
1.3 Orbits in the Kerr Metric
symmetry of the metric, which led to the conservation
of angular momentum. This meant that every orbits
was in a plane, with the direction of the angular mo-
mentum perpendicular to the plane. We introduced the
concept of an effective potential, from which the nature
of possible orbits could be easily studied. We found
that for a given angular momentum, depending on the
energy of the particle (1) the orbits could be parabolic
or hyperbolic in which case the particles or photons
come from infinite distance and return to it, (2) orbits
which come from infinity and spiral into the black hole,
(3) for particles with mass bound precessing elliptical
orbits are possible, a special case of which was a stable
circular orbit when the particle was at the minimum
of the effective potential and (4) an unstable circular
orbit when the particle was at the maximum of the ef-
fective potential. For photons, the only closed orbits
were unstable circular orbits at the maximum of the
effective potential for photons, since the potential had
no minimum.
For massive particles in the Schwarzschild geom-
etry, we found in BHS-8 that as the angular momentum
of the particle decreases, the minimum in the effec-
tive potential moves closer to the black hole, so that
the radius of the corresponding circular orbit becomes
smaller and the particle is more tightly bound. The
closest circular orbit occurs when the angular momen-
tum has the value L = 2
√
3GM. The radius of this in-
nermost stable circular orbit is r
ISCO
= 6GM/c
2
= 3R
S
.
Orbits with smaller radii are not possible because for
lesser angular momentum, the effective potential has
no minimum. When a particle is in an elliptical orbit in
the Schwarzschild geometry, if energy and angular mo-
mentum are extracted from it, the orbit becomes more
circular and moves closer to the black hole. The pro-
cess can continue until r
ISCO
is reached, but no further
energy extraction is possible. If the angular momen-
tum decreases further, the particle will plunge into the
black hole. We mentioned in BHS-7 that a maximum
5.7 percent of the rest mass energy of the particle can
be extracted in the process.
The Kerr black hole has no spherical symmetry,
therefore the total angular momentum is not in general
conserved, and so an orbit does not have to lie in a
plane. The orbits can in general have very complex
behaviour and have been studied in detail by using
analytical and numerical techniques. One simplifying
factor is that the rotation of the black hole defines an
axis around which there is symmetry. The component
of the angular momentum in the direction of the spin
axis is therefore conserved and orbits in the equatorial
plane, which is perpendicular to the spin axis, will
remain confined to it. For such orbits, the component of
the angular momentum in the direction of the black hole
spin is in fact the total angular momentum. The study
of these orbits is relatively simple and yet produces
very useful insights.
For equatorial orbits, taking into account the con-
servation of energy and angular momentum, the equa-
tion for the radial coordinate can be written in much
the same manner as for the Schwarzschild orbits we
considered in BHS-6 and 7. The equation can be
written in terms of the effective potential, which, as
in the Schwarzschild case, has three terms: one pro-
portional to -1/r, which is the Newtonian part which
applies at large r, a second term proportional to 1/r
2
which is the centrifugal term, and a third term propor-
tional to -1/r
3
which is the general relativistic term. In
the Schwarzschild case the effective potential depends
only on the square of the angular momentum of the
particle and not on the energy. But in the Kerr case
the potential depends on the angular momentum and
its square, as well as energy. The orbits therefore de-
pend on whether the initially the particle is moving in
the spin direction, i.e., corotating or counter to it, i.e.,
counterrotating. An interesting feature is that a parti-
cle which falls radially towards a Kerr black hole from
infinity, confined to the equatorial plane, sweeps out
an angle ∆ϕ, which is due to the frame dragging we
mentioned above. In the Schwarzschild case, an initial
radial orbit, which has zero angular momentum, would
always remain radial.
For the Kerr metric, the radial coordinate of the
innermost stable circular orbit in the equatorial plane
depends on the ratio of the angular momentum param-
eter a to GM/c
2
. This is shown in Figure 2.
In the diagram the dependence r
ISCO
/M on a/M is
shown for counterrotating as well as corotating orbits.
10