[yafmvelb] Spherical potential functions

We imagine the experience of a ghostlike neutrino traveling around a universe populated by a single spherical mass of uniform density.

Outside the spherical mass, it feels a gravitational potential energy function proportional to -1/r.  Explanation: Newton's universal law of gravitation: force ~= 1/r^2.  Work: integral (force * dr) = -r^(-1).

Assuming a 3D universe, inside the spherical mass (within which the neutrino moves unimpeded), the potential energy function is a parabola, quadratic.  Newton again: Force=GMm/r^2 = G (4/3) Pi r^3 rho m / (r^2) ~= r.  Work = integral r dr ~= r^2.  Incidentally, the quadratic potential function induces simple harmonic motion, assuming you don't exit the well.  This yields the famous problem of a ball oscillating within a tube drilled through the center of the earth.

Together, you get a piecewise potential function of a broad sheet of -1/r in empty space capped by a r^2+C bowl inside the spherical mass.  The bowl prevents -1/r from going to negative infinity.

Generalizing Newton's law to other dimensions, the potential function remains 1/r outside the mass.  However, inside, it changes.  2D: U = abs(r). 1D: U = log(abs(r)).  In 1D, we weirdly have a singularity at the center of the 1D ball, the midpoint of a line segment.  4D: U = abs(r)^3.  Depict test masses moving within these potential wells.  It's no longer simple harmonic motion.