In the present paper, we have studied - type and -type plane wave solutions of Einstein's field equations in general theory of relativity in the case where the zero mass scalar field coupled with gravitational field and zero mass scalar field coupled with gravitational & electromagnetic field and established the existence of these two types of plane wave solutions in . Furthermore we have considered the case of massive scalar field and shown that the non-existence of these two types of plane wave solutions in GR theory.

1. Research Inventy: International Journal Of Engineering And Science
Vol.5, Issue 2 (February 2015), PP 30-38
Issn (e): 2278-4721, Issn (p):2319-6483, www.researchinventy.com
30
])(
2
1
[ tzy  -type and ])(2[ zyt  -type four dimensional plane wave
solutions of Einstein’s field equations in general relativity
B. G. Ambatkar , J K Jumale*
Head Department of mathematics, R.S. Bidkar Arts, Comm. and Science College, Hinganghat,
* R.S. Bidkar Arts, Comm. and Science College, Hinganghat, Distt. Wardha*, jyotsnajumale21@gmail.com
ABSTRACT: In the present paper, we have studied ])(
2
1
[ tzy  - type and ])(2[ zyt  -type plane
wave solutions of Einstein’s field equations in general theory of relativity in the case where the zero mass scalar
field coupled with gravitational field and zero mass scalar field coupled with gravitational & electromagnetic
field and established the existence of these two types of plane wave solutions in 4V . Furthermore we have
considered the case of massive scalar field and shown that the non-existence of these two types of plane wave
solutions in GR theory.
Keywords : Einstein’s field equations in general relativity in case where the zero mass scalar field coupled with
gravitational field and zero mass scalar field coupled with gravitational & electromagnetic field, four
dimensional space-time, etc.
I. INTRODUCTION
H.Takeno (1961) has obtained the non-flat plane wave solutions ijg of the field equations 0ijR and
established the existence of )( tz  -type and )( zt -type plane waves in purely gravitational case in four-
dimensional empty region of space-time. On the lines of Takeno (1961), Kadhao and Thengane (2002) have also
obtained the plane wave solutions ijg of the field equations 0ijR in purely gravitational case by
reformulating Takeno's (1961) definition of plane wave as follows:
Definition A plane wave ijg is a non-flat solution of field equations
0ijR , ]5,4,3,2,1,[ ji (1.1)
in an empty region of space-time such that
)(Zgg ijij  , )( i
xZZ  , where tzyxuxi
,,,, (1.2)
in some suitable co-ordinate system such that
,0,, ji
ij
ZZg ii
x
Z
Z


, , (1.3)
),,,( tzyZZ  ,0,2 Z ,0,3 Z 0,4 Z . (1.4) In this
definition, the signature convention adopted is
,0rrg 3,2,1r
0
sssr
rsrr
gg
gg
,
333231
232221
131211
ggg
ggg
ggg
<0, 044 g (1.5)
[not summed for r,s =1,2,3] and accordingly 0)det(  ijgg (1.6)

2. ])(
2
1
[ tzy  -type and ])(2[ zyt  -type four…
31
and in the paper refer it to [2], we have established the existence of ])(
2
1
[ tzy  -type and
])(2[ zyt  -type plane waves in 5V .
Furthermore, in the paper [3] by investigating the plane symmetric line elements
][ 22222
dtdzdyBAdxds  (1.7)
and
2222222
BdtBdzZBdyZAdxds  (1.8)
we have obtained the relations of non-vanishing components of Ricci tensor as
342423443322 22222 RRRRRRP  , (1.9)
and 232
2
44
2
332
2
222
2
)(
2
)()()(
R
Z
zy
R
zy
R
Z
zy
R
Z
zy
P








