Chapter 9: Fundamental Physics

Section 15: The Phenomenon of Gravity

Cylinder volumes

In any d-dimensional space, the volume of a cylinder of length x and radius r whose direction is defined by a unit vector v turns out to be given by

s[d-1] r^(d-1) x (1 - (d-1)/(d+1)(RicciScalar-RicciTensor . v .v) r^2 + …)

Note that what determines the volume of the cylinder is curvature orthogonal to its direction—and this is what leads to the combination of Ricci scalar and tensor that appears.

From Stephen Wolfram: A New Kind of Science [citation]