁣畲= 慴ar攠捯汬楮敡瑩潮献⁈敲攬⁷攠慲攠楮瑥牥獴敤t
楮⁴桯獥⁣慳敳⁷桥牥⁴e攠牡湫映瑨攠
66
×
⁒楥=慮渠aa瑲楸⁩猠汥ts⁴桡渠潲⁥=畡氠瑯⁴桲敥e=
We⁲敭i湤⁴桡琠瑨nr攠ere,⁡汴潧=瑨敲 Ⱐ景牴礭潮攠灯獳楢楬,瑩敳⁦o爠瑨攠牡湫r=
66×
Riemann
matrix to be (
3≤
), that is, twenty for rank three, fifteen for two and six for one. Suppose
the rank of the
66×
Riemann matrix is three. Then there exist only three non zero rows
or columns in (4.3.2). If we set three rows or columns identically zero in (4.3.2) then
there exist twenty possibilities when the rank of the
66
×
⁒楥=慮渠aa瑲楸⁩猠瑨牥攮⁉n=

81
these twenty possibilities fifteen gives contrdiction and only five will survive. For
example, the case when the rank of the
66
×
⁒楥= 慮渠aa瑲楸⁩猠瑨te攬=
椮攮,0
9854321
=
=
=====
α
α
α
α
α
α
α
0,0
76


α
α
⁡湤=.0


α
⁔桥⁣潮獴牡楮t猠
0
9854321
=
==
=
===
α
α
α
α
α
α
α
imply that
0),,(
=
θ
rtB
rr
and
.0),,(
=
θ
θ
rtB

Substitution of this information in (4.3.2) we get 0
7
=
α
⁷桩捨⁧楶敳⁣潮瑲慤楣瑩潮=
扥捡畳攠睥⁡獳畭e搠瑨慴≤.0
7

α
⁓漠瑨楳⁩猠湯琠灯獳楢汥= ⁎潷⁣潮獩摥爠慮瑨敲=
灯獳楢楬楴p⁷桥渠瑨攠牡湫= ⁴桥⁡扯癥a瑲楸⁩i⁴桲敥= =
椮攮,0
9654321
=
=
=====
α
α
α
α
α
α
α
0,0
87


α
α
⁡湤=.0


α
⁔桥⁣潮獴牡楮瑳=
0
9654321
=
==
=
===
α
α
α
α
α
α
α
imply that
,0),,(

θ
rtA
t

,0),,(
=
θ
rtA
r

,0),,( =
θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,(

θ
rtB
r

,0),,(

θ
θ
rtB

0),,(2),,(),,(
=

θ
θ
θ
θθ
rtBrtBrtB
rr
and .0),,(),,(

+
θ
θ
θθ
rtBrtB
rr
This is the case
P1. Now suppose rank of the
66
×
⁒楥=慮渠浡瑲楸⁩猠潮攮⁔桥渠瑨敲攠楳湬礠潮攠湯渠
穥牯⁲潷爠捯汵≠n⁩渠=4⸳⸲⤮⁉映睥⁳整⁦楶攠牯睳爠捯汵浮猠楤敮瑩捡汬礠穥牯⁩渠⠴.3⸲⤠
瑨敮⁴桥牥⁥硩獴⁳楸⁰潳獩扩汩瑩敳⁷桥渠瑨攠牡湫映瑨攠
66
×
⁒楥=慮渠aa瑲楸⁩猠潮攮⁉t=
瑨敳攠獩砠灯獳楢楬楴楥猠瑷漠杩癥⁣潮瑲≤楣瑩潮i 慮搠潮汹⁦潵爠睩汬⁳畲癩l攮⁆潲⁥硡e灬攬⁴桥p
捡獥⁷桥渠瑨攠牡湫映瑨攠
66
×
⁒楥= 慮渠aa瑲楸⁩猠潮攬e
椮攮 0
㄰98754321
=
=
=
=
=====
α
α
α
α
α
α
α
α
α
and .0
6

α
⁔桥⁣潮s瑲慩湴t=
0
㄰98754321
=
=
=
==
=
===
α
α
α
α
α
α
α
α
α
imply that
0),,( =
θ
rtB
r
and
.0),,( =
θ
θ
rtB
Substituting this information in (4.3.2) we get 0
6
=
α
which gives
contradiction because we assumed that .0
6

α
⁔桥牥景牥Ⱐ瑨楳⁣慳攠楳潴⁰潳獩el攮⁎潷e
捯湳楤敲⁡n瑨敲⁰o獳楢楬楴礠睨敮⁴s 攠牡湫映瑨攠慢潶攠ea瑲楸⁩猠潮攬t
椮攮 0
㄰97654321
=
=
=
=
=====
α
α
α
α
α
α
α
α
α
and .0
8

α
⁔桥獥⁣潮= 瑲慩湴t=
杩癥Ⱐ
,0),,( ≠
θ
rtA
t

,0),,(
=
θ
rtA
r

,0),,(
=
θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,(
=
θ
rtB
r

0),,( ≠
θ
θ
rtB
and
.0),,( =
θ
θθ
rtB
This is the case Q3. By the similar analysis we
summarized that there are altogether thirty nine surviving possibilities when the rank of
the
66×
Riemann matrix is three or less which are:
(H1) Rank=3,
,0),,( ≠
θ
rtA
t

,0),,(
=
θ
rtA
r

,0),,(

θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( ≠
θ
rtB
r

,0),,( =
θ
θ
rtB

0),,(
=
θ
rtB
rr
and .0),,(2),,(
2
=+
θθ
θθθ
rtArtA

82
(H2) Rank=3,
,0),,( ≠
θ
rtA
t

,0),,(
=
θ
rtA
r

,0),,(

θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( ≠
θ
rtB
r

,0),,( =
θ
θ
rtB

0),,(
=
θ
rtB
rr
and .0),,(2),,(
2
≠+
θθ
θθθ
rtArtA
(H3) Rank=3,
,0),,( ≠
θ
rtA
t