34
2
24
2
2
)(
2
)(
R
Z
zy
R
Z
zy 


 . (1.10)
for ])(
2
1
[ tzy  -type and ])(2[ zyt  -type plane waves respectively.
In this paper, we investigate whether these two types of plane wave solutions exist in the case where zero
mass scalar field coupled with gravitational field and the zero mass scalar field coupled with gravitational &
electromagnetic field. Furthermore we consider the coupling of massive scalar field with gravitational field
and the massive scalar field with gravitational & electromagnetic field in 4V to investigate the existence of
these two types of plane wave solutions of Einstein's field equation
]
2
1
)[8( TgTR ijijij  , ]4,3,2,1,[ ji (1.11)
where ijR is the Ricci tensor of the space-time ,
ijT is the energy momentum tensor ,
ij
g is the fundamental tensor of the space-time
and ij
iji
i TgTT  .
We study these two plane wave solutions separately.
Case I ])(
2
1
[ tzy  -type plane wave solutions in 4V
II. ZERO MASS SCALAR FIELD COUPLED WITH GRAVITATIONAL FIELD
The energy momentum tensor of zero mass scalar field is given by
]
2
1
[
4
1 k
kijjiij VVgVVT 

, ]4,3,2,1[ k (2.1)
where V is scalar function of Z and jj
x
V
V


 , ),,,( tzyxx j
 .

3. ])(
2
1
[ tzy  -type and ])(2[ zyt  -type four…
32
Thus VVVV  432 22 021  VV (2.2)
where (-) bar denotes partial derivative w. r. to Z
From line element (1.7) we have
,22 Bg  ,33 Bg  Bg 44 , (2.3)
,
122
B
g  ,
133
B
g 
B
g
144
 (2.4)
 0k
kVV . (2.5)
Therefore equation (2.1) becomes
][
4
1
jiij VVT

 . (2.6)
And from equation (2.6), in zero mass scalar field
0 ij
iji
i TgTT (2.7)
Then using equations (2.6) and (2.7), Einstein’s field equation (1.11) becomes
][2 jiij VVR  (2.8)
which further gives
2
,22 VR 
2
,33 VR 
2
,44 2VR 
2
,23 VR 
2
,24 2 VR 
2
34 2VR  . (2.9)
Thus non-vanishing components of Ricci tensor are related as
342423443322 22222 RRRRRR  . (2.10)
It is observed that the equation (2.10) is compatible with the equation (1.9) which is obtained in the case of
purely gravitational field. Hence we have
Conclusion ])(
2
1
[ tzy  -type plane wave solutions exist in the case where zero mass scalar field is
coupled with the gravitational field.
III. Zero Mass scalar field coupled with gravitational & electromagnetic field
In this case the energy momentum tensor is given by
ij
k
kijjiij EVVgVVT  ]
2
1
[
4
1

(3.1)
where ijE denotes electromagnetic energy momentum tensor.
But as in the previous case for zero mass scalar field
0k
kVV and 0 ij
iji
i TgTT . (3.2)
Therefore equation (3.1) becomes
ijjiij EVVT 

 ][
4
1
. (3.3)
Hence Einstein’s field equation (1.11) becomes
]4[2 ijjiij EVVR  . (3.4)
Then equation (3.4) yields

4. ])(
2
1
[ tzy  -type and ])(2[ zyt  -type four…
33
]4
2
[2 22
2
22 E
V
R  , ]4
2
[2 33
2
33 E
V
R  , ]4[2 44
2
44 EVR  ,
]4
2
[2 23
2
23 E
V
R  , ]4
2
[2 24
2
24 E
V
R  , ]4
2
[2 34
2
34 E
V
R  . (3.5)
All above non-vanishing components of Ricci tensor are related as
342423443322 22222 RRRRRR  . (3.6)
It is observed that the equation (3.6) is compatible with equation (1.9) which is obtained in the case of purely
gravitational field. Hence we have
Conclusion ])(
2
1
[ tzy  -type type plane wave solutions exists in the case where zero mass scalar field
is coupled with gravitational & electromagnetic field.
IV. MASSIVE SCALAR FIELD COUPLED WITH GRAVITATIONAL FIELD
The energy momentum tensor of massive scalar field is given by
)](
2
1
[
4
1 22
VmVVgVVT k
kijjiij 

 , ]4,3,2,1[ k (4.1)
where V is scalar function of Z and
jj
x
V
V


 , ),,,( tzyxx j
 ,
m is mass associated with the massive scalar field.
Thus VVVV  432 22 01 V (4.2)
where (-) bar denotes partial derivative w. r. to Z.
From line element (1.7) we have
,22 Bg  Bg 33 Bg 44 (4.3)
,
122
B
g  ,
133
B
g 
B
g
144
 (4.4)
 0k
kVV . (4.5)
Therefore equation (4.1) implies
]
2
1
[
4
1 22
VmgVVT ijjiij 

 . (4.6)
Equation (4.6) yields
]2[
4
1 22
VmTgTT ij
iji
i

 . (4.7)
Using (4.6) and (4.7), Einstein’s field equation (1.11) becomes
]
2
1
[2 22
VmgVVR ijjiij  (4.8)
which further yields
][ 222
22 VBmVR  , ][ 222
33 VBmVR  , ]2[ 222
44 VBmVR  ,
2
,23 VR  2
2
24 VR  , 2
2
34 VR  . (4.9)

5. ])(
2
1
[ tzy  -type and ])(2[ zyt  -type four…
34
But from the line element (1.7) we have the relation of non-vanishing components of Ricci tensor as
342423443322 22222 RRRRRR  . (4.10)
It is to be noted that here the equation (4.9) is incompatible with the equation (4.10) which is obtained in the
case of purely gravitational field. Hence we have
Conclusion ])(
2
1
[ tzy  -type plane wave solutions doesn’t exist in the case where massive scalar field
is coupled with gravitational field.
Remark If 02
m then the equation (4.9) is compatible to (4.10) and we have a result as in the case where
the zero mass scalar field is coupled with the gravitational field.
V. Massive scalar field coupled with gravitational & electromagnetic field
In this case the energy momentum tensor is given by
ij
k
kijjiij EVmVVgVVT 

 )](
2
1
[
4
1 22
(5.1)
where ijE denotes electromagnetic energy momentum tensor.
But as in the previous case for massive scalar field
0k
kVV and ]2[
4
1 22
VmTgTT ij
iji
i

 . (5.2)
Therefore equation (5.1) becomes
ijijjiij EVmgVVT 

 ]
2
1
[
4
1 22
. (5.3)
Hence Einstein’s field equation (1.11) becomes
]4
2
1
[2 22
ijijjiij EVmgVVR  . ( 5.4)
Then from equation (5.4) we have
]4
22
[2 22
222
22 E
VBmV
R  , ]4
22
[2 33
222
33 E
VBmV
R  ,
]4
2
2[2 44
22
2
44 E
VBm
VR  , ]4
2
[2 23
2
23 E
V
R  ,
]4
2
[2 24
2
24 E
V
R  , ]4
2
[2 34
2
34 E
V
R  . (5.5)
But line element (1.7) gives the relation of non-vanishing components of Ricci tensor as
22222 342423443322 RRRRRR  . (5.6)
Here it has been observed that the equation (5.5) is incompatible with the equation (5.6) which is obtained in
the case of purely gravitational field. Hence we have
Conclusion ])(
2
1
[ tzy  -type plane wave solutions of Einstein’s field equation in general relativity
doesn’t exist in the case where massive scalar field is coupled with gravitational & electromagnetic field.
Remark If 02
m then the equation (5.5) is compatible to (5.6) and we have a result as in the case where the
zero mass scalar field is coupled with the gravitational & electromagnetic field .

6. ])(
2
1
[ tzy  -type and ])(2[ zyt  -type four…
35
Case II we consider the case of ])(2[ zyt  -type plane wave solutions.
VI. ZERO MASS SCALAR FIELD COUPLED WITH GRAVITATIONAL FIELD
The energy momentum tensor of zero mass scalar field is given by
]
2
1
[
4
1 k
kijjiij VVgVVT 

, ]4,3,2,1[ k (6.1)
where V is scalar function of Z and
jj
x
V
V


 , ),,,( tzyxx j
 .
Thus 01 V and 2V
)(2
43
zy
ZVZ
VV

 (6.2)
here ( - ) bar denotes partial derivative w. r. to Z
From line element (1.8) we have
,, 44
2
33
2
22 BgBZgBZg  , (6.3)
B
g
BZ
g
BZ
g
1
,
1
,
1 44
2
33
2
22
 (6.4)
 0k
kVV . (6.5)
Therefore equation (6.1)
 ][
4
1
jiij VVT

 (6.6)
Equation (6.6) yields
0 ij
iji
i TgTT . (6.7)
Then from (6.6) and (6.7), Einstein’s field equation (1.11) becomes
][2 jiij VVR  (6.8)
which further gives
]
)(
[2 2
22
22
zy
ZV
R

 , ]
)(
[2 2
22
33
zy
ZV
R

 , ]
)(
2
[2 2
2
44
zy
V
R


]
)(
[2 2
22
23
zy
ZV
R

 , ]
)(
[22 2
2
24
zy
ZV
R

 , ]
)(
[22 2
2
34
zy
ZV
R

 . (6.9)
Thus non-vanishing components of Ricci tensor are related as
232
2
44
2
332
2
222
2
Z
z)(y
2
z)(y
Z
z)(y
Z
z)(y
RRRR







34
2
24
2
2Z
z)(y
2Z
z)(y
RR



 . (6.10)
It is observed that the equation (6.10) is compatible with equation (1.10) which is obtained in the case of purely
gravitational field. Hence we have
Conclusion ])(2[ zyt  -type plane wave solutions exist in the case where zero mass scalar field is
coupled with the gravitational field.

7. ])(
2
1
[ tzy  -type and ])(2[ zyt  -type four…
36
VII. ZERO MASS SCALAR FIELD COUPLED WITH GRAVITATIONAL &
ELECTROMAGNETIC FIELD
In this case the energy momentum tensor is given by
ij
k
kijjiij EVVgVVT 

 ]
2
1
[
4
1
(7.1)
where ijE denotes electromagnetic energy momentum tensor.
But as in the previous case for zero mass scalar field
0k
kVV and 0 ij
iji
i TgTT . (7.2)
Therefore equation (7.1) becomes
ijjiij EVVT 

 ][
4
1
. (7.3)
Hence Einstein’s field equation )11.1( becomes
]4[2 ijjiij EVVR  (7.4)
which further yields
]4
)(
[2 222
22
22 E
zy
ZV
R 

 , ]4
)(
[2 332
22
33 E
zy
ZV
R 

 , ]4
)(
2
[2 442
2
44 E
zy
V
R 

 ,
]4
)(
[2 232
22
23 E
zy
ZV
R 

 , ]4
)(
2
[2 242
2
24 E
zy
ZV
R 


 ,
]4
)(
2
[2 342
2
34 E
zy
ZV
R 


 .(7.5)
All above non-vanishing components of Ricci tensor are related as
232
2
44
2
332
2
222
2
)(
2
)()()(
R
Z
zy
R
zy
R
Z
zy
R
Z
zy 






34
2
24
2
2
)(
2
)(
R
Z
zy
R
Z
zy 


 . (7.6)
It is observed that the equation (7.6) is compatible with equation (1.10) which is obtained in the case of purely
gravitational field. Hence we have
Conclusion ])(2[ zyt  -type plane wave solutions exist in the case where zero mass scalar field is
coupled with gravitational & electromagnetic field.
VIII. MASSIVE SCALAR FIELD COUPLED WITH GRAVITATIONAL FIELD
The energy momentum tensor of massive scalar field is given by
)](
2
1
[
4
1 22
VmVVgVVT k
kijjiij 

 , ]4,3,2,1[ k (8.1)
where V is scalar function of Z and
jj
x
V
V


 , ),,,( tzyxx j
 ,
m is mass associated with the massive scalar field.

8. ])(
2
1
[ tzy  -type and ])(2[ zyt  -type four…
37
Thus 2V
)(2
43
zy
ZVZ
VV

 , 01 V (8.2)
where ( - ) bar denotes partial derivative w. r. to Z.
From line element (1.8) we have
,, 44
2
33
2
22 BgBZgBZg  , (8.3)
B
g
BZ
g
BZ
g
1
,
1
,
1 44
2
33
2
22
 (8.4)
 0k
kVV . (8.5)
Therefore equation (8.1) becomes
]
2
1
[
4
1 22
VmgVVT ijjiij 

 . (8.6)
From equation (8.6) we have
]2[
4
1 22
VmTgTT ij
iji
i

 . (8.7)
Then from (8.6) and (8.7), Einstein’s field equation )11.1( becomes
]
2
1
[2 22
VmgVVR ijjiij  (8.8)
which further yields
]
2)(
[2
222
2
22
22
VBmZ
zy
ZV
R 

