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, $\frac{\delta \nu }{\nu }=\frac{\delta U(1+\xi )}{c^2}$, $\xi \ne 0$ indicating violation, and nonuniversality of free-fall rates which can occur in nonmetric theories of gravity.