DICTIONARY OF GEOPHYSICS, ASTROPHYSICS, and ASTRONOMY

ing the gravitational field outside a rotating axisymmetric,
electrically charged source. When
the electric charge is zero, the metric becomes
the Kerr metric.
The line element in Boyer–Lindquist coordinates(t,r,θ,φ)is
ds 2 =−ρ
+ 2
ρ 2
2
dt2
2
dφ 2 − 2aMr
2 dt
+ ρ2
dr2 +ρ 2 dθ 2 ,
2
whereρ 2 =r 2 +a 2 cos 2 θ,=(r −r+)(r −
r−), and 2 =(r 2 +a 2 ) 2 −a 2 sin 2 θ. The
quantityM is the ADM mass of the source,a≡
J/M its angular momentum per unit mass, and
Q its electric charge. WhenQ= 0 one also has
an electromagnetic field given by
Frt = Q r2 −a 2 cos 2 θ
ρ 4
Fφr = a sin 2 θFrt
Ftθ = Q a2 r sin 2θ
ρ 4
Fθφ = r2 +a 2
a
Ftθ .
For M 2 < a 2 +Q 2 the metric describes
a naked singularity at r = 0 and θ = π/2.
For M 2 ≥ a 2 +Q 2 the metric describes
a charged, spinning black hole. See also
Reissner–Nordström metric, black hole, naked
singularity.
Kerr–Schild space-times The collection of
metrics
ds 2 = λℓaℓbdx a dx b + ds 2 0
where λ is the parameter and ds2 0 is the “seed”
metric, often assumed to be the Minkowski metric.
The vector ℓ is null with respect to all metrics
of the collection. Many space-times, e.g.,
the Kerr metric and the plane-fronted gravitational
waves, are in the Kerr–Schild class.
Keulegan–Carpenter Number A dimensionless
parameter used in the study of waveinduced
forces on structures. The Keulegan–
Carpenter Number is defined as ūmaxT/D,
© 2001 by CRC Press LLC
Killing horizon
where ūmax is the maximum wave-induced velocity,
averaged over the water depth, T is wave
period, and D is structure diameter.
K-function Diffuse attenuation coefficient.
Kibble mechanism Process by which defects
(strings, monopoles, domain walls) are
produced in phase transitions occurring in the
matter. In the simplest cases there is a field
φ (called the Higgs field) whose lowest-energy
state evolves smoothly from an expected value
of zero at high temperatures to some nonzero
φ at low temperatures which gives minimum
energies, but there is more than one such minimum
energy (vacuum) configuration. Both thermal
and quantum fluctuations influence the new
value taken by φ, leading to the existence of domains
wherein φ is coherent and regions where it
is not. The coherent regions have typical size ξ,
the coherence length. Thus, points separated by
r ≫ ξ will belong to domains with, in principle,
arbitrarily different orientations of the field. It is
the interfaces of these different regions (sheets,
strings, points) which become the topological
defects. In cosmology, this is viewed as occurring
in the early universe. Because of the finite
speed of light, ξ is bounded by the distance light
can travel in one Hubble time: ξ ∼