,0),,(

θ
rtA
r

,0),,(
=
θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( =
θ
rtB
r

,0),,( ≠
θ
θ
rtB

0),,(
=
θ
θθ
rtB
and .0),,(2),,(
2
≠+
θθ
rtArtA
rrr

(H4) Rank=3,
,0),,( ≠
θ
rtA
t

,0),,(

θ
rtA
r

,0),,(
=
θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( =
θ
rtB
r

,0),,( ≠
θ
θ
rtB

0),,(
=
θ
θθ
rtB
and .0),,(2),,(
2
≠+
θθ
rtArtA
rrr

(H5) Rank=3,
,0),,( =
θ
rtA
t

,0),,(
=
θ
rtA
r

,0),,(

θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( ≠
θ
rtB
r

,0),,( =
θ
θ
rtB

0),,(
=
θ
rtB
rr
and
.0),,(2),(
2
=+ θθ
θθθ
rtArtA

(H6) Rank=3,
,0),,( =
θ
rtA
t

,0),,(
=
θ
rtA
r

,0),,(

θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( ≠
θ
rtB
r

,0),,( =
θ
θ
rtB

0),,(
=
θ
rtB
rr
and
.0),(2),(
2
≠+ rtArtA
θθθ

(H7) Rank=3,
,0),,( =
θ
rtA
t

,0),,(

θ
rtA
r

,0),,(
=
θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( =
θ
rtB
r

,0),,( ≠
θ
θ
rtB

0),,(
=
θ
θθ
rtB
and .0),(2),(
2
=+ rtArtA
rrr

(H8) Rank=3,
,0),,( =
θ
rtA
t

,0),,(

θ
rtA
r

,0),,(
=
θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( =
θ
rtB
r

,0),,( ≠
θ
θ
rtB

0),,(
=
θ
θθ
rtB
and .0),(2),(
2
≠+ rtArtA
rrr


(P1)

