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The time cone when

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 Previous: A rederivation of
Wed Feb 14 17:48:17 EST 1996