Geodesic motion in Kerr spacetime

Transcription

1 Chapte 20 Geodesic motion in Ke spacetime Let us conside a geodesic with affine paamete λ and tangent vecto u µ = dxµ dλ ẋµ. (20.1) In this section we shall use Boye-Lindquist s coodinates, and the dot will indicate diffeentiation with espect to λ. The tangent vecto u µ is solution of the geodesic equations u µ u ν ;µ =0, (20.2) which, as shown in Chapte 11, is equivalent to the Eule-Lagange equations associated to the Lagangian d L dλ ẋ = L α x α (20.3) L (x µ, ẋ µ )= 1 2 g µνẋ µ ẋ ν. (20.4) By defining the conjugate momentum p µ as the Eule-Lagange equations become p µ L ẋ µ = g µνẋ ν, (20.5) d dλ p µ = L x. (20.6) µ Note that, if the metic does not depend on a given coodinate x µ, the conjugate momentum p µ is a constant of motion and coincides with the constant of motion associated to the Killing vecto tangent to the coesponding coodinate lines. The Ke metic in Boye-Lindquist coodinates is indepentent of t and φ, theefoe p t =ẋ t u t = const and p φ =ẋ φ u φ = const; (20.7) these quantities coincide with the constant of motion associated to the Killing vectos k µ = (1, 0, 0, 0) and m µ = (0, 0, 0, 1), i.e. k µ u µ = u t and m µ u µ = u φ. 295

2 CHAPTER 20. GEODESIC MOTION IN KERR SPACETIME 296 Theefoe, geodesic motion in Ke geomety is chaacteized by two constants of motion, which we indicate as: E k µ u µ = u t = p t constant along geodesics L m µ u µ = u φ = p φ constant along geodesics. (20.8) (20.9) As explained in Section 11.2, fo massive paticles E and L ae, espectively, the enegy and the angula momentum pe unit mass, as measued at infinity with espect to the black hole. Fo massless paticles, E and L ae the enegy and the angula momentum at infinity. Equations (20.2) (o, equivalently, (20.3)) in Ke spacetime ae vey complicate to solve diectly. To simplify the poblem we hall use the conseved quantities, as we did in Chapte 11 in when we studied geodesic motion in Schwazschild s spacetime. Fo this, we need fou algebaic elations involving u µ. Futhemoe g µν u µ u ν = κ (20.10) whee κ = 1 κ = 1 fo timelike geodesics fo spacelike geodesics κ = 0 fo null geodesics. (20.11) Eqs. (20.8), (20.9), (20.10) give thee algebaic elations involving u µ, but they ae not sufficient to to detemine the fou unknowns u µ. In Schwazschild spacetime a fouth equation is povided by the planaity of the obit (u θ = 0 if θ(λ = 0) = π/2); in Ke spacetime obits ae plana only in the equatoial plane theefoe, in geneal, geodesic motion cannot be studied in a simple way, using eqs. (20.8), (20.9), (20.10) only, as we did fo Schazschild. Howeve, as we shall biefly explain in the last section of this chapte, thee exists a futhe conseved quantity, the Cate constant, which allows to find the tangent vecto u µ using algebaic elations Equatoial geodesics In this section we study geodesic motion in the equatoial plane, i.e. geodesics with θ π 2. (20.12) Fist of all, let us pove that such geodesics exist, i.e. that eq. (20.12) is solution of the Eule-Lagange equations. The Lagangian is L = 1 2 g µνẋ µ ẋ ν = 1 2 [ + Σ θ 2 + { ( 1 2M Σ 2 + a 2 + 2Ma2 Σ ) ṫ 2 2M ] } sin 2 θ φ 2 sin2 θ Σ a sin2 θ ṫ φ + Σ ṙ2 (20.13)