Rank=3,
,0),,( ≠
θ
rtA
t

,0),,(
=
θ
rtA
r

,0),,(
=
θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( ≠
θ
rtB
r

,0),,( ≠
θ
θ
rtB
0),,(2),,(),,( =

θ
θ
θ
θθ
rtBrtBrtB
rr
and
.0),,(),,( ≠+
θ
θ
θθ
rtBrtB
rr

(P2) Rank=3,
,0),,( =
θ
rtA
t

,0),,(
=
θ
rtA
r

,0),,(
=
θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( ≠
θ
rtB
r

,0),,( ≠
θ
θ
rtB
0),,(2),,(),,( ≠

θ
θ
θ
θθ
rtBrtBrtB
rr
and
.0),,(),,( ≠+
θ
θ
θθ
rtBrtB
rr

(P3) Rank=3,
,0),,( =
θ
rtA
t

,0),,(
=
θ
rtA
r

,0),,(
=
θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( =
θ
rtB
r

0),,( ≠
θ
θ
rtB
and .0),,(

θ
θθ
rtB
(P4) Rank=3,
,0),,( =
θ
rtA
t

,0),,(
=
θ
rtA
r

,0),,(
=
θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( =
θ
θ
rtB

0),,( ≠
θ
rtB
r
and
.0),,(

θ
rtB
rr


83
(P5) Rank=2,
,0),,( ≠
θ
rtA
t

,0),,(

θ
rtA
r

,0),,(

θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( =
θ
rtB
r

,0),,(
=
θ
θ
rtB
,0),,(2),,(
2
≠+
θθ
rtArtA
rrr

0),,(2),,(
2
≠+ θθ
θθθ
rtArtA
and .0),,(2),,(),,(

+
θ
θ
θ
θθ
rtArtArtA
rr

(P6) Rank=2,
,0),,( =
θ
rtA
t

,0),,(

θ
rtA
r

,0),,(

θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( =
θ
rtB
r

,0),,(
=
θ
θ
rtB
,0),,(2),,(
2
≠+
θθ
rtArtA
rrr

0),,(2),,(
2
≠+ θθ
θθθ
rtArtA
and .0),,(2),,(),,(

+
θ
θ
θ
θθ
rtArtArtA
rr

(P7) Rank=2,
,0),,( =
θ
rtA
t

,0),,(

θ
rtA
r

,0),,(

θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( =
θ
rtB
r

0),,( =
θ
θ
rtB
and .0),,(2),,(),,(
=
+
θ
θ
θ
θθ
rtArtArtA
rr

(P8) Rank=2,
,0),,( ≠
θ
rtA
t

,0),,(

θ
rtA
r

,0),,(

θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( =
θ
rtB
r

0),,( =
θ
θ
rtB
and .0),,(2),,(),,(
=
+
θ
θ
θ
θθ
rtArtArtA
rr

(P9) Rank=2,
,0),,( =
θ
rtA
t

,0),,(

θ
rtA
r

,0),,(

θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( =
θ
rtB
r

,0),,(
=
θ
θ
rtB
,0),,(2),,(
2
=+
θθ
rtArtA
rrr

0),,(2),,(
2
≠+ θθ
θθθ
rtArtA
and .0),,(2),,(),,(

+
θ
θ
θ
θθ
rtArtArtA
rr

(P10) Rank=2,
,0),,( ≠
θ
rtA
t

,0),,(

θ
rtA
r

,0),,(

θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( =
θ
rtB
r

,0),,(
=
θ
θ
rtB
,0),,(2),,(
2
=+
θθ
rtArtA
rrr

0),,(2),,(
2
≠+ θθ
θθθ
rtArtA
and .0),,(2),,(),,(

+
θ
θ
θ
θθ
rtArtArtA
rr

(P11) Rank=2,
,0),,( =
θ
rtA
t

,0),,(

θ
rtA
r

,0),,(

θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( =
θ
rtB
r

,0),,(
=
θ
θ
rtB
,0),,(2),,(
2
≠+
θθ
rtArtA
rrr

0),,(2),,(
2
=+ θθ
θθθ
rtArtA
and .0),,(2),,(),,(

+
θ
θ
θ
θθ
rtArtArtA
rr

(P12) Rank=2,
,0),,( ≠
θ
rtA
t

,0),,(

θ
rtA
r

,0),,(
=
θ
θ
rtB

,0),,(
=
θ
rtB
t

,0),,( =
θ
rtB
r
,0),,(2),,(
2
≠+
θθ
rtArtA
rrr

,0),,(2),,(
2
=+ θθ
θθθ
rtArtA

0),,( ≠
θ
θ
rtA
and .0),,(2),,(),,(

+
θ
θ
θ
θθ
rtArtArtA
rr

(P13) Rank=2,
,0),,( =
θ
rtA
t

,0),,(

θ
rtA
r

,0),,(
=
θ
θ
rtB

,0),,(
=
θ
rtB
t

,0),,( =
θ
rtB
r
,0),,(2),,(
2
=+
θθ
rtArtA
rrr

,0),,(2),,(
2
=+ θθ
θθθ
rtArtA

0),,( ≠
θ
θ
rtA
and .0),,(2),,(),,(

+
θ
θ
θ
θθ
rtArtArtA
rr


84
(P14) Rank=2,
,0),,( ≠
θ
rtA
t

,0),,(

θ
rtA
r

,0),,(
=
θ
θ
rtB

,0),,(
=
θ
rtB
t

,0),,( =
θ
rtB
r
,0),,(2),,(
2
=+
θθ
rtArtA
rrr

,0),,(2),,(
2
=+ θθ
θθθ
rtArtA

0),,( ≠
θ
θ
rtA
and .0),,(2),,(),,(

+
θ
θ
θ
θθ
rtArtArtA
rr

(P15) Rank=2,
,0),,( =
θ
rtA
t

,0),,(
=
θ
rtA
r

,0),,(
=
θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( ≠
θ
rtB
r

,0),,( ≠
θ
θ
rtB

,0),,(
=
θ
rtB
rr

0),,( =
θ
θθ
rtB
and
.0),,(2),,(),,(


θ
θ
θ
θθ
rtBrtBrtB
rr


(P16) Rank=3,
,0),,( ≠
θ
rtA
t

,0),,(
=
θ
rtA
r

,0),,(
=
θ
θ
rtA

,0),,(

θ
rtB
t

,0),,(),,(),,(2),,(
2
=−+
θθθθ
rtBrtArtBrtB
ttttt

0),,(
=
θ
θ
rtB
and
.0),,( =
θ
rtB
r

(P17) Rank=3,
,0),,( =
θ
rtA
t

,0),,(
=
θ
rtA
r

,0),,(
=
θ
θ
rtA

,0),,(

θ
rtB
t

,0),,(2),,(
2
=+
θθ
rtBrtB
ttt

0),,(
=
θ
θ
rtB
and
.0),,(
=
θ
rtB
r

(P18) Rank=3,
,0),,( =
θ
rtA
t

,0),,(

θ
rtA
r

,0),,(
=
θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( ≠
θ
rtB
r
,0),,(),,(),,(2),,(
2
=−+
θθθθ
rtBrtArtArtA
rrrrr

0),,( =
θ
θ
rtB
and
.0),,( =
θ
rtB
rr

(P19) Rank=3,
,0),,( ≠
θ
rtA
t

,0),,(

θ
rtA
r

,0),,(
=
θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( ≠
θ
rtB
r

,0),,( =
θ
rtB
rr

,0),,(
=
θ
θ
rtB
0),,(2),,(
2
≠+
θθ
rtArtA
rrr
and
.0),,(),,(),,(2),,(
2
=−+
θθθθ
rtBrtArtArtA
rrrrr

(P20) Rank=3,
,0),,( ≠
θ
rtA
t

,0),,(
=
θ
rtA
r

,0),,(

θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( =
θ
rtB
r

,0),,( ≠
θ
θ
rtB

,0),,(
=
θ
θθ
rtB
0),,(2),,(
2
≠+
θθ
θθθ
rtArtA and
.0),,(),,(),,(2),,(
2
=−+
θθθθ
θθθθθ
rtBrtArtArtA
(P21) Rank=3,
,0),,( =
θ
rtA
t

,0),,(
=
θ
rtA
r

,0),,(

θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( =
θ
rtB
r

,0),,( ≠
θ
θ
rtB

,0),,(
=
θ
θθ
rtB

0),,(2),,(
2
≠+
θθ
θθθ
rtArtA
and
.0),,(),,(),,(2),,(
2
=−+
θθθθ
θθθθθ
rtBrtArtArtA

(Q1) Rank=1,
,0),,( ≠
θ
rtA
t

,0),,(

θ
rtA
r

,0),,(
=
θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( =
θ
rtB
r

0),,( =
θ
θ
rtB
and .0),,(2),,(
2
≠+
θθ
rtArtA
rrr


85
(Q2) Rank=1,
,0),,( =
θ
rtA
t

,0),,(

θ
rtA
r

,0),,(
=
θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( =
θ
rtB
r

0),,( =
θ
θ
rtB
and .0),,(2),,(
2
≠+
θθ
rtArtA
rrr

(Q3) Rank=1,
,0),,( =
θ
rtA
t

,0),,(
=
θ
rtA
r

,0),,(

θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( =
θ
rtB
r

0),,( =
θ
θ
rtB
and
.0),,(2),,(
2
≠+ θθ
θθθ
rtArtA

(Q4) Rank=1,
,0),,( ≠
θ
rtA
t

,0),,(
=
θ
rtA
r

,0),,(

θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( =
θ
rtB
r

0),,( =
θ
θ
rtB
and
.0),,(2),,(
2
≠+ θθ
θθθ
rtArtA


(Q5) Rank=1,
,0),,( ≠
θ
rtA
t

,0),,(
=
θ
rtA
r

,0),,(
=
θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( =
θ
rtB
r

0),,( ≠
θ
θ
rtB
and
.0),,(
=
θ
θθ
rtB

(Q6) Rank=1,
,0),,( =
θ
rtA
t

,0),,(
=
θ
rtA
r

,0),,(
=
θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,( ≠
θ
rtB
r

0),,( =
θ
θ
rtB
and
.0),,(
=
θ
rtB
rr

(Q7) Rank=1,
,0),,( ≠
θ
rtA
t

,0),,(
=
θ
rtB
t

0),,(
=
θ
θ
rtB

,0),,(
=
θ
rtB
r

,0),,(2),,(
2
≠+
θθ
rtArtA
rrr

,0),,(2),,(
2
=+ θθ
θθθ
rtArtA

,0),,(

θ
θ
rtA

0),,( ≠
θ
rtA
r
and .0),,(2),,(),,(
=
+
θ
θ
θ
θθ
rtArtArtA
rr

(Q8) Rank=1,
,0),,( =
θ
rtA
t

,0),,(
=
θ
rtB
t

0),,(
=
θ
θ
rtB

,0),,(
=
θ
rtB
r

,0),,(2),,(
2
=+
θθ
rtArtA
rrr

,0),,(2),,(
2
≠+ θθ
θθθ
rtArtA

,0),,(

θ
θ
rtA

0),,( ≠
θ
rtA
r
and .0),,(2),,(),,(
=
+
θ
θ
θ
θθ
rtArtArtA
rr

(Q9) Rank=1,
,0),,( ≠
θ
rtA
t

,0),,(
=
θ
rtB
t

0),,(
=
θ
θ
rtB

,0),,(
=
θ
rtB
r

,0),,(2),,(
2
=+
θθ
rtArtA
rrr

,0),,(2),,(
2
≠+ θθ
θθθ
rtArtA

,0),,(

θ
θ
rtA

0),,( ≠
θ
rtA
r
and .0),,(2),,(),,(
=
+
θ
θ
θ
θθ
rtArtArtA
rr

(Q10) Rank=1,
,0),,( =
θ
rtA
t

,0),,(
=
θ
rtB
t

0),,(
=
θ
θ
rtB

,0),,(
=
θ
rtB
r

,0),,(2),,(
2
≠+
θθ
rtArtA
rrr

,0),,(2),,(
2
=+ θθ
θθθ
rtArtA

,0),,(

θ
θ
rtA

0),,( ≠
θ
rtA
r
and .0),,(2),,(),,(
=
+
θ
θ
θ
θθ
rtArtArtA
rr

We will discuss each case separately.



86
Case H1
In this case
,0),,( ≠
θ
rtA
t

,0),,(
=
θ
rtA
r

,0),,(

θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,(

θ
rtB
r

,0),,( =
θ
θ
rtB

0),,( =
θ
rtB
rr
and ,0),,(2),(
2
=+
θθ
θθθ
rtArtA which imply that
2
21
))()(ln(),( tDtDtA +=
θθ
and ,)( barrB
+
=
where
)(
1
tD
is no where zero function
of integration and
)(
2
tD
is function of integration and ).0(,


aRba Here, the rank of
the
66×
Riemann matrix is three and there exists no non trivial solution of equation
(2.6.1). According to the above information, the line element (4.3.1) can be written in the
form

).())()((
222)(22
21
2
φθθ
dddredttDtDds
bra
++++−=
+

(4.3.3)
This case belongs to the class A. In this class the rank of the
66
×
⁒楥=慮渠aa瑲楸⁩猠㘬i
㔬‴Ⱐ㌠潲′
數捬畤楮朠瑨攠捬慳猠䈩⁡湤⁴5 敲攠ex楳瑳漠湯i⁴物= 楡氠獯汵瑩潮映敱畡瑩潮i
⠲⸶⸱⤠嬸崮⁃潮獥煵敮瑬礠瑨攠䍃匠慲攠瑨攠桯(o 瑨整楣⁶散t潲⁦i敬摳⁛8Ⱐ㙝⸠䡥湣攠楮⁴桩.=
捡獥漠灲潰敲⁃䍓⁥硩獴⸠e
Cases (H2-H8)

In all cases from H2 to H8 the rank of
66
×
⁒楥=慮渠浡瑲楸⁩猠瑨牥攠慮搠扥汯湧⁴漠
捵牶慴′r攠捬慳猠䄮⁃䍓⁩渠瑨敳攠捡獥猠慲攠瑨 攠桯eo瑨整楣⁶散瑯爠晩 敬摳⸠周畳漠灲潰敲e
䍃匠數楳琠楮⁴桥獥⁣a獥献†
Case P1
In this case
,0),,( ≠
θ
rtA
t

,0),,(
=
θ
rtA
r

,0),,(
=
θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,(

θ
rtB
r

,0),,( ≠
θ
θ
rtB
0),,(2),,(),,(
=

θ
θ
θ
θθ
rtBrtBrtB
rr
and .0),,(),,(

+
θ
θ
θθ
rtBrtB
rr

Here, the rank of the
66×
Riemann matrix is three and there exists a unique (up to
scaling) no where zero vector field
aa
tt
,
=
⁳畣栠瑨慴=
.0
;
=
ba
t
From the Ricci identity
.0=
d
abcd
tR The above constraints give
2
))()(ln(),(

+=
θθ
qrprB and ),(tAA = where
)(rp and )(
θ
q are no where zero functions of integration. After a suitable rescaling of
,t

the line element can written in the form

).())()((
222222
φθθ
dddrqrpdtds ++++−=


(4.3.4)
The above spacetime is clearly 1+3 decomposable and belongs the curvature class C.
CCS in this case are [8] of the form

87

,)( X
t
tNX

+


=
(4.3.5)
where )(tN is an arbitrary function of t and
X

⁩猠愠桯=o瑨整楣⁶散瑯爠tiel≤⁩渠瑨攠
楮摵捥搠ieom整特渠敡捨映瑨攠瑨牥攭ei me湳楯湡氠獵扭an楦潬摳映捯湳瑡湴i.t The
induced metric
αβ
g (where 3,2,1,
=
β
α
⤠睩瑨潮⁺敲漠捯)灯湥湴猠楳⁧楶敮†
=
2
㌳㈲ㄱ
⤩()((

+===
θ
qrpggg
(4.3.6)
A vector field
X

is called homothetic vector field if it satisfies

,2
αβαβ
gcgL
X
=

where
.Rc


(4.3.7)
One can expand the above equation (4.3.6) using (4.3.7) to get

)),()((,))()(()()(
1
121
θθθ
θ
qrpcXqrpXqXrp
r
+=+++
(4.3.8)

,0,,
1
2
2
1
=+ XX
(4.3.9)

)),()((,))()(()()(
2
221
θθθ
θ
qrpcXqrpXqXrp
r
+=+++
(4.3.10)

,0,,
1
3
3
1
=+ XX
(4.3.11)

,0,,
2
3
3
2
=+ XX
(4.3.12)

)),()((,))()(()()(
3
321
θθθ
θ
qrpcXqrpXqXrp
r
+=+++
(4.3.13)
Differentiating equations (4.3.11) and (4.3.12) with respect to
θ
⁡湤=
φ
⁲敳=散瑩e敬礬⁡e搠
瑨敮⁳畢t牡r瑩湧⁴桥tⰠ睥⁧整,.0,,

2

1
=− XX Differentiation of equation (4.3.9) with
respect to
φ
⁧楶敳=.0,,

2

1
=+ XX From these two equations, one has ,0,
23
1
=X
0,
13
2
=X and which imply ),,(),(
211
φθ
rArAX +=

),,(),(
432
φθθ
ArAX += where
),,(
1
θ
rA ),,(
2
φ
rA ),(
3
θ
rA and ),(
4
φθ
A are the functions of integration. Using the
above information in equation (4.3.11), we have ),,(),(
523
φθφ
φ
AdrrAX +−=

where
),(
5
φθ
A is function of integration. Combining the values of ,
1
X
2
X
and ,
3
X we have

),,(),(
),,(),(),,(),(
523
432211
φθφ
φθθφθ
φ
AdrrAX
ArAXrArAX
+−=
+=+=


(4.3.14)
In order to find the solution of the above system of equations we need to find ),,(
1
θ
rA
),,(
2
φ
rA ),,(
3
θ
rA ),(
4
φθ
A and ).,(
5
φθ
A If one proceeds further and after some
lengthy calculation the solution of the above equations from (4.3.8) to (4.3.13) is

88

,
1
,
1
,
1
3
3
2
2
1
1
cc
k
Xcc
k
Xcc
k
r
X +
+
=+
+
=+
+
=
φ
θ

(4.3.15)
where
,
1
1
)(
1
k
cc
k
r
ck
k
rp






+
+
+
=

,
1
1
)(
2
k
cc
kck
k
q






+
+
+
=
θ
θ
Rkccc ∈,,,
321
and
.1,0 −≠k The subcase when
0=k
or
1

=
k
will be discussed later. In this case induced
geometry on each of the three dimensional submanifolds of constant t admits proper
homothetic vector fields. CCS in this case by the use of equation (4.3.15) in equation
(4.3.5) are given by


