Dyadic Analysis of Curvature and Weyl Tensors Bharti Saxena*, Basant Kumar Singh Department of Mathematics, AISECT University Bhopal (MP)

INTERNATIONAL JOURNAL OF SCIENTIFIC PROGRESS AND RESEARCH (IJSPR) ISSN: 2349-4689 Issue 88, Volume 31, Number 02, 2017

… … … . . Abstract-In the present paper some results are obtained Dyadic Now being a world scalar …… components of the curvature tensor also Dyadic analysis is … . . . … . …. ……..,[ ]= 𝑡𝑡 done for Weyl Tensors. The obtained results are generalized Hence successive𝑡𝑡 intrinsic differentiation gives, 𝑇𝑇 𝑇𝑇 𝑢𝑢 𝜇𝜇𝜇𝜇 𝑐𝑐 form of well known results in the field of general theory of …….. 𝑢𝑢 …..Ʃ ……. …… relativity. ….. ……,[ ] = [ ] …. …., = . …… ….., … … … 1.2 𝑡𝑡 𝑡𝑡 𝑞𝑞 𝑡𝑡 Keywords: Dyadic Analysis, Weyl Tensor. Intrinsic𝑇𝑇 𝑢𝑢 differentiation𝑟𝑟𝑟𝑟 ⎾ 𝑠𝑠𝑠𝑠 𝑇𝑇 of𝑢𝑢 the𝑞𝑞 relation𝛬𝛬 𝑟𝑟𝑟𝑟 𝑇𝑇 𝑢𝑢 𝑞𝑞 I. INTRODUCTION 1 . = 2 ; ; 𝜇𝜇 𝜈𝜈 In the present work an attempt has been made to study 𝑟𝑟 𝑙𝑙 𝑙𝑙 𝑠𝑠𝑠𝑠 𝑟𝑟 𝑟𝑟 𝛬𝛬 � 𝜇𝜇 𝜈𝜈 − 𝜈𝜈 𝜇𝜇 � 𝑡𝑡 𝑠𝑠 certain problems of general theory of relativity in terms of and complete antisymmetrication𝑙𝑙 𝑙𝑙 with respect to Lorentz Dyadics. These dyadics are expressed with respect to the indices give orthogonal space triad of an orthonormal space timetetrad .[ ] = 2 … … … … … … … … … … … … … … . .1.3 set up at a point. Under certain orthogonal transformation . . , 𝑞𝑞 𝑟𝑟 𝑟𝑟 of the tetrad field for which the time-like congruence do 𝑠𝑠𝑠𝑠𝑠𝑠 �𝑟𝑟𝑟𝑟𝛬𝛬 𝑡𝑡�𝑞𝑞 not change the three space-time congruences change so 𝛬𝛬 𝛬𝛬 that the 3-dyadics and 3-vectors are introduced retain their a set of 16 differential identities which are integrability character and interpretation. The equations set up in this conditions of . as the world - lines of a whole set of observers. It has 𝑟𝑟been shown that vacuum space-times with formation bear a close relationship to corresponding 𝑠𝑠𝑠𝑠 𝛬𝛬 equations in Newman- Penrose Spinor formalism with petrov types I II or D weyl tensors can not be complex respect to a nulltetrad. According to following theorem by recurrent. Geroch namely "A necessary and sufficient condition for In chapters IV an V we consider the motion of currents and a non-compact space time to admit of a spin structure break up the strangled components of tensor that has a global field of orthonormal tetrads", it is 1 = always possible to introduce𝑀𝑀 orthonormal tetrads and𝐿𝐿 [ ; ] 6 , hence𝑀𝑀 3- dyadics and 3-vectors as suggested above, 𝑅𝑅 𝐽𝐽𝜆𝜆𝜆𝜆𝜆𝜆 𝑅𝑅𝜈𝜈 𝜆𝜆 𝜇𝜇 − 𝑔𝑔𝜈𝜈�𝜆𝜆 𝜇𝜇� whenever null-tetrads exist. We begin by introducing the in two 3-vectors and and two traceless 3-dyauics and notation of a dyadic and usual operation on it. for gravitational fields interacting with matter or with electromagnetic 𝐽𝐽fields.𝐾𝐾 In the former case, we 𝐽𝐽have 1.1 Differentiation in 3-space of auxiliary triad. considered𝐾𝐾 the perfect null and in the latter, both null and The intrinsic derivatives along the congruence of the non-null electromagnetic fields. orthonormal tetrad are given by It is convenient to introduce a 3-vector operator to denote . . …. covariant differentiation in the 3-space of the auxiliary ……. … … … = …… + + ……. ; ………. ; … space-like congruences from equation 2.1: 𝛼𝛼 𝑡𝑡 𝜇𝜇 𝜎𝜎 𝑡𝑡 … . . 𝑙𝑙… . 𝑙𝑙 𝑙𝑙 𝑡𝑡 𝑞𝑞 𝑇𝑇 𝜇𝜇 𝜎𝜎 𝑇𝑇 𝜇𝜇 𝑠𝑠 𝑇𝑇 𝑢𝑢 . . � � 𝑋𝑋 … … …𝑌𝑌 …𝑠𝑠 … … … … … … … … … … … . . … … .1.1 = + . + … … … … … … … 1.4 …… ⎾𝑠𝑠𝑞𝑞 . ℎ ℎ 𝑞𝑞 𝑡𝑡 𝑘𝑘 𝑖𝑖𝑖𝑖 𝑘𝑘 𝑖𝑖𝑖𝑖 𝑖𝑖 ℎ𝑗𝑗 𝑖𝑖ℎ ⋯ 𝑇𝑇 𝑞𝑞 Where∇ 𝛿𝛿 are𝐷𝐷 𝛿𝛿the ordinary𝛿𝛿𝑘𝑘 𝐶𝐶 derivatives𝛿𝛿𝑘𝑘 𝑗𝑗 𝐶𝐶 in the 3-space and ⎾𝑠𝑠𝑠𝑠 Which are world scalars in which play the same role are components of the 3-vector operator which define 𝐷𝐷𝑘𝑘 as the coefficients of affine connections𝑡𝑡 in Riemannian gradient, divergence and cunl. For the operator , we have space. But while the latter are symmetric⎾𝑟𝑟𝑟𝑟 in the first two the following formulae. If is a 3-vector𝐷𝐷 and be a indices (for holonomic co-ordinates), are symmetric ∇ antisymetric in the last two indices and have only𝑡𝑡 24 𝑉𝑉 𝑀𝑀 components compared to 40 in the case of former. ⎾𝑟𝑟𝑟𝑟 3-dyadics, 1 = ( ) × 1 × … … … . .1.5 2 ∇ 𝜈𝜈. =𝐷𝐷 𝜈𝜈 − �2𝑁𝑁 − . …𝑡𝑡…𝑟𝑟 𝑁𝑁…𝑖𝑖…−…𝐿𝐿… … �… …𝜈𝜈… … … … 1.6

∇ 𝜈𝜈 𝐷𝐷 𝜈𝜈 − 𝐿𝐿 𝜈𝜈 × = × . × … … … … … … … … .1.7

∇ × 𝜈𝜈 = 𝐷𝐷 𝑣𝑣 − 𝑁𝑁 𝜈𝜈 − 𝐿𝐿 𝜈𝜈 1 × 2 . × + × 1 + ( ) + ( ) + ( . )1 2 1∇ 𝑀𝑀 ( )( )1 … … … … … … … … … … … … … … . …𝑟𝑟… .1.8 𝑟𝑟 2𝐷𝐷 𝑀𝑀 − 𝑀𝑀𝑀𝑀 − 𝑁𝑁 𝑀𝑀 − 𝐿𝐿 𝑀𝑀 𝐿𝐿 𝑀𝑀 𝑡𝑡 𝑁𝑁 𝑀𝑀 𝑡𝑡 𝑀𝑀 𝑁𝑁 𝑁𝑁 𝑀𝑀 − 𝑡𝑡𝑟𝑟 𝑁𝑁 𝑡𝑡𝑟𝑟 𝑀𝑀 www.ijspr.com IJSPR | 197 INTERNATIONAL JOURNAL OF SCIENTIFIC PROGRESS AND RESEARCH (IJSPR) ISSN: 2349-4689 Issue 88, Volume 31, Number 02, 2017

