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Chapter 22 Geodesic Motion in Kerr Spacetime

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Chapter 22 Geodesic Motion in Kerr Spacetime Chapter 22 Geodesic motion in Kerr spacetime Let us consider a geodesic with affine parameter λ and tangent vector dxµ uµ = x˙ µ. (22.1) dλ ⌘ In this section we shall use Boyer-Lindquist’s coordinates, and the dot will indicate di↵eren- tiation with respect to λ.Thetangentvectoruµ is solution of the geodesic equations µ ⌫ u u ;µ =0, (22.2) which, as shown in Chapter 11, is equivalent to the Euler-Lagrange equations d @ @ L = L (22.3) dλ @x˙ ↵ @x↵ associated to the Lagrangian 1 (xµ, x˙ µ)= g x˙ µx˙ ⌫ . (22.4) L 2 µ⌫ By defining the conjugate momentum pµ as @ p L = g x˙ ⌫ , (22.5) µ ⌘ @x˙ µ µ⌫ the Euler-Lagrange equations become d @ p = L . (22.6) dλ µ @xµ Note that, if the metric does not depend on a given coordinate xµ,theconjugatemomentum pµ is a constant of motion and coincides with the constant of motion associated to the Killing vector tangent to the corresponding coordinate lines. The Kerr metric in Boyer-Lindquist coordinates is indepentent of t and φ,therefore p =˙x u = const and p =˙x u = const;(22.7) t t ⌘ t φ φ ⌘ φ these quantities coincide with the constant of motion associated to the Killing vectors kµ = µ µ µ (1, 0, 0, 0) and m =(0, 0, 0, 1), i.e. k uµ = ut and m uµ = uφ. 316 CHAPTER 22. GEODESIC MOTION IN KERR SPACETIME 317 Therefore, geodesic motion in Kerr geometry is characterized by two constants of motion, which we indicate as: E kµu = u = p constant along geodesics ⌘ µ − t − t (22.8) L mµu = u = p constant along geodesics . ⌘ µ φ φ (22.9) As explained in Section 11.2, for massive particles E and L are, respectively, the energy and the angular momentum per unit mass, as measured at infinity with respect to the black hole. For massless particles, E and L are the energy and the angular momentum at infinity. Equations (22.2) (or, equivalently, (22.3)) in Kerr spacetime are very complicate to solve directly. To simplify the problem we hall use the conserved quantities, as we did in Chapter 11 in when we studied geodesic motion in Schwarzschild’s spacetime. For this, we need four algebraic relations involving uµ. Furthermore µ ⌫ gµ⌫u u = (22.10) where = 1fortimelikegeodesics − =1 forspacelikegeodesics =0 fornullgeodesics. (22.11) Eqs. (22.8), (22.9), (22.10) give three algebraic relations involving uµ, but they are not suf- ficient to to determine the four unknowns uµ.InSchwarzschildspacetimeafourthequation is provided by the planarity of the orbit (u✓ =0if✓(λ =0)=⇡/2); in Kerr spacetime orbits are planar only in the equatorial plane therefore, in general, geodesic motion cannot be studied in a simple way, using eqs. (22.8), (22.9), (22.10) only, as we did for Scharzschild. However, as we shall briefly explain in the last section of this chapter, there exists a further conserved quantity, the Carter constant,whichallowstofindthetangentvectoruµ using algebraic relations. 