This is because higher temperatures cause higher RMS speeds of the bouncing atoms, and this causes greater accelerations when the bouncing atoms change directions. But there are also additional factors that contribute to atomic oscillation rate changes, such as temperature changes within the collection of atoms that are being monitored inside the clock.
The work of both Lorentz and Einstein had already predicted that moving atomic clocks and atomic clocks that change altitudes would change their oscillation rates, their “tick rates”, so this was no surprise when the GPS satellites were first sent up. Ok, so, since atomic oscillation rates tend to be very steady, if a collection of atoms are not moving rapidly through fields, not changing RMS acceleration rates due to temperature changes, and not changing altitudes or gravitational potentials, atomic clocks eventually became our “time base standard” for measuring very accurate and short time durations. We can’t say “atoms resting in a gravity field” since atoms don’t “rest”, they move around rather quickly because of heat energy, i.e. In 1911 Einstein deduced that motion-related acceleration was quite similar to gravitational acceleration, so he extended Lorentz theory to include a slow-down in the oscillation rates of atoms that existed in a gravity field. Lorentz hypothesized in his 1895 book, “Versuch Einer Theorie Der Elektrischen Und Optischen Erscheninungen In Bewegten Körpen,” that the oscillation frequency of an atom could slow down if the atom moved through certain fields and if it was subjected to acceleration. Atomic time is based on the frequency of light given off by an oscillating atom, and the frequency of that light is determined by the oscillation frequency of the atom. You can find it mentioned on page 3 of the first volume of his famous “Treatise on Electricity and Magnetism”. The concept of “atomic time” first came from Maxwell in 1873.
Lorentz, not Einstein, who first invented the concept of “atomic” time dilation. Well, Russ, it’s kind of complicated and difficult to explain briefly. simply a matter of mechanical force and not real time dilation? If so, how do you reconcile that with the fact that the GPS system works and is based on Einsteinian time dilation, not mechanical clock rate issues? I maybe haven't been paying enough attention to your other posts (maybe you've discussed this before), but are you saying that an atomic clock sees its rate change for the same reason as a pendulum, ie. If the motion is geodesic, then one may also refer to the equation of geodesic motion which will in some few cases yield the result strait away or eliminate coordinate variables from the expression for the line element. Finding exact solutions isn't always feasible so usually a linearized weak field approximation is made in which case one can simply input the Newtonian gravitational potential into places in the metric and get an approximate answer. T= \frac for arbitrary numbers of gravitational sources is related by second order nonlinear differential equations to those sorces through Einstein's field equations.
The gravitational time dilation formula is
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