References
97
Also (4.117) is equivalent to the equations:
H 11 = σ 13,2 − σ 12,3 , H 12 = σ 23,2 − σ 22,3 , H 13 = σ 33,2 − σ 23,3 ,
H 22 = σ 12,3 − σ 23,1 , H 23 = σ 13,3 − σ 33,1 , H 33 = σ 23,1 − σ 13,2 .
(4.236)
Now using (4.221)–(4.226), and the very useful Eqs. (4.227) and (4.228), we find
that
(σ 13 + i λ 13 ) ,2 − (σ 12 + i λ 12 ) ,3 = −
i (A + i C)
8 r
Y
Y −
1
Y
2
= H 11 − i E 11 ,
(4.237)
(σ 23 + i λ 23 ) ,2 − (σ 22 + i λ 22 ) ,3 =
(A + i C)
8 r
Y
Y
2
−
1
Y 2
= H 12 − i E 12 ,
(4.238)
(σ 33 + i λ 33 ) ,2 − (σ 23 + i λ 23 ) ,3 =
i (A + i C)
4 r
Y
Y −
1
Y
= H 13 − i E 13 ,
(4.239)
(σ 12 + i λ 12 ) ,3 − (σ 23 + i λ 23 ) ,1 =
i (A + i C)
8 r
Y
Y +
1
Y
2
= H 22 − i E 22 ,
(4.240)
(σ 13 + i λ 13 ) ,3 − (σ 33 + i λ 33 ) ,1 = −
(A + i C)
4 r
Y
Y +
1
Y
= H 23 − i E 23 ,
(4.241)
(σ 23 + i λ 23 ) ,1 − (σ 13 + i λ 13 ) ,2 = −
i (A + i C)
2 r
Y = H 33 − i E 33 .
(4.242)
Comparing the real parts on either side of these equations reveals that (4.236) are
satisfied.
References
1. L. Mariot, C. R. Acad. Sci 238, 2055 (1954)
2. I. Robinson, J. Math. Phys. 2, 290 (1961)
3. H. Bateman, The Mathematical Analysis of Electrical and Optical Wave-Motion (Dover, New
York, 1955)
4. P.A. Hogan, Proc. R. Soc. Lond. A 396, 199 (1984)
5. D. Cox, E.J. Flaherty, Communs. Math. Phys. 47, 75 (1976)
6. G.F.R. Ellis, Relativistic Cosmology (Gordon and Breach, London, 1971)
7. R. Penrose, W. Rindler, Spinors and Space-Time, vol. 1 (Cambridge University Press, Cambridge, 1984)
97
Also (4.117) is equivalent to the equations:
H 11 = σ 13,2 − σ 12,3 , H 12 = σ 23,2 − σ 22,3 , H 13 = σ 33,2 − σ 23,3 ,
H 22 = σ 12,3 − σ 23,1 , H 23 = σ 13,3 − σ 33,1 , H 33 = σ 23,1 − σ 13,2 .
(4.236)
Now using (4.221)–(4.226), and the very useful Eqs. (4.227) and (4.228), we find
that
(σ 13 + i λ 13 ) ,2 − (σ 12 + i λ 12 ) ,3 = −
i (A + i C)
8 r
Y
Y −
1
Y
2
= H 11 − i E 11 ,
(4.237)
(σ 23 + i λ 23 ) ,2 − (σ 22 + i λ 22 ) ,3 =
(A + i C)
8 r
Y
Y
2
−
1
Y 2
= H 12 − i E 12 ,
(4.238)
(σ 33 + i λ 33 ) ,2 − (σ 23 + i λ 23 ) ,3 =
i (A + i C)
4 r
Y
Y −
1
Y
= H 13 − i E 13 ,
(4.239)
(σ 12 + i λ 12 ) ,3 − (σ 23 + i λ 23 ) ,1 =
i (A + i C)
8 r
Y
Y +
1
Y
2
= H 22 − i E 22 ,
(4.240)
(σ 13 + i λ 13 ) ,3 − (σ 33 + i λ 33 ) ,1 = −
(A + i C)
4 r
Y
Y +
1
Y
= H 23 − i E 23 ,
(4.241)
(σ 23 + i λ 23 ) ,1 − (σ 13 + i λ 13 ) ,2 = −
i (A + i C)
2 r
Y = H 33 − i E 33 .
(4.242)
Comparing the real parts on either side of these equations reveals that (4.236) are
satisfied.
References
1. L. Mariot, C. R. Acad. Sci 238, 2055 (1954)
2. I. Robinson, J. Math. Phys. 2, 290 (1961)
3. H. Bateman, The Mathematical Analysis of Electrical and Optical Wave-Motion (Dover, New
York, 1955)
4. P.A. Hogan, Proc. R. Soc. Lond. A 396, 199 (1984)
5. D. Cox, E.J. Flaherty, Communs. Math. Phys. 47, 75 (1976)
6. G.F.R. Ellis, Relativistic Cosmology (Gordon and Breach, London, 1971)
7. R. Penrose, W. Rindler, Spinors and Space-Time, vol. 1 (Cambridge University Press, Cambridge, 1984)