-type and -type four dimensional plane wave solutions of Einstein's

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 [

1 2

( y  z )  t ] -type

and

[t 2 ( y  z )] -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 [

1 2

( y  z )  t ] - type and [t 2 ( y  z )] -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 V 4 . 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 g ij of the field equations Rij  0 and established the existence of ( z  t ) -type and (t z ) -type plane waves in purely gravitational case in fourdimensional empty region of space-time. On the lines of Takeno (1961), Kadhao and Thengane (2002) have also obtained the plane wave solutions g ij of the field equations Rij  0 in purely gravitational case by reformulating Takeno's (1961) definition of plane wave as follows: Definition A plane wave g ij is a non-flat solution of field equations

Rij  0 ,

[i, j  1,2,3,4,5]

(1.1)

in an empty region of space-time such that

g ij  g ij (Z ) ,

Z  Z (xi ) ,

where x  u, x, y, z , t i

(1.2)

in some suitable co-ordinate system such that

Z ,i 

g ij Z ,i Z , j  0, Z  Z ( y, z, t ),

Z

x i

Z , 2  0,

,

(1.3)

Z ,4  0 .

Z , 3  0,

(1.4) In this

definition, the signature convention adopted is

g rr  0, r  1,2,3

g rr g sr

g rs  0, g ss

g11

g12

g13

g 21

g 22

g 31

g 32

g 23 <0, g 33

g 44  0

[not summed for r,s =1,2,3] and accordingly g  det( g ij )  0

30

(1.5)

(1.6)

[

1 2

( y  z )  t ] -type and [t 2 ( y  z )] -type four…

and in the paper refer it to [2], we have established the existence of [

1 2

( y  z )  t ] -type and

[t 2 ( y  z )] -type plane waves in V 5 .

Furthermore, in the paper [3] by investigating the plane symmetric line elements

ds 2   Adx 2  B[dy 2  dz 2  dt 2 ]

(1.7)

ds 2   Adx 2  Z 2 Bdy 2  Z 2 Bdz 2  Bdt 2

(1.8)

and we have obtained the relations of non-vanishing components of Ricci tensor as

P  2R22  2R33  R44  2R23   2R24   2R34 , and

P 

( y  z) 2 ( y  z) 2 ( y  z) 2 ( y  z) 2 R  R  R  R23 22 33 44 2 Z2 Z2 Z2



for [

1 2

(1.9)

( y  z) 2 Z 2

R24  

( y  z) 2 Z 2

R34 .

(1.10)

( y  z )  t ] -type and [t 2 ( y  z )] -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 V 4 to investigate the existence of these two types of plane wave solutions of Einstein's field equation

Rij  (8)[Tij 

1 g ijT ] , 2

[i, j  1,2,3,4]

(1.11)

where Rij is the Ricci tensor of the space-time ,

Tij is the energy momentum tensor ,

g ij is the fundamental tensor of the space-time and

T  Tii  g ijTij .

We study these two plane wave solutions separately.

Case I [

1 2

( y  z )  t ] -type plane wave solutions in V 4

II. ZERO MASS SCALAR FIELD COUPLED WITH GRAVITATIONAL FIELD The energy momentum tensor of zero mass scalar field is given by

Tij 

1 1 [ViV j  g ijVkV k ] , 4 2

where V is scalar function of Z and

Vj 

[k  1,2,3,4]

V , x j

( x j  x, y , z , t ) .

31

(2.1)

[

2V2  2V3  V4  V

Thus

1 2

( y  z )  t ] -type and [t 2 ( y  z )] -type four…

 V1  V2  0

(2.2)

where (-) bar denotes partial derivative w. r. to Z From line element (1.7) we have

g 22   B, 1 g 22   , B k  VkV  0 .

g 44  B , 1 g 44  B

g 33   B, 1 g 33   , B

(2.3) (2.4) (2.5)

Therefore equation (2.1) becomes

1 [ViV j ] .

4

Tij 

(2.6)

And from equation (2.6), in zero mass scalar field

T  Tii  g ijTij  0

(2.7)

Then using equations (2.6) and (2.7), Einstein’s field equation (1.11) becomes

Rij  2[ViV j ]

(2.8)

which further gives 2

2

2 R22  V , R33  V, R44  2V, R23   V, 2

2

2

R24  2 V, R34  2V .

(2.9)

Thus non-vanishing components of Ricci tensor are related as

2R22  2R33  R44  2R23   2R24   2R34 .

(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

1 2

( y  z )  t ] -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

Tij  where

Eij

1 1 [ViV j  g ijVkV k ]  Eij 4 2

(3.1)

denotes electromagnetic energy momentum tensor.

But as in the previous case for zero mass scalar field

VkV k  0 and T  Ti i  g ij Tij  0 .

(3.2)

Therefore equation (3.1) becomes

Tij 

1 [ViV j ]  Eij .

4

(3.3)

Hence Einstein’s field equation (1.11) becomes Rij  2[ViV j  4Eij ] .

(3.4)

Then equation (3.4) yields

32

[ 2

R22

V  2[  4E22 ] , 2

R23

V  2[  4E23 ] , 2

1 2

( y  z )  t ] -type and [t 2 ( y  z )] -type four…

2

V R33  2[  4E33 ] , 2

2

R24  2[

V

2

R44  2[V  4E44 ] ,

2

2

 4E 24 ] ,

R34  2[

V

2

2

 4E34 ] . (3.5)

All above non-vanishing components of Ricci tensor are related as

2R22  2R33  R44  2R23   2R24   2R34 .

(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 [

1 2

( y  z )  t ] -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

1 1 [ViV j  g ij (Vk V k  m 2V 2 )] , [k  1,2,3,4] 4 2 V ( x j  x, y , z , t ) , where V is scalar function of Z and V j  , j x Tij 

(4.1)

m is mass associated with the massive scalar field.

Thus

V2 2  V3 2  V4  V

V1  0

(4.2)

where (-) bar denotes partial derivative w. r. to Z. From line element (1.7) we have

g 22   B , 1 g 22   , B k  VkV  0 .

g 44  B 1 g 44  B

g 33   B 1 g 33   , B

(4.3) (4.4) (4.5)

Therefore equation (4.1) implies

Tij 

1 1 [ViV j  g ij m 2V 2 ] .

4 2

(4.6)

Equation (4.6) yields

T  Tii  g ijTij 

1 [2m 2V 2 ] .

4

(4.7)

Using (4.6) and (4.7), Einstein’s field equation (1.11) becomes

Rij  2[ViV j 

1 g ij m 2V 2 ] 2

(4.8)

which further yields 2

R22  [V  Bm2V 2 ] , R23   V,

2

2

R33  [V  Bm2V 2 ] ,

R24  V

2

2

R44  [2V  Bm2V 2 ] , R34  V

2, 33

2

2.

(4.9)

[

1 2

( y  z )  t ] -type and [t 2 ( y  z )] -type four…

But from the line element (1.7) we have the relation of non-vanishing components of Ricci tensor as

2R22  2R33  R44  2R23   2R24   2R34 .

(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

1

Conclusion [

2

( y  z )  t ] -type plane wave solutions doesn’t exist in the case where massive scalar field

is coupled with gravitational field.

Remark If m  0 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.

2

V. Massive scalar field coupled with gravitational & electromagnetic field In this case the energy momentum tensor is given by

Tij  where

1 1 [ViV j  g ij (Vk V k  m 2V 2 )]  Eij 4 2

(5.1)

Eij denotes electromagnetic energy momentum tensor.

But as in the previous case for massive scalar field

VkV k  0 and T  Tii  g ijTij 

1 [2m 2V 2 ] .

4

(5.2)

Therefore equation (5.1) becomes

Tij 

1 1 [ViV j  g ij m 2V 2 ]  Eij .

4 2

(5.3)

Hence Einstein’s field equation (1.11) becomes

Rij  2[ViV j 

1 g ij m 2V 2  4Eij ] .

2

( 5.4)

Then from equation (5.4) we have 2

R22

V Bm2V 2  2[   4E22 ] , 2 2

R44

Bm 2V 2  2[2V   4E 44 ] , 2

R24

V  2[  4E24 ] , 2

2

2

V Bm2V 2 R33  2[   4E33 ] , 2 2 2

R23

2

R34  2[

V

V  2[  4E23 ] , 2 2

2

 4E34 ] .