1 × × = × × 3 . 3 . × + × + ( ) + 2( ) + 2( . )1 2 ( )( )1 … … … … … … … … … … … … … … … … … 1.9 ∇ 𝑀𝑀 − 𝑀𝑀 ∇ 𝐷𝐷 𝑀𝑀 − 𝑀𝑀 𝐷𝐷 − 𝑀𝑀 𝑁𝑁 − 𝑁𝑁 𝑀𝑀 − 𝑀𝑀 𝑀𝑀 𝐿𝐿 𝑡𝑡𝑟𝑟 𝑁𝑁 𝑀𝑀 𝑡𝑡𝑟𝑟 𝑀𝑀 𝑁𝑁 𝑁𝑁 𝑀𝑀 − 𝑡𝑡 𝑟𝑟 𝑁𝑁 𝑡𝑡 𝑟𝑟 𝑀𝑀 . = . 3 . × + ( ) … … . 1.10

∇ 𝑀𝑀 𝐷𝐷 𝑀𝑀 − 𝐿𝐿 𝑀𝑀 − 𝑁𝑁 𝑀𝑀 𝑡𝑡𝑟𝑟 𝑀𝑀 2.2. Dyadic components of the curvature tensor: The strangled components of the curvature tensor can be split into following manner.

Containing two indices equal to zero.𝑅𝑅 𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆

𝑅𝑅𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 Containing one index equal to zero.

𝑅𝑅𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 Containing no index equal to zero. Here indices𝑅𝑅ℎ𝑖𝑖𝑖𝑖 𝑖𝑖are Lorentz indices denoting components with respect to O N T. And , , , take the values 1, 2, 3 strangled components can but be calculated using 1 1 1 ℎ 𝑖𝑖 𝑗𝑗 𝑘𝑘[ | |, ] = + [ ] .. + … … … … … … … … … .1.11 2 .. 2 .. 2 𝑡𝑡 𝑠𝑠𝑠𝑠 𝑘𝑘 𝑝𝑝𝑝𝑝𝑝𝑝 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡 𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝 𝑞𝑞 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 Where [ | | ] means⎾ and r are⎾ excluded⎾ 𝑞𝑞 − from⎾ antisymmetrisation.⎾ 𝑞𝑞 ⎾ ⎾𝑞𝑞 𝑅𝑅

Here follow𝑡𝑡 𝑠𝑠𝑠𝑠 the𝑝𝑝 component𝑠𝑠 of curvature Tensor. 0101 001,1 101,0 112 00 113 00 012 10 013 10 101 .01 102 .01 103 .01 = 2 3 + 2 + ..3 1 2 3 012 .01 013 .01 + 2 + 3 𝑅𝑅 ⎾ − ⎾ − ⎾ ⎾− − ⎾ ⎾− ⎾ ⎾− ⎾ ⎾ − ⎾ ⎾ − ⎾ ⎾ − ⎾ ⎾ = + + 2 + 2( 3 2 ) + 2 + ( 2)2 + 0101 11 ⎾ 1,1⎾ 2( 13⎾ 2) ⎾ 3( 12 + 3) 11 12 12 13 13 12 13 1 3 2 ( ) = 11 𝑅𝑅 −𝑠𝑠 𝑎𝑎 𝑎𝑎 𝑁𝑁 −𝑙𝑙 − 𝑎𝑎 𝑁𝑁 𝑙𝑙 − 𝑠𝑠 − 𝑠𝑠 𝑠𝑠 − 𝑠𝑠 𝑠𝑠 𝜔𝜔 𝑠𝑠 − 𝜔𝜔 𝑠𝑠 𝑎𝑎 𝛬𝛬 1 = = + + + ( ) ( ) + ( + ) 𝛬𝛬 𝑄𝑄 0203 0302 23 2 2,3 3,2 1 22 33 2 12 3 3 12 2 12 13 ( + ) + 2 1 3 + 1 + 2 3 = 23 22 𝑅𝑅 33 𝑅𝑅12 −22𝑠𝑠 �𝑎𝑎 13 𝑎𝑎 33 𝑎𝑎 𝑁𝑁2 3 − 𝑁𝑁 − 𝑎𝑎 23𝑁𝑁 − 𝑙𝑙 𝑎𝑎 𝑁𝑁 𝑙𝑙 � − 𝑠𝑠 𝑠𝑠 − 1 = + ( + + ) + ( + + ) ( + + 𝑠𝑠 𝑠𝑠 𝑠𝑠0123 𝑠𝑠 12𝜔𝜔,3 − 𝑠𝑠13,2𝜔𝜔 − 𝜔𝜔11𝑠𝑠 11 𝜔𝜔12𝑠𝑠 12 𝑎𝑎 13𝑎𝑎 −13𝛬𝛬 𝛬𝛬 2 13𝑄𝑄 3 12 2 11 22 33 11 11 22 33 11− 11 11+2 12 12+2 13 13+ 22 22+2 23 23+ 33 33+12 11+ 22+ 33 11+ 22+ 33+ 1,− 12+ 3 3𝑅𝑅+( 13−�𝑠𝑠2) 2−+2𝑠𝑠 1� 1−𝑠𝑠1 𝑁𝑁1+ 2𝑠𝑠 2𝑁𝑁+ 3 3𝑠𝑠=𝑁𝑁11 𝑙𝑙 𝑆𝑆 − 𝑙𝑙 𝑠𝑠 − 𝑁𝑁 𝑁𝑁 𝑁𝑁 𝑠𝑠 − 𝑆𝑆 𝑆𝑆 𝑆𝑆 𝑁𝑁 𝑆𝑆 𝑁𝑁 𝑆𝑆 𝑁𝑁 𝑆𝑆 𝑁𝑁 𝑆𝑆 𝑁𝑁 𝑆𝑆 𝑁𝑁 𝑆𝑆 𝑁𝑁 𝑆𝑆 𝑆𝑆 𝑆𝑆 𝑁𝑁 𝑁𝑁 𝑁𝑁 𝛬𝛬 𝑁𝑁 1 𝑙𝑙 𝛬𝛬 𝑁𝑁 = 𝑙𝑙 𝛬𝛬 𝑎𝑎 +𝛬𝛬 3𝑎𝑎+𝛬𝛬( 𝑎𝑎 +𝛬𝛬 )𝑎𝑎( 𝛬𝛬 + 𝐵𝐵3) + ( + )( 2) ( ) + 0112 11,2 12,1 11 23 1 12 2 11 22 33 13 13 2 22 1 ( + )( + 1) ( 3) + ( + )( 3) + ( + )( + 2) + ( + 3) + 12 𝑅𝑅3 23 𝑆𝑆 − 𝑆𝑆1 12 𝛬𝛬 𝑁𝑁 23 𝐿𝐿 1 𝑆𝑆 12 𝛬𝛬 2𝑁𝑁 11− 𝑁𝑁 22 𝑁𝑁 33 𝑆𝑆 13− 𝛬𝛬 − 𝑁𝑁 1 −12𝐿𝐿 𝑆𝑆 1 ( ) + ( + + )( + 2) = + 2 𝑁𝑁13 𝐿𝐿2 𝑆𝑆11 2𝛬𝛬 11− 𝑎𝑎 22𝑆𝑆 − 𝛬𝛬33 13 𝑁𝑁 𝐿𝐿 𝑆𝑆13 − 𝛬𝛬 𝑁𝑁 − 𝑁𝑁 𝑁𝑁 𝑆𝑆 𝛬𝛬 𝑎𝑎 𝑆𝑆 𝛬𝛬 1 𝑁𝑁 − 𝐿𝐿 𝑆𝑆 = 𝑁𝑁 𝑁𝑁 𝑁𝑁3 𝑆𝑆 ( 𝛬𝛬 )(𝐵𝐵 𝑡𝑡 3) + ( + )( + 1) + ( + ) 0221 22,1 12,2 12 13 2 12 2 11 22 33 23 23 1 11 1 ( )( 2) ( + 3) ( )( + 3) + ( + )( 1) = + 1 12 3𝑅𝑅 13 𝛿𝛿 − 𝛿𝛿2 12− 𝛬𝛬 − 𝑁𝑁13 − 𝑙𝑙2 𝑆𝑆12 − 𝛬𝛬 2 𝑁𝑁11 − 𝑁𝑁22 𝑁𝑁33 𝑆𝑆23 𝛬𝛬 𝑁𝑁 23 𝑙𝑙 𝑆𝑆 − 1 𝑁𝑁 − 𝑙𝑙 𝑆𝑆 =− 𝛬𝛬 − 𝑎𝑎 𝑆𝑆 + 𝛬𝛬2 +− (𝑁𝑁 +− 𝑙𝑙 )(𝑆𝑆 + 𝛬𝛬2) (𝑁𝑁 − 𝑁𝑁 𝑁𝑁 )(𝑆𝑆 − 𝛬𝛬1) (−𝐵𝐵 𝑡𝑡) + 0331 33,1 13,3 13 12 3 13 2 11 22 33 23 23 1 11 1 ( + )( + 3) ( 2) + ( + )( 2) ( )( + 1) + ( + 2) + 13 2𝑅𝑅 12 𝛿𝛿 − 𝛿𝛿3 13 𝛬𝛬 𝑁𝑁12 𝑙𝑙3 𝑆𝑆13 𝛬𝛬 − 2 𝑁𝑁11 − 𝑁𝑁22 − 𝑁𝑁33 𝑆𝑆23 − 𝛬𝛬 − 𝑁𝑁3 13− 𝑙𝑙 𝑆𝑆 1 ( ) + ( + )( + 1) = + 1 𝑁𝑁23 𝑙𝑙1 𝑆𝑆33 2𝛬𝛬 11− 𝑎𝑎 22𝑆𝑆 − 𝛬𝛬33 23 𝑁𝑁 𝑙𝑙 𝑆𝑆23 − 𝛬𝛬 − 𝑁𝑁 − 𝑁𝑁 − 𝑁𝑁 𝑆𝑆 𝛬𝛬 𝑎𝑎 𝑆𝑆 𝛬𝛬

