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2.4.4 Spherical Coordinates

Given the spherical symmetry of gravitational structure, it is convenient to formulate it using spherically symmetrical coordinates. That is, instead of expressing gradient and curvature in terms of xyz coordinates, they are expressed as functions of radius, \(r\) and displacement, \(N(r)\). This considerably simplifies the calculation of gravitational parameters and provides valuable insights.

In flat, \((x,y,z)\), coordinates, gradient and curvature are defined as \begin{align} Gradient\ \nabla R(x,y,z) = i \frac{\partial R}{\partial x} +j \frac{\partial R}{ \partial y}+k \frac{\partial R}{\partial z} \nonumber \\ Curvature\
\nabla^2 R(x,y,z) = (\frac{\partial^2 R}{\partial x^2} + \frac{\partial^2 R}{\partial y^2} + \frac{\partial^2 R}{\partial z^2}) \nonumber \end{align}

Because the curved, xyz coordinates of the ideal universe are substantially flat over regions of the continuum containing a gravitational system, it is reasonable to use the standard conversion formula to provide spherically symmetrical coordinates. This allows a single coordinate, radius \(r\), to be used in place of the three xyz coordinates, such that \(r^2=(x^2+y^2+z^2)\).

The standard conversion formula give the Euclidean gradient and curvature in terms of radius \(r\), as \begin{align}
Gradient\ \nabla R(r) =&\ \frac{\partial R}{\partial r} \nonumber \\ Curvature\ \nabla^2 R(r) =&\ \frac{1}{r^2}\frac{\partial}{\partial r}(r^2 \frac{\partial R}{\partial r}) \nonumber \\ = \frac{\partial ^2 R}{\partial r^2} + \frac{2}{r} \frac{\partial R}{\partial r} \nonumber \end{align}

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