3 CHAPTER 20. GEODESIC MOTION IN KERR SPACETIME 297 and the θ component of Eule-Lagange s equations is d dλ (g θµẋ µ )= d dλ (Σ θ) = Σ θ +Σ,µ ẋ µ 1 θ = 2 g µν,θẋ µ ẋ ν. (20.14) The ight-hand side is 1 2 g µν,θẋ µ ẋ ν = 1 { ( ) (ṙ) 2 Σ,θ 2 +( θ) sin θ cos θ( 2 + a 2 )( φ) 2 2M Σ Σ ( 2,θ a sin 2 θ φ ṫ ) 2 4M ( + a sin 2 θ Σ φ ṫ ) φ} 2a sin θ cos θ (20.15) whee Σ,θ = 2a 2 sin θ cos θ and Σ, =2. It is easy to check that, when θ = π/2, equation (20.14) educes to θ = 2 ṙ θ. (20.16) Theefoe, if θ = 0 and θ = π/2 at λ = 0, then fo λ> 0 θ 0 and θ π/2. Thus, a geodesic which stats in the equatoial plane, emains in the equatoial plane at late times. This also occus in Schwazschild spacetime, and in that case, due to the spheical symmety, it is possible to genealize the esult to any obit, and pove that all geodesics ae plana. This genealization is not possible fo the Ke metic which is axially symmetic. In this case only equatoial geodesics ae plana. On the equatoial plane, Σ = 2, theefoe ( g tt = 1 2M ) g tφ = 2Ma and g = 2 g φφ = 2 + a 2 + 2Ma2 ( E = g tµ u µ = 1 2M L = g φµ u µ = 2Ma ṫ + To solve eqs. (20.18), (20.19) fo ṫ, φ we define ) ( ṫ + 2Ma 2 + a 2 + 2Ma2 (20.17) φ (20.18) ) φ. (20.19) A 1 2M B 2Ma C 2 + a 2 + 2Ma2 (20.20)

4 CHAPTER 20. GEODESIC MOTION IN KERR SPACETIME 298 and wite eqs. (20.18), (20.19) as E = Aṫ + B φ (20.21) L = Bṫ + C φ. (20.22) Futhemoe, the following elation can be used ( AC + B 2 = 1 2M ) ( ) 2 + a 2 + 2Ma2 + 4M 2 a 2 = 2 2M + a 2 =. (20.23) 2 Theefoe, CE BL = [AC + B 2 ]ṫ =ṫ AL + BE = [AC + B 2 ] φ = φ (20.24) i.e. ṫ = 1 [( φ = a 2 + 2Ma2 [( 1 2M ) ) E 2Ma ] L L + 2Ma ] E. (20.25) The quantity C defined in eq. (20.20) can be witten in a diffeent fom, which will be useful in the following: ( 2 + a 2 ) 2 a 2 2 = 1 2 [(2 + a 2 )( 2 + a 2 ) a 2 ( 2 + a 2 2M)] = 1 2 [(2 + a 2 ) 2 +2Ma 2 ]= 2 + a 2 + 2Ma2 C. (20.26) Note that C is always positive. Let us now deive the equation fo the adial component of the fou-velocity. Equation (20.10) can be witten in tems of A, B, C: g µν u µ u ν = κ = Aṫ 2 2Bṫ φ + C φ ṙ2 = [Aṫ + B φ]ṫ +[ Bṫ + C φ] φ + 2 ṙ2 whee we have used eqs. (20.21), (20.22). Theefoe, = Eṫ + L φ + 2 ṙ2 (20.27) ṙ 2 = 2 (Eṫ L φ + κ) = 1 2 [ CE 2 2BLE AL 2] + κ 2. (20.28)