𝑁𝑁2323−=𝑙𝑙 𝛿𝛿223,3 𝑁𝑁 323𝑁𝑁,2 +− 𝑁𝑁330 𝑆𝑆220 𝛬𝛬 331𝐵𝐵 221𝑡𝑡 23 320 + 231 321 + 320 023 321 123 32 323 23 023 + 23 123 + 232 223 𝑅𝑅 ⎾ − ⎾ ⎾ ⎾ − ⎾ ⎾ − ⎾ ⎾ ⎾ ⎾ ⎾ ⎾1 − ⎾ ⎾ − = ( ) + ( + ) + + ( )( + ) ( 1)( + 1) ( + ⎾ ⎾ −12⎾ 3⎾13 ⎾13 ⎾2 12 ⎾22 33 ⎾ 23 1 23 1 23 23 4 11 22 33 11+ 22− 33− 1 23− 1−14 11+ 22− 33 11− 22− 33− 13+ 22+ 1 23+ 1−14 11− 22+ 33 11− 22−−𝑁𝑁 33−−𝑙𝑙 12− 32𝑁𝑁 𝑙𝑙 𝑆𝑆 𝑆𝑆 𝑁𝑁 − 𝑙𝑙 𝑁𝑁 𝑙𝑙 − 𝑆𝑆 − 𝛬𝛬 𝑆𝑆 𝛬𝛬 − 𝑁𝑁 − 𝑁𝑁 𝑁𝑁 𝑁𝑁 𝑁𝑁 𝑁𝑁 𝜔𝜔 𝑆𝑆 𝛬𝛬 𝑁𝑁 𝑁𝑁 𝑁𝑁 𝑁𝑁 𝑁𝑁 𝑁𝑁 𝑁𝑁 𝑙𝑙 𝜔𝜔 𝑆𝑆 𝛬𝛬 𝑁𝑁 𝑁𝑁 𝑁𝑁 𝑁𝑁 1 𝑁𝑁 𝑁𝑁 𝑁𝑁 𝑙𝑙 1 1 = { × × ( + )} + + + ( × ) + 1 1 + 1 1 + 1 1 + 2 2 2 + 2 ,1 1 1 2 11 4 11 22 33 2 22 33− 232 ∇ 𝑁𝑁 − 𝑁𝑁 ∇ − ∇𝐿𝐿 𝐿𝐿∇ ∇𝐿𝐿 𝐿𝐿 𝐿𝐿 𝑆𝑆 𝑆𝑆 𝛬𝛬 𝛬𝛬 𝜔𝜔 𝛬𝛬 𝛬𝛬 𝜔𝜔 �𝑁𝑁 − 𝑁𝑁 − 𝑁𝑁 1 1 1 1 = { × × ( + )} + + ( × ) + 1 1 + 2 1 1 + ( ) + ( × ) + . 𝑁𝑁2 𝑁𝑁 𝑁𝑁 11 1 1 2 × 11 2 11 2 × 11 1 2 ( ) = 𝑟𝑟 4 ∇ 𝑁𝑁 − 𝑁𝑁11 ∇ − ∇𝐿𝐿 𝐿𝐿∇ 𝐿𝐿 𝐿𝐿 𝑆𝑆 𝑆𝑆 𝛬𝛬 𝛬𝛬 𝜔𝜔 𝛬𝛬 𝑡𝑡 𝑁𝑁 𝑁𝑁 𝑁𝑁 𝑁𝑁 �∇ 𝐿𝐿 − 𝑡𝑡𝑟𝑟 𝑁𝑁 � −𝑃𝑃 www.ijspr.com IJSPR | 198 INTERNATIONAL JOURNAL OF SCIENTIFIC PROGRESS AND RESEARCH (IJSPR) ISSN: 2349-4689 Issue 88, Volume 31, Number 02, 2017

1 1 = ( + ) ( + ) + ( 3)( 2) + ( + )( ) 3112 12 3 ,2 2 11 22 33 ,1 12 13 2 23 1 11 22 33 1 𝑅𝑅 − 𝑁𝑁 𝐿𝐿 (− +𝑁𝑁 1−) 𝑁𝑁( 𝑁𝑁 )( 𝑆𝑆 −)𝛬𝛬 𝑆𝑆2( − 𝛬𝛬 3) + 𝑁𝑁( 𝐿𝐿+ 𝑁𝑁)( − 𝑁𝑁 − 𝑁𝑁+ ) 11 12 13 2 12 3 12 2 23 1 11 22 33 1 − 𝑆𝑆 ( 𝑆𝑆 𝛬𝛬 +− 𝑁𝑁)( − 𝐿𝐿 𝑁𝑁) +− 𝐿𝐿2( − 𝜔𝜔+ 𝑆𝑆3) −(𝛬𝛬 )𝑁𝑁( +𝐿𝐿 )𝑁𝑁 − 𝑁𝑁 𝑁𝑁 2 11 22 33 23 1 12 13 2 12 3 1 −+ (𝑁𝑁 − 𝑁𝑁 )( 𝑁𝑁 𝑁𝑁 − 𝐿𝐿 ) = 𝜔𝜔 𝑆𝑆 𝛬𝛬 − 𝑁𝑁 − 𝐿𝐿 𝑁𝑁 𝐿𝐿 2 23 1 11 22 33 23 1 1 = ( ) + ( 𝑁𝑁 +− 𝐿𝐿 𝑁𝑁 −) 𝑁𝑁+ (− 𝑁𝑁+ 2).−(𝑃𝑃 + 3) + ( )( ) 2113 13 2 ,3 2 11 22 33 ,1 13 12 2 23 1 11 22 33 1 1 ( 1) ( + )( + ) + 3( + 2) + ( )( + ) ( + )( + 𝑅𝑅11 23 𝑁𝑁 − 𝐿𝐿 12 3 𝑁𝑁 13 𝑁𝑁 2− 𝑁𝑁 13 𝑆𝑆 𝛬𝛬 2 𝑆𝑆23 𝛬𝛬1 11 𝑁𝑁 22− 𝐿𝐿 33𝑁𝑁 −2 𝑁𝑁23 − 𝑁𝑁1 −11 22 33− 3 13− 2− 12+ 3 13− 2+12 23+ 1( 11− 22− 33) 𝑆𝑆 𝑆𝑆 − 𝛬𝛬 − 𝑁𝑁 𝐿𝐿 𝑁𝑁 𝐿𝐿 𝜔𝜔 𝑆𝑆 𝛬𝛬 𝑁𝑁 − 𝐿𝐿 𝑁𝑁 𝑁𝑁 − 𝑁𝑁 − 𝑁𝑁 𝐿𝐿 𝑁𝑁 𝑁𝑁 − 1 1 1 𝑁𝑁 𝜔𝜔= 𝑆𝑆( 𝛬𝛬 + 𝑁𝑁 )𝐿𝐿=𝑁𝑁 { ×𝐿𝐿 𝑁𝑁× 𝐿𝐿 ( 𝑁𝑁 + 𝑁𝑁)} +𝑁𝑁 + ( × ) + 2 3 + 2 3 + 2 2 + 3112 2 3112 2113 2 2 3 2 × 23 1 ( + + ) + ( ) = 2𝑅𝑅 11 22𝑅𝑅 33 𝑅𝑅 23 12 ∇13 𝑁𝑁 −11𝑁𝑁 23 ∇ − ∇𝐿𝐿23 𝐿𝐿∇ 𝐿𝐿 𝐿𝐿 𝑆𝑆 𝑆𝑆 𝛬𝛬 𝛬𝛬 𝜔𝜔 𝜔𝜔 𝛬𝛬 𝜔𝜔

