DICTIONARY OF GEOPHYSICS, ASTROPHYSICS, and ASTRONOMY

curve of growth
classical) field theory, it is sometimes useful to
formulate the fields and their interactions on the
background of an arbitrary metric which is not a
priori a solution of the concrete field equations.
First, one can achieve many results working on
general curved space-time without specifying
the equations for the metric itself. These results
include, in particular, the covariant formulation
of matter fields and their renormalization,
calculation of the leading terms in the quantum
corrections to the (undefined) classical action of
gravity, and some applications of these. Second,
the originally unspecified geometry of the
space-time may be defined by the quantum effects
of matter fields (or string). In this case, the
action of gravity does not have a classical part,
and gravitational interaction is induced by quantum
effects of other fields. See general relativity
(which defines a curved spacetime as a solution
to Einstein’s equations). See induced gravity,
nonminimal coupling, quantum field theory in
curved spacetime, quantum gravity.
curve of growth A curve giving the optical
density of a spectral line as a function of the
atomic column density. Thus, densities can be
estimated from the strength of emission or absorption
lines.
curvilinear coordinates A coordinate system
in which the coordinate lines are not
straight, and in which the metric tensor expressing
Pythagorean theorem
ds 2 =gij(x k )dx i dx j
hassignificantlynonconstantmetrictensorcomponents
gij(x k ).
curvilinear coordinates [in a plane] A coordinate
system is a relationship that is established
between the points of the plane and pairs of ordered
numbers called coordinates. Curvilinear
coordinates are a coordinate system that is not
Cartesian (see Cartesian coordinates). While
Cartesian coordinates may be visualized as the
determination of points in a plane by the intersection
of two straight lines perpendicular to
each other (each line corresponding to one of
the pair of numbers or coordinates), curvilinear
coordinates may be visualized as the determination
of points by the intersection of two curves
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in general (each curve corresponding to one of
the coordinates). The two curves are not necessarily
perpendicular to each other. In curvilinear
coordinates, the relationship between points in
a plane and pairs of numbers is not necessarily
nonsingular at all points. A frequently used
curvilinear coordinate is the polar coordinates
(r,θ)defined by the expressions: x =r cos(θ)
and y=r sin(θ). Polar coordinates can be visualized
as the determination of points by the
intersection of circles of radiusr with rays starting
at the origin of the coordinate system and
extending outward at an angleθ with respect to
the x-axis. They are, however singular atr= 0,
and in principle, a different coordinate system
has to be used at this point.
curvilinear coordinates [in space] A coordinate
system is a relationship that is established
between the points in space and trios of
ordered numbers called coordinates. Curvilinear
coordinates are a coordinate system that is
not Cartesian (see Cartesian coordinates), while
Cartesian coordinates may be visualized as the
determination of points in space by the intersection
of three planes perpendicular to each other
(each plane corresponding to one of the trio of
numbers or coordinates). Curvilinear coordinates
may be visualized as the determination of
points by the intersection of three surfaces in
general (each surface would correspond to one
of the coordinates). Although the relation between
curvilinear and rectangular coordinates
is required to be nonsingular, in typical cases
there are isolated points (e.g., at r= 0) where
the relation is singular. In this case, in principle,
a different coordinatization should be used.
cusp (cosmic string) Small localized regions
on cosmic strings which attain velocities close to
that of light. These regions are highly energetic
with enormous string curvature that may favor
the emission of Higgs constituent particles (i.e.,
the scalar field making up the string) away from
the string core as well as gravitational radiation.
See cosmic string.
cusp, polar One of two points or regions in
the magnetosphere where, in the noon-midnight
meridional surface, field lines swept back into
the tail part company with the ones closing near