5 CHAPTER 20. GEODESIC MOTION IN KERR SPACETIME 299 The polynomial [CE 2 2BLE AL 2 ] has zeos V ± = BL ± B 2 L 2 + ACL 2 C Consequently, eq. (20.28) can be witten as = L C [B ± ]. (20.29) ṙ 2 = C 2 (E V +)(E V )+ κ 2. (20.30) Using eq. (20.26), eqs. (20.30) and (20.29) finally become ṙ 2 = (2 + a 2 ) 2 a 2 (E V 4 + )(E V )+ κ, (20.31) 2 and V ± = 2Ma ± 2 ( 2 + a 2 ) 2 a 2 L. (20.32) In the Schwazschild limit a 0 and V + + V a 0, V + V L2 4 (20.33) theefoe, if we define V V + V, eqs. (20.31), (20.32) educe to the well known fom ṙ 2 = E 2 V (), whee V () = κ ( + L2 = 1 2M ) ( ) κ + L2 (20.34) whee we ecall that κ = 1 fo timelike geodesics, κ = 0 fo null geodesics, κ = 1 fo spacelike geodesics The potentials behaviou V ± = 2MLa ± L2 ( 2 + a 2 ) 2 a 2. (20.35) In pinciple we would have fou possibilities, coesponding to L positive and negative and a positive and negative. In pactice, thee ae only two inteesting cases: La > 0 and La < 0, i.e. the test paticle is eithe cootating o counteotating with the black hole. If the signs of L and a change simultaneously, the potentials V ± intechange: V + becomes V and vicevesa. To avoid this, it is bette to edefine the names of the potentials as follows so that the following inequality is always tue In geneal, we find that: V ± = 2MLa ± L 2 ( 2 + a 2 ) 2 a 2, (20.36) V + V. (20.37)

6 CHAPTER 20. GEODESIC MOTION IN KERR SPACETIME 300 V + and V coincide fo = 0, i.e. fo while fo > +, > 0 and then V + >V. Futhemoe, which is positive if La > 0, negative if La < 0. In the limit, V ± 0. = + = M + M 2 a 2 (20.38) V + ( + )=V ( + )= 2M +La ( a 2 ) 2, (20.39) If La > 0 (cootating obits), the potential V + is definite positive; V (which is positive at + ) vanishes when which gives =2 Ma 2 ( 2 2M + a 2 )=4M 2 a 2 (20.40) 4 2M 3 + a 2 2 4M 2 a 2 =( 2M)( 3 + a 2 +2Ma 2 ) = 0; (20.41) thus V vanishes at =2M, which is the location of the egosphee in the equatoial plane. If La < 0 (counteotating obits), the potential V is definite negative and V (which is positive at + ) vanishes at =2M. The study of the deivatives of V ±, which is too long to be epoted hee, shows that both potentials, V + and V, have only one stationay point. In summay, V + () and V () have the shapes shown in Figue 20.1 whee the uppe and lowe panels efe, espectively, to the case La > 0 and La < 0 cases Null geodesics In the case of null geodesics the adial equation (20.31) becomes ṙ 2 = C 2 (E V +)(E V )= (2 + a 2 ) 2 a 2 4 (E V + )(E V ) (20.42) Since ṙ 2 must be positive, fom eq. (20.42) we see that, and being ( 2 + a 2 ) 2 a 2 > 0, null geodesics ae possible fo massless paticle whose constant of motion E satisfies the following inequalities E < V o E > V +. (20.43) Thus, the egion V fobidden. < E < V +, coesponding to the dashed egions in Figue 20.1, is

7 CHAPTER 20. GEODESIC MOTION IN KERR SPACETIME 301 Figue 20.1: The potentials V + () and V (), fo cootating (al > 0) and counteotating (al < 0) obits. The shadowed egion is not accessible to the motion of photons o othe massless paticles. In ode to study the obits, it is useful to compute the adial acceleation. By diffeentiating eq. (20.42) with espect to the affine paamete λ, we find 2ṙ = [ ( C 2 ) (E V + )(E V ) C 2 V +(E V ) C 2 V (E V + ) ] ṙ (20.44) i.e. = 1 ( ) C (E V )(E V ) C [ V (E V )+V (E V +) ], (20.45) whee the pime indicates diffeentiation with espect to. Let us evaluate the adial acceleation in a point whee the adial velocity ṙ is zeo, i.e. when E = V + o E = V : Since = = C 2 V +(V 2 + V ) if E = V + = = C 2 V (V 2 V + ) if E = V. (20.46) 2 L 2 V + V = ( 2 + a 2 ) 2 a 2 = 2 L, (20.47) C