𝑁𝑁 𝑁𝑁 𝑁𝑁 𝑁𝑁 𝑁𝑁 𝑁𝑁 − 𝑁𝑁 𝑁𝑁 −𝑃𝑃 = , 1 = + + . + + × × 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 + 𝑗𝑗𝑗𝑗( . ) 1 2 𝑅𝑅 𝑄𝑄 𝑄𝑄Is a symmetric−𝑠𝑠 �∇ dyadic.𝑎𝑎 𝑎𝑎 ∇ � − 𝑠𝑠 𝑠𝑠 𝑎𝑎 𝑎𝑎 𝑠𝑠 𝜔𝜔 − 𝜔𝜔 𝑠𝑠 − 𝛬𝛬 𝛬𝛬 𝛬𝛬 𝛬𝛬 3 2 0123 = 11, 0131 = 12 , 0112 = 13 + 3 1 1 2 0233 = 12 + , 0231 = 22 , 0212𝑅𝑅 = 23𝐵𝐵 𝑅𝑅 0323 =𝐵𝐵 13− 𝑡𝑡+ 𝑅𝑅, 0331 𝐵𝐵= 23𝑡𝑡+ , 0312 = 33 𝑅𝑅Where 𝐵𝐵 𝑡𝑡 𝑅𝑅 𝐵𝐵 𝑅𝑅 𝐵𝐵 − 𝑡𝑡 𝑅𝑅 𝐵𝐵 𝑡𝑡 𝑅𝑅 𝐵𝐵 𝑡𝑡 𝑅𝑅 𝐵𝐵 1 1 = × × + + + + . 1 2 2 Is a traceless symmetric dyadic𝐵𝐵 and− �∇ 𝑆𝑆 − 𝑆𝑆 ∇� �∇ 𝛬𝛬 𝛬𝛬 ∇� 𝑎𝑎 𝛬𝛬 𝛬𝛬𝑎𝑎 − �𝑎𝑎 𝛬𝛬� 1 1 = + ( ) × + 2( × ) 2 2 𝑟𝑟 𝑡𝑡 2323− =∇ 𝑆𝑆 11, ∇2331𝑡𝑡 𝑆𝑆=− ∇12 ,𝛬𝛬 2312 =𝛬𝛬 𝑎𝑎13

𝑅𝑅3131 = −𝑃𝑃22 ,𝑅𝑅3112 = −𝑃𝑃23 ,𝑅𝑅1212 = −𝑃𝑃33 Where 𝑅𝑅 −𝑃𝑃 𝑅𝑅 −𝑃𝑃 𝑅𝑅 −𝑃𝑃 1 1 1 1 1 1 = × × + + ( ) × × ( )2 1 2 2 2 2 × 2 × 4 𝑃𝑃Is a symmetric− �∇ 𝑁𝑁 dyadic.− 𝑁𝑁 We∇� can� write∇ 𝐿𝐿 𝐿𝐿 ∇� − 𝑡𝑡𝑟𝑟 𝑁𝑁 𝑁𝑁 − 𝑁𝑁 𝑁𝑁 − 𝑆𝑆 𝑆𝑆 − 𝛬𝛬 𝛬𝛬 − 𝜔𝜔 𝛬𝛬 − 𝛬𝛬 𝜔𝜔 − 𝐿𝐿 𝐿𝐿 − �∇ 𝛬𝛬 − 𝑡𝑡𝑟𝑟 𝑁𝑁 � 1 = + × + + + ) , 2 × 1 1 1 1 1 Where = × × 𝑃𝑃+ − �+𝐸𝐸 ( 𝑆𝑆 )𝑆𝑆�+ 𝛬𝛬 𝛬𝛬× 𝜔𝜔+ 𝛬𝛬 +𝛬𝛬 𝜔𝜔 ( )2 1 2 2 2 2 × 4 1 1 𝑟𝑟 𝑟𝑟 We have𝐸𝐸 =�∇ 𝑁𝑁: − 𝑁𝑁 ( ∇� )−2 �∇2 𝐿𝐿. 𝐿𝐿 ∇ � 𝑡𝑡 𝑁𝑁 2𝑁𝑁 𝑁𝑁 𝑁𝑁 𝐿𝐿 𝐿𝐿 �∇ 𝐿𝐿 − 𝑡𝑡 𝑁𝑁 � 2 4 In case 𝑡𝑡𝑟𝑟de𝑝𝑝 termine𝑁𝑁 𝑁𝑁 a− normal𝑡𝑡𝑟𝑟 𝑁𝑁 congruence− ∇ 𝐿𝐿 − ( 𝐿𝐿=𝐿𝐿 −0)𝛬𝛬 𝛬𝛬 − 𝜔𝜔 𝛬𝛬 𝜇𝜇 𝑒𝑒 1 𝑜𝑜 𝛬𝛬 = × 2 × and its six components give the six components of curvature𝐸𝐸 −𝑃𝑃 tensor− 𝑆𝑆 of𝑆𝑆 the Riemannian 3-space orthogonal to . We also 𝜇𝜇 have, in this case 𝑒𝑒 𝑜𝑜 . E = 0

3. Representation∇ of curvature tensor in bi-vector space : Strangled components of the Ricci tensor are given by = . 𝑡𝑡 these can𝑅𝑅 be𝑟𝑟𝑟𝑟 spilt𝑅𝑅 into𝑟𝑟𝑟𝑟𝑟𝑟 , 0 , 00 ; , = 1,2,3 0 2 3 11 = .110𝑅𝑅+𝑖𝑖𝑖𝑖 𝑅𝑅.112𝑖𝑖 𝑅𝑅+ .113𝑖𝑖 𝑗𝑗 = 0110 + 2112 + 3113 www.ijspr.com𝑅𝑅 𝑅𝑅 𝑅𝑅 𝑅𝑅 − 𝑅𝑅 𝑅𝑅 𝑅𝑅 IJSPR | 199 INTERNATIONAL JOURNAL OF SCIENTIFIC PROGRESS AND RESEARCH (IJSPR) ISSN: 2349-4689 Issue 88, Volume 31, Number 02, 2017

