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At time 
the radius of a grain that nucleated at time 
is
If the magnitude of the grain
radius 
exceeds the distance between 
and 
,
the point 
will have transformed before time . The time
cone is the set of all points that satisfy this inequality.
The surface of the time cone, which is the event horizon are the
nuclei points 
that grow as grains that
reach exactly at time 
. They are given by the points in
for which the inequality (2.3) is an equality.
This allows the introduction of the method of characteristic applied
to geometric growth [23][22][21] into the time cone theory. Such
growth theories are based on the time 
of arrival of a moving
surface at a point . Thus is a function of . The surface
at any time is a level surface of 
. If the surface is
smooth, the direction of the gradient of is parallel to the
normal to the surface,
and its magnitude is 
Thus 
, and 
. In a subsequent paper,
we will use characteristics to construct time cones when 
is a
function of
and 
with only a few restrictions, and derive relations
for the kinetics for specimens that are anisotropic and inhomogeneous
in space and time. One crucial corollary of the theory is that it can
identify the assumptions necessary so that nuclei from points outside
the time cone never grow into the time cone. [15]
If is constant, equation(2.3) becomes the equation of the
points in a cone,
Compare this equation with one for the time horizon for the point
in the theory of relativity, 
where 
is the velocity of light.
At any time the spaces in which any nucleation event will
affect 
are thus: in 1-d, a line segment of length 
in 2-d
a circle with radius 
in 3-d a sphere with radius
; etc., all centered on , and increasing in size
the further back in time we go.
With growth rates that depend on time, the event horizon is a
conical surface of revolution in this space with apex at
with its axis parallel to the time axis,
and given by
The time cone can be considered unbounded in the negative time
direction, to times before the onset of nucleation, if a value is
assigned to 
. Because 
there, the results do not
depend on what is chosen for .
Next: The expected number
Up: A rederivation of
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