.
1
,
1
,
1
),(
3
3
2
2
1
10
cc
k
Xcc
k
Xcc
k
r
XtNX +
+
=+
+
=+
+
==
φ
θ

(4.3.16)
One can write the above equation (4.3.16) after subtracting homothetic vector fields

).0,0,0),((
tNX
=

(4.3.17)
CCS clearly form an infinite dimensional vector space.
Consider the subcase when
.1

=
k
In this case homothetic vector field is a
killing vector field which are given below

,,0
4
321
cXXX ===
(4.3.18)
where
.
4
Rc ∈
Proper CCS in this case are given in (4.3.17).
Considering the sub case when
.0
=
k
In this case induced geometry on each of
the three dimensional submanifolds of constant t admit proper homothetic vector fields
which are

,,,
3
3
2
2
1
1
ccXccXccrX +=+=+=
φθ

(4.3.19)
where
( )
,ln)(
1
1
c
ccrrp +=
( )
,ln)(
1
2
c
ccq +=
θθ
and .,,
321
Rccc

CCS in this are given
by use of equation (4.3.19) in (4.3.5) as

.,,),(
3
3
2
2
1
10
ccXccXccrXtNX +=+=+==
φθ

(4.3.20)
Proper CCS in this case are given in (4.3.17).
Cases (P2-P15)

In the cases P2 to P4 the rank of the
66
×
⁒楥=慮渠aa瑲楸⁩猠瑨=敥⁷桩汥⁩e⁣慳e猠sr潭⁐㔠
瑯⁐ㄵ⁴桥⁲a湫n⁴桥=
66×
Riemann matrix is two. All these cases belong to curvature
class C and under the given constraints spacetime (4.3.1) becomes 1+3 decomposable.
CCS for such cases can be found precisely in the similar way as described in case.P1.

89
Case P16
In this case
,0),,( ≠
θ
rtA
t

,0),,(
=
θ
rtA
r

,0),,(
=
θ
θ
rtA

,0),,(

θ
rtB
t

,0),,(),,(),,(2),,(
2
=−+
θθθθ
rtBrtArtBrtB
ttttt

0),,(
=
θ
θ
rtB
and
,0),,( =
θ
rtB
r
the
rank of the
66×
Riemann matrix is three and there exists a unique (up to multiple)
aa
tt
,
= solution of equation (2.6.1) but
a
t is not covariantly constant. From the above
constraints, we have
)(
tAA
=
and
,)ln()(
2
2
)(
bdteatB
tA
+=

where
).0(,


aRba

Now the line element after a suitable rescaling of t can be written as

).()(
222222
φθ
dddrbatdtds ++++−=
(4.3.21)
The above spacetime (4.3.21) become special class of FRW K=0 model. Substituting the
above informations into CC equations from (2.7.5) to (2.7.19) and after simplification,
we see that
,0,,,
3
0
2
0
1
0
=== XXX

,0,,
1
1
0
1
== XX

,0,,
2
2
0
2
== XX

,0,,
3
3
0
3
== XX

,0,,
1
2
2
1
=+ XX ,0,,
1
3
3
1
=+ XX .0,,
2
3
3
2
=+ XX It follows from [26] that the CCS are

,
,,),(
642
3
321
1
321
10
ccrcX
cccXcccXtNX
+−=
+−−=+−−==
θ
φθφθ

(4.3.22)
where Rcccccc ∈
654321
,,,,, and )(tN is an arbitrary function. After subtracting Killing
vector fields (4.3.22) can be written in the form given by equation (4.3.17).
Cases (P17-P21)

In cases P17 to P21 the rank of the
66
×
⁒楥=慮渠aa瑲楸⁩猠瑨牥攠慮搠瑨敳攠捡獥r⁡汳==
扥汯湧⁴漠捵牶慴畲攠捬慳猠䌮⁕獩湧⁴桥⁩b 景牭慴楯渠灲潶楤敤⁢礠瑨敳攠捡獥猠楮⁃䌠
敱畡瑩潮s⁦牯=
㈮㜮㔩⁴漠⠲⸷⸱㤩n攠捡渠敡e楬礠晩湤⁣畲癡瑵牥⁣潬汩湥慴楯湳⸠

Case Q1
Here, we have
,0),,( ≠
θ
rtA
t

,0),,(

θ
rtA
r

,0),,(
=
θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,(
=
θ
rtB
r

0),,( =
θ
θ
rtB
and ,0),,(2),,(
2
≠+
θθ
rtArtA
rrr
the rank of the
66
×
⁒楥=慮渠aa瑲楸⁩t=
潮攮⁈敲攬⁴桥牥⁥硩= 琠瑷漠汩湥a牬r⁩湤数敮摥湴⁳潬n瑩潮t=
aa,
θ
θ
=
⁡湤=
aa,
φ
φ
=
of
equation (2.6.1) and satisfying 0
;
=
ba
θ
and .0
;
=
ba
φ
From the above constraints, we get
),( rtAA =
and ,mB = where
.Rm

The line element after rescaling
θ
⁡湤=,
φ
⁣慮⁢攠
睲楴瑥渠慳†

90

.
2222),(2
φθ
dddredteds
mrtA
+++−=
(4.3.23)
The above spacetime (4.3.23) is clearly 1+1+2 decomposable and belongs to the
curvature class D. CCS in this case are [8]

,),(),( YJIX +


+


=
φ
φθ
θ
φθ

(4.3.24)
where ),(
φ
θ
J and ),(
φ
θ
I are arbitrary functions of
θ
⁡湤=
φ
⁡湤=
Y
is a CC on each of
two dimensional submanifolds of constant
θ
⁡湤=.
φ
⁔桥數琠獴数⁩猠瑯⁷潲欠=×琠瑨攠䍃匠
楮⁴桥⁩湤畣敤⁧敯i整特映瑨 攠獵扭慮楦潬摳映捯湳瑡湴a
θ
⁡湤=.
φ
⁔桥e瑨潤⁦潲=
晩湤楮朠䍃匠楮′ⵤ業敮獩潮慬⁳畢f慮楦潬摳 ⁩猠杩癥渠楮⁳散瑩潮′⸹⸠周攠湯渠穥牯=
捯′p潮敮瑳映瑨攠楮摵捥搠oe瑲楣渠敡捨t 潦⁴桥⁴睯⁤業e湳楯湡氠獵扭慮楦潬摳映
捯湳瑡湴′
θ
⁡湤=
φ
⁡牥⁧楶敮⁢礠=
=
),(

rtA
eg −= .
11
m
eg =
(4.3.25)
Calculation shows that the nonzero components of the Ricci tensor turn out to be

