Long ago when I was a kid, one of my first teacher's professors at the New England Conservatory of Music in Boston had studied with Albion Metcalf, himself a student of Tobias Matthay, an exponent of relaxation in performance and prolific author on the subject. So a necessary amount of relaxation method came my way, as you can imagine, especially relaxed arm weight.
The relaxed arm weight concept cannot be taken literally though. The idea is NOT for the arms to be so totally relaxed that they go into free-fall due to gravity, thereby allowing the hands and fingers to crash into the keyboard. Relaxed arm weight actually needs to be directed by the upper arm (for accuracy), and secondly there has to be just sufficient enough control during the descent that the hands and fingers drop and sink into the keys so as to produce a rich tone. Key velocity which transfers to hammer velocity is the main element in tone production. But it goes deeper than that. Consider a would-be pianist who brutally forces the keys down leading to fast hammer velocity yet results in unpleasant "pounding" or "banging" as we say. That is not a rich tone! If that same person is then shown how to relax the arm and allow it to descend with gravity--but with a modicum of control for accuracy in depressing--not crashing into--the correct notes--the banging will, with practice, be replaced by a rounder and richer tone. So there is a bit of deliberate tension mixed in with the relaxation, which is why "relaxed arm weight" is figurative, not literal. Obviously this is not an objective scientific principle; rather, its more subjective. It's not so much quantitative as it is qualitative. But I believe it to be essential in attaining a beautiful tone.
Back to key/hammer velocity for a moment, Ortmann believed that the same key velocity achieved by a finger, pencil, or umbrella tip would create the same tone quality. He was probably right as far as that statement goes. But the major fallacy is that the pianist plays not a single, isolated tone all the time, but usually many tones including chords. The piano, of course, is essentially a percussive instrument. The job of the pianist then is truly to be an illusionist. We have to create the illusion of a connected melodic line through rich tone production, phrasing, legato touch in cantilena or bel canto passages, pedaling, dynamics, rubato, and nuances, etc. Scientists can poke a key with a pencil, a huge oversimplification, but they cannot account for the numerous variables or complexities in creating the illusion that I have just mentioned. Indeed that's the kind of empirical situation that separates science from art.
Here is another aspect of it. During a grueling practice session, who here has never experienced a buildup in tension affecting the whole playing apparatus? Probably nobody! If ignored, it will soon spread to the neck, which will make the discomfort even more noticeable. While tension builds, concentration, accuracy and artistry are all diminished. Now if the pianist gets up, moves away from the piano, swings the loose arms parallel to the body, then swings the arms in front of the body such that they cross one another forming an X, and then bends over from the waist a bit, dangles the arms while shaking and rotating them like two ropes being blown randomly about by the wind, like magic all tension is released immediately, and back at the piano relaxed arm weight is resumed leading to restored artistry. So this is an empirical effect that we can directly observe and feel thereby informing us that relaxation benefits artistry.
One qualifier: Although the fingers are part of the playing apparatus, they can NEVER be relaxed. Instead they always need to be taut, otherwise articulation will sound more like wet noodles or cotton.
There is usually an exception to every rule, and the fingers are the exception to relaxed arm weight. Relaxed arm weight is truly a paradox--the collision of two truths. First, the arm must be relaxed enough to respond to gravity; second, there must be enough residual tension to allