60		3 Rotating Black Holes and Thermodynamics
function:
S(τ, q, P ) =	1	μ2τ + E t − Lφ + σ (θ ) + (r)	(3.5.20)
	2

where σ (θ ) is some function of the variable θ and (r) some function of the variable r. The ansatz (3.5.20) is consistent with the definitions:

		E =	∂S	;	−L =		∂S			(3.5.21)
			∂t				∂φ
and yields:
pθ =	∂S	= ∂θ σ (θ );			pr =	∂S			= ∂r (r)	(3.5.22)
	∂θ					∂r

Furthermore, upon insertion into the Hamilton Jacobi equation (3.5.15), the chosen ansatz reproduces the constraint:

1	μ2 = H	q,	∂S		(3.5.23)
2			∂q

provided the following equation holds:

Hθ (θ ) + Hr (r) = 0

(3.5.24)

where we have introduced the following two functions of the declination angle θ and of the radius r, respectively:

Hθ (θ ) = α2μ2 cos2 θ + (αE sin θ − L csc θ )2 + σ (θ )2								(3.5.25)
	((r2	α2)E	−	Lα)2
Hr (r) = −	+				+ r2μ2	+ r2	− 2mr + α2	(r)2
	r2	2mr	+	α2
	−

Since Hθ depends only on the θ variable and Hr (r) depends only on the r variable, (3.5.24) can be true if and only if both functions are constant throughout the geodesic motion and their constant values are opposite. Namely we must have:

K = Hθ (θ ) = −Hr (r)

(3.5.26)

The constant K, named the Carter constant is the fourth missing integral of motion which ensures full-integrability of the mechanical system.

3.5.3 Reduction to First Order Equations

Thanks to the above introduced Carter constant the geodesic equations can be completely reduced to a first order system. The procedure is straightforward. Varying

3.5 Geodesics of the Kerr Metric

61

˙

θ

we find the explicit form of

pr and

the Lagrangian (3.5.1) with respect to r and ˙

pθ , respectively:

ρ2

pr =

r˙

(3.5.27)

p

θ = ρ

˙

(3.5.28)

2θ

Solving (3.5.26) for (r) = pr

and σ (θ ) = pθ and equating the results to those of

(3.5.28), we obtain:

ρ2r˙ = ±

E 2 − μ2

(3.5.29)

p(r)

ρ

2θ

α2

μ

2

2

2

+ K)

1/2

(3.5.30)

where

˙ = ±(−

cos θ − (L csc

θ − αε sin θ )

p(r) ≡

1

(E 2

− μ2 r4

+ 2mμ2r3

− K + α −2αE 2 + 2LE + αμ2 r2

E 2

μ2

−

+ 2Kmr + α2 (L − αE )2 − K)

(3.5.31)

is a quartic polynomial in the radial variable r whose coefficients depend algebraically on the first integrals of motion E , L, K, μ2.

Changing variable in the second of equations (3.5.30) by setting u = cos θ we

can rewrite it as follows:
ρ2u˙ = ±α μ2 − E 2 q(u)	(3.5.32)

where also q(u) is a quartic polynomial, but it as the special property that it contains only the even powers of u:

q(u)

=

u4

+

(K + α(2Lε + α(μ2 − 2ε2)))u2

(L − αε)2 − K

(3.5.33)

α2(ε2 − μ2)

α2(ε2 − μ2)

+

Hence we have the differential system:

ρ2r˙ = ±

p(r)

(3.5.34)

E 2 − μ2

ρ2u˙ = ±α

μ2 − E 2

(3.5.35)

q(u)

Let us name ei , (i = 1, . . . , 4) the roots of the polynomial, p(r), namely let us set:

4
p(r) = *(r − ei )	(3.5.36)
i=1

62 3 Rotating Black Holes and Thermodynamics

and let us name g1, g2 the two independent roots of the polynomial q(u) which is necessarily of the form:

2
i 1
q(u) = * u2 − gi2	(3.5.37)
=

Eliminating τ from (3.5.35) we conclude that the relation between the variables r and u is reduced to quadratures, namely:

	dr		= iα	du	+ cost
	√			√q(u)		(3.5.38)
		p(r)

One finds that the relevant integrals appearing in the above relation can be analytically evaluated and expressed in terms of the elliptic integral function:

F (ξ |m) ≡ ξ

dφ

(3.5.39)

1 − m sin2 φ

0

Indeed we find:

dr

F arcsin

| (e2−−e3)(e1

−−e4)

(r−−e1)(e2−−e4)

P(r, ei ) ≡

(r

e2)(e1

e4)

(e1

e3)(e2

e4)

√

= −2

√

(3.5.40)

(r)

(e

2

−

e

)(e

1

−

e

)

p

g12

3

4

F arcsin

u

≡

√q(u)

| g22

=

g2

W(u, gi )

du

g1

(3.5.41)

and the final relation between u and r along the geodesics is implicitly given by:

P(r, ei ) − iαW(u, gi ) = c1

(3.5.42)

where c1 is the first found of the remaining four integration constants.