8 CHAPTER 20. GEODESIC MOTION IN KERR SPACETIME 302 we find = L 2 V ± if E = V ±. (20.48) Unstable cicula obits If E = V + ( max ), whee max is the stationay point of V + (i.e. V + ( max) = 0), the adial acceleation vanishes; since when E = V + ( max ) the adial velocity also vanishes, a massless paticle with that value of E can be captued on a cicula obit, but the obit is unstable, as it is the obit at =3M fo the Schwazschild metic. It is possible to show that max is solution of the equation ( 3M) 2 4Ma 2 =0. (20.49) Note that the value of max is independent of L. The solution of (20.49) is a deceasing function of a, and, in paticula, max = 3M fo a =0 max = M fo a = M max = 4M fo a = M. (20.50) Theefoe, while fo a Schwazschild black hole the unstable cicula obit of a photon is at =3M, fo a Ke black hole it can be much close; in paticula, in the extemal case a = M, fo cootating obits max = M coincides with the oute hoizon. Radial captue A photon falling fom infinity with constant of motion E > V + ( max ), cosses the hoizon and falls towad the singulaity. Deflection If 0 > E > V + ( max ), the paticle eaches the tuning point whee E = V + () and ṙ = 0; eq. (20.48) shows that at the tuning point >0, theefoe the paticle evets its motion and escapes fee at infinity. In this case the paticle is deflected. In the above cases the constant of motion E associated to the timelike Killing vecto is assumed to be positive. It emains to conside the case E < V, and in paticula to see whethe negative values of E, admitted in pinciple admitted by eq. (20.43), have a physical meaning How do we measue the enegy of a paticle The enegy of a paticle is an obseve-dependent quantity. In special elativity, the enegy of a paticle with fou-momentum P µ, measued by an obseve with fou-velocity u µ, is defined as E (u) = η µν u µ P ν = u µ P ν. (20.51) Fo instance, the enegy measued by a static obseve u µ st = (1, 0, 0, 0) is E (ust) = P 0. (20.52)

9 CHAPTER 20. GEODESIC MOTION IN KERR SPACETIME 303 A negative enegy would coespond to a paticle moving backwads in time, and causality would be violated. Thus, enegy is always positive; if measued by a diffeent obseve it will be diffeent, but still positive. Eq. (20.51) is a tenso equation; it holds in a locally inetial fame, whee g µν η µν, theefoe it can be witten as E (u) = g µν u µ P ν = u µ P µ. (20.53) Thus, by the pinciple of geneal covaiance, eq. (20.53), is the definition of enegy valid in any fame, and consequently E must be positive in any fame. Let us now conside a static obseve with u µ st = (1, 0, 0, 0), in Ke spacetime, located at adial infinity, whee such obseve can exist. Accoding to the definition (20.53), the enegy measued by the static obseve is E (ust) = P 0. Let us now compae this quantity with the constant of motion E = u 0 given in eq. (20.8). If the paticle is massless we can always paametize the geodesic in such a way that P 0 u 0. Thus: E (ust) = P 0 = E (20.54) We conclude that fo a paticle stating (o ending) its motion at adial infinity with espect to the black hole, the constant of motion E is the paticle enegy, as measued by a static obseve located at infinity 1. Fo such paticles obits with negative values of E ae not allowed. Thus, efeing to Figue 20.1, obits with E < V and E negative impinging fom adial infinity ae fobidden, even though fo such values ṙ 2 > 0 (see eq. (20.42)). Let us now conside a massless paticle which stats its motion in the egoegion, i.e. between + and 0 (see Figue 20.1). In this egion static obseves cannot exist, theefoe we need to conside a diffeent obseve, fo instance a ZAMO, whose fou-velocity can be witten as u µ ZAMO = const(1, 0, 0, Ω) (20.55) whee the ZAMO angula velocity Ω on the equatoial plane is (see eq. (19.27)) Ω= 2Ma ( 2 + a 2 ) 2 a 2 (20.56) and the constant is found by imposing g µν u µ u ν = 1. The constant must be positive, othewise the ZAMO would move backwads in time. The paticle enegy measued by the ZAMO is E ZAMO = P µ u µ ZAMO = const(e ΩL), (20.57) whee we have used eqs. (20.8) and (20.9). Thus, the equiement E ZAMO > 0 is equivalent to E>ΩL. (20.58) By compaing (20.56) with the expession of the potentials V ± given by eq. (20.36) we find that V < ΩL < V +. (20.59) 1 similaly, fo massive paticles E is the enegy pe unit mass as measued by a static obseve at infinity.

10 CHAPTER 20. GEODESIC MOTION IN KERR SPACETIME 304 Theefoe, geodesics with E > V + (20.60) satisfy the positive enegy condition (20.58), and ae allowed, wheeas those with E < V ae fobidden, since do not satisfy eq. (20.58). Thus, efeing to Figue 20.36: a cootating paticle (La > 0) can move within the egoegion only if the costant of motion E is positive and is in the ange V + ( + ) < E < V + ( max ). (20.61) If E > V + ( max ) the paticle can coss the egosphee and escape at infinity. Fo counteotating paticles (La < 0), since in the egoegion V + is negative the equiement E > V + (necessay and sufficient to ensue that E > 0) allows negative values of the constant of motion E. Thus, counteotating paticles moving in the egoegion can have negative E, povided V + ( + )(= V ( + )) < E < 0. (20.62) As we shall show in the next section, this possibility has an inteesting consequence. It should be stessed that this is not a contadiction, because it is only at infinity that E epesents the paticle enegy; the geodesics we ae consideing neve each infinity The Penose pocess In this section we will use a slightly diffeent notation fo the constants of motion E, L, which have been shown to be the enegy and angula momentum pe unit mass, fo massive paticles, and the enegy and angula momentum fo massless paticles, as measued by a static obseve at infinity. Hee we define E and L to be the enegy and angula momentum at infinity, both fo massive and massless paticles, so that eqs. (20.8) and (20.9) become E = k µ P µ, L = m µ P µ. (20.63) This simply means that, fo massive paticles, E and L have been multiplied by the paticle mass m. We shall now show that since paticles with negative E can exist in the egoegion, we can imagine a pocess though which it may be possible to extact otational enegy fom a Ke black hole; this is named Penose s pocess. In what follows we shall set a>0. Assuming a<0 would lead to the same conclusions. Suppose that we shoot a massive paticle with enegy E and angula momentum L fom infinity, so that it falls towads the black hole in the equatoial plane. Its fou-momentum covaiant components ae P µ =( E,0, 0,L). (20.64)

