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

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, and axial , 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 . 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. The presence of this double sign− manifestly induces a problem: since in the second member of equation (11) only asymptotic quantities appear, it follows that in a gravitational scattering this equation predicts a net variation of the angular 2 2 2 momentum 1JQ JQ JQ 4a`m√0. In this case, however, such a variation is 0 0 out 0 ∞ = ∞ − ∞in = not to be connected to a physical cause, but rather to a coordinate effect.

3. The physical significance of the separation constant

The above procedure was aimed at effacing the ϑ dependence of equation (7), to give an expression for the separation constant involving constant quantities alone; this has been done using Newtonian considerations. In the following we shall try instead to explain the ‘additional’ terms in equation (7), making no use whatever of any Newtonian counterpart of the Kerr field. Let us therefore turn to equation (7), and try to determine the physical origin of its various terms, so that the fourth constant could be identified with something ‘physical’, and not just ‘mathematical’, loosely speaking. First of all, we easily recognize in the first two terms of equation (7) the declination and azimuthal component of the angular motion, with respect to the origin O of the coordinate frame; they are exactly the same terms appearing in the Schwarzschild case, where they (alone) define the separation constant and allow us to identify it physically with the square of the particle’s angular momentum, as is well known. It is the remaining terms whose meaning is to be explained: they are characteristic of the Kerr case and constitute a sort of ‘mysterious’ addition when a 0. We shall now see that these terms are generated by the not too obvious fact that a purely radial6= motion (i.e. with the angular variables kept constant) has itself ‘angular implications’, since it contributes with a non-zero angular momentum. Such a thing, as we will now show, is a direct consequence of the structure of the coordinate frame used to describe the spacetime. If we rewrite equation (7) in the following way:

`2 E 2 p2 + + m2a2 1 sin2 ϑ 3, (12) ϑ sin2 ϑ m − =    where we have redefined the separation constant as

3 K + m2a2, (13) = it is easy to see that the square-bracketed term of equation (12) is the Lorentzian expression of the square modulus of a free particle’s linear momentum; since the Kerr metric is asymptotically flat, this is the meaning that a static observer will assign to that term, when the particle is located at infinity with respect to the source. At infinity, we can express the spatial metric of the asymptotic form of the Kerr field (2), using the appropriate limit of transformation (10) to the oblate spheroidal coordinates, namely x (r2 + a2)1/2 sin ϑ cos ϕ = y (r2 + a2)1/2 sin ϑ sin ϕ (14) = z r cos ϑ, = to obtain 6 dσ 2 dr2 + 6 dϑ2 + (r2 + a2) sin2 ϑ dϕ2. (15) = r2 + a2 The separation constant in the Kerr metric 2933

The square modulus of the particle’s linear momentum under these conditions is given by

2 2 2 2 rr 2 ϑϑ 2 ϕϕ 2 p pr + pϑ + pϕ g pr + g pϑ + g pϕ, (16) =| | | | | | = ∞ ∞ ∞ where gij indicate the inverse of the asymptotic metric (15). Analysing∞ the various terms one by one, recalling equation (6) and considering its asymptotic limit, we find that

2 r E rr 2 →∞ 2 2 g pr m 1 p (17) ∞ −→ m − = ;    then, from equation (7)

r ϑϑ 2 →∞ g pϑ 0 , (18) ∞ −→ and, finally,

r ϕϕ 2 →∞ g pϕ 0 . (19) ∞ −→ 2 2 Hence in asymptotic conditions, the contribution to p comes from the radial term pr only, so that equation (12) can be written, for the particle at infinity† | |

2 2 ` 2 2 2 p + + pr a sin ϑ 3. (20) ϑ sin2 ϑ | | = Equation (20) implies that if a particle has at infinity a non-zero linear momentum in the radial direction with modulus pr , it also has a non-zero angular momentum with respect to the origin of the coordinate frame| | taken as the pole (namely the centre O of the r 0 disc). Let us in fact make a parallel transport of the initial linear momentum along the ϑ =constant, ϕ constant surfaces down to the point of maximum approach to the selected pole;= clearly, if the= distance to the pole is not null, there will be a net angular momentum. In oblate spheroidal coordinates (such as the Boyer–Lindquist ones) the ϑ constant surfaces of metric (2) are not cones with vertices at the origin, but hyperboloids= of rotation, which cross the z 0 plane orthogonally into a circle within the r 0 disc, at a distance a sin ϑ from the origin= (see [3]). At infinity the radial direction coincides= with that of the asymptotes of the hyperboloid, and the vector pr lies in the plane which is tangential to it. A parallel transport along ϑ constant means maintainingE the direction of the shift tangent to the surface of the hyperboloid.= The point of maximum approach to the selected pole O will then be reached when one crosses the r 0 disc, to which pr will be orthogonal. So its = E 2 2 2 momentum with respect to the centre O has a square modulus given by pr a sin ϑ which is just the term appearing in equation (20). | | When hyperbolic orbits are considered, the separation constant 3 in Kerr spacetime has the meaning of the square of the ‘extended’ angular momentum, in the sense that it is given by the sum of independent contributions by the angular motion associated with the angles ϑ and ϕ, and also by the ‘angular implications’ of purely radial motion. This last contribution is null in three cases only: when a goes to zero (in this case we are led back to the Schwarzschild metric), when the particle has zero linear momentum at infinity (i.e. when parabolic motion is concerned), and finally in the case of stable motion along the symmetry axis.

2 2 † Here we should make a brief clarification: at infinity, both pϑ and pϕ , whose physical meaning is that of square of the ϑ and, respectively, azimuthal component of the particle’s| | linear| | momentum, reduce to zero; what, in 2 2 contrast, do not reduce to zero are pϑ and pϕ , whose physical meaning is that of square of the ϑ and, respectively, azimuthal component of the particle’s angular momentum. 2934 F de Felice and G Preti

Up to this point we have been confining our attention to unbound orbits, either hyperbolic or parabolic, when 0 0, because we were seeking an asymptotic interpretation of the constants of motion. If the→ orbits are bounded but ‘sufficiently’ extended, the situation can still be that of the nonrelativistic weak-field limit; we could therefore follow the same derivation as before, provided the expression ‘r going to infinity’ is replaced by ‘at distances much higher than the dimensions of the source’. Nevertheless, in this case the special relativistic formula connecting the particle’s linear momentum to its energy and rest mass does not allow us to draw such conclusions as those attained before. For a bounded orbit it is never possible to consider the particle as ‘free’ (in the special relativistic sense), and so interpret the term 0 appearing in equation (12) (i.e. the square bracketed term) as just kinetic; on the contrary, we must take into account the potential which ‘bounds’ the particle. Let the bounded orbit be, as stated, extended enough that a part of it is in the weak-field region; we can then appeal to the virial theorem (the scalar version, in this case). Thus we are able to express the particle’s 2 0 factor in the weak-field region as a function of the kinetic term alone, namely 0wf p 1 =− (from T + V 2 0, since in the weak-field region 0 equals twice the kinetic energy per unit mass of the particle,= the potential going to zero, and 2T + V 0). We shall thus find that the ‘additional’ term appearing in the definition of 3 (equation (12))= turns out to have the same form as in the case of unbounded orbits, but with an opposite sign, namely `2 p2 + p 2a2sin2 ϑ 3. (21) ϑ sin2 ϑ −| | = We have now obtained the physical meaning of all the terms appearing in the definition of the separation constant 3, which can therefore be interpreted as follows: it is not the square modulus of the ‘usual’ angular (i.e. deriving from ϑ and ϕ motion) momentum of the particle, as in the spherically symmetric case, but it is the sum of the squared angular momenta associated with the three variables, ϑ, ϕ and also r, bearing in mind that in the bounded orbits case the contribution coming from the radial term has a negative sign. Evidently, the radial contribution to the total angular momentum arises from the particular choice of coordinates which in turn ensures the separability of the Hamilton–Jacobi equation. Moreover, if we notice that equation (12) does not contain M, i.e. the metric ‘source’, the same result is reached in Minkowski spacetime (M 0), provided one uses oblate spheroidal coordinates (14), together with t. This implies that the= physical interpretation of the separation constant in this flat spacetime remains valid when extended to the Kerr case (even if a caveat is due). In flat spacetime then, if we consider ϑ constant,ϕ constant motions, we are left with = =

2 2 2 3 pr a sin ϑ (22) =| | (from equation (12); compare with equation (20), but this time no r limits are involved), and the same line of reasoning as used above can be followed, to recognize→∞ in the contribution (22) to 3 a purely radial-motion-generated angular momentum. This interpretation, as we have noted, still holds when applied to the separation constant in the Kerr case, but—here comes the caveat—we must keep track of the different sign when bounded or unbounded orbits are considered. If we apply the above flat spacetime result to the Kerr case, we should conclude that 3 is given by the sum of the squared angular momenta in ϑ and ϕ, plus (markedly, ‘+’) the squared radial-angular momentum, no ‘ ’ sign appearing. This incongruency with the previous result is easily effaced when we consider− that no bounded motion is possible for a (neutral) test particle in the absence of sources of curvature; hence, while the analysis in flat spacetime does provide us with the physical meaning of the right-hand side of equation (22), it cannot bear any information about the difference between bounded and unbounded orbits, The separation constant in the Kerr metric 2935 since only the latter can exist in the M 0 case; the distinction can be made only when M 0, so this is where attention must be paid= in extending flat spacetime results to the original6= Kerr problem. One more observation should be made, to avoid any misunderstanding: in obtaining the physical explanation of the ‘additional’ term in equation (12), and hence of the 3 constant, we have considered the weak-field region of the Kerr metric; this is the same procedure followed to identify the physical meaning of the other constants E and ` in ‘total energy’ and ‘axial angular momentum’ (and of the separation constant in the spherically symmetric spacetime, identified as the square modulus of the angular momentum). It is clear that all these identifications are made by the ‘far-away’ observer and that E and `, as well as 3, maintain their thus established meaning—corresponding to that of their nonrelativistic, far-field counterparts— only with respect to this observer.

4. Conclusions

We have reconsidered the question of the separability of the Hamilton–Jacobi equation for the Kerr problem in oblate spheroidal coordinates, and introduced a new separation constant, 3, different from that originally used by Carter [4], and have seen that it is possible to give an adequate physical interpretation of it (section 3). We have seen that 3 can be interpreted as an algebraic sum of the squared contributions to the particle’s total angular momentum, which come not only from the angular motions, but also from the purely radial motion (which is not that obvious, i.e. that a purely radial—ϑ constant, ϕ constant—motion should itself generate an angular momentum) and this due= to the particular= structure of the coordinate frame used to describe the Kerr field. While the squares of the ϑ and ϕ components of the particle’s angular momentum are taken with the ‘+’ sign, the square of the ‘radial-angular’ term has the positive sign only when the motion is unbounded; in contrast, it is negative if the particle describes a bounded orbit. It follows that the separation constant 3 can also have negative values, in contrast with the corresponding separation constant in the Schwarzschild case which instead has properly the meaning of the square modulus (always positive) of the particle’s angular momentum; in particular, a negative 3 necessarily implies bounded motions.

References

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