 , ]
2)(
[2
222
2
22
33
VBmZ
zy
ZV
R 

 , ]
)(
[2 2
22
23
zy
ZV
R

 ,
]
2)(
2
[2
22
2
2
44
VBm
zy
V
R 

 , ]
)(
2
[2 2
2
24
zy
ZV
R


 , ]
)(
2
[2 2
2
34
zy
ZV
R


 . (8.9)
But from the line element (1.8) we have the relation of non-vanishing components of Ricci tensor as
232
2
44
2
332
2
222
2
)(
2
)()()(
R
Z
zy
R
zy
R
Z
zy
R
Z
zy 






34
2
24
2
2
)(
2
)(
R
Z
zy
R
Z
zy 


 (8.10)
Here we observed that the equation (8.9) is incompatible with the equation (8.10) which is obtained in the case
of purely gravitational field. Hence we have
Conclusion ])(2[ zyt  -type plane wave solutions of Einstein’s field equation in general theory of
relativity doesn’t exist in the case where massive scalar field is coupled with the gravitational field.
Remark If 02
m then the equation (8.9) is compatible to (8.10) and we have a result as in the case where
the zero mass scalar field is coupled with the gravitational field.
IX. MASSIVE SCALAR FIELD COUPLED WITH GRAVITATIONAL &
ELECTROMAGNETIC FIELD
In this case the energy momentum tensor is given by
ij
k
kijjiij EVmVVgVVT 

 )](
2
1
[
4
1 22
(9.1)

9. ])(
2
1
[ tzy  -type and ])(2[ zyt  -type four…
38
where ijE denotes electromagnetic energy momentum tensor.
But as in the previous case for massive scalar field
0k
kVV and ][
4
1 22
VmTgTT ij
iji
i

 . (9.2)
Therefore equation (9.1) becomes
ijijjiij EVmgVVT 

 ]
2
1
[
4
1 22
. (9.3)
Hence Einstein’s field equation )11.1( becomes
]4
2
1
[2 22
ijijjiij EVmgVVR  . (9.4)
And from equation (9.4) we have
]4
2)(
[2 22
222
2
22
22 E
VBmZ
zy
ZV
R 

 , ]4
2)(
[2 33
222
2
22
33 E
VBmZ
zy
ZV
R 

 ,
]4
2)(
2
[2 44
22
2
2
44 E
VBm
zy
V
R 

 , ]4
)(
[2 232
22
23 E
zy
ZV
R 

 ,
]4
)(
2
[2 242
2
24 E
zy
ZV
R 


 , ]4
)(
2
[2 342
2
34 E
zy
ZV
R 

 . (9.5)
But line element (1.8) gives the relation of non-vanishing components of Ricci tensor as
232
2
44
2
332
2
222
2
)(
2
)()()(
R
Z
zy
R
zy
R
Z
zy
R
Z
zy 






24
2
2
)(
R
Z
zy 

. 34
2
2
)(
R
Z
zy 
 . (9.6)
Here we observed that the equation (9.5) is incompatible with the equation (9.6) which is obtained in the case
of purely gravitational field. Hence we have
Conclusion ])(2[ zyt  -type plane wave solutions of Einstein’s field equation doesn’t exist in the case
where massive scalar field is coupled with gravitational & electromagnetic field.
Remark If 02
m then the equation (9.5) is compatible to (9.6) and we have a result as in the case where the
zero mass scalar field is coupled with the gravitational & electromagnetic field.
Acknowledgement The authors are thankful to Professor K. D. Thengane of India for his constant inspiration .
REFERENCES
[1] Kadhao S R and : “Some Plane wave solutions of field equations 0ijR in four
Thengane K D (2002) dimensional space-time.” Einstein Foundation
International Vol.12,2002, India.
[2] Jumale J K and : “ ])(
2
1
[ tzy  -type and ])(2[ zyt  -type plane waves
Thengane K D (2004) in four dimensional space-time” Tensor New Series Japan,
vol.65, No.1, pp.85-89.
[3] H Takeno(1961) : “The Mathematical theory of plane gravitational wave in general relativity”. Scientific report of the Research
Institute to theoretical physics. Hiroshima, University, Takchara, Hiroshima-ken, Japan-1961.