DICTIONARY OF GEOPHYSICS, ASTROPHYSICS, and ASTRONOMY

Killing tensor
Killing tensor A totally symmetric n-index
tensor fieldKa1...an which satisfies the equation
∇(bKa1...an) = 0 .
(the round brackets denote symmetrization). A
trivial Killing tensor is a product of lower-rank
Killing tensors. See Killing vector.
Killing vector The vector generator ξ associated
with an isometry. When the Killing
vector exists it may be viewed as generating an
infinitesimal displacement of coordinatesx µ ↦→
x µ +ɛξ µ (|ɛ| ≪1), with the essential feature
that this motion is an isometry of the metric and
of any auxiliary geometrical objects (e.g., matter
fields). Such a vector satisfies the Killing
condition
Lξ ,g= 0 .
the vanishing of the Lie derivative of the metric
tensor. In coordinates this is equivalent to
ξµ;ν +ξν;µ = 0 where ; denotes a covariant
derivative with respect to the metric. Similarly,
other geometrical objects, T , satisfy LξT= 0.
In a coordinate system adapted to the Killing
vector, ξ = ∂
∂xa , the metric and other fields do
not depend on the coordinate xa . The Killing
vectors of a given D-dimensional metric space
form a vector space whose maximum possible
dimension is D(D + 1)/2. Wilhelm Killing
(1888). See isometry.
K index The K index is a 3-hourly quasilogarithmic
local index of geomagnetic activity
based on the range of variation in the observed
magnetic field measured relative to an assumed
quiet-day curve for the recording site. The range
oftheindexisfrom0to9. TheKindexmeasures
the deviation of the most disturbed of the two
horizontal components of the Earth’s magnetic
field. See geomagnetic indices.
kinematical invariants The quantities associated
with the flow of a continuous medium:
acceleration, expansion, rotation, and shear. In
a generic flow, they are nonzero simultaneously.
kinematics The study of motion of noninteracting
objects, and relations among force, position,
velocity, and acceleration.
© 2001 by CRC Press LLC
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kinematic viscosity Molecular viscosity divided
by fluid density, ν = µ/ρ, where ν =
kinematic viscosity,µ = molecular (or dynamic
or absolute) viscosity, andρ = fluid density. See
eddy viscosity, dynamic viscosity.
kinetic energy The energy associated with
moving mass; since it is an energy, it has the
units of ergs or Joules in metric systems. In nonrelativistic
systems, for a point mass, in which
position is described in rectangular coordinates
{xi i = 1 ···N, where N = the dimension of
the space, the kinetic energy T is
T = 1
2 m
v i2
= 1
2 mδij v i v j
δij = 1ifi = j; δij = 0 otherwise. Since
δij are the components of the metric tensor in
rectangular coordinates, it can be seen that T is
m
2 times the square of velocity v, computed as a
vector T = m 2 (v · v). This can be computed in
any frame, and in terms of components involves
the components gij of the metric tensor as
T = m
2 gij v i v j ,
where vi are now the components of velocity
expressed in the general non-rectangular frame.
In systems involving fluids, one can assign a
kinetic energy density. Thus, if ρ is the mass
velocity, the kinetic energy density (Joules/m3 )
is t = ρvv.
For relativistic motion, the kinetic energy is
the increase in the relativistic mass with motion.
We use E = γmc2 , where m is the rest mass of
the object, c is the speed of light, c ∼ = 3 × 10 10
cm/sec, and γ =
1
1− v2
c2 is the relativistic dila-
tion factor and T = E −mc2 .Ifvis small compared
to c, then Taylor expansion of γ around
v = 0gives
T = m 2
c 1 + 1 v
2
2 3 v
+
c2 8
4
··· − mc
c4 2
which agrees with the nonrelativistic definition
of kinetic energy, and also exhibits the first relativistic
correction. Since v ≪ c in everyday experience,
relativistic corrections are not usually