Authors: C. A. Laforet
In this paper, the integration of a function over a curved manifold is examined in the case where the curvature of the manifold results in a varying density of coordinates over which the function is being integrated where the upper bound of the of integration is infinity. It is shown that when the coordinate density varies in such a case, the true area under the curve is not correctly calculated by traditional techniques of integration, but must account for the varying coordinate density. This integration technique is then applied to the Schwarzschild metric of General Relativity to examine the proper time taken for a freefalling observer to reach the event horizon of a black hole.
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