,)2(
4
1
),(2
00
mrtA
eAAR

′′
+

=
)2(
4
1
2
11
AAR
′′
+

−=

(4.3.26)
and the Ricci scalar is given by
.)2(
2
1
2 m
eAAR

′′
+

−= Accordingly, the tensor
1,0,;
2
=≡
βα
αβαβ
g
R
G
has nonzero components

,)2(
4
1
),(2
00
mrtA
eAAG

′′
+

=
)2(
4
1
2
11
AAG
′′
+

−=
(4.3.27)
It follows from [8] CCS in the two dimensional submanifolds of constant
θ
⁡湤=
φ
⁡牥=
瑨攠獯汵瑩潮= ⁴桥⁥煵慴楯渠.0
=
αβ
GL
Y

Expanding the previous equation and using
equation (4.3.27), one has

,0)2(2
))2(())2((
0
0,
),(2
1),(20),(2
=
′′
+

+
′′′
+

+
′′
+

YeAA
YeAAYeAA
rtA
rtArtA
&

(4.3.28)

,0
0
1,
),(1
0,
=− YeYe
rtAm

(4.3.29)

.0)2(2)2()2(
1
1,
21202
=
′′
+

+
′′′
+

+
′′
+

YAAYAAYAA
&

(4.3.30)
After some calculation, one finds that CCS in this induced geometry of 2-dimensional
submanifolds are Killing vector fields which are

91

,0
10
==
YY

(4.3.31)
Proper CCS in this case can be written as

)),,(),,(,0,0(
φ
θ
φ
θ
JIX
=
(4.3.32)
Clearly CCS in this case form an infinite dimensional vector space.
Case Q2

Here, we have
,0),,( =
θ
rtA
t

,0),,(

θ
rtA
r

,0),,(
=
θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,(
=
θ
rtB
r

0),,( =
θ
θ
rtB
and ,0),,(2),,(
2
≠+
θθ
rtArtA
rrr
the rank of the
66
×
⁒楥=慮渠aa瑲楸⁩t=
潮攠慮搠瑨敲攠數楳琠瑷漠汩o 敡牬礠楮摥灥湤敮琠獯汵瑩潮猠
aa,
θ
θ
=
⁡湤=
aa,
φ
φ
= of equation
(2.6.1) and satisfying 0
;
=
ba
θ
and .0
;
=
ba
φ
From the above constraints, one has
)(rAA =
and ,
mB
= where
.Rm

The line element after rescaling
θ
⁡湤=,
φ
⁣慮⁢攠
睲楴瑥渠慳†
=
.
2222)(2
φθ
dddredteds
mrA
+++−=
(4.3.33)
The above spacetime (4.3.33) is clearly 1+1+2 decomposable and belongs to the
curvature class D. CCS in this case are of the form

,),(),(
XNMX

+


+


=
φ
φθ
θ
φθ

(4.3.34)
where
),(
φ
θ
M
and
),(
φ
θ
N
are arbitrary functions of
θ
⁡湤=
φ
⁡湤=
X

is a CC on each
of two dimensional submanifolds of constant
θ
⁡湤=
.
φ
⁔桥數琠獴数⁩猠瑯⁷潲欠潵琠瑨攠
䍃匠楮⁴桥⁩湤畣敤⁧敯C整特映瑨攠獵扭慮楦潬摳映捯湳瑡湴≤
θ
⁡湤=
.
φ
⁔桥e瑨潤⁦潲t
晩湤楮朠䍃匠楮′-摩≤e湳楯湡氠獵nm慮楦潬摳⁩a ⁧楶敮⁩= ⁳散瑩潮′⸹= ⁉映潮攠灲= 捥敤猠
晵牴桥爠瑨攠湯渠穥牯⁣潭灯湥湴猠 潦⁴桥⁩湤畣敤e瑲楣渠t 慣栠潦⁴桥⁴睯⁤業e湳楯湡n=
獵扭慮楦潬摳映捯湳瑡湴a
θ
⁡湤=
φ
⁡牥⁧楶敮⁢礠=
=
)(

rA
eg
−= .
11
m
eg
=
(4.3.35)
Calculation shows that the nonzero components of the Ricci tensor turn out to be

,)2(
4
1
)(2
00
mrA
eAAR

′′
+

=
)2(
4
1
2
11
AAR
′′
+

−=
(4.3.36)
and the Ricci scalar is given by
.)2(
2
1
2 m
eAAR

′′
+

−= Accordingly, the tensor
3,1,;
2
=≡
βα
αβαβ
g
R
G
has nonzero components

92

,)2(
4
1
)(2
00
mrA
eAAG

′′
+

=
)2(
4
1
2
11
AAG
′′
+

−=
(4.3.37)
It follows from [8] CCS in the two dimensional submanifolds of constant
θ
⁡湤=
φ
⁡牥=
瑨攠獯汵瑩潮= ⁴桥⁥煵慴楯渠
.0
=
αβ
GL
X

Expanding the previous equation and using
equation (4.3.37), we get

,0))2((2))2((
0
0,
)(21)(2
=
′′
+

+
′′′
+

XeAAXeAA
rArA

(4.3.38)

,0
3
1,
)(1
0,
=− XeXe
rAm

(4.3.39)

.0)2(2)2(
1
1,
212
=
′′
+

+
′′′
+

XAAXAA

(4.3.40)
From the above equations (4.3.38) to (4.3.40) as explained in case B1, there exist two
possibilities:
(
α

η

(
β

,
η

Ω

睨敲攠
.)2()
)2(2
))2((
(
2
1
2
2
3
2
)(2
Ω=
′′
+
′′
′′
+

′′′
+


AAm
rA
eAAe
AA
eAA
and .
R

η

First consider the subcase (
α
⤬⁩渠睨楣栠晵牴桥爠瑨牥攠ro牥 ⁰潳獩扩汩瑩敳⁥=楳琠ih楣栠慲i†
⡩⤠( 0>
η
(ii) ,0
=
η
† ⡩楩⤠.0
<
η

We⁷楬氠捯is楤敲⁥慣栠ao獳楢楬楴礠楮⁴畲渮s
(
α
椩i 䍃匠楮⁴桩猠捡獥⁡猠Ci獣畳獥搠楮⁣慳e⁂ㄠ慲攠=
=
,)2⤨獩湨捯獨(
,)2()獩湨捯獨(
2
1
2
78
1
9
2
1
2
87
0



