Existence Of Z = (zt) Type Plane Gravitational Waves ...

Global Journal of Science Frontier Research

P a g e |20 Vol. 10 Issue 8(Ver1.0),December 2010

Existence Of Z = (z-t) Type Plane Gravitational Waves Carrying Some Energy In Six-Dimensional GJSFR-A (FOR) Space-Time Classification: 020105

S R Bhoyar1, S N Bayaskar2 , A G Deshmukh3 Abstract : This paper, which asserts the existence of Z=(z - t) type plane gravitational waves carrying some energy and momentum in the direction of their propogation in six dimensional space-time.

Keywords : Metric tensor, Plane Gravitational Waves, Energy Momentum Pseudo tensor of Einstein and energy momentum tensor of Landau and Lipschitz. I. INTRODUCTION

In general relativity one of the problem is to be solved is whether or not there exist

gravitational waves carrying energy. H.Takeno [1] mathematically investigated two types of purely plane gravitational waves of Z=(z - t) and Z =  t  type and obtained the line elements for z

both waves respectively in four dimensional space time V4 as, ds 2 = − Adx 2 − 2 Ddxdy − Bdy 2 − dz 2 + dt 2 ,

(1.1)

ds 2 = − Adx 2 − 2 Ddxdy − Bdy 2 − Z 2 (C − E )dz 2 − 2 ZEdzdt + (C + E )dt 2 .

(1.2)

and Furthermore, by calculating non vanishing components of energy momentum pseudo tensors of Einstein and Landua and Lipschitz, Takeno [1], conclude that both the waves carries some energy and momentum in the direction of their propogation. In this paper we propose to obtain the existence of Z=(z - t) type plane gravitational waves carrying some energy and momentum in the direction of their propogation in the sense of Takeno [1] by extending to Six-dimensional spacetime deduced by Adhav and Karade[3]. II.

DEFINITION

A plane wave gij is defined as non flat solution of the field equation Rij = 0 , i, j = 1,2,3,4,5,6.

(2.1)

Vol.10 Issue 8(Ver1.0),December 2010 P a g e | 21


In an empty region of space-time with, g ij = g ij (Z ), Z = Z ( x i ) where x i = ( x, y, u , v, z , t )

(2.2)

In some suitable co-ordinate system such that, g ij Z ,i Z , j = 0

∂Z    Z ,i =  , ∂xi  

(2.3) (2.4)

Z = Z ( z , t ), Z ;5 ≠ 0, Z , 6 ≠ 0 The signature convention adopted is, g aa < 0

g aa g ba

g ab > 0, g bb

g11

g12

g13

g14

g 21

g 22

g 23

g 24

g 31

g 32

g 33

g 34

g 41

g 42

g 43

g 44

> 0,

g11 g 21 g 31

g12 g 22 g 32

g13 g 23 < 0 g 33

g11

g12

g13

g14

g15

g 21

g 22

g 23

g 24

g 25

g 31

g 32

g 33

g 34

g 35 < 0,

g 41

g 42

g 43

g 44

g 45

g 51

g 52

g 53

g 54

g 55

g 66 > 0 ,

(2.5)

(Not summed for a & b; a,b=1,2,3,4,5) , and accordingly g = det(gij) < 0.

(2.6) III.

THE PLANE WAVE METRIC

We adopt the space-time deduced by Adhav and Karade [3] for Z=(z-t) type plane gravitational waves in six-dimensional space-time, ds 2 = −dl 2 − (C − E )dz 2 + 2 Edzdt + (C + E )dt 2 ,

(3.1)

(

dl 2 = − Adx 2 − 2 Ddxdy − 2Gdxdu − 2 Idxdu − 2 Idxdv − Bdy 2 − 2 Hdydu − 2 Idydv − Edu 2 − 2 Kdudv − Ldv 2

)

.

It is known that E in (3.1) can be transformed away by suitable transformation of co-ordinate, so in new system. ds 2 = −dl 2 − Cdz 2 + Cdt 2 .