22.1 Equatorial geodesics In this section we study geodesic motion in the equatorial plane, i.e. geodesics with ⇡ ✓ . (22.12) ⌘ 2 First of all, let us prove that such geodesics exist, i.e. that eq. (22.12) is solution of the Euler-Lagrange equations. The Lagrangian is 1 µ ⌫ 1 2Mr 2 4Mr 2 ⌃ 2 = gµ⌫x˙ x˙ = 1 t˙ a sin ✓ t˙φ˙ + r˙ (22.13) L 2 2 ⇢− ✓ − ⌃ ◆ − ⌃ ∆ 2Mra2 +⌃✓˙2 + r2 + a2 + sin2 ✓ sin2 ✓ φ˙2 " ⌃ # ) CHAPTER 22. GEODESIC MOTION IN KERR SPACETIME 318 and the ✓ component of Euler-Lagrange’s equations is d d 1 (g x˙ µ)= (⌃✓˙)=⌃✓¨ +⌃ x˙ µ✓˙ = g x˙ µx˙ ⌫ . (22.14) dλ ✓µ dλ ,µ 2 µ⌫,✓ The right-hand side is 2 1 µ ⌫ 1 (˙r) 2 2 2 2 gµ⌫,✓x˙ x˙ = ⌃,✓ +(✓˙) +2sin✓ cos ✓(r + a )(φ˙) 2 2 ( ∆ ! 2Mr 2 4Mr ⌃ a sin2 ✓φ˙ t˙ + a sin2 ✓φ˙ t˙ 2a sin ✓ cos ✓φ˙ − ⌃2 ,✓ − ⌃ − ⇣ ⌘ ⇣ ⌘ (22.15) 2 where ⌃,✓ = 2a sin ✓ cos ✓ and ⌃,r =2r.Itiseasytocheckthat,when✓ = ⇡/2, equation (22.14) reduces− to 2 ✓¨ = r˙✓˙ . (22.16) −r Therefore, if ✓˙ =0and✓ = ⇡/2atλ =0,thenforλ>0 ✓˙ 0and✓ ⇡/2. Thus, a geodesic which starts in the equatorial plane, remains in the equatorial⌘ plane⌘ at later times. This also occurs in Schwarzschild spacetime, and in that case, due to the spherical sym- metry, it is possible to generalize the result to any orbit, and prove that all geodesics are planar. This generalization is not possible for the Kerr metric which is axially symmetric. In this case only equatorial geodesics are planar. On the equatorial plane, ⌃= r2,therefore 2M gtt = 1 − ✓ − r ◆ 2Ma g = tφ − r r2 g = rr ∆ 2Ma2 g = r2 + a2 + (22.17) φφ r and µ 2M 2Ma E = gtµu = 1 t˙ + φ˙ (22.18) − ✓ − r ◆ r 2 µ 2Ma 2 2 2Ma L = gφµu = t˙ + r + a + φ˙ . (22.19) − r r ! To solve eqs. (22.18), (22.19) for t˙, φ˙ we define 2M A 1 ⌘ − r 2Ma B ⌘ r 2Ma2 C r2 + a2 + (22.20) ⌘ r CHAPTER 22. GEODESIC MOTION IN KERR SPACETIME 319 and write eqs. (22.18), (22.19) as E = At˙ + Bφ˙ (22.21) L = Bt˙ + Cφ˙ . (22.22) − Furthermore, the following relation can be used 2 2 2 2 2M 2 2 2Ma 4M a 2 2 AC + B = 1 r + a + + 2 = r 2Mr + a =∆. (22.23) ✓ − r ◆ r ! r − Therefore, CE BL =[AC + B2]t˙ =∆t˙ − AL + BE =[AC + B2]φ˙ =∆φ˙ (22.24) i.e. 1 2Ma2 2Ma t˙ = r2 + a2 + E L ∆ " r ! − r # . (22.25) 1 2M 2Ma φ˙ = 1 L + E ∆ ✓ − r ◆ r The quantity C defined in eq. (22.20) can be written in a di↵erent form, which will be useful in the following: (r2 + a2)2 a2∆ 1 − = [(r2 + a2)(r2 + a2) a2(r2 + a2 2Mr)] r2 r2 − − 1 2Ma2 = [(r2 + a2)r2 +2Mra2]=r2 + a2 + r2 r C. (22.26) ⌘ Note that C is always positive. Let us now derive the equation for the radial component of the four-velocity. Equation (22.10) can be written in terms of A, B, C: µ ⌫ gµ⌫u u = r2 = At˙2 2Bt˙φ˙ + Cφ˙2 + r˙2 − − ∆ r2 = [At˙ + Bφ˙]t˙ +[ Bt˙ + Cφ˙]φ˙ + r˙2 − − ∆ r2 = Et˙ + Lφ˙ + r˙2 (22.27) − ∆ where we have used eqs. (22.21), (22.22). Therefore, ∆ 1 ∆ r˙2 = (Et˙ Lφ˙ + )= CE2 2BLE AL2 + . (22.28) r2 − r2 − − r2 h i CHAPTER 22. GEODESIC MOTION IN KERR SPACETIME 320 The