3.5 Geodesics of the Kerr Metric

If gpp(r, Ω) never changes sign (namely it is positive definite) then the effective 2- dimensional metric displays no horizon. If gpp(r, Ω) goes through zero, then in the p, r plane there is a horizon. However if there is a horizon for a certain time p(Ω) light can still escape to infinity along some other time p(Ω ) for which gpp(r, Ω ) is positive-definite. In other words we look for the norm of the Killing vectors χ (Ω):

If all the possible vectors χ (Ω) have negative norm then we are below the horizon. This implies that we are below the horizon when the discriminant of the quadratic form (3.4.8) is negative, so that the horizon is indeed given by the condition (3.4.3) as we claimed. On the horizon r = r+ the equation:

The above quantity ΩH can be interpreted as the angular velocity of the eventhorizon in the sense that any physical test-body sitting on the horizon necessarily rotates with such a velocity with respect to the fixed stars.

The Horizon Area We can now easily calculate the area of the horizon. By definition we have:

and by comparison with (3.4.11) we obtain the following very interesting relation of the horizon area with the mass m and the angular momentum J ≡ mα of the black-hole:

J

AreaH = 4π (3.4.13)

ΩH m

3.5 Geodesics of the Kerr Metric

The Kerr metric was discovered at the beginning of the sixties of the XXth century but it took several years before the problem of integrating its geodesics equations

was solved. For the Schwarzschild field the geodesics equations are almost immediately reduced to quadratures by regarding them as Euler Lagrange equations of a mechanical problem with 4 Lagrangian coordinates qμ = (t, r, θ, φ) and exploiting two facts:

1.There are three first integrals of the motion respectively given by the energy E , the angular momentum L and the mass μ of the particle

2.One Lagrangian coordinate can be eliminated from start, since all orbits are pla-

nar and the declination angle θ can be conventionally fixed to the value θ = π2 without loss of generality.

In this way, after elimination of θ we have a number of conserved charges equal to the number of effective Lagrangian coordinates and the mechanical system is necessarily reduced to the quadratures. The really crucial point, therefore, is the elimination of θ which, in the Schwarzschild case might be seen as a consequence of the full-spherical symmetry, absent in the Kerr case. At α = 0 there is dynamics also in the declination angle θ , while at first glance, the integrals of motion seem to be just three as at α = 0. Hence integrability seem to be lost for the Kerr metric.

As Carter2 discovered, the truth is more subtle and the Kerr geodesic system is still fully integrable. The reason for that is the existence of a fourth hidden integral of motion, the Carter constant K, which exists at all values of α and is, in the limit α → 0, the real source for the trivialization of the θ motion.

In order to discover the Carter constant one has to reformulate the geodesic problem within the framework of the Hamilton Jacobi approach to classical mechanics and this is what we shall do in the present section. As a preparation to this task let us first review the construction of the three integral of motion associated with manifest symmetries.

3.5.1 The Three Manifest Integrals, E , L and μ

The two first integrals E and L are associated with symmetries of the metric via Noether theorem (see Sect. 1.7 in Chap. 1 of Volume 1). They exist just because the two Lagrangian coordinates t and φ are cyclic. On its turn this cyclicity follows from

are true for the Kerr metric as much as for the Schwarzschild one. Hence also the Kerr metric admits the first integrals E and L.

Defining the Lagrangian according to the conventions of used in Chaps. 3 and 4 of Volume 1 and using the form (3.2.10) of the Kerr metric in Boyes-Lindquist coordinates, namely:

2Brandon Carter is an Australian born theoretical physicist working at Meudon (CNRS), France.

Equation (3.5.3) replace the homologous ones of the Schwarzschild case (see Chap. 3 of Volume One). In the limit α → 0 the Kerr metric degenerates into the Schwarzschild metric and the definitions (3.5.3) of the energy and angular momen-

tum of a test particle flow to the Schwarzschild ones. This is easily checked, noting that at α = 0 we have ρ2 = = r2 and Σ2 = r4.

Equation (3.5.3) can be effectively interpreted in the following matrix form:

is function only of the coordinates r and θ . The same, obviously is true also of the inverse matrix.

Hence if the geodesic flow of the coordinates r, θ has already been determined in terms of the first integral of motion, namely if we have the two proper-time functions: