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On the Meaning of the Separation Constant in the Kerr Metric Class. Quantum Grav. 16 (1999) 2929–2935. Printed in the UK PII: S0264-9381(99)02524-1 On the meaning of the separation constant in the Kerr metric Fernando de Felice and Giovanni Preti Dipartimento di Fisica ‘G Galilei’, Universita` di Padova and INFN Sezione di Padova, Italy E-mail: [email protected] and [email protected] Received 8 March 1999 Abstract. The existence of a fourth constant of motion, beyond rest mass, energy and axial angular momentum, for a free particle in a Kerr spacetime has been shown by Carter through the separability of the Hamilton–Jacobi equation using oblate spheroidal coordinates. This fourth constant of motion is connected to a second-rank Stackel–Killing tensor related to the symmetries of the Kerr solution; while this mathematical aspect is clear, the physical meaning of the separation constant is not. In this note we solve this problem, showing how the separation constant can be interpreted physically. PACS numbers: 0420, 0420C, 0420M 1. Introduction The Kerr metric is stationary and axisymmetric, hence in coordinates adapted to symmetries the motion of a free particle is characterized by having the moments, pt and pφ, constant. The interpretation of these quantities, labelled, respectively, as E and `, is that of (minus) the total energy of the particle and axial component of its angular− momentum, as measured by a ‘far-away’ observer, i.e. one ‘not influenced’ by the metric curvature; this is possible since the Kerr metric is asymptotically flat. A further evident constant of motion is the rest mass m of the particle; thus we have three ‘obvious’ constants of motion, yet not as many as are needed to reduce the solution of the problem of the particle’s motion to quadratures. To solve the geodesic equations some additional conditions are imposed, aiming to reduce the number of the system’s degrees of freedom—for instance, studying equatorial motions or those along the symmetry axis (see [1–3]). In order to study the most general case it is necessary to have a fourth integral of motion. Indeed, Carter [4] proved the separability of the Hamilton–Jacobi equation for a free particle in a Kerr field, when the metric is described by oblate spheroidal coordinates (such as Kerr or Boyer–Lindquist coordinates); hence the fourth integral of motion is thus identified with the constant of separation itself. As the constancy of E and ` is due to the existence of two Killing vectors ξt ∂t and ≡ ξϕ ∂ϕ, in connection with time-translation and azimuthal-rotation invariance of the Kerr metric,≡ the separation constant is also related to the metric symmetries, not through Killing vectors but through a quadratic Stackel–Killing tensor, i.e. a second-rank symmetric tensor Kij satisfying K(ij k) 0, (1) ; = 0264-9381/99/092929+07$30.00 © 1999 IOP Publishing Ltd 2929 2930 F de Felice and G Preti the Carter constant being found by invariant contraction of Kij with the particle’s 4- momentum pi . However, even if the correlation between the fourth constant of motion and this Stackel–Killing tensor has been thoroughly analysed and clarified (see, for instance, [5–8]), what remains unclear is the physical meaning to be attached to this constant. One could confidently say that in the a 0 (where a is the Kerr parameter) case, i.e. in Schwarzschild spacetimes—where the Hamilton–Jacobi→ equation can be easily separated using simple spherical coordinates—the separation constant has the patent meaning of the ‘square modulus of the total angular momentum of the particle’. Therefore, one could argue that in the Kerr case its meaning should not be unrelated to this: similar, but definitely not strictly the same. Unfortunately, we are not led too far this way, since we just get a gist about the sought- for explanation, and something more definitive is needed. In the forthcoming discussion we will consider the algebraic expression of the separation constant, in order to specify its various terms, thus obtaining—through simple geometrically based considerations—the desired physical interpretation. 