3.5.4The Exact Solution of the Schwarzschild Orbit Equation as an Application

The Schwarzschild metric is a particular limit of the Kerr metric for α → 0. Hence the above formal integration of the geodesic equations in the Kerr case should provide, as a by-product, also the exact analytic equation of the Schwarzschild orbit equation, which in Chap. 4 of Volume 1 we treated only perturbatively. As an illustration of the method, in this section we derive the complete analytic form of the orbit for a massive test-particle moving around a spherical symmetric Schwarzschild black-hole.

3.5 Geodesics of the Kerr Metric

63

In the Schwarzschild case the equation for the derivatives of the time and azimuthal coordinates (3.5.9) reduce to:

φ		L
	= r2			(3.5.43)
˙
t˙ =			r2E	(3.5.44)
		r2	− 2m

while the equation for the derivative of the declination angle θ is:

r	2		2	2	(θ )	(3.5.45)
		˙ = K − L		csc
		θ

which follows from (3.5.30) by setting α = 0. From the above relation we conclude that we can always impose the vanishing of the θ -derivative for any value of θ by choosing the Carter constant K appropriately. Since in a spherical symmetric field the actual value of θ is purely conventional, we can just choose to confine all motions to the equatorial plane by setting:

θ =	π	; K = L2	(3.5.46)
	2

Fixing α = 0 and K = L2 the quartic polynomial (3.5.31) becomes:

p(r) = r4		2mr3	L2r2		2L2mr
	+		− E 2 − 1	+		(3.5.47)
		E 2 − 1			E 2 − 1

which is still quartic but has the property that one of its roots is r = 0. Hence we can write:

3
p(r) = r *(r − ei )	(3.5.48)
i=1

and the relation between the three non-trivial roots ei and the physical first integrals is the following:

L2		=	e1e2e3			(3.5.49)
			e1 + e2 + e3
m =			e1e2e3			(3.5.50)
			2(e2e3 + e1(e2 + e3))
E	2	=	(e1 + e2)(e1 + e3)(e2 + e3)			(3.5.51)
			(e2 + e3)e12 + (e22 + 3e3e2 + e32)e1 + e2e3(e2 + e3)

At this point we can directly obtain the analytic form of the orbit eliminating dτ from the two equations:

	r2	dr	= E 2 − 1		(3.5.52)
				p(r)

dτ

64

3 Rotating Black Holes and Thermodynamics

dφ

r2

= L

(3.5.53)

dτ

In this way we get:

=

√p(r)

L

E 2 −

1

dφ

dr

(3.5.54)

From which we immediately get:

E

− 1

φ

+

+

(e1−e2)e3

,

(3.5.55)

e1−e3

,|

2F

arcsin

(

e1

−1)e3

e2(e1−e3)

2

r

= −

√

L

(e2 − e1)e3

which can be rewritten as:

√

F arcsin X |z = Y

(3.5.56)

X ≡

( er1

− 1)e3

(3.5.57)

e1 − e3

z

≡

e2(e1 − e3)

(3.5.58)

(e1 − e2)e3

ϕ

e3

e1

+e2

+−e3

e1e2(e2 e1)

Y ≡ −

(3.5.59)

2 (e2e3 + e1(e2 + e3))

The first of (3.5.59) can be analytically inverted in terms of special functions since, by very definition, we have:

√				√
F arcsin X |z = Y					X = sn(Y |z)		(3.5.60)

where sn(Y |z) is the Jacobi special elliptic function sn while F (t|z) denotes the elliptic integral of the first kind, whose definition we have already recalled in (3.5.39).

In this way we obtain the final explicit analytic form of the Schwarzschild orbit for a massive particle depending on the three integration constants e1, e2, e3 which parameterize the angular momentum L, the energy E and the Schwarzschild emiradius m. We find:

r(φ) =

e1e3

(3.5.61)

e3

e1e2(e2

e1)

+ -−

−

| (e1−e2)e3 .,

2

e1 e2

e3

1

−

3

2

+

3

(e2e3

+e1(e2

+e3))

(e

e

)

sn

ϕ

+ +

e2(e1−e3)

e

Equation (3.5.61) contains both closed and open orbit depending on whether the energy E 2 is less or larger than one. Two examples of orbits described by formula (3.5.61) are displayed in Figs. 3.5 and 3.6.