polynomial [CE2 2BLE AL2]haszeros − − BL pB2L2 + ACL2 L V = ± = [B p∆] . (22.29) ± C C ± Consequently, eq. (22.28) can be written as 2 C ∆ r˙ = (E V+)(E V )+ . (22.30) r2 − − − r2 Using eq. (22.26), eqs. (22.30) and (22.29) finally become 2 2 2 2 2 (r + a ) a ∆ ∆ r˙ = − (E V+)(E V )+ , (22.31) r4 − − − r2 and 2Mar r2p∆ V = ± L . (22.32) ± (r2 + a2)2 a2∆ − In the Schwarzschild limit a 0and ! L2∆ V+ + V a 0 ,V+V (22.33) − / ! − ! r4 therefore, if we define V V+V ,eqs.(22.31),(22.32)reducetothewellknownform ⌘ − 2 2 2 2 ∆ L ∆ 2M L r˙ = E V (r), where V (r)= 2 + 4 = 1 + 2 (22.34) − − r r ✓ − r ◆ − r ! where we recall that = 1fortimelikegeodesics, =0fornullgeodesics, =1for spacelike geodesics. − 22.1.1 Kerr’s potentials for equatorial geodesics 2MLar Lr2p∆ V = ± . (22.35) ± (r2 + a2)2 a2∆ − In principle we would have four possibilities, corresponding to L positive and negative and a positive and negative. In practice, there are only two interesting cases: La > 0andLa < 0, i.e. the test particle is either corotating or counterrotating with the black hole. If the signs of L and a change simultaneously, the potentials V interchange: V+ becomes V and viceversa. ± − To avoid this, it is better to redefine the names of the potentials as follows 2MLar L r2p∆ V = ±| | , (22.36) ± (r2 + a2)2 a2∆ − so that the following inequality is always true V+ V . (22.37) ≥ − In general, we find that: CHAPTER 22. GEODESIC MOTION IN KERR SPACETIME 321 V+ and V coincide for ∆= 0, i.e. for • − r = r = M + pM 2 a2 (22.38) + − while for r>r+,∆> 0andthenV+ >V . Furthermore, − 2Mr+La V+(r+)=V (r+)= 2 2 2 , (22.39) − (r+ + a ) which is positive if La > 0, negative if La < 0. In the limit r , V 0. • !1 ± ! If La > 0(corotatingorbits),thepotentialV+ is definite positive; V (which is positive • − at r+)vanisheswhen rp∆=2Ma r2(r2 2Mr + a2)=4M 2a2 (22.40) ) − which gives r4 2Mr3 + a2r2 4M 2a2 =(r 2M)(r3 + a2r +2Ma2)=0; (22.41) − − − thus V vanishes at r =2M,whichisthelocationoftheergosphereintheequatorial − plane. If La < 0(counterrotatingorbits),thepotentialV is definite negative and V+ (which • − is negative at r+)vanishesatr =2M. The study of the derivatives of V ,whichistoolongtobereportedhere,showsthat • ± both potentials, V+ and V ,haveonlyonestationarypoint. − In summary, V+(r)andV (r) have the shapes shown in Figure 22.1 where the upper and − lower panels refer, respectively, to the case La > 0andLa < 0cases. 22.1.2 Null geodesics In the case of null geodesics the radial equation (22.31) becomes 2 2 2 2 2 C (r + a ) a ∆ r˙ = (E V+)(E V )= − (E V+)(E V )(22.42) r2 − − − r4 − − − Sincer ˙2 must be positive, from eq. (22.42) we see that, and being (r2 + a2)2 a2∆ > 0, null geodesics are possible for massless particle whose constant of motion E satisfies− the following inequalities E<V or E>V+ . (22.43) − Thus, the region V <E<V+, corresponding to the dashed regions in Figure 22.1, is − forbidden.
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