l µ l µ l 0 (1, x r, y r, z r ) 1 r (1, x r, y r, z r ) l µ g µν η µν 2ml µ l ν 1 2m r 2mx r 2 2my r 2 2mz r 2 2mx r 2 1 2mx2 2mxy 2mxz 2my r 2mz 2 r

2 1 (7a)(7b) λ i( w w ) + [ w + w ] 1 + w w l 2 0 Re(γ) α (7a)(7b) 2 γ 0, ( w) 2 1, w 1 γ (1) l µ, λ j γ l 2 0 Re(γ) α, λ w + w i( w w ) 1 + w w γ γ 1 w 1 r [x2 + y 2 + z 2 ] 1/2 ( w) 2 x2 + y 2 + z 2 w 2 1 γ α + iβ β 0 l 2 0 λ i 1 (7a)(7b) l 2 0 Re(γ) 1 r λ w + w i( w w ) 1 + w w w ( x w, y w, z w ), w ( x w, y w, z w ) λ 1 2 x w yz i( w zy 2 w ) 2 x 1 + 1 w x r λ 2 y r λ 3 z r 1

l µ l µ l 0 (1, x r, y r, z r ) 1 r (1, x r, y r, z r ) l µ g µν η µν 2ml µ l ν 1 2m r 2mx r 2 2my r 2 2mz r 2 2mx r 2 1 2mx2 2mxy 2mxz 2my r 2mz 2 r 2 2mxy 1 2my2 2myz 2mxz 2myz 1 2mz2 ds 2 (dx 0 ) 2 (dx) 2 2m r (dx0 + xdx + ydy + zdz ) 2 r m γ γ 1 r [(x a 1) 2 + (y a 2 ) 2 + (z a 3 ) 2 ] 1/2 a i const (1) a a γ γ 1 r [x2 + y 2 + (z ia) 2 ] 1/2 γ γ l 2 0 λ i (7a)(7b) w w + iσ [x 2 + y 2 + (z ia) 2 ] 1/2 (r 2 a 2 2iaz) 1/2 (2) r 2 x 2 + y 2 + z 2 σ σ 2

( + iσ) 2 2 σ 2 + 2iσ r 2 a 2 2iaz 2 σ 2 r 2 a 2 σ az 4 2 (r 2 a 2 ) a 2 z 2 0 2 2 r2 a 2 ± (r 2 a 2 ) 2 + 4a 2 z 2 2 r2 a 2 2 ± ( (r2 a 2 ) 2 4 + a 2 z 2 ) 1/2 ± + r a 2 r2 2 + r2 2 r2 r (2) a + iσ (r 2 a 2 2iaz) 1/2 r l0 2 γ γ 1 + iσ iσ ( + iσ)( iσ) 2 + σ 2 iσ 2 + σ 2 l 2 0 Re(γ) 2 + σ 2 2 + a2 z 2 2 3 4 + a 2 z 2 3

(7a) λ w w (2) w (r 2 a 2 2iaz) 1/2 ( x(r 2 a 2 2iaz) 1/2, y(r 2 a 2 2iaz) 1/2, (z ia)(r 2 a 2 2iaz) 1/2) r iak w w r + iak w r r (x, y, z) k k (0, 0, 1) z (7a) w w r2 + a 2 ww r2 + a 2 w 2, w w ( 2iya w 2, 2ixa w 2, 0) λ w + w i( w w ) 1 + w w w (r iak) + w(r + iak) + 2a(r k) w 2 + r 2 + a 2 ( iσ)(r iak) + ( + iσ)(r + iak) + 2a(r k) w 2 + r 2 + a 2 2[r aσk + a(r k)] w 2 + r 2 + a 2 w w 2 (r 2 a 2 2iaz) 1/2 (r 2 a 2 + 2iaz) 1/2 ((r 2 a 2 ) 2 + 4a 2 z 2 ) 1/2 (( 2 σ 2 ) 2 + 4σ 2 2 ) 1/2 (( 2 + σ 2 ) 2 ) 1/2 2 2 r 2 a 2 + 2a 2 λ λ 2[r aσk + a(r k)] w 2 + r 2 + a 2 [r + ak az + a(r k)] ( 2 + a 2 ) 2 + a 2 [r + a2 z 2 k + a (r k)] λ 4

λ 1 λ 2 λ 3 x + ay 2 + a 2 y ax 2 + a 2 2 + a 2 [z + a2 z 2 ] z r a l µ l 0 (1, x + ay y ax 2, + a2 2 + a 2, z ), 3 l2 0 4 + a 2 z 2 g µν η µν 2ml µ l ν g 00 1 2m 3 4 + a 2 z 2, g 11 1 2m 3 + ay 4 + a 2 (x z2 2 + a 2 )2 g 22 1 2m 3 ax 4 + a 2 (y z2 2 + a 2 )2, g 33 1 2m 3 4 + a 2 z 2 (z )2 g 01 2m 3 x + ay 4 + a 2 z 2 2 + a 2, g 02 2m 3 y ax 4 + a 2 z 2 2 + a 2 g 03 2m 3 z 4 + a 2 z 2, g 12 2m 3 (x + ay)(y ax) 4 + a 2 z 2 ( 2 + a 2 ) 2 ds 2 g 13 2m 3 z(x + ay) 4 + a 2 z 2 ( 2 + a 2 ), g 23 2m 3 z(y ax) 4 + a 2 z 2 ( 2 + a 2 ) ds 2 (dx 0 ) 2 dx 2 2m3 4 + a 2 z 2 [(dx0 ) 2 x + ay + ( 2 + a 2 )2 dx 2 y ax + ( 2 + a 2 )2 dy 2 + ( z )2 dz 2 + 2(x + ay) 2 + a 2 dx 0 dx + 2(x + ay)(y ax) + ( 2 + a 2 ) 2 dxdy + 2(y ax) 2 + a 2 dx 0 dy + 2z dx0 dz 2z(x + ay) 2z(y ax) ( 2 + a 2 dxdz + ) ( 2 + a 2 ) dydz] 2m 3 /( 4 + a 2 z 2 ) (a + b + c + d) 2 a 2 + b 2 + c 2 + d 2 + 2ab + 2ac + 2ad + 2bc + 2cd + 2bd 5

[ ds 2 (dx 0 ) 2 dx 2 2m3 4 + a 2 z 2 dx 0 + (dx 0 ) 2 dx 2 2m3 4 + a 2 z 2 x + ay y ax 2 dx + + a2 2 + a 2 dy + z ] 2 dz [ dx 0 (xdx + ydy) a(ydx xdy) + 2 + a 2 + 2 + a 2 + z dz ] 2 (3) (Kerr) 1963 1918 θ cos θ z φ x + iy ( sin θ) cos φ + i( sin θ) sin φ sin θ(cos φ + i sin φ) ( sin θ)e iφ ia ( ia) sin θe iφ x + iy φ u x 0 + (3) dz 2 dz 2 (cos θd sin θdθ) 2 dx 2 + dy 2 dx 2 + dy 2 d(x + iy) 2 [sin θe iφ d + ( ia) cos θe iφ dθ + i( ia) sin θe iφ dφ] [sin θe iφ d + ( + ia) cos θe iφ dθ i( + ia) sin θe iφ dφ] sin 2 θd 2 + 2 cos 2 θdθ 2 + a 2 cos 2 θdθ 2 + 2 sin 2 θdφ 2 + a 2 sin 2 θdφ 2 + 2 sin θ cos θddθ + 2a sin 2 θddφ (sin θd + cos θdθ + a sin θdφ) 2 + ( sin θdφ a cos θdθ) 2 6

xdx + ydy xdx + ydy 1 2 (2xdx + 2ydy) 1 2 d(x2 + y 2 ) 1 d x + iy 2 2 1 2 d[(2 + a 2 ) sin 2 θ] sin 2 θd + ( 2 + a 2 ) sin θ cos θdθ xdy ydx xdy ydx Im[(x + iy)d(x iy)] Im[( ia)e iφ sin θd{( + ia)e iφ sin θ}] Im[( ia)e iφ sin θ{sin θe iφ d + ( + ia)e iφ cos θdθ i( + ia)e iφ sin θdφ}] Im[( ia) sin 2 θd i( 2 + a 2 ) sin 2 θdφ + ( 2 + a 2 ) sin θ cos θdθ] ( 2 + a 2 ) sin 2 θdφ + a sin 2 θd zdz zdz cos θ(cos θd sin θdθ) cos 2 θd 2 cos θ sin θdθ xdy ydx 2m 3 4 + a 2 z 2 2m 3 4 + a 2 2 cos 2 θ 2m 2 + a 2 cos 2 θ dx 0 du d (x ) 7

