Geodesics of the Kerr metric

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• If La < 0(counterrotatingorbits),thepotentialV − is definite negative and V + (which is negative at r + )vanishesatthe ergosphere r = r S+ =2M. • The study of the derivatives of V ± , which is too long to be reported here, tells us that both potentials, V + and V − ,have only one stationary point. In summary, V + (r) andV − (r) have the shapes shown in Figure 4.1 where in the two panels we show the cases La > 0(up)andLa < 0 (down), respectively. Once we have drawn the curves V + (r),V − (r) L V V r + r S+ La > 0 V + V La < 0 V + r r + S+ r r L V Figure 4.1: The potentials V + (r) andV − (r), for corotating (La > 0) and counterrotating (La < 0) orbits. The shadowed region is not accessible to the motion of photons or other massless particles. for assigned values of a, M and L, wecanmakeaqualitativestudy 74
of the orbits. To this purpose it is useful to compute the radial acceleration. By differentiating eq. (4.34) (ṙ) 2 = C r (E − V +)(E − V 2 − ) (4.42) with respect to the geodesic parameter λ, wefind 2ṙ¨r = [( ) ′ C (E − V r 2 + )(E − V − ) − C r V +(E ′ − V 2 − ) − C ] r V ′ 2 − (E − V +) ṙ (4.43) i.e. ¨r = 1 ( ) ′ C (E − V 2 r 2 + )(E − V − ) − C [ V ′ 2r 2 + (E − V −)+V − ′ (E − V +) ] . (4.44) Let us consider the radial acceleration in a point where the radial velocity ṙ is zero, i.e. when E = V + or E = V − : Since we find ¨r = − C 2r V ′ 2 + (V + − V − ) if E = V + ¨r = − C 2r V ′ 2 − (V − − V + ) if E = V − . (4.45) V + − V − = 2|L|r 2√ ∆ (r 2 + a 2 ) 2 − a 2 ∆ = 2|L|√ ∆ , (4.46) C ¨r = ∓ |L|√ ∆ r 2 V ′ ± if E = V ± . (4.47) This result shows that if, for example, E = V + (r max )wherer max is astationarypointforV + (i.e. V ′ + (r max) =0),theradialacceleration vanishes; since when E = V + (r max )theradialvelocityalsovanishes, a massless particle with that value of E can be captured on a circular orbit, but the orbit is unstable, as it is the orbit at r =3M for the Schwarzschild metric. It is possible to show that r max is the solution of the equation r(r − 3M) 2 − 4Ma 2 =0. (4.48) 75
Page 1 and 2: Chapter 4 Geodesics of the Kerr met
Page 3 and 4: κ =1 forspacelikegeodesics κ =0 f
Page 5 and 6: B ≡ 2Ma r C ≡ r 2 + a 2 + 2Ma2
Page 7: 4.1.1 Null geodesics In the case of
Page 11 and 12: ackwards in time. Furthermore, from
Page 13 and 14: 4.1.3 The Penrose process A short c
Page 15 and 16: If we compute the integral at a tim
Page 17 and 18: Figure 4.2: Geodesics of Massive pa
Page 19 and 20: we have defined the conjugate momen
Page 21 and 22: [ ] (r 2 + a 2 ) 2 − − a 2 sin
Page 23: Then we have dθ dλ ′ = ± √

Geodesics of the Kerr metric

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