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.
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