11 CHAPTER 20. GEODESIC MOTION IN KERR SPACETIME 305 Along the geodesic the paticle fou-momentum changes, but the covaiant components P t = k µ P µ = E, and P φ = m µ P µ = L emain constant, i.e., P µ =( E, P, 0,L). (20.65) When the paticle entes the egoegion, it decays in two photons, with momenta P 1 µ =( E 1,P 1, 0,L 1 ) P 2 µ =( E 2,P 2, 0,L 2 ). (20.66) Since the fou-momentum is conseved in this decay, we have fom which it follows that P µ = P 1 µ + P 2 µ o equivalently P µ = P 1 µ + P 2 µ, E = E 1 + E 2, L = L 1 + L 2. (20.67) Let us assume that ṙ 1 < 0, so that the photon 1 falls into the black hole, and that it has negative constants of motion, i.e. E 1 < 0 and L 1 < 0, with (see eq. (20.62)) V + ( + )(= V ( + )) <E 1 < 0. We futhe assume that ṙ 2 > 0, i.e. the photon 2 comes back to infinity. Note that, as explained in section , this is possible only if Its enegy and angula momentum ae E 2 >V + ( max ). E 2 = E E 1 >E L 2 = L L 1 > L, (20.68) thus, at the end of the pocess the paticle we find at infinity is moe enegetic than the one we sent in. It is possible to show that, since E 1 < 0,L 1 < 0, the captue of photon 1 by the black hole educes its mass-enegy M and its angula momentum J = Ma; indeed thei values M fin, J fin ae espectively: M fin = M + E 1 <M (20.69) J fin = J + L 1 < J. (20.70) To pove the inequality (20.69), we note that, as shown in Chapte 17, the total mass-enegy of the system is Ptot 0 = d 3 x( g)(t 00 + t 00 ), (20.71) V whee V is the volume of a t = const. thee-suface. If we neglect the gavitational field geneated by the paticle, t 00 is due to the black hole only, thus P 0 tot = V d 3 x( g)tpaticle 00 + M. (20.72)

12 CHAPTER 20. GEODESIC MOTION IN KERR SPACETIME 306 Let us compute this integal when the pocess stats, i.e. at a time when the massive paticle is shoot into the black hole; the spacetime is flat, the paticle enegy is E, and the 00-component of the stess-enegy tenso of a point paticle with enegy E, in Minkowskian coodinates is Tpaticle 00 = Eδ3 (x x(t)). (20.73) Thus eq. (20.72) gives P 0 tot in = E + M in. (20.74) Repeating the computation at the end of the pocess, namely when the photon 2 eaches infinity, we find Ptot 0 fin = E 2 + M fin. (20.75) Due to the stationaity of the Ke metic, if we neglect the outgoing gavitational flux geneated by the paticle, Ptot 0 is a conseved quantity; theefoe by equating the initial and final momentum we find P 0 tot in = P 0 tot fin M fin = M in +(E E 2 ) M fin = M in + E 1 <M in. (20.76) This poves the elation (20.69), and eq. (20.70) can be poved accodingly. In conclusion, by this pocess we have extacted otational enegy fom the black hole The innemost stable cicula obit fo timelike geodesics The study of timelike geodesics is much moe complicate, because equation (20.30) which, when κ = 1, becomes ṙ 2 = C 2 (E V +)(E V ) 2, (20.77) does not allow a simple qualitative study as in the case of null geodesics. Theefoe, hee we only epot some esults of a detailed study of geodesics equation in this geneal case. A vey elevant quantity (of astophysical inteest) is the location of the innemost stable cicula obit (ISCO), which, in the Schwazschild case, is at = 6M. In Ke spacetime, the expession fo ISCO is quite complicate, but its qualitative behaviou is simple: thee ae two solutions ± ISCO(a), (20.78) one coesponding to cootating and counteotating obits. Fo a = 0, the two solutions coincide to 6M, as expected; by inceasing a, the ISCO moves close to the black hole fo cootating obits, and fathe fo counteotating obits. When a = ±M, the cootating ISCO coincides with the oute hoizon, at = + = M. This behaviou is vey simila to that we have aleady seen in the case of unstable cicula obits fo null geodesics. In Figue we show (fo a 0) the locations of the last stable and unstable cicula obits fo timelike geodesics, and of the unstable cicula obit fo null geodesics. This figue is taken fom the aticle whee these obits have been studied (J. Badeen, W. H. Pess, S. A. Teukolsky, Astophys. J. 178, 347, 1972).