[ ds 2 (dx 0 ) 2 dx 2 2m3 4 + a 2 z 2 dx 0 (xdx + ydy) a(xdy ydx) + 2 + a 2 + 2 + a 2 + z ] 2 dz (du 2 + d 2 2dud) [(sin θd + cos θdθ + a sin θdφ) 2 + ( sin θdφ a cos θdθ) 2 + (cos θd sin θdθ) 2 ] [ 2m 2 + a 2 cos 2 du d + ( sin2 θd + ( 2 + a 2 ) sin θ cos θdθ) θ 2 + a 2 + a{(2 + a 2 ) sin 2 θdφ + a sin 2 θd} 2 + a 2 + cos2 θd 2 ] 2 cos θ sin θdθ (du 2 + d 2 2dud) (d 2 + 2 dθ 2 + (a 2 + 2 ) sin 2 θdφ 2 + 2a sin 2 θddφ + a 2 cos 2 θdθ 2 ) 2m [ du d + d + a sin 2 2 + a 2 cos 2 θdφ ] 2 θ (du 2 2dud) ( 2 dθ 2 + (a 2 + 2 ) sin 2 θdφ 2 + 2a sin 2 θddφ + a 2 cos 2 θdθ 2 ) (1 2m 2 + a 2 cos 2 θ [du2 + a 2 sin 4 θdφ 2 + 2a sin 2 θdφdu] 2m 2 + a 2 cos 2 θ )du2 ( 2 + a 2 cos 2 θ)dθ 2 [(a 2 + 2 ) sin 2 θ + 2ma2 sin 4 θ 2 + a 2 cos 2 θ ]dφ2 2dud 4ma sin2 θ 2 + a 2 cos 2 θ dφdu 2a sin2 θddφ (4) φ (4) ω ds 2 (c 2 ω 2 r 2 )dt 2 (dr 2 + r 2 dφ 2 + 2ωr 2 dφdt + dz 2 ) dφdt (4) ddφ, dud ds 2 g 00 du 2 + g 22 dθ 2 + g 33 dφ 2 + 2g 03 dudφ + 2g 01 dud + 2g 13 ddφ (5) cˆt u A(), du cdˆt + A d ˆφ φ B(), dφ d ˆφ + B d A B (11) 8

ds 2 g 00 du 2 + g 22 dθ 2 + g 33 dφ 2 + 2g 03 dudφ + 2g 01 dud + 2g 13 ddφ g 00 (cdˆt + A d) 2 + g 22 dθ 2 + g 33 (d ˆφ + B d) 2 + 2g 03 (cdˆt + A d)(d ˆφ + B d) + 2g 01 (cdˆt + A d)d + 2g 13 (d ˆφ + B d)d g 00 [(cdˆt) 2 + A 2 d 2 + 2A cdˆtd] + g 22 dθ 2 + g 33 (d ˆφ 2 + B 2 d 2 + 2B d ˆφd) + 2g 03 (cdˆtd ˆφ + A dd ˆφ + A B d 2 + B cdˆtd) + 2g 01 (cdˆt + A d)d + 2g 13 (d ˆφ + B d)d g 00 (cdˆt) 2 + (g 00 A 2 + g 33 B 2 + 2g 03 A B + 2g 01 A + 2g 13 B )d 2 + g 22 dθ 2 + g 33 d ˆφ 2 + 2g 03 cdˆtd ˆφ + 2(g 33 B + g 03 A + g 13 )d ˆφd + 2(g 00 A + g 03 B + g 01 )cdˆtd (6) d ˆφd, cdˆtd 0 A B g 33 B + g 03 A + g 13 0 g 00 A + g 03 B + g 01 0 { A g03b +g 01 g 00 B g 03A +g 13 g 33 B g 13g 00 g 03 g 01 g 2 03 g 00g 33 A g 01g 33 g 03 g 13 g 2 03 g 33g 00 g µν A g 00 g 33 (1 2m 2 + a 2 cos 2 θ )[(a2 + 2 ) sin 2 θ + 2ma2 sin 4 θ 2 + a 2 cos 2 θ ] (a 2 + 2 ) sin 2 θ 2ma2 sin 4 θ 2 + a 2 cos 2 θ + (a2 + 2 ) sin 2 2m θ 2 + a 2 cos 2 θ + (2m)2 a 2 sin 4 θ ( 2 + a 2 cos 2 θ) 2 g 2 03 (2ma sin2 θ) 2 ( 2 + a 2 cos 2 θ) 2 9