(5.5)

But line element (1.7) gives the relation of non-vanishing components of Ricci tensor as

2R22  2R33  R44  2R23  R24 2  R34 2 .

(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 [

1 2

( y  z )  t ] -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 m  0 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 .

2

34

[

1 2

( y  z )  t ] -type and [t 2 ( y  z )] -type four…

Case II we consider the case of [t 2 ( y  z )] -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

Tij 

1 1 [ViV j  g ijVkV k ] , 4 2

[k  1,2,3,4]

where V is scalar function of Z and V j 

V1  0 and V2  V3  V4

Thus

Z 2

V x

j



, (x

j

(6.1)

 x , y, z, t ) .

VZ ( y  z)

(6.2)

here ( - ) bar denotes partial derivative w. r. to Z From line element (1.8) we have

g 22  Z 2 B , g 22  

1 2

Z B

g33  Z 2 B , g 33  

,

1 2

Z B

g 44  B , g 44 

,

(6.3)

1 B

(6.4)

 VkV k  0 .

(6.5)

Therefore equation (6.1)

 Tij 

1 [ViV j ] 4

(6.6)

Equation (6.6) yields

T  Tii  g ijTij  0 .

(6.7)

Then from (6.6) and (6.7), Einstein’s field equation (1.11) becomes

Rij  2[ViV j ]

(6.8)

which further gives 2

R22

V Z2  2[ ], ( y  z) 2

R23

V Z2  2[ ], ( y  z) 2

2

R33

V Z2  2[ ], ( y  z) 2

R24

V Z  2 2[ ], ( y  z) 2

2

2

R44

2V  2[ ] ( y  z) 2

R34

V Z  2 2[ ]. ( y  z) 2

2

2

(6.9)

Thus non-vanishing components of Ricci tensor are related as

(y  z) 2 Z2

R22 

(y  z) 2 Z2

(y  z) (y  z) (y  z) 2 (y  z) 2 R24   R34 . (6.10) R44  R23   2 2 Z Z 2 Z 2 2

R33 

2

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 [t 2 ( y  z )] -type plane wave solutions exist in the case where zero mass scalar field is coupled with the gravitational field.

35

[

1 2

( y  z )  t ] -type and [t 2 ( y  z )] -type four…

VII. ZERO MASS SCALAR FIELD COUPLED WITH GRAVITATIONAL & ELECTROMAGNETIC FIELD In this case the energy momentum tensor is given by

Tij  where

1 1 [ViV j  g ijVk V k ]  Eij 4 2

(7.1)

Eij denotes electromagnetic energy momentum tensor.

But as in the previous case for zero mass scalar field

VkV k  0 and

T  Tii  g ijTij  0 .

(7.2)

Therefore equation (7.1) becomes

Tij 

1 [ViV j ]  Eij .

4

(7.3)

Hence Einstein’s field equation (1.11) becomes

Rij  2[ViV j  4Eij ]

(7.4)

which further yields 2

2

2

2

2

R22

V Z2 V Z2 2V  2[  4E 22 ] , R33  2[  4E33 ] , R44  2[  4E 44 ] , 2 2 ( y  z) ( y  z) ( y  z) 2

R23

V Z2 V Z 2  2[  4E 23 ] , R24  2[  4E 24 ] , 2 ( y  z) ( y  z) 2

R34

V Z 2  2[  4E34 ] .(7.5) ( y  z) 2

2

All above non-vanishing components of Ricci tensor are related as

( y  z) 2 ( y  z) 2 ( y  z) 2 ( y  z) 2 R  R  R  R23 22 33 44 2 Z2 Z2 Z2 ( y  z) 2 ( y  z) 2  R24   R34 .

Z 2 Z 2

(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 [t 2 ( y  z )] -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

1 1 [ViV j  g ij (Vk V k  m 2V 2 )] , [k  1,2,3,4] 4 2 V j where V is scalar function of Z and V j  , ( x  x , y, z, t ) , j x m is mass associated with the massive scalar field.

Tij 

36

(8.1)

[

Thus V2  V3  V4

Z 2



1 2

( y  z )  t ] -type and [t 2 ( y  z )] -type four…

VZ , V1  0 ( y  z)

(8.2)

where ( - ) bar denotes partial derivative w. r. to Z. From line element (1.8) we have

g 22  Z 2 B , 1 g 22   2 , Z B

g33  Z 2 B , 1 g 33   2 , Z B

g 44  B , 1 g 44  B

(8.3) (8.4)

 VkV k  0 .

(8.5)

Therefore equation (8.1) becomes

Tij 

1 1 [ViV j  g ij m 2V 2 ] .

4 2

(8.6)

From equation (8.6) we have

T  Tii  g ijTij 

1 [2m 2V 2 ] .

4

(8.7)

Then from (8.6) and (8.7), Einstein’s field equation (1.11) becomes

Rij  2[ViV j 

1 g ij m 2V 2 ] 2

(8.8)

which further yields 2

R22

V Z2 Z 2 Bm2V 2  2[  ], 2 ( y  z) 2

R44

2V Bm2V 2  2[  ], 2 ( y  z) 2

2

2

R33

V Z2 Z 2 Bm2V 2 V Z2 , R23  2[  2[  ] ], 2 ( y  z) 2 ( y  z) 2

R24

V Z 2 V Z 2  2[ ] , R34  2[ ].

2 ( y  z) ( y  z) 2

2

2

2

(8.9)

But from the line element (1.8) we have the relation of non-vanishing components of Ricci tensor as

( y  z) 2 ( y  z) 2 ( y  z) 2 ( y  z) 2 R  R  R  R23 22 33 44 2 Z2 Z2 Z2  ( y  z) 2  ( y  z) 2  R24  R34 Z 2 Z 2

(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 [t 2 ( y  z )] -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 m  0 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.

2

IX. MASSIVE SCALAR FIELD COUPLED WITH GRAVITATIONAL & ELECTROMAGNETIC FIELD In this case the energy momentum tensor is given by

Tij 

1 1 [ViV j  g ij (Vk V k  m 2V 2 )]  Eij 4 2 37

(9.1)

[

where

Eij

1 2

( y  z )  t ] -type and [t 2 ( y  z )] -type four…

denotes electromagnetic energy momentum tensor.

But as in the previous case for massive scalar field

VkV k  0 and T  Tii  g ijTij 

1 [m 2V 2 ] .

4

(9.2)

Therefore equation (9.1) becomes

1 1 [ViV j  g ij m 2V 2 ]  Eij .

4 2 Hence Einstein’s field equation (1.11) becomes 1 Rij  2[ViV j  g ij m 2V 2  4Eij ] .

2 Tij 

(9.3)

(9.4)

And from equation (9.4) we have 2

R22

V Z2 Z 2 Bm2V 2  2[   4E 22 ] , 2 ( y  z) 2

R44

2V Bm2V 2  2[   4E 44 ] , 2 ( y  z) 2

2

R33

2

V Z2 Z 2 Bm2V 2  2[   4E33 ] , 2 ( y  z) 2 2

R23

V Z2  2[  4E23 ] , ( y  z) 2

2

R24  2[

V Z 2  4E 24 ] , ( y  z) 2

2

R34  2[

V Z 2  4E34 ] . ( y  z) 2

(9.5)

But line element (1.8) gives the relation of non-vanishing components of Ricci tensor as

( y  z) 2 ( y  z) 2 ( y  z) 2 ( y  z) 2  ( y  z) 2 R  R  R  R  R24 22 33 44 23 2 Z2 Z2 Z2 Z 2  ( y  z)2  R34 . .

(9.6) Z 2 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 [t 2 ( y  z )] -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 m  0 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.

2

Acknowledgement The authors are thankful to Professor K. D. Thengane of India for his constant inspiration .

REFERENCES [1] Kadhao S R and Thengane K D (2002)

[2] Jumale J K and Thengane K D (2004) [3] H Takeno(1961)

: “Some Plane wave solutions of field equations

Rij  0

in four

dimensional space-time.” Einstein Foundation International Vol.12,2002, India. :“

[

1 2

( y  z )  t ] -type and [t 2 ( y  z )] -type plane waves

in four dimensional space-time” Tensor New Series Japan, vol.65, No.1, pp.85-89. : “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.

38