EIGEN VALUES OF SIX-DIMENSIONAL SYMMETRIC TENSORS OF CURVATURE AND WEYL TENSORS FOR Z = (z-t)-type PLANE FRONTED WAVES IN PERES SPACE-TIME S.R. BHOYAR, A.G. DESHMUKH * Department o Mathematics, College o Agriculture, Darwha 445 0 (India) E-mail: sbhoyar68@yahoo.com *Ex.Reader and Head, Department o Mathematics, Govt. Vidarbha Institute o Science and Humanities, Amravati, Joint Director, Higher Education, Nagpur Division, Nagpur - 440 0 (India), E-mail: dragd00@yahoo.co.in. Received June, 009 In this paper, we have shown the existence o eigen values o six-dimensional symmetric tensor obtained rom Curvature tensor, weyl conormal curvature tensor and projective curvature tensor respectively or plane ronted waves in peres space-time. Key words: Plane pronted waves, metric tensor, Rimannian curvature tensor, Ricci tensor (R ijkl ), Weyl conormal curvature tensor (C ijkl ) and Projective curvature tensor (W ijkl ).. INTRODUCTION The wave like exact solutions o the Einstein s ield equation are plane, spherical or cylindrical gravitational waves. The plane gravitational waves g ij (Z) are mathematically exposed by H. Takeno (96), in general relativity Takeno [] has deduced the line elements or (z t) and ( t z) - type waves as ds = Adx Ddxdy Bdy dz + dt. (.) and ds = Adx Ddxdy Bdy Z ( C E) dz ZEdzdt + ( C + E) dt (.) respectively. Peres [] deduced the space-time or Z = (z-t) type plane ronted waves as, ds = dx dy dz + dt ( x, y, Z)( dz dt) (.) where Z= (z t) and g ij = g ij (), = (x,y,z), Z = (z,t). Rom. Journ. Phys., Vol. 56, Nos., P. 8 44, Bucharest, 0
Eigen values o six-dimensional symmetric tensors 9 The necessary and suicient condition that space-time (.) satisies ield equation R ij and = +, =, = x x y y. Takeno (96) denote (O) system the original co-ordinate system in which metric is (.) and 0- system is a co-ordinate system in which the metric is o the orm as (.) and calculate non vanishing components o R ijkl and R ij are as ollows R = R = R = (.5) ab ab4 a4b4 ab and R = R4 = R44 =. (a,b =,). Deshmukh and Karade [] have shown the existence o eigen values o sixdimensional symmetric tensor obtained rom Riemannian curvature tensor (R ijkl ), rom Weyl conormal curvature tensor (C ijkl ) and projective curvature tensor (W ijkl ) t or both the (z t) and type plane gravitational waves with the metric (.) z and (.) respectively. In this paper, we want to deduce the same or Z = (z t) type plane ronted waves in peres space-time whose metric is (.).. EIGEN VALUES OF THE SIX-DIMENSIONAL SYMMETRIC TENSOR R αβ = R ijkl FOR Z = (z t)- TYPE WAVES WITH METRIC (.) and Non-vanishing components o R ijkl and R ij rom (.) are R = R4 = R44 = R = R4 = R44 = R = R = R = R = 4 4 44 (.) R = R4 = R44 =. (.) To calculate non vanishing components o symmetric tensor R αβ or (.), let renumbering the indices as () =, () =, (4) =, () = 4, (5) = 5, (4) = 6 The components o (R αβ ) using (.) are R = R = R = R4 = R5 = R4 = R5 = (.) R = R = R =. 44 45 55
40 S.R.Bhoyar, A.G.Deshmukh Also non-vanishing independent component o g αβ are g = g = αβ g g g g g = g = + 45 g = g = 55 g = g = 45 g = g =. 66 ijkl ik jl il kj (.4) The eigen values o six-dimensional symmetric tensor R αβ are given by determinantal equation Rαβ gαβ, + + + +. + ( ) Ater simpliying, the above equation becomes, { ( )( ) ( ) } ( + ) ( ) + ( ) 6. Six-dimensional symmetric tensor R αβ has six zero eigen values. Hence existence o (z t)-type plane ronted waves imply six zero eigen values o six-dimensional symmetric tensor o curvature tensor R αβ R ijkl.. EIGEN VALUES OF SIX-DIMENSIONAL SYMMETRIC TENSOR C αβ = C ijkl FOR Z = (z t) TYPE WAVES FOR METRIC (.) We have, R Cijkl = Rijkl + gkirjl gijrjk g jl Rik g jk R il gil gjk gik g jl. + + 6 (.) ij Now, R = g R ij 4 44 = g R + g R + g R + g R + g R4 + g R44 = ( + 4 + + ). The non-zero component o R ij or (.) are R = R4 = R44 = [Takeno (96) (55.5)]. (.)
