To establish the relationship between the metric of the General Theory and
the characteristics of existence in D1 it is necessary to extend (2.15)
into the second spatial plane thus
where b is an angle in the second spatial plane. This is the temporal
rate for three-dimensional motion in D1. From (2.17), (4.12), and
(4.18), (E.1) can be transformed to a temporal distance in D0 thus
where dx0 is the distance moved along the temporal axis in an element of time dt in D0.
Incorporating (4.7), with (4.18) incorporated therein, converts (E2) to
as derived in , pp194, Eq(57.64) for the metric of the space-time of the General Relativity in the co-ordinate axes of Pseudo-Euclidean Space-Time. The above process shows that the metric of the General Theory is directly proportional to the temporal rate of a gravitating mass in D1. This suggests that the metric of the General Theory is a temporal metric rather than one of a true space-time interval.
Nevertheless, however (E.3) is interpreted, from the above it clearly involves three-dimensional spatial terms and, as such, can only represent the metric of the General Theory for the case in which three-dimensional spatial variation in position is involved. If this variation is put to zero, then a metric for a spatially stationary point in the co-ordinate system of the General Theory is obtained. Thus by putting dr, df and db to zero in (E.3) gives
Re-inserting (4.7) and (4.18) then gives
so that the proper time of this point relative to Pseudo-Euclidean Space-Time is then
as derived in (2.18).
Thus the proper time of a spatially stationary point in the space-time of the General Theory, relative to Pseudo-Euclidean Space-Time, is identical to the proper time of D1 relative to D0.
G1 Version 2.2.4 Ó P.G.Bass, November 2009
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