= 0101 + 2112 + 3113 = 11 + 22 + 33 = 11 11 + 0 3 12 =𝑅𝑅 .120 +𝑅𝑅 .123 =𝑅𝑅 0102 +𝑄𝑄 3123𝑃𝑃= 12𝑃𝑃 12𝑄𝑄 − 𝑃𝑃 𝑡𝑡𝑟𝑟 𝑃𝑃 2 3 01𝑅𝑅= .𝑅𝑅012 + .𝑅𝑅013 = .2012𝑅𝑅 + 3013𝑅𝑅 = 1221𝑄𝑄 +− 𝑃𝑃0331 = 2 1 1 2 3 𝑅𝑅00 = 𝑅𝑅 .001 + 𝑅𝑅 .002 + 𝑅𝑅 .003 = 𝑅𝑅1001 + 𝑅𝑅2012 + 𝑅𝑅3003 𝑡𝑡 ( ) ( ) 𝑅𝑅 =𝑅𝑅 0101𝑅𝑅+ 0202𝑅𝑅+ 0303 𝑅𝑅= 11𝑅𝑅 + 22𝑅𝑅+ 33 = Hence the ten strangled− 𝑅𝑅 components𝑅𝑅 𝑅𝑅 of the Ricci− 𝑄𝑄 tensor𝑄𝑄 correspond𝑄𝑄 to− a𝑡𝑡 symmetric𝑟𝑟 𝑄𝑄 3-dyadic + ( )1 𝑄𝑄 − 𝑃𝑃 a𝑡𝑡 3𝑟𝑟-𝑃𝑃vector and a scalar ( ) 𝑡𝑡 The scalar curvature R can be expressed− 𝑡𝑡𝑟𝑟 𝑄𝑄 as

= . = = 00 + 11 + 22 + 33 = 2( + ) 𝑟𝑟 𝑟𝑟𝑟𝑟 In the Orthonormal𝑅𝑅 𝑅𝑅 𝑟𝑟 tetrad𝜂𝜂 𝑅𝑅 𝑠𝑠frame,𝑠𝑠 − 𝑅𝑅Einstein's𝑅𝑅 equations𝑅𝑅 𝑅𝑅take the form𝑡𝑡𝑟𝑟 𝑃𝑃 𝑡𝑡𝑟𝑟 𝑄𝑄 1 2 = 2 Hence, 𝑇𝑇𝑟𝑟𝑟𝑟 𝑅𝑅𝑟𝑟𝑟𝑟 − 𝑅𝑅𝜂𝜂𝑟𝑟𝑟𝑟 1 2 = = ,2 = = 11 11 2 11 11 12 12 12 12 𝑟𝑟 2𝑇𝑇01 = 𝑅𝑅01 −= 2𝑅𝑅1 𝑄𝑄 − 𝑃𝑃 − 𝑡𝑡 𝑄𝑄 𝑇𝑇 𝑅𝑅 𝑄𝑄 − 𝑃𝑃 1 2 = + = + + = 𝑇𝑇00 𝑅𝑅00 2 𝑡𝑡 1 Hence the 𝑇𝑇energy𝑅𝑅 - momentum𝑅𝑅 − tensor𝑡𝑡𝑟𝑟 𝑄𝑄 breaks𝑡𝑡𝑟𝑟 𝑃𝑃 𝑡𝑡up𝑟𝑟 𝑄𝑄 into𝑡𝑡 symmetric𝑟𝑟 𝑃𝑃 3-dyadic = [ ( )1] 2 3 - vector 𝑇𝑇 𝑄𝑄 − 𝑃𝑃 − 𝑡𝑡𝑟𝑟 𝑄𝑄 1 and scalar = ( ) 𝑡𝑡 2 in the 3-space of auxiliary𝑃𝑃 𝑡𝑡 𝑟𝑟congruences𝑃𝑃 we know that curvature tensor is represented in a 6-dimentional bivector space by introducing the collective indices

01 1, 02 𝑅𝑅2𝜆𝜆 𝜆𝜆𝜆𝜆𝜆𝜆,03 3 23 ⟵⟶ 4 , 31⟵⟶ 5 , 12⟵⟶ 6 ( ) In terms of ⟵, ⟶, and the⟵ ⟶matrix ⟵⟶, , = 1,2,3, 6 has the representation 3 2 𝑄𝑄11𝑃𝑃 𝐵𝐵 12𝑡𝑡 13 𝑅𝑅𝐴𝐴𝐴𝐴 11𝐴𝐴 𝐵𝐵 12 − −13−+ 3 1 12 22 23 12 + 22 23 𝑄𝑄 𝑄𝑄 𝑄𝑄 𝐵𝐵 2 𝐵𝐵 − 𝑡𝑡1 𝐵𝐵 𝑡𝑡 13 23 33 13 23 + 33 = 𝑄𝑄 𝑄𝑄 3 𝑄𝑄 2 𝐵𝐵 𝑡𝑡 𝐵𝐵 ⎛ 11 12 + 13 11 12 𝐵𝐵 −13𝑡𝑡 ⎞ 𝑄𝑄 3 𝑄𝑄 𝑄𝑄 1 𝐴𝐴𝐴𝐴 ⎜ 12 22 23 + 𝐵𝐵 −12𝑡𝑡 𝐵𝐵 22𝑡𝑡 𝐵𝐵 23 ⎟ 𝑅𝑅 𝐵𝐵 2 𝐵𝐵 𝑡𝑡1 𝐵𝐵 − 𝑡𝑡 ⎜ 13 + 23 33 −𝑃𝑃13 −𝑃𝑃23 −𝑃𝑃33 ⎟ 𝐵𝐵 − 𝑡𝑡 𝐵𝐵 𝐵𝐵 𝑡𝑡 −𝑃𝑃 −𝑃𝑃 −𝑃𝑃 i.e. ⎝=𝐵𝐵 𝑡𝑡 𝐵𝐵 where− 𝑡𝑡 =𝐵𝐵 + ×−1𝑃𝑃 −𝑃𝑃 −𝑃𝑃 ⎠ 𝑄𝑄 𝜇𝜇 𝑅𝑅𝐴𝐴𝐴𝐴 � 𝑇𝑇 � 𝜇𝜇 𝐵𝐵 𝑡𝑡 Introducing 𝜇𝜇the traceless−𝑃𝑃 symmetric dyadic 1 1 = + ( + )1 2 3 we have 𝐴𝐴 �𝑃𝑃 𝑄𝑄 − 𝑡𝑡𝑟𝑟 𝑃𝑃 𝑡𝑡𝑟𝑟 𝑄𝑄 � 1 1 = + 1, = 1 3 6 where 𝑄𝑄 𝑇𝑇 𝐴𝐴 − �𝑃𝑃 − 𝑅𝑅� −𝑃𝑃 𝑇𝑇 − 𝐴𝐴 − �𝑃𝑃 − 𝑅𝑅� = 2( )

( ) 𝑟𝑟 Hence 𝑅𝑅may be𝑡𝑡 represented𝑇𝑇 − 𝑃𝑃 by , and as 𝑅𝑅𝐴𝐴𝐴𝐴 𝑇𝑇 𝐴𝐴 𝐵𝐵 www.ijspr.com IJSPR | 200 INTERNATIONAL JOURNAL OF SCIENTIFIC PROGRESS AND RESEARCH (IJSPR) ISSN: 2349-4689 Issue 88, Volume 31, Number 02, 2017

1 + 1 + × 1 3 = 1 × 1 + 1 𝐴𝐴 𝑇𝑇 − �𝑃𝑃 − 𝑅𝑅� 𝐵𝐵 𝑡𝑡 6 𝑅𝑅𝐴𝐴𝐴𝐴 � � 2.4. Dyadic analysis of 𝐵𝐵W−eyl𝑡𝑡 tensor: 𝐴𝐴 − 𝑇𝑇 �𝑃𝑃 − 𝑅𝑅� In terms of the Ortho-normal tetrad corresponding to the decomposition 1 = + + [ ] [ ] 3 [ ] We have 𝑅𝑅 𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆 𝐶𝐶𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆 𝑔𝑔𝜆𝜆 𝑟𝑟 𝑅𝑅𝑃𝑃 𝜇𝜇 𝑅𝑅𝜆𝜆 𝑟𝑟 𝑔𝑔𝑃𝑃 𝜇𝜇 − 𝑅𝑅𝑔𝑔𝜆𝜆 𝑟𝑟 𝑔𝑔𝑃𝑃 𝜇𝜇 1 = + + [ ] [ ] 12 [ ] [ ] Where 𝑅𝑅𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 are the𝐶𝐶𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 strangled�𝜂𝜂𝑟𝑟 𝑢𝑢components𝐸𝐸𝑡𝑡 𝑠𝑠 − 𝜂𝜂𝑠𝑠 𝑢𝑢 𝐸𝐸 of𝑡𝑡 𝑟𝑟 Weyl� tensor𝑅𝑅�𝜂𝜂𝑟𝑟 𝑢𝑢and𝜂𝜂𝑡𝑡 𝑠𝑠 − 𝜂𝜂𝑠𝑠 𝑢𝑢 𝜂𝜂𝑡𝑡 𝑟𝑟 � 1 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 = 𝐶𝐶 4 we have 𝐸𝐸𝑟𝑟𝑟𝑟 𝑅𝑅𝑟𝑟𝑟𝑟 − 𝑅𝑅𝜂𝜂𝑟𝑟𝑟𝑟 1 1 1 = + + 0101 0101 2 11 2 00 24 1 1 1 𝑅𝑅 𝐶𝐶= +𝐸𝐸 − 𝐸𝐸 𝑅𝑅 0101 2 11 2 00 6 𝐶𝐶 𝑅𝑅 − 𝑅𝑅 −1 𝑅𝑅 1 1 = + ( + ) + ( + ) 0101 2 11 11 2 3 1 𝐶𝐶 1 𝑄𝑄 − 𝑃𝑃 𝑡𝑡𝑟𝑟1𝑃𝑃 𝑡𝑡𝑟𝑟 𝑄𝑄 − 𝑡𝑡1𝑟𝑟 𝑃𝑃 𝑡𝑡𝑟𝑟 𝑄𝑄 1 = ( ) + ( + ) ( + ) = ( + ) + ( + ) 0101 11 2 11 11 3 2 2 11 11 6 1 1 𝐶𝐶 𝑄𝑄 − =𝑄𝑄 (− 𝑃𝑃 + ) 𝑡𝑡𝑟𝑟 𝑃𝑃 ( 𝑡𝑡𝑟𝑟 𝑄𝑄+ − ) 𝑡𝑡𝑟𝑟 𝑃𝑃 𝑡𝑡𝑟𝑟 𝑄𝑄 𝑃𝑃 𝑄𝑄 𝑡𝑡𝑟𝑟 𝑃𝑃 𝑡𝑡𝑟𝑟 𝑄𝑄 2 11 11 6 𝑃𝑃 𝑄𝑄 − 𝑡𝑡𝑟𝑟 𝑃𝑃 𝑡𝑡𝑟𝑟 𝑄𝑄 1 = + 0203 0203 2 23 𝑅𝑅 𝐶𝐶 1 𝐸𝐸 = + 0203 2 23 𝐶𝐶 1 𝑅𝑅 = + ( ) 23 0203 2 23 23 1 = ( + ) 𝑄𝑄 𝐶𝐶 𝑄𝑄 − 𝑃𝑃 0203 2 23 23

𝐶𝐶0123 = 0123𝑃𝑃 𝑄𝑄

𝑅𝑅0123 = 𝐶𝐶11 1 1 = + = + ∴ 𝐶𝐶0221 𝐵𝐵0221 2 10 0221 2 01

𝑅𝑅0221 = 𝐶𝐶1 23 𝐸𝐸 1 = 𝐶𝐶 23 𝑅𝑅 1 1 1 = ∴ 𝐶𝐶2323 𝑡𝑡 2323− 𝐵𝐵 2− 33𝑡𝑡 2−𝐵𝐵22 12 1 1 1 1 𝑅𝑅 = 𝐶𝐶 − 𝐸𝐸 − 𝐸𝐸 +− 𝑅𝑅 2323 2 33 2 22 4 12 1 1 1 = 𝐶𝐶 − 𝑅𝑅 − 𝑅𝑅 + 𝑅𝑅 − 𝑅𝑅 2323 2 33 2 22 6 1 1 1 = 𝐶𝐶 +− (𝑅𝑅 − 𝑅𝑅 + 𝑅𝑅) + ( + ) ( + ) 2323 11 2 33 33 2 22 22 3 𝐶𝐶 −𝑃𝑃 𝑄𝑄 − 𝑃𝑃 𝑡𝑡𝑟𝑟 𝑃𝑃 1 𝑄𝑄 −1 𝑃𝑃 1𝑡𝑡𝑟𝑟 𝑃𝑃 − 1 𝑡𝑡𝑟𝑟 𝑃𝑃 𝑡𝑡𝑟𝑟 𝑄𝑄 1 = + + + ( + ) 11 2 11 2 2 11 2 3 1 1 𝑟𝑟 𝑟𝑟 𝑟𝑟 𝑟𝑟 𝑟𝑟 = ( + ) + ( −𝑃𝑃+ )𝑃𝑃 − 𝑡𝑡 𝑃𝑃 − 𝑄𝑄 𝑡𝑡 𝑄𝑄 𝑡𝑡 𝑃𝑃 − 𝑡𝑡 𝑃𝑃 𝑡𝑡 𝑄𝑄 2 11 11 3 1 𝑟𝑟 1 𝑟𝑟 = − +𝑃𝑃 𝑄𝑄= 𝑡𝑡+𝑃𝑃 𝑡𝑡 𝑄𝑄 3112 3112 2 23 3112 2 23 1 𝑅𝑅 =𝐶𝐶 + 𝐸𝐸( 𝐶𝐶 ) 𝑅𝑅 3112 2 23 23 1 1 = 𝐶𝐶 (𝑄𝑄 − 𝑃𝑃 ) = ( + ) 3112 23 2 23 23 2 23 23 𝐶𝐶 −𝑃𝑃 − 𝑄𝑄 − 𝑃𝑃 − 𝑃𝑃 𝑄𝑄 www.ijspr.com IJSPR | 201 INTERNATIONAL JOURNAL OF SCIENTIFIC PROGRESS AND RESEARCH (IJSPR) ISSN: 2349-4689 Issue 88, Volume 31, Number 02, 2017

1 1 = = ( + ) ( + ) = 0101 2323 2 11 11 6 11 𝑟𝑟 𝑟𝑟 ∴ 𝐶𝐶0203 = −𝐶𝐶3112 = 23𝑃𝑃 𝑄𝑄 − 𝑡𝑡 𝑃𝑃 𝑡𝑡 𝑄𝑄 𝐴𝐴 1 1 = ( + ) ( + )1 𝐶𝐶 2−𝐶𝐶 𝐴𝐴6 Hence𝐴𝐴 the symmetric𝑃𝑃 𝑄𝑄 −dyadic𝑡𝑡𝑟𝑟 𝑃𝑃 and𝑡𝑡𝑟𝑟 𝑄𝑄 together with = = 0 give the ten independent strangled𝐴𝐴 components𝐵𝐵 of weyl𝑡𝑡𝑟𝑟 𝐴𝐴 tensor.𝑡𝑡𝑟𝑟 𝐵𝐵 5. Petrov type classification for and : The five dyad components of weyl spinor 0 , 𝐴𝐴1 , 2 ,𝐵𝐵3 , 4 corresponding to are 𝜓𝜓 𝜓𝜓 𝜓𝜓 𝜓𝜓 𝑎𝑎𝑎𝑎𝑎𝑎 𝜓𝜓

𝐶𝐶𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆 = 0 𝛼𝛼 𝛽𝛽 𝛾𝛾 𝛿𝛿 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼 𝜓𝜓 =−𝐶𝐶 𝑙𝑙 𝑚𝑚 𝑙𝑙 𝑚𝑚 1 𝛼𝛼 𝛽𝛽 𝛾𝛾 𝛿𝛿 1𝜓𝜓 −𝐶𝐶𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼 𝑙𝑙 𝑛𝑛 𝑙𝑙 𝑚𝑚 = ( + ) 2 2 𝛼𝛼 𝛽𝛽 𝛾𝛾 𝛿𝛿 𝛼𝛼 𝛽𝛽 𝛾𝛾 𝛿𝛿 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼 𝜓𝜓 − 𝐶𝐶 = 𝑙𝑙 𝑛𝑛 𝑙𝑙 𝑛𝑛 𝑙𝑙 𝑛𝑛 𝑚𝑚 𝑚𝑚 3 𝛼𝛼 𝛽𝛽 𝛾𝛾 𝛿𝛿 𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼 𝜓𝜓= 𝐶𝐶 𝑙𝑙 𝑛𝑛 𝑛𝑛 𝑚𝑚 4 𝛼𝛼 𝛽𝛽 𝛾𝛾 𝛿𝛿 From these components, we have 𝜓𝜓 −𝐶𝐶𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼 𝑛𝑛 𝑚𝑚 𝑛𝑛 𝑚𝑚