4 Eigen values o six-dimensional symmetric tensors 4 C 44 4 Also g = ( ), g = ( + ), g = (.) The non-zero components o R ijkl are R = R4 = R44 =. R = R4 = R44 =. [by (55.4), Takeno (96)]. (.4) R = R4 = R4 = R44 = = Hence using the above results the non-zero components o C ijkl are C = R + [ gr gr + gr gr ] + 0 = + [ gr 0+ 0 0] + 0 = + ( ) =. C = R + g R g R + g R g R [ ] 4 4 4 4 4 4 = =. Similarly, we get 44, [ ] = C =, C4 = C4 =, C = C44 = C 4 = C44 =. (.5) The non-vanishing components o C αβ using (.5) are given as, C = C = C =, C = C = C = C = 4 5 4 5 C44 = C45 = C55 =. (.6)
4 S.R.Bhoyar, A.G.Deshmukh 5 Also non-vanishing components o g αβ are given by (.4). The eigen values or the six-dimensional conormal tensor C αβ are given by determinant equation C g αβ αβ i.e. ( ) 0 + + = + + + Ater simpliying the above equation becomes 6 Six-dimensional symmetric tensor C αβ has six zero eigen values. 4. EIGEN VALUES OF SIX-DIMENSIONAL SYMMETRIC TENSOR W αβ = W ijkl FOR Z = (z t) TYPE WAVES WITH METRIC (.) are, =, the non-vanishing components o W ijkl Using Wijkl Rijkl ( gil Rjk gik Rjl ) W = R g R g R [ ] = 0 ( ) = [ ] W = R g R g R [ ] 4 4 4 4 = [ ] =. Similarly, other component o W αβ are given as
6 Eigen values o six-dimensional symmetric tensors 4 W =, W =, W44 = W4 =, W44 =, W44 =. The non-vanishing components o W αβ are given as, W = W = W =, W4 = W5 = W4 = W5 = W44 = W45 = W55 =. (4.) Also non-vanishing components o g αβ are given by (.4). Eigen values o the six-dimensional symmetric tensor W αβ are given by the determinant equation as, Wαβ gαβ ( + ) + + + + + + + + + + Solving the above determinant equation we obtain 6 Thus the six-dimensional symmetric tensor W αβ has six zero eigen values. 5. CONCLUSIONS. The existence o Z = (z t)-type plane ronted waves in Peres space-time are guranteed by the six zero eigen values o six-dimensional symmetric tensors o Riemannian curvature tensor, Weyl conormal curvature tensor and projective curvature tensor respectively.. This is geometrical aspect or conormation o Z = (z t)-type waves as a plane ronted gravitational waves and g ij o (.) posses plane (wave-like) character.
44 S.R.Bhoyar, A.G.Deshmukh 7. Furthermore this is analogus to one obtain the same or Takeno s spacetime (.)and (.) as waves are purely plane gravitational waves. 4. So geometrically both the space times o Peres and Takeno related to plane wave-like and plane wave solutions o Einstein ield equations o General Relativity or Z = (z t)-type waves. REFERENCES. H. Takeno, The mathematical theory o plane gravitational waves in general relativity, Scientiic report o research institute o theoretical physics, Hiroshima University, Japan, 96.. A. Peres, Some gravitational waves, Phys. Rev. Let., (57 57), 959.. A.G. Deshmukh and T.M. Karade, Some aspects o plane symmetry in general theory o relativity, Ph. D. thesis submitted to Amravati University, Amravati [Page No. 86 98], 00.