13 CHAPTER 20. GEODESIC MOTION IN KERR SPACETIME 307

14 CHAPTER 20. GEODESIC MOTION IN KERR SPACETIME The 3 d Keple law Let us conside a cicula timelike geodesic in the equatoial plane. We emind that the Lagangian (20.4) is and the -component of the Eule-Lagange equation is L = 1 2 g µνẋ µ ẋ ν (20.79) Being g µ = 0 if µ, we have d L dλ ṙ = L. (20.80) Fo cicula geodesic, ṙ = = 0, and this equation educes to The angula velocity is ω = φ/ṫ, thus We emind that on the equatoial plane then The equation has disciminant and solutions 2 ( d dλ (g ṙ) = 1 2 g µν,ẋ µ ẋ ν. (20.81) g tt, ṫ 2 +2g tφ, ṫ φ + g φφ, φ2 =0. (20.82) g φφ, ω 2 +2g tφ, ω + g tt, =0. (20.83) ( g tt = 1 2M g tφ = 2Ma ) g φφ = 2 + a 2 + 2Ma2 Ma2 2 ), (20.84) ω 2 + 4Ma 2 ω 2M 2 =0. (20.85) ( 3 Ma 2 )ω 2 +2Maω M = 0 (20.86) M 2 a 2 + M( 3 Ma 2 )=M 3 (20.87) ω ± = Ma ± M 3 3 Ma 2 = ± M 3/2 a M 3 Ma 2 = ± 3/2 a M M ( 3/2 + a M)( 3/2 a M) M = ± 3/2 ± a M. (20.88)

15 CHAPTER 20. GEODESIC MOTION IN KERR SPACETIME 309 This is the elation between angula velocity and adius of cicula obits, and educes, in Schwazschild limit a = 0, to M ω ± = ±, (20.89) 3 which is Keple s 3 d law Geneal geodesic motion: the Cate constant To study geodesics in Ke spacetime, it is convenient to use the Hamilton-Jacobi appoach, which allows to indentify a futhe constant of motion. It should be stessed that this constant is not associated to a spacetime symmety. Given the Lagangian of the system and given the conjugate momenta 2 L(x µ, ẋ µ )= 1 2 g µνẋ µ ẋ ν (20.90) p µ = L ẋ µ = g µνẋ ν, (20.91) by inveting eq. (20.91), we can expess ẋ µ in tems of the conjugate momenta: ẋ µ = g µν p ν. (20.92) The Hamiltonian is a functional of the coodinate functions x µ (λ) and of thei conjugate momenta p µ (λ), defined as H(x µ,p ν )=p µ ẋ µ (p ν ) L (x µ, ẋ µ (p ν )). (20.93) Thus, in ou case H = 1 2 gµν p µ p ν. (20.94) Geodesic equations ae equivalent to the Eule-Lagange equations fo the Lagangian functional (20.90), which ae equivalent to the Hamilton equations fo the Hamiltonian functional: ẋ µ = H p µ ṗ µ = H x µ. (20.95) Solving eqs. (20.95) pesents the same difficulties as solving Eule-Lagange s equations. Howeve, in the Hamilton-Jacobi appoach, which we biefly ecall, the futhe constant of motion emeges quite natually. 2 Not to be confused with the fou-momentum of the paticle, which we denote with P µ.

16 CHAPTER 20. GEODESIC MOTION IN KERR SPACETIME 310 In the Hamilton-Jacobi appoach, we look fo a function of the coodinates and of the cuve paamete λ, S = S(x µ,λ) (20.96) which is solution of the Hamilton-Jacobi equation ( H x µ, S ) x µ + S λ =0. (20.97) In geneal such solution depends on fou integation constants. It can be shown that, if S is a solution of the Hamilton-Jacobi equation, then S x µ = p µ. (20.98) Theefoe, once eq. (20.97) is solved, the expessions of the conjugate momenta (and of ẋ µ ) follows in tems of the fou constants, and allows to wite the solutions of geodesic equations in a closed fom, though integals. Fist of all, we can use what we aleady know, i.e. These conditions equie that H = 1 2 gµν p µ p ν = 1 2 κ p t = E constant p φ = L constant. (20.99) S = 1 2 κλ Et + Lφ + S(θ) (, θ) (20.100) whee S (θ) is a function of and θ to be detemined. Futhemoe, we look fo a sepaable solution, by making the ansatz S = 1 2 κλ Et + Lφ + S() ()+S (θ) (θ). (20.101) Substituting (20.101) into the Hamilton-Jacobi equation (20.97), and using the expession (19.14) fo the invese metic, we find κ + Σ [ 1 ( ds () d ) Σ 2 + a 2 + 2Ma2 Σ ( ds (θ) dθ ] sin2 θ ) 2 E 2 + 4Ma Σ EL + a2 sin 2 θ Σ sin 2 θ L2 =0. (20.102) Using the elation (19.26) ( 2 + a 2 )+ 2Ma2 Σ sin2 θ = 1 Σ [ ( 2 + a 2 ) 2 a 2 sin 2 θ ] (20.103)