′′
+

+=
+
′′
+

+=

AAtctcX
cdreAAtctcX
Am
ηη
ηηη

(4.3.41)
provided that ,
η
=Ω where .,,
987
Rccc


(
α
楩i

In this case
0
=
η
and CCS are

,)2()(
,)
2
()2(
2
1
2
87
1
98
2
7
2
1
2
7
0



′′
+

+=
++−
′′
+

=

AActcX
ctct
c
MdreAAcX
Am

(4.3.42)
provided that
,
)2(2
))2((
1
2
3
2
)(2
Me
AA
eAA
Am
rA
=
′′
+

′′′
+


where ).0(,,,
11876


MRMccc


93
(
α
楩椩⁈e牥r
.0
<
η
We utilize the substitution
,
σ
η

=
⁷桥牥=
R

σ

( )
.0>
σ
CCS in
this case are

,)2)(sincos(
,)2()sincos(
2
1
2
78
1
9
2
1
2
87
0



′′
+

+=
+
′′
+

−=

AAtctcX
cdreAAtctcX
Am
σσ
σσσ

(4.3.43)
provided that ,
σ
−=Ω where .,,
987
Rccc


(
β
⤠ 䥮⁴桩猠捡獥⁃䍓⁡牥†
=
,0,
1
9
0
==
XcX

(4.3.44)
where .
9
Rc
∈ CCS in this case are given in equation (4.3.34), either of (4.3.41), (4.3.42),
(4.3.43) or (4.3.44).
Case (Q3-Q4)

In cases Q3 and Q4 spacetime (4.3.1) becomes 1+1+2 decomposable and belong to
curvature D. Following the same procedure as described in cases Q1 or Q2, CCS can
easily be calculated.
Case Q5
In this case
,0),,( ≠
θ
rtA
t

,0),,(
=
θ
rtA
r

,0),,(
=
θ
θ
rtA

,0),,(
=
θ
rtB
t

,0),,(
=
θ
rtB
r

0),,( ≠
θ
θ
rtB
and
,0),,( =
θ
θθ
rtB
the rank of the
66
×
⁒楥m慮渠aa瑲楸⁩t湥⸠䙲=m⁴桥=
慢潶攠捯湳瑲慩湴猬⁷攠桡癥a
)(tAA =
and
,baB
+
=
θ
where
).0(,≠

aRba
Here, there
exist two linearly independent solutions
aa
tt
,
=
⁡湤=
aa,
θ
θ
=
映敱畡瑩潮
㈮㘮ㄩ⁡湤⁴桥=
癥捴潲⁦楥汤v
a
t
is covariantly constant where as
a
θ
is not covariantly constant. The line
element, after rescaling of
t
can be written as


).(
22222
φθ
θ
dddredtds
ba
+++−=
+

(4.3.45)
The above spacetime is clearly 1+3 decomposable and belongs to the curvature class D.
Substituting the above informations into CC equations from (2.7.5) to (2.7.19) and after
simplification, we get
,0
0
3,
0
1,
== XX

,0
1
2,
1
1,
1
0,
=== XXX

,0
2
3,
2
1,
== XX

,0
3
3,
3
2,
3
0,
=== XXX

.0
3
1,
1
3,
=+ XX

From the above system of equations, one can easily find that the CCS are

,),,(,),,(
31
32
21
10
crcXtNXccXtMX
+−==+==
θφθ

(4.3.46)

94
where ),(
θ
tM
and ),(
θ
tN
are the arbitrary functions and .,
32,1
Rccc

One can write
the above equation (4.3.46) after subtracting the Killing vector fields as

).0),,(,0),,((
θ
θ
tNtMX =

(4.3.47)
CCS clearly form an infinite dimensional vector space.
Cases (Q6-Q10)
In these cases the rank of the
66
×
⁒楥=慮渠aa瑲楸⁩猠潮攠慮e⁴桥=攠數楳琠瑷漠汩湥arl礠
楮摥灥湤敮琠癥捴潲⁦楥汤猠睨楣栠慲攠獯汵瑩潮猠 潦⁥煵慴楯渠⠲⸶⸱⤠慮搠潮攠潦⁴桥m⁩猠湯琠
捯癡物慮瑬礠捯湳瑡′t⸠卵扳瑩瑵瑩潮t⁣潮獴= 慩at猠潦⁴桥獥⁣慳敳a步猠瑨攠獰k′e瑩te=
⠴⸳⸱⤠扥捯me猠ㄫ㌠摥捯s灯獡扬攠慮搠扥汯湧猠瑯⁣畲癡瑵r攠捬慳猠䐮⁓畢獴楴s瑩湧⁴桥=
慢潶攠楮景牭慴楯渠楮瑯⁃䌠敱畡瑩潮猠晲潭
㈮㜮㔩⁴漠⠲⸷⸱㤩Ⱐ潮攠捡渠敡獩汹⁦楮搠=C匠楮S
捡獥猠儶⁴漠儱〠慳⁩湴敲灲整敤⁩渠捡′攠儵⸠e
㐮4 Conclusions
In this chapter a study of non-static cylinderically symmetric and special non-static
axially symmetric spacetimes according to their proper CCS is given. An approach is
adopted to study the above spacetimes by using the rank of the 66
×
⁒楥=慮渠aa瑲楸⁡湤t
慬獯⁵獩湧⁴桥⁴桥潲敭⁧楶敮⁩渠嬸崬⁷桩捨= 獵杧敳瑥搠睨敲攠灲潰敲⁃䍓⁥硩獴⸠䙲潭⁴桥=
慢潶攠獴畤礠睥扴慩渠瑨 攠景汬潷楮朠牥獵汴献e
周攠牥獵汴θ扴慩湥搠批⁳瑵摹楮g⁰牯灥爠 捵牶慴′r攠捯汬楮敡瑩潮e⁦潲潮⵳瑡瑩挠
捹汩湤r楣慬汹⁳祭浥瑲楣⁳p慣整業敳⁡牥⁡猠景汬潷猺†
⡩⤠ 周攠捡獥⁷桥渠瑨攠牡θk映瑨攠 6 6
×
⁒楥=慮渠aa瑲楸⁩猠瑨牥r⁡湤⁴桥r攠數楳瑳⁡e
畮楱略漠睨敲攠穥牯⁩湤数敮摥湴⁴業敬楫攠癥捴 潲⁦楥汤⁷h楣栠楳⁡⁳潬畴楯渠潦⁥煵慴楯渠
⠲⸶⸱⤠慮搠楳⁣潶慲楡湴汹⁣潮獴慮琮⁔桩猠楳 ⁴桥⁳灡捥瑩=攠e4⸲⸳ㄩ⁡湤⁩琠慤.楴猠灲潰敲i
䍃匠睨楣栠景Cm⁡渠楮晩湩瑥⁤ime湳楯湡氠癥捴潲⁳p慣攠⡳敥⁣慳攠䌱⤮e
⡩椩( 周攠捡獥⁷桥渠瑨攠牡θk映瑨攠 6 6
×
⁒楥=慮渠aa瑲楸⁩猠瑨牥r⁡湤⁴桥r攠數楳瑳⁡e
畮楱略潷桥牥⁺敲漠楮摥灥湤敮琠瑩×敬楫攠癥捴 潲⁦楥汤⁷桩捨⁩猠愠獯汵瑩潮映敱畡瑩潮o
⠲⸶⸱⤠慮搠楳潴⁣潶慲楡湴汹⁣潮獴慮琮⁔桩猠 楳⁴i攠獰慣e瑩t攠⠴⸲⸵㜩⁡湤⁩琠慤=楴猠灲潰敲o
䍃匠睨楣栠景Cm⁡渠楮晩湩瑥⁤ime湳楯湡氠癥捴潲⁳p慣攠⡳敥⁣慳攠䌴㌩⸠e
⡩楩⤠ 周攠捡獥⁷桥渠瑨攠牡θk映瑨攠 6 6
×
⁒楥=慮渠aa瑲楸⁩猠潮攠慮搠瑨敲攠數楳琠瑷漠
湯睨敲攠穥牯⁩湤数敮摥湴⁶散瑯爠晩敬摳⁷桩n 栠慲攠獯汵瑩潮猠潦⁥煵慴楯渠⠲⸶⸱⤠慮搠慲攠

