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Variable Extra Dimensional Spacetime and Solution to Initial Singularity Paradox of Our Universe From Extra Dimensions

Considering a n-dimensional general spacetime, we deduce its 4-dimensional Einstein equation and Friedman equations, and discover a general dual relation between the scale factor $a(t)$ of our universe and the scale factor $B(t)$ of extra dimensions. Based on the dual relation equation, predictions of shrinking of extra dimensions and free of singularity problem of our universe are given. Therefore, solution to initial singularity paradox of our universe is achieved. Because the dual relation is general, this Letter discovers that it is just the extra dimensional shrinking contribution that results in our universe's expanding in terms of the dual relation in the bulk space, and actually the dual relation is deduced doesn't depend on the 4-dimensional matter concrete Lagrangian, these are key important for a lot of future relative investigations.