
1.3 The Kerr Metric
space-time around it is spherically symmetric, so it is
best described in terms of the coordinates t, r, θ, φ.
As described in Story 5, t is the time coordinate, and
r, θ, φ indicate the position of a point in space. The
two angular coordinates a θ and φ are similar to the
two angles from the spherical polar coordinates used
to describe flat 3-dimentional space, but the radial co-
ordinate r is somewhat different. Because the space is
curved, r no longer is the distance from the origin, but
it helps to fix the position in space. The mass M is
located at the origin r=0. If we take a fixed value of
r and vary the angular coordinates over their ranges, a
spherical surface is generated. The area of this sphere
is 4πr
2
as in flat space.
The spherical surface with radius R
S
= 2GM/c
2
is
known as the event horizon. This has the property
that no particle or light ray can travel from inside the
event horizon to the outside. The region inside the
event horizon is cut off from the rest of the Universe
and therefore we have a black hole. At the position of
the point mass M, the matter density is infinitely large
and so is the curvature of space-time, and so we have
a space-time singularity. The outside world cannot
see the singularity because of the event horizon. It is
possible for matter and light to fall into the black hole
from the outside the event horizon.
As described in Story 5, the motion of particles
with mass in a gravitational field is described by time-
like geodesics, while that of a light ray is described by
a null geodesic. There are two symmetries associated
with the Schwarzschild metric: it is independent of
time and is spherically symmetric. Therefore the en-
ergy and angular momentum of a particle or light ray in
orbit around a Schwarzschild black hole are conserved,
that is they remain constant. It is therefore possible to
analyse the nature of the geodesics in a simple manner.
In Story 6, we have described how the nature of time
like geodesics is studied using an effective potential
V
eff
. For a particle with a given angular momentum,
the effective potential depends only on the radial coor-
dinate r . In general it has a maximum and minimum,
which produces a potential well. Depending on its en-
ergy, (1) a particle can come in from large distances,
swing around the centre and recede again to large dis-
tances, (2) it can fall into the black hole, or (3) move
in a bound orbit around the black hole with shape cor-
responding to a precessing ellipse. When the energy
of the particle is equal to the minimum of the effective
potential, the orbit is circular in shape. As described
in Story 8, the behaviour of light rays, i.e. photons is
somewhat different. They can have orbits as in (1) and
(2), but the only bound orbits occur at a fixed value of
r=1.5r
S
. These orbits are circular and unstable.
1.3 The Kerr Metric
The Kerr metric provides the structure of space-
time around a particle which has mass and angular
momentum or spin. The angular momentum defines a
direction around which the particle spins. That is easy
to visualise for an extended body like the Earth, but
the same physics applies to a point particle too. Be-
cause the mass and angular momentum are constant,
the metric is constant in time. The spin axis is also
a symmetry axis, in the sense that the metric remains
the same for all points in a plane perpendicular to the
spin axis (this and other such concepts can be mathe-
matically defined for the curved space-time of general
relativity, but I am using simple expressions for qual-
itative understanding). Roy Kerr obtained an exact
solution for Einstein’s equations for the special case of
a spinning, massive particle.
It is convenient to express the Kerr solution in
terms of coordinate system t, r, θ, φ known as Boyer-
Lindquist coordinates. Here t is the time coordinate as
usual; the other three coordinates have the appearance
of the spherical polar coordinates used in Schwarzschild
metric, but the appearance is deceptive. For example,
the coordinate r does not have the same meaning as in
the Schwarzschild case. There a surface with r constant
has a spherical shape with area 4πr
2
, though r is not the
distance from the origin, which is at r = 0. This inter-
pretation is no longer applicable in the Boyer-Lindquist
coordinates. The angle φ goes round the axis defined
by the direction of the spin, while the interpretation of
angle θ is the familiar one only in the special cases we
will consider below.
The Kerr metric depends on two parameters, the
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