(3.2)


By using transformation z1 + t 1 = z + t


and z 1 + t 1 = ∫ C ( z )dz , (3.2) can be transformed into

simpler form

ds 2 = −dl 2 − dz 2 + dt 2 .

(3.3)

The components of metric tensor gij and gij for (3.3) are as follows − A − D − D − B  − G − H g ij =  −I − J  0 0  0  0

−G −H −F −K 0 0

[ ]

−I −J −K −L 0 0

0 0 0 0 −1 0

0 0 0  , 0 0  1

(3.4)

where A > 0, B > 0, F > 0, L > 0 and m = ABFL − ABK 2 − AH 2 L + 2 AHJK = AJ 2 F − D 2 FL + D 2 K 2 + 2 DGHL − 2 DHIK − 2 DGJK _ 2 DFIJ − BG 2 L + 2 BGIK

(3.5)

+ G 2 J 2 − 2GHIJ − BFI 2 + H 2 I 2 n = -1  − A′ − D′ − G′ − I ′ 0  − D′ − B ′ − H ′ − J ′ 0   − G′ − H ′ − F ′ − K ′ 0 g ij =   − I ′ − J ′ − K ′ − L′ 0  0 0 0 0 −1  0 0 0 0  0 where, A′ = B′ =

D′ =

(− BFL + B K + H 2

2

0 0 0  , 0 0  1

L − 2 HJK + FJ 2 )

m

(− AFL + AK

(DFL + DK

2

2

+ G 2 L − 2GJK + FI 2 ) , m

)

− HGL + HIK + GJK , m

,

(3.6)



G′ =

(− DHK + DJK + BGL − BIK − GJ

2

+ HIJ )

m

,

H′ =

(AHL − AJK − DGL + DIK + GIJ − HI ) ,

I′ =

(DHK − DFJ − BJK + BFI + HGI − H I ) ,

J′ =

(− AHK + AFI + DGK − DFI − G I + GHI ) ,

K′ =

(ABK − AHJ − D K + DHI + DGJ − BGI ) ,

F′ = L′ =

2

m

2

m

2

m

2

m

(− ABL + AJ

+ D 2 L − DIK + BI 2 ) , m

2

(− ABF + AH

2

+ D 2 F − 2GDH + BG 2 ) . m

The non-vanishing components of christoffel symbols for metric (3.3) are, 1 1   = −  = P , 16 15

3 3   = −   = P′ 16 15

1 1   = −  = Q , 26 25

3 3   = −   = Q′ 25 26

1 1   = −  = R , 36 35

3 3   = −   = R′ 36 35

1 1   = −  = S , 46 45

3 3   = −  = S ′ 46 45

2 2   = −  = T , 16 15

4 4   = −  = T ′ 16 15

2 2   = −  = U , 26 25

4 4   = −  = U ′ 25 26

2 2   = −  = V , 36 35

4 4   = −  = V ′ 36 35

(3.7)


2 2   = −  = W , 46 45


4 4   = −  = W ′ 46 45

6 5 A   = −  = − , 2 11 11

6 5 D   = −  = − 2 12 12

6 5 G   = −  = − , 2 13 13

6 5 I   = −  = − 2 14 14

6 5 B   = −  = − , 2 22 22

6 5 H   = −  = − 2 23 23

6 5 J   = −  = − , 2 24 24

6 5 F   = −  = − 2 33 33

6 5 K   = −  = − , 2 34 34

6 5 L   = −  = − . 2 44 44

Where P=

A′A + D′D + G′G + I ′I , 2m

P′ =

F ′G + K ′I + G′A + H ′D 2m

Q=

A′D + D′B + G′H + I ′J , 2m

Q′ =

F ′H + G′D + H ′B + K ′J 2m

R=

A′G + D′H + G′F + I ′K , 2m

R′ =

F ′F + G′G + H ′H + K ′K 2m

S=

A′I + D′J + G′K + I ′L , 2m

S′ =

F ′K + K ′L + H ′J + G′I 2m

T=

B′D + H ′G + J ′I + D′A , 2m

T′ =

L′I + I ′A + J ′D + K ′G 2m

U=

B′B + D′D + H ′H + J ′J , 2m

U′ =

L′J + I ′D + J ′B + K ′H 2m

V=

B′H + D′G + H ′F + J ′K , 2m

V′ =

L′K + K ′F + J ′H + I ′G 2m

W=

B′J + D′I + H ′K + J ′K , 2m

W′ =

L′L + I ′I + J ′J + K ′K . 2m



Pawar et.al [5] investigated coexistence of Z= (z - t) type plane gravitational waves with electromagnetic waves in six-dimensional space-time and obtained exact solution of field equation (2.1) as,

{

N = 8π σ 2 A′ − 2σρD′ − 2η G′ − 2σξ I ′ + ρ 2 B′ + 2 ρηH ′ + 2 ρξ J ′ +

η 2 F ′ + 2ηξK ′ + ξ 2 L′} . IV.

(3.8)

SOME USEFUL FORMULAE

For metric (3.3) and from (3.7) we made following formulae which are useful to calculate the components of pseudo tensors.

− g = m,

(

)

m m   ,− − g , i =  0,0,0,0,  2m 2m  

i  m i  1   1   i   i  ,   =   =   =   = 0,   = −  = 6i  2m 5i  1i  2i  3i  4i  a b   = − , b5 a5

r  g pq   = 0,  pq 

(4.1) (4.2)

 r  s  g pq    = 0  ps qr 

 b  q   b  a   b  a   p  d     = 0,    = −   =   , 5a 6b  6a 6b  5a 5b  ac bp 

(4.3)

 q  p   q  p   q  p     = −    =    . 5 p 6q  6 p 6q  5 p 5q  Where, m = {− A′A − 2 D′D − 2G′G − 2 I ′I + B′B − 2 H ′H − J ′J − F ′F − 2 K ′K − L′L } and

i = 1,2,3,……. ,6 a, b, c, d = 1, 2, 3, 4; p, q, r, s = 5,6 and summation convention is used

with respect to these indices. V.

PSEUDO-TENSOR OF EINSTEIN

We calculate the components of energy-momentum Pseudo-tensor ti j introduced by Einstein which satisfies the conservation relation in the form



{ − g (T

j

i

}

∂   + ti j ) , j = 0  , j = j , ∂x  

(i

). j =, 1 ,23 , 6 ,. . ...

(5.1)

When Ti j is energy momentum tensor of matter coincides with Ei j According to Einstein theory the ti j expresses the energy momentum due to gravitational field and is given by

(

) (

 j  16π − g ti j =   − g g mn ,i − log − g mn

)

,m

+

(

)

  h  k   h δ i j    g mn − g − g mn   − g ,h  . mn mk nh  

(5.2)

If we substitute (3.3) and (3.7) into (5.2) and using (4.1), (4.2) and (4.3) we obtain following result; t55 =

β , 16πm

t56 = t65 = t66 =

−β , 16πm

(5.3)

β . 16πm

Where

β =β = {12{D G ′−−2 I2′II −I ′2−J ′J2 J− 2JK′ −′K2−KAK ′ +′ − ′L }F F ′ + L L′} 2 DD′D′ −− 2G′G A ′′−−BAB A F FB′ +B ′L+ 1 22

And other ti j = 0 . From above, we have t55 = −t56 = −t65 = t66 = φ , where

16πφ =

β m

.

Here φ is a function of Z and does not Vanish in general.

(5.4) (5.5)


Global Journal of Science Frontier Research VI.

. Pseudo-tensor of Landau and Lifshitz

Next we calculate the components of the symmetric energy momentum pseudo-tensor t*ij proposed by Landau and Lipshitz which satisfies the conservation relation in the form

{(− g )(T

ij

+ t *ij )}ij = 0,

(i,j=1,2,3,….6)

and it is given by expression   h  m   m  h   h  m  16π t *ij = g ik g jl − g ij g kl 2   −    −     kl hm khlm khlm

(

)

  i  h   i  h   j  h   j  h  + g ik g mn    +    −    −    khmn mnkh nhkm kmnh

(6.1)

  i  h   i  h   i  h   i  h  + g jk g mn    +    −    −    khmn mnkh nhkm kmnh   i  j   i  j  − g hk g mn    −   , hmkn hk mn

(t

*ij

= t * ji

)

.