2. The separation constant In order to set the scene for the following discussion, we shall now recall the procedure leading to the determination of the separation constant. Let us consider the Kerr metric in Boyer–Lindquist coordinates [3]: dr2 2Mr ds2 6 +dϑ2 +(r2 + a2) sin2 ϑ dϕ2 dt2 + a sin2 ϑ dϕ dt 2, (2) = 1 − 6 − where 6 r2 + a2 cos2 ϑ, 1 r2 2Mr + a2, with M the mass of the source and a its specific angular= momentum, expressed= − in units of length; ϑ is the zenithal distance and ϕ is the azimuthal angle. The inverse metric gij is given by ∂ 2 1 ∂ 2 ∂ 2 1 a2 sin2 ϑ ∂ 2 4Mra ∂ ∂ A ∂ 2 1 + + − , ∂s = 6 ∂r ∂ϑ 1 sin2 ϑ ∂ϕ − 1 ∂t ∂ϕ − 1 ∂t (3) where we have set A (r2 + a2)2 a21 sin2 ϑ, hence the Hamilton–Jacobi equation for a free particle with mass=m, namely − ij 2 g pi pj + m 0,i,j0,1,2,3 (4) = = i (where pi (∂S/∂x ), with S the Hamilton–Jacobi integral function) reads ≡ 2 2 2 2 (r + a ) 2 2 2 2 2 1 a 2 4Mra 2 + a sin ϑ p + 1p + p + p ptpϕ + 6m 0, − 1 t r ϑ sin2 ϑ − 1 ϕ − 1 = (5) where pi are the conjugate momenta. Since t and ϕ are ignorable coordinates for the metric (2), we have, as stated, pt constant E,pϕ constant `, and the separability of equation (5) manifestly emerges.= Thus we≡− obtain the= two following≡ equations: 1 1p 2 + r 2 m2 [(r2 + a2)2E2 + a2`2 4MarE`] K, (6) r − 1 − =− `2 p2 + + m2a2 cos2 ϑ + E2a2 sin2 ϑ K, (7) ϑ sin2 ϑ = The separation constant in the Kerr metric 2931 where K is the separation constant. Notice that K does not coincide with the used by Carter [4]; they are related by K +2aE`. K Let us examine equation= K (7): observe that as a 0 it reduces to the one obtained for the variable ϑ by the separation of the Hamilton–Jacobi→ equation for geodesic motion in the Schwarzschild metric. It is well known that in this case the separation constant has the physical meaning of the square modulus of the total angular momentum, as measured by the far-away observer; hence, we presume that K has a similar meaning, yet bearing with it some kind of corrections connected to the fact that a 0. However, what precisely are these corrections and what is their physical origin? 6= To bring the question into focus, it is possible to calculate the particle’s angular momentum explicitly. This trail has been followed by de Felice [9] as a way to obtain physical information on the separation constant. To this end we shall use the Newtonian definition of angular α αβγ momentum, J ε xβ pγ (Greek indices run from 1 to 3), working in the Newtonian Kerr- equivalent potential= [10, 11]. Such a potential describes all the characteristics of the relativistic metric, with the only exception being dragging effects (the Lense–Thirring effect); however, since this effect vanishes asymptotically faster than the potential, the use of the latter to study asymptotic quantities is justified. In this context de Felice has shown that the square modulus of the particle’s angular momentum (with respect to the origin of the coordinates) in the limit r ,isgivenby →∞ J2 L+m2a20cos2 ϑ, (8) ∞ = where 0 (E/m)2 1, and L is a new separation constant defined as = − L K a2E2. (9) = − From equation (8) it follows that L is the square of the particle’s angular momentum when a 0, while in a general case it appears an angular dependence which, since the metric is asymptotically= flat, cannot be due to the metric source, but has to be interpreted as a ‘coordinate’ anomaly. In the cited article it is shown that with the choice of a suitable pole, different from the origin of the coordinate frame, one can express J 2 as a function of coordinate-independent quantities alone. Recalling that a generic point Q∞has Cartesian coordinates related to the Boyer–Lindquist ones by the relation x (r2 + a2)1/2 sin ϑ cos(ϕ arctan(a/r)) = − y (r2 + a2)1/2 sin ϑ sin(ϕ arctan(a/r)) (10) = − z r cos ϑ, = we find that Q is mapped into a point Q0 lying in the r 0 disc, with coordinates (x a sin ϑ sin ϕ, y asin ϑ cos ϕ, z 0), obtained through= a parallel transport along the=ϑ constant and=−ϕ constant hyperplanes.= In the generic case a 0, it is the point = = 6= Q0, and not the origin O of the Cartesian frame (namely the centre of the r 0 disc), which plays the role of the centre of symmetry for all those points characterized by= the same values of ϑ and ϕ. Evidently, Q0 reduces to O when a 0, and in this case the Cartesian origin is the centre of symmetry for all points, independent= of their coordinates. Hence, if in the a 0 case we calculate the asymptotic expression for the square modulus of the particle’s angular6= momentum with respect to the Q0 pole, the dependence on the angular variable ϑ present in equation (8) is removed and an expression containing constants only is obtained [9]: J 2 L + m2a20 2am`√0. (11) Q0 ∞ = ± (Equation (11), with the sign, corrects the original formula in the cited article; the double ± 2 sign derives from considering both the positive and the negative values of pr .) Since we are p 2932 F de Felice and G Preti considering asymptotic situations, and therefore unbound orbits, the sign ‘ ’ refers to ingoing particles, while ‘+’ refers to outgoing ones.
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