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2.4.3 Displacement

In visualizing how a part of the continuum may be displaced from the normally flat, three dimensional shape, a useful analogy is that of an area on the Earth's surface which is small enough to be regarded as smooth, flat land. Since the Earth's sphere is very large, a small indentation into its surface can be considered as a 'crater' in an otherwise flat surface. The depth of this crater at each point can be represented as a scalar variable in the two coordinates of the original flat surface. In this two dimensional case, the symmetry of the crater will be circular.

Similarly, the most convenient way to visualize displacement of the three dimensional continuum is to view it as a displacement of the practically flat, three dimensional continuum into a fourth direction. In this case, a symmetrical 'crater' is spherical and its 'depth' can be represented by a displacement function, \(N(r)\), at each point. The variable, \(N\), represents the depth of this hypothetical 'spherical crater'. If \(R_0\) is the radius of the ideal continuum, and \(R\) is its radius within the 'crater', then displacement \(N=(R-R_0)\). That is, the displacement function, \(N(r)\), can be treated as a scalar variable in three dimensional, Euclidean space.

In this way, a spherically symmetrical gravitational system can be represented by the displacement function, \(N(r)\), where the coordinate axis, \(r\), is the radius of the spherical crater. In this analysis, the coordinate system is to be aligned so that its origin, (\(r\)=0, and \(N(r)\)=0) are at the same point from which the center of the gravitational structure will be displaced in a direction opposite to that of expansion, (\(N(r)\) is negative).

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