g 2 03 g 00 g 33 (a 2 + 2 ) sin 2 θ + 2ma2 sin 4 θ 2 + a 2 cos 2 θ (a2 + 2 ) sin 2 2m θ 2 + a 2 cos 2 θ (2m)2 a 2 sin 4 θ ( 2 + a 2 cos 2 θ) 2 + (2ma sin2 θ) 2 ( 2 + a 2 cos 2 θ) 2 (a2 + 2 )( 2 + a 2 cos 2 θ) sin 2 θ + 2ma 2 sin 4 θ 2m(a 2 + 2 ) sin 2 θ 2 + a 2 cos 2 θ A g 33 g 01 [(a 2 + 2 ) sin 2 θ + 2ma2 sin 4 θ 2 + a 2 cos 2 θ ] g 03 g 13 a sin 2 θ 2ma sin2 θ 2 + a 2 cos 2 θ g 33 g 01 g 03 g 13 (a2 + 2 )( 2 + a 2 cos 2 θ) sin 2 θ + 2ma 2 sin 4 θ 2ma 2 sin 4 θ 2 + a 2 cos 2 θ (a2 + 2 )( 2 + a 2 cos 2 θ) sin 2 θ 2 + a 2 cos 2 θ A A (a 2 + 2 )( 2 + a 2 cos 2 θ) sin 2 θ (a 2 + 2 )( 2 + a 2 cos 2 θ) sin 2 θ + 2ma 2 sin 4 θ 2m(a 2 + 2 ) sin 2 θ (a 2 + 2 )( 2 + a 2 cos 2 θ) (a 2 + 2 )( 2 + a 2 cos 2 θ) + 2ma 2 (1 cos 2 θ) 2m(a 2 + 2 ) (a 2 + 2 )( 2 + a 2 cos 2 θ) ( 2 + a 2 cos 2 θ)[(a 2 + 2 ) 2m( 2 + a 2 cos 2 θ)/( 2 + a 2 cos 2 θ)] a 2 + 2 a 2 + 2 2m B g 00 g 13 a sin 2 θ(1 2m 2 + a 2 cos 2 θ ) g 03 g 01 2ma sin2 θ 2 + a 2 cos 2 θ 10

g 00 g 13 g 03 g 01 a(2 + a 2 cos 2 θ) sin 2 θ 2 + a 2 cos 2 θ B a( 2 + a 2 cos 2 θ) sin 2 θ (a 2 + 2 )( 2 + a 2 cos 2 θ) sin 2 θ + 2ma 2 sin 4 θ 2m(a 2 + 2 ) sin 2 θ a( 2 + a 2 cos 2 θ) (a 2 + 2 )( 2 + a 2 cos 2 θ) + 2ma 2 (1 cos 2 θ) 2m(a 2 + 2 ) a a 2 + 2 2m (6) d 2 g 00 A 2 + g 33 B 2 + 2g 03 A B + 2g 01 A + 2g 13 B A (g 00 A + g 03 B + g 01 ) + B (g 03 A + g 33 B + g 13 ) + g 01 A + g 13 B d ˆφd, cdˆtd ( ) 0 d 2 g 01 A + g 13 B A B (4) (6) g 01 A a 2 + 2 a 2 + 2 2m, g 13B a 2 sin 2 θ a 2 + 2 2m a2 a 2 cos 2 θ a 2 + 2 2m ds 2 g 00 (cdˆt) 2 + (g 01 A + g 13 B )d 2 + g 22 dθ 2 + g 33 d ˆφ 2 + 2g 03 cdˆtd ˆφ (1 2m 2 + a 2 cos 2 θ )(cdˆt) 2 2 + a 2 cos 2 θ a 2 + 2 2m d2 ( 2 + a 2 cos 2 θ)dθ 2 [(a 2 + 2 ) sin 2 θ + 2ma2 sin 4 θ 2 + a 2 cos 2 θ ]d ˆφ2 2 2ma sin2 θ 2 + a 2 cos 2 θ cdˆtd ˆφ (Boyer-Lindquist) a 0 e 0 11

dˆtd ˆφ dˆt, d ˆφ dˆtd ˆφ ( ) ds 2 a2 sin 2 θ (cdt) 2 Σ Σ d2 (2 + a 2 ) 2 a 2 sin 2 θ Σ sin 2 θdφ 2 Σdθ 2 2 2ma sin2 θ cdtdφ Σ ( Σ 2 + a 2 cos 2 θ 2 + a 2 2m 12