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Additional info for General relativity and gravitation Vol. 29
As an example, we put L = v + iX , (83) and write out the equat ion for the special case m = 1. W hen we subst itute (83) int o (81) , for this case, we obt ain b = [ (v + iX ) 2 ( 2i (v + iX ) XÈ ± 3(1 + i XÇ ) 2 ) ] / [2( 1 + XÇ 2 ) ( (v ± i X ) 3 (1 + i XÇ ) 2 ± (v + iX ) 3 (1 ± i XÇ ) 2 ) ], (84) where overdot denot es diŒerentiat ion with respect t o v . When this is in turn subst ituted int o (82) , which for m = 1 simpli® es to . (v + i X ) 2 (1 + i XÇ ) (2(v + iX ) ( bÅ ± (1 + i XÇ ) 3 bÅ 2 ± (1 ± i XÇ ) 3 b bÅ ) + 9(1 + i XÇ ) bÅ ) iX ) 2 (1 ± i XÇ ) (2(v ± iX ) ( bÇ ± (1 + i XÇ ) 3 b bÅ ) + 9( 1 ± i XÇ ) b ) + (v ± ± (1 ± i XÇ ) 3 b 2 = ± (v 2 + X 2 ) / 2 .
We could have derived these equat ions for much more general classes of spacetimes, using a generalisat ion of the geometric argum ent used by Held in ; however, this would involve us in deeper quest ions t han we wish to consider here, and the above equat ions are su cient for our purposes. R EFER ENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. , an d E dgar, S. B . ( 1996) . G e n . Rel. G rav. 2 8 , 707. , an d P en rose, R. (1973) . J . Math. P hys . 1 4 , 874. New m an, E .
12. 13. 14. 15. 16. 17. , an d E dgar, S. B . ( 1996) . G e n . Rel. G rav. 2 8 , 707. , an d P en rose, R. (1973) . J . Math. P hys . 1 4 , 874. New m an, E . , and Unt i, T . ( 1962) . J. Math. P hys . 3 , 891. New m an, E . , and Unt i, T . ( 1963) . J. Math. P hys . 4 , 1467. New m an, E . , and P en rose, R. ( l962) . J . Math. P hys. 3 , 566. Held, A. ( 1974) . Com m u n . Ma th . P h ys . 3 7 , 311. Held, A. ( 1975) . Com m u n . Ma th . P h ys . 4 4 , 211. Held, A. ( 1976) . G e n . Re l.
General relativity and gravitation Vol. 29