If we substitute (3.3) and (3.7) into expression (6.1) and using (4.1), (4.2) and (4.3) we obtain following result : t *55 = t *56 = t *65 = t *66 = φ *, other t *ij = 0,

(6.2)

where m2    β + m  16π φ * = − , m and

(6.3)

) (FL − K ) + (A F − G )(BL − J ) + (A L − I )(BF − H ) (F B − D )(AL − I ) + (L B − J )(AF − G ) + (F L − K )(AB − D ) (

β = [ AB − D

2

2

2

2

2

2

2

2

2

2

2

2

) ) ) ( ( ( + 2(B G − H D ) (IK − LG ) + 2(B I − D J ) (GK − FI ) + 2(B K − J H ) (IG − AK ) + 2(F D − H G ) ( JI − LD ) + 2(F I − K G ) ( JD − BI ) + 2(F J − H K ) (ID − AJ ) + 2(L D − I J ) (GH − FD ) + 2(L H − K J ) (GD − AH ) + 2(L G − I K ) (HD − BG ) + 2 A B − DG ( JK − LH ) + 2 A J − D I ( JF − KH ) + 2 A K − I G ( JH − BK )

+ 2 H I (2 HI − DK − JG ) + 2G J (2GJ − DK − HI ) + 2 D K (2 DK − GJ − HI ) ].

Again φ * is not zero in general. To compare results (5.4) with (6.2) we introduce the quantity tij in the coordinate system, as, g ik t kj = t ij = t ij .

(6.5)



Then we have from (5.4) =t 55. t 56 = t 65 = t 66 = φ =

−β 16πm

(6.6)

Here we find that tij and t*ij are composed of similar components but they are not same. When the field equations (3.8), is satisfied, we have N=

m m2 β − − = 8π σ 2 A′ − 2σρD′ − 2ηG′ − 2σξI ′ + ρ 2 B′ + 2 ρηH ′ + 2 2m 4m 2m

(

2 ρξJ ′ + η 2 F ′ + 2ηξK ′ + ξ 2 L′

)

(6.7)

.

Hence we have,  m m2  , − 16πφ = 2 N − + 2m 4m 2  

(6.8)

m   and 16πφ* = 2 N −  . 2m   Also when space-time is empty, we have N=0, φ and φ* becomes respectively m2   − m +  2m   , − 16πφ = m 16πφ* = −

(6.9)

m . 2m

Again these values are functions of Z and do not vanish in general.

VII.

CONCLUSION

We conclude that, the energy momentum pseudo tensors (5.1) or (6.1) which expresses the energy momentum due to gravitational field hence the gravitational waves given by (3.3) carry some energy and momentum in the direction of their propogation.

VIII.

REFERENCES

[1] Takeno H (1961) : ‘The mathematical theory of plane gravitational waves in general relativity’, Scientific report of research institute of theoretical physics, Hiroshima University, Japan.



[2] Landau L. and Lifshitz E. (1951) : The classical theory of fields (Cambridge, 1951). P.No.318. [3] Adhav K.S. and Karade T.M. (1994) : Plane gravitational waves in higher dimensional space-time-II : POST RAAG REPORT NO,279, October (1994). [4] Moller C. (1952) : The theory of relativity, (Oxford, 1952), P. 341. [5] Pawar D.D. Bhaware S.W. and Deshmukh A.G. (2006) : Co-existence of plane gravitational waves electromagnetic waves for Z=(z-t) in higher dimensional space-time-I PROC. NAT. ACAD. SCI. INDIA, 76(A), 11, 2006. P.No. (133-138) [6] Rosen N. (1956) : Gravitational waves, Helv. Phys. Acta, Supplementum IV. [7] Takeno H. (1961) : On the space time of Peres, Tensor, New Series, 11.