17 R() [ C + κ 2 (L ae) 2] + [ E( 2 + a 2 ) La ] 2, (20.109) CHAPTER 20. GEODESIC MOTION IN KERR SPACETIME 311 and multiplying by Σ = 2 + a 2 cos 2 θ, we get ( ) ds κ( 2 + a 2 cos 2 () 2 ( ) ds (θ) 2 θ)+ + d dθ [ ( 2 + a 2 ) 2 ] a 2 sin 2 θ E 2 + 4Ma ( ) 1 EL + sin 2 θ a2 L 2 =0 i.e. = ( ) ds () 2 κ 2 (2 + a 2 ) 2 d ( ds (θ) dθ E 2 + 4Ma EL a2 ) 2 + κa 2 cos 2 θ a 2 sin 2 θe 2 1 sin 2 θ L2. L2 (20.104) (20.105) We eaange equation (20.105) by adding to both sides the constant quantity a 2 E 2 + L 2 : ( ) ds () 2 κ 2 (2 + a 2 ) 2 E 2 + 4Ma a2 EL d L2 + a 2 E 2 + L 2 ( ) ds (θ) 2 = + κa 2 cos 2 θ + a 2 cos 2 θe 2 cos2 θ dθ sin 2 θ L2. (20.106) In equation (20.106), the left-hand side does not depend on θ, and is equal to the ight-hand side which does not depend on ; theefoe, this quantity must be a constant C: ( ) ds (θ) 2 [ cos 2 θ (κ + E 2 )a 2 1 ] dθ sin 2 θ L2 = C ( ) ds () 2 κ 2 (2 + a 2 ) 2 E 2 + 4Ma a2 EL d L2 + E 2 a 2 + L 2 ( ) ds () 2 = κ 2 +(L ae) 2 1 [ E( 2 + a 2 ) La ] 2 = C. d (20.107) Note that in eaanging the tems in the last two lines, we have used the elation 2aLE +2aLE 2 + a 2 If we define the functions R() and Θ(θ) as [ Θ(θ) C + cos 2 θ (κ + E 2 )a 2 1 ] sin 2 θ L2 = 4aM LE. (20.108)

18 CHAPTER 20. GEODESIC MOTION IN KERR SPACETIME 312 then ( ) ds (θ) 2 = Θ dθ ( ds () d ) 2 = R 2 (20.110) and the solution of the Hamilton-Jacobi equation has the fom S = 1 2 κλ Et + Lφ + R d + Θdθ. (20.111) Thus, the constant C, which is called Cate s constant, fom its discovee B. Cate, emeges as a sepaation constant and chaacteize, togethe with E and L, geodetic motion in Ke spacetime. We stess again that, unlike E and L, it is not associated to a spacetime symmety. Once we have the solution of the Hamilton-Jacobi equations, depending on fou constants (κ, E, L, C), it is possible to find the paticle tajectoy. Indeed, fom (20.98) we know the expessions of the conjugate momenta theefoe p 2 θ = (Σ θ) 2 = Θ(θ) p 2 = ( ) Σ 2 = R() ṙ 2 (20.112) θ = 1 Σ Θ which can be solved by numeical integation. ṙ = 1 Σ R (20.113)