95
covariantly constant. This is the spacetime (4.2.60) and it admits proper CCS, which form
an infinite dimensional vector space (see case D1).
(iv) The case when the rank of the 66
×
⁒楥=慮渠aa瑲楸⁩猠潮攠慮搠瑨敲攠數楳琠瑷漠
湯睨敲攠穥牯⁩湤数敮摥湴⁳n 汵瑩潮猠潦⁥煵慴楯渠⠲⸶⸱⤠扵琠潮汹湥映瑨lm⁩猠
捯癡物慮瑬礠捯湳瑡′t⸠周楳⁩猠瑨攠獰.捥瑩′攠 ⠴⸲⸶㈩⁡湤⁩琠慤(楴猠灲潰敲⁃䍓⁷桩捨⁦潲m=
慮⁩湦楮楴攠摩ae湳楯湡氠癥捴潲⁳n慣攠⡳敥⁣慳攠䐹⤮†
⡶⤠ 周攠捡獥⁷桥渠瑨攠牡θk映瑨攠 6 6
×
⁒楥=慮渠aa瑲楸⁩猠潮攠慮搠瑨敲攠數楳琠瑷漠
湯睨敲攠穥牯⁩湤数敮摥湴⁳潬畴楯湳映敱畡瑩 潮
㈮㘮ㄩ⁢畴⁡牥潴⁣潶慲楡湴汹⁣潮獴慮琮o
周楳⁩θ⁴桥⁳灡捥瑩t攠⠴⸲⸶㤩⁡湤⁩琠慤n 楴猠灲潰敲⁃䍓⁷h楣栠io牭⁡渠楮= i湩瑥=
摩≤e湳楯湡氠癥捴潲⁳n慣攠⡳敥⁣慳攠䐱㤩⸠a
周攠景汬潷楮朠牥獵汴猠慲攠潢瑡楮敤⁦牯θ⁳瑵摹= 潦⁳灥捩慬潮⵳瑡瑩挠慸楡汬礠獹浭整物挠
獰慣攠瑩se猺†
⡩⤠ 周攠捡獥⁷桥渠瑨攠牡θk映瑨攠
66
×
⁒楥=慮渠aa瑲楸⁩t⁴=牥攠慮搠瑨敲攠數楳瑳⁡i
畮楱略漠睨敲攠穥牯⁩湤数敮摥湴⁴業敬楫攠癥捴 潲⁦楥汤⁷h楣栠楳⁡⁳潬畴楯渠潦⁥煵慴楯渠
⠲⸶⸱⤠慮搠楳⁣潶慲楡湴汹⁣潮獴慮琮⁔桩猠楳 ⁴=攠ep慣整業攠⠴⸳⸴⤠慮搠楴⁡摭楴猠灲潰敲⁃䍓i
睨楣栠景∂m⁡渠楮晩湩瑥⁤業e湳楯湡氠癥捴潲⁳n慣攠⡳敥⁣慳攠倱⤮†
⡩椩( 周攠捡獥⁷桥渠瑨攠牡θk映瑨攠
66
×
⁒楥=慮渠aa瑲楸⁩t⁴=牥攠慮搠瑨敲攠數楳瑳⁡i
畮楱略漠睨敲攠穥牯⁩湤数敮摥湴⁴業敬楫攠癥捴 潲⁦楥汤⁷h楣栠楳⁡⁳潬畴楯渠潦⁥煵慴楯渠
⠲⸶⸱⤠慮搠楳潴⁣潶慲楡湴汹⁣潮獴慮琮⁔桩猠 楳⁴i攠獰慣e瑩t攠⠴⸳⸲ㄩ⁡湤⁩琠慤=楴猠灲潰敲o
䍃匠睨楣栠景Cm⁡渠楮晩湩瑥⁤ime湳楯湡氠癥 捴潲⁳p慣攠⡦潲⁤整慩汳⁳敥⁣慳攠倱㘩⸠e
⡩楩⤠ 周攠捡獥⁷h敮⁴桥⁲慮e映瑨攠
66
×
⁒楥=慮渠aa瑲楸⁩猠潮攠慮t⁴桥牥⁥硩=瑳⁴=漠
湯⁷桥牥⁺敲漠楮摥灥湤敮琠癥捴潲⁦楥汤猠睨楣 栠慲攠獯汵瑩潮猠潦⁥煵慴楯渠⠲⸶⸱⤠慮搠扯瑨h
慲攠捯癡物慮瑬礠捯湳瑡湴t⁔桥獥⁡牥⁴桥⁳灡捥瑩= 敳
㐮㌮㈳⤠慮搠⠴⸳⸳㌩⁡湤⁴桥礠慤ei琠
灲潰敲⁃䍓Ⱐ睨楣栠景pm⁡渠楮晩湩瑥⁤業e湳n 潮慬⁶散瑯爠獰慣攠⡳敥⁣慳敳⁑ㅡ湤⁑㈩⸠a
⡩瘩( 周攠捡獥⁷h敮⁴桥⁲慮e映瑨攠
66
×
⁒楥=慮渠aa瑲楸⁩猠潮攠慮t⁴桥牥⁥硩=瑳⁴=漠
湯⁷桥牥⁺敲漠楮摥灥湤敮琠癥捴潲⁦楥汤猠睨楣 栠慲攠獯汵瑩潮猠潦⁥煵h 瑩潮
㈮㘮ㄩ⁢畴湬礠
潮攠楳⁣潶=物慮瑬礠捯湳瑡湴t⁔桩猠楳⁴桥⁳= 慣整業e
㐮㌮㐵⤠慮搠 楴⁡摭楴猠ir潰o爠rC匬S
睨楣栠景∂m⁡渠楮晩湩瑥⁤業e湳楯湡氠癥捴潲⁳n慣攠⡳敥⁣慳攠儵⤮†