DICTIONARY OF GEOPHYSICS, ASTROPHYSICS, and ASTRONOMY

which the dilaton plays the role of a scalar component
of gravity and can also be thought of as
a variable (in both space and time) Newton’s
“constant” (see Brans–Dicke theory). The dilaton
couples to matter fields, e.g., Yang–Mills
SU(N) fields and Maxwell’sU(1) electromagnetic
field. Because of this, the presence of a
non-trivial dilaton in general spoils the principle
of equivalence. See dilaton.
dilatonicblackhole Infieldtheoryappliedto
cosmology, a dilaton is an additional scalar field
associated with gravity. A few dilatonic gravity
solutions are known in four space-time dimensions
which represent black holes with a nontrivial
dilaton field. When the latter field couples
to the electromagnetic tensor, the black holes
must be electrically charged. One exact solution
is given by the analog of the spherically symmetric
general relativistic black hole with charge,
the Reissner–Nordström metric. When angular
momentum is present (see Kerr–Newman metric)
only perturbative solutions are known for
either small angular momentum or small electric
charge. All the solutions have in common a
dilaton which decreases and vanishes at a large
distance from the center of the hole. However,
near the event horizon the dilaton is nonzero and
can possibly affect the scattering of passing radiation.
Other solutions have been found when the
dilatoncouplestoYang–MillsSU(N)fieldsgiving
rise to richer structures. See black hole,
dilaton gravity, future/past event horizon, Kerr–
Newman metric, Reissner–Nordström metric.
dilution Defined as the total volume of a sample
divided by the volume of effluent (contaminant)
contained in the sample.
dimension A statement of the number of independent
parameters necessary to uniquely define
a point in the space under consideration.
Everyday experience with space indicates that
it is 3-dimensional, hence specifying thex,y,z
labels (for instance) of a point uniquely defines
the point. If it is wished to specify the locations
of two mass points, it is convenient to introduce
a 6-dimensional space, giving, for instance,
the x,y,z labels of the location of each
of the mass points. In special and general rela-
© 2001 by CRC Press LLC
dimensional transmutation
tivity, time is considered a separate dimension,
and so events are given by specifying x,y,z,
and t (time); thus, spacetime is 4-dimensional.
In many mathematical operations, e.g., in integration,
the dimension of the space enters explicitly
and as a consequence the solution to
standard equations depends on the dimension
of the space in which the solution is found. It is
found effective theoretically to allow noninteger
dimensions in those cases. Also, since the ratio
of the “volume” to the “surface” depends on the
dimension, this concept has been generalized to
a “fractal dimension” which is defined in terms
of the ratio of these quantities in some suitable
sense. See fractal.
dimensional analysis In usual physical descriptions,
quantities are assigned units, e.g.,
centimeter for length, gram for mass, second for
time. In dimensional analysis, one constructs a
combination of known quantities which has the
dimension of the desired answer. Because specific
systems have typical values for dimensional
quantities, results constructed in this way are
usually close to the correctly computed result.
Typically dimensional analysis omits factors of
order one, or of order π; the results then differ
from exactness by less than an order of magnitude.
For instance, the typical length associated
with a sphere is its radius r (measured in centimeters,
say). Its volume by dimensional analysis
is then r 3 (cm 3 ), whereas the exact value
including π and factors of order unity is
4π/3r 3 ∼ 4.18r 3 .
dimensional transmutation (Coleman
Weinberg, 1973) In field theory, dimensional
transmutation occurs in a first or second order
phase transition in an originally massless theory.
The classical potential of the massless scalar
field ϕ has single minima at ϕ = 0 of a particular
shape. Due to the quantum effects, the potential
acquires the second minima at ϕc. In turn, the
existence of the critical point ϕc leads to spontaneous
symmetry breaking (the field is nonzero,
even though the underlying theory does not pick
out a nonzero value for ϕ: it could have ended
up in the other minimum). Additionally, quantum
effects can modify the shape of the potential