Geodesics of the Kerr metric

where V ± (r) are the solutions of the equation in E
i.e.
V ± = BL ± √ B 2 L 2 + ACL 2
C
CE 2 − 2BLE − AL 2 =0, (4.27)
= 1 C (BL ±|L|√ ∆) . (4.28)
Some authors write this formula without the modulus (which is
equivalent to exchange the definitions of V + and V − when L 0, C>0.
In the Schwarzschild limit a → 0, we have
V + + V − ∝ a → 0 ,
V + V − →− L2 ∆
r 4 (4.32)
therefore, if we define V ≡−V + V − − κ∆/r 2 ,Eqns. (4.30),(4.31)
reduce to the well known form
ṙ 2 = E 2 − V (r)
V (r) = − κ∆
r + L2 ∆
(
2 r 4
= 1 − 2M )
)(−κ + L2
r
r 2
(4.33)
where we recall that κ = −1 fortimelikegeodesics,κ = 0 for null
geodesics, κ =1forspacelikegeodesics.
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