= ( 1) 0 𝛼𝛼 𝛽𝛽 𝛾𝛾 𝛿𝛿 −𝜓𝜓 𝐶𝐶𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼 𝑙𝑙 𝑚𝑚 𝑙𝑙 𝑚𝑚 𝐿𝐿 i.e.= 4 = + + 0 0𝛼𝛼 1𝛼𝛼 2𝛽𝛽 3𝛽𝛽 0𝛾𝛾 1𝛾𝛾 𝑒𝑒 𝑒𝑒 𝑒𝑒 𝑒𝑒 𝑒𝑒 𝑒𝑒 − 𝜓𝜓 𝐶𝐶𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼 � � � − 𝑖𝑖 � � � = + + + + + + 0𝛼𝛼 2𝛽𝛽 0𝛾𝛾 2𝛿𝛿 0𝛼𝛼 2𝛽𝛽 1𝛾𝛾 2𝛿𝛿 1𝛼𝛼 2𝛽𝛽 0𝛾𝛾 2𝛿𝛿 1𝛼𝛼 2𝛽𝛽 1𝛾𝛾 2𝛿𝛿 0𝛼𝛼 3𝛽𝛽 0𝛾𝛾 3𝛿𝛿 0𝛼𝛼 3𝛽𝛽 1𝛾𝛾 3𝛿𝛿 1𝛼𝛼 3𝛽𝛽 0𝛾𝛾 3𝛿𝛿 1 3 1 3𝛼𝛼𝛼𝛼𝛼𝛼−𝛼𝛼 𝑒𝑒0 𝑒𝑒 2𝑒𝑒 𝑒𝑒0 3𝑒𝑒 +𝑒𝑒 0𝑒𝑒 𝑒𝑒2 1𝑒𝑒 𝑒𝑒3 𝑒𝑒+𝑒𝑒1 2𝑒𝑒 𝑒𝑒0𝑒𝑒 3𝑒𝑒 + 1𝑒𝑒 𝑒𝑒2 𝑒𝑒 1𝑒𝑒 3 𝑒𝑒+ 𝑒𝑒0 𝑒𝑒 3𝑒𝑒 0 𝑒𝑒 2𝑒𝑒 +𝑒𝑒 0𝑒𝑒 3 1 2 𝐶𝐶 � − � + 1 3 0 2 + 1 3 1 2 𝑒𝑒 𝛼𝛼𝑒𝑒 𝛽𝛽𝑒𝑒 𝛾𝛾𝑒𝑒 𝛿𝛿 𝑖𝑖𝑒𝑒 𝛼𝛼𝑒𝑒 𝛽𝛽𝑒𝑒 𝛾𝛾𝑒𝑒 𝛿𝛿 𝑒𝑒 𝛼𝛼𝑒𝑒 𝛽𝛽𝑒𝑒 𝛾𝛾𝑒𝑒 𝛿𝛿 𝑒𝑒 𝛼𝛼𝑒𝑒 𝛽𝛽𝑒𝑒 𝛾𝛾𝑒𝑒 𝛿𝛿 𝑒𝑒 𝛼𝛼𝑒𝑒 𝛽𝛽𝑒𝑒 𝛾𝛾𝑒𝑒 𝛿𝛿 𝑒𝑒 𝛼𝛼𝑒𝑒 𝛽𝛽𝑒𝑒 𝛾𝛾𝑒𝑒 𝛿𝛿 𝑒𝑒 𝛼𝛼𝑒𝑒 𝛽𝛽𝑒𝑒 𝛾𝛾𝑒𝑒 𝛿𝛿 𝑒𝑒 𝛼𝛼𝑒𝑒 𝛽𝛽𝑒𝑒 𝛾𝛾𝑒𝑒 𝛿𝛿 𝑒𝑒 𝛼𝛼𝑒𝑒 𝛽𝛽𝑒𝑒 𝛾𝛾𝑒𝑒 𝛿𝛿

= 0202 + 2 0212 + 1212 0203 2 0313

𝐶𝐶 1313𝐶𝐶 (2 0203𝐶𝐶 +−2𝐶𝐶0213 +− 2𝐶𝐶1203 + 2 1213 ) = 0202−𝐶𝐶+ 2 −0212𝑖𝑖 𝐶𝐶 2 0313 𝐶𝐶 0303 +𝐶𝐶 1212 𝐶𝐶 ( ) 𝐶𝐶 1313 2𝐶𝐶 0203− +𝐶𝐶 0213−+𝐶𝐶 1203 +𝐶𝐶 1213

2 = ( 22 + 33 2 23) + ( 22 + 33 + 2 23) −𝐶𝐶 − 𝑖𝑖 𝐶𝐶 𝐶𝐶 𝐶𝐶0 𝐶𝐶 Similarly, 𝜓𝜓 −𝐴𝐴 𝐴𝐴 − 𝐵𝐵 𝑖𝑖 −𝐵𝐵 𝐵𝐵 𝐴𝐴

2 = ( 12 + 13) + ( 13 12) … … … … . ( 2) 1

2𝜓𝜓 = − 𝐴𝐴11 𝐵𝐵11 … …𝑖𝑖 …𝐴𝐴… …−…𝐵𝐵… … … … … . . (𝐿𝐿3) 2

2𝜓𝜓 = −(𝐴𝐴 12− 𝑖𝑖𝐵𝐵13) ( 13 + 12) … … … … ( 𝐿𝐿4) 3

𝜓𝜓 − 𝐴𝐴 − 𝐵𝐵 − 𝑖𝑖 𝐴𝐴 2 =𝐵𝐵( 22 + 33 𝐿𝐿+ 2 23) + ( 22 + 33 2 23). . ( 5) 4 Hence 𝜓𝜓 −𝐴𝐴 𝐴𝐴 𝐵𝐵 𝑖𝑖 −𝐵𝐵 𝐵𝐵 − 𝐴𝐴 𝐿𝐿

1 11 = + , 12 = ( + + + ) 2 2 2 1 1 3 3 𝐴𝐴 − �𝜓𝜓 𝜓𝜓 � 𝐴𝐴 − 𝜓𝜓 𝜓𝜓 𝜓𝜓 𝜓𝜓 = + , 13 2 𝑖𝑖 1 1 3 3 𝐴𝐴 −1 �𝜓𝜓 − 𝜓𝜓 −1𝜓𝜓 𝜓𝜓 � 22 = + + + , 2 2 2 4 0 0 4 4 𝐴𝐴 �𝜓𝜓 𝜓𝜓 � − �𝜓𝜓 𝜓𝜓 − 𝜓𝜓 𝜓𝜓 � www.ijspr.com IJSPR | 202 INTERNATIONAL JOURNAL OF SCIENTIFIC PROGRESS AND RESEARCH (IJSPR) ISSN: 2349-4689 Issue 88, Volume 31, Number 02, 2017

= + , 23 4 𝑖𝑖 0 0 4 4 𝐴𝐴 1− �𝜓𝜓 − 𝜓𝜓 1− 𝜓𝜓 𝜓𝜓 � 33 = ( + 2) + + + + , 2 2 4 0 0 4 4 𝐴𝐴 𝜓𝜓 𝜓𝜓� �𝜓𝜓 𝜓𝜓 𝜓𝜓 𝜓𝜓 � 11 = , 2 2 𝐵𝐵 𝑖𝑖 �𝜓𝜓 − 𝜓𝜓 � = + , 12 2 𝑖𝑖 1 1 3 3 𝐵𝐵 �1𝜓𝜓 − 𝜓𝜓 𝜓𝜓 − 𝜓𝜓 � 13 = + , 2 1 1 3 3 𝐵𝐵 − �𝜓𝜓 𝜓𝜓 − 𝜓𝜓 − 𝜓𝜓 � = + + , 22 2 4 𝑖𝑖 2 2 𝑖𝑖 0 0 4 4 𝐵𝐵 − 1 �𝜓𝜓 − 𝜓𝜓 � �𝜓𝜓 − 𝜓𝜓 𝜓𝜓 − 𝜓𝜓 � 23 = + , 4 0 0 4 4 𝐵𝐵 − �𝜓𝜓 𝜓𝜓 − 𝜓𝜓 − 𝜓𝜓 � = ( + ) 33 2 4 𝑖𝑖 2 2 𝑖𝑖 0 0 4 4 We𝐵𝐵 can choose− �𝜓𝜓 the− 𝜓𝜓null� tetrad− 𝜓𝜓 so− tha𝜓𝜓t 𝜓𝜓= 0− 𝜓𝜓 0 ( is a principal null vector of Weyl tensor).𝜓𝜓 Then setting 𝜇𝜇 𝑙𝑙 + = , + = 4 , + 2 2 4 4 1 1 3 3 𝜓𝜓 𝜓𝜓 𝑎𝑎� 𝜓𝜓 𝜓𝜓 𝑎𝑎 𝑖𝑖 �𝜓𝜓 − 𝜓𝜓 𝜓𝜓 − 𝜓𝜓 � = 2 , = 4 1 1 3 3 4 4 𝑖𝑖 ��𝜓𝜓 − 𝜓𝜓 � − �𝜓𝜓 − 𝜓𝜓 �� − 𝑏𝑏 𝑖𝑖 �𝜓𝜓 − 𝜓𝜓 � 𝑐𝑐 = 2 , + + + = 2 2 2 1 1 3 3 𝑖𝑖 �𝜓𝜓 − 𝜓𝜓 � − 𝑐𝑐̃ 𝜓𝜓 𝜓𝜓 𝜓𝜓 𝜓𝜓 − 𝑑𝑑 + = 2 , 1 1 3 3 𝜓𝜓we have𝜓𝜓 the− 𝜓𝜓 following− 𝜓𝜓 representation− 𝑑𝑑̃ for A and B. = 2 + + + + + ( ) + + + ( + ) ; = 2 + + + + + ( + ) + + + ( ) 𝐴𝐴 − 𝑎𝑎�𝑢𝑢𝑢𝑢 𝑑𝑑�𝑢𝑢𝑣𝑣 𝑣𝑣 𝑢𝑢� 𝑏𝑏��𝑢𝑢 𝜔𝜔 𝜔𝜔 𝑢𝑢� 𝑎𝑎� − 𝑎𝑎 𝑣𝑣 𝑣𝑣 𝑐𝑐�𝑣𝑣 𝜔𝜔 𝜔𝜔 𝑣𝑣� 𝑎𝑎� 𝑎𝑎 𝜔𝜔 𝜔𝜔 𝐵𝐵 − 𝑐𝑐̃ 𝑢𝑢 𝑢𝑢 For𝑏𝑏�𝑢𝑢 𝑣𝑣different𝑣𝑣𝑢𝑢� Petrov𝑑𝑑�𝑢𝑢 𝜔𝜔types𝜔𝜔 of𝑢𝑢� Weyl𝑐𝑐 ̃ tensor,𝑐𝑐 𝑣𝑣 𝑣𝑣we have𝑎𝑎�𝑣𝑣 𝜔𝜔for 𝜔𝜔 and𝑣𝑣� , 𝑐𝑐̃ − 𝑐𝑐 𝜔𝜔 𝜔𝜔 Type -1: = = 0(Two of the principal null vectors of are and ) 0 4 𝐴𝐴 𝐵𝐵 𝜇𝜇 𝜇𝜇 𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆 𝜓𝜓= 2𝜓𝜓 + + + + + 𝐶𝐶 + 𝑙𝑙 𝑛𝑛

𝐴𝐴 = −2𝑎𝑎� 𝑢𝑢 𝑢𝑢 +𝑑𝑑�𝑢𝑢𝑣𝑣 +𝑣𝑣𝑢𝑢 � +𝑏𝑏��𝑢𝑢 𝜔𝜔 +𝜔𝜔 𝑢𝑢 � +𝑎𝑎�𝑣𝑣 𝑣𝑣 +𝑎𝑎� 𝜔𝜔 𝜔𝜔 𝐵𝐵 − 𝑐𝑐̃ 𝑢𝑢 𝑢𝑢 𝑏𝑏�𝑢𝑢 𝑣𝑣 𝑣𝑣 𝑢𝑢� 𝑑𝑑�𝑢𝑢 𝜔𝜔 𝜔𝜔 𝑢𝑢� 𝑐𝑐̃𝑣𝑣 𝑣𝑣 𝑐𝑐̃ 𝜔𝜔 𝜔𝜔 Type-2: = = = 0( A repeated principal null vector and a second principal null vector). 0 1 4 𝜇𝜇 𝜇𝜇 𝜓𝜓 𝜓𝜓 𝜓𝜓 𝑙𝑙 = 2 + + + 𝑛𝑛+ + +

= 2 + +𝐴𝐴 − 𝑎𝑎�𝑢𝑢 𝑢𝑢 𝑑𝑑+�𝑢𝑢 𝑣𝑣 +𝑣𝑣 𝑢𝑢� +𝑏𝑏�𝑢𝑢 𝜔𝜔 𝜔𝜔 𝑢𝑢� 𝑎𝑎� 𝑣𝑣 𝑣𝑣 𝑎𝑎� 𝜔𝜔 𝜔𝜔 Type-D:𝐵𝐵 −= 𝑐𝑐̃𝑢𝑢𝑢𝑢= 𝑏𝑏�𝑢𝑢=𝑣𝑣 =𝑣𝑣 𝑢𝑢0� (principal− 𝑑𝑑�𝑢𝑢 𝜔𝜔 null𝜔𝜔 direction𝑢𝑢� 𝑐𝑐̃𝑣𝑣 𝑣𝑣of 𝑐𝑐̃𝜔𝜔 𝜔𝜔 are , , , ). 0 1 3 4 𝜇𝜇 𝜇𝜇 𝜇𝜇 𝜇𝜇 = 2 𝜓𝜓 +𝜓𝜓 𝜓𝜓+ 𝜓𝜓 ; = 2 + + 𝐶𝐶𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆 𝑙𝑙 𝑙𝑙 𝑛𝑛 𝑛𝑛

𝐴𝐴Type−-3:𝑎𝑎 �𝑢𝑢 𝑢𝑢= 𝑎𝑎�𝑣𝑣=𝑣𝑣 𝑎𝑎�=𝜔𝜔 𝜔𝜔 =𝐵𝐵0 (principal− 𝑐𝑐𝑢𝑢𝑢𝑢 null𝑐𝑐̃𝑣𝑣 direction𝑣𝑣 𝑐𝑐̃𝜔𝜔 𝜔𝜔s of are , , , ). 0 1 2 4 𝜇𝜇 𝜇𝜇 𝜇𝜇 𝜇𝜇 𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆 = 𝜓𝜓 + 𝜓𝜓 +𝜓𝜓( 𝜓𝜓+ ) ; = + ( 𝐶𝐶 + )𝑙𝑙 𝑙𝑙 𝑙𝑙 𝑛𝑛

Type𝐴𝐴 𝑑𝑑-N�𝑢𝑢: 𝑣𝑣 =𝑣𝑣 𝑢𝑢� =𝑏𝑏 𝑢𝑢=𝜔𝜔 𝜔𝜔=𝑢𝑢0 (principal𝐵𝐵 𝑏𝑏�𝑢𝑢 null𝑣𝑣 direction𝑣𝑣 𝑢𝑢� − 𝑑𝑑 of𝑢𝑢 𝜔𝜔 𝜔𝜔 are𝑢𝑢 , , , ). 0 1 2 3 𝜇𝜇 𝜇𝜇 𝜇𝜇 𝜇𝜇 𝜓𝜓 𝜓𝜓 𝜓𝜓 𝜓𝜓 𝐶𝐶𝜆𝜆𝜆𝜆𝜆𝜆𝜆𝜆 𝑙𝑙 𝑙𝑙 𝑙𝑙 𝑙𝑙

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