Quantitative relationship between clock gravitational "red-shift" violations and nonuniversality of free-fall rates in nonmetric theories of gravity

When we assume conservation of energy (absence of perpetual motion of the first kind), a generally valid quantitative relationship is derived between possible violations of the clock gravitational "red-shift" effect, δνν=δU(1+ξ)c², ξ≠0 indicating violation, and nonuniversality of free-fall rates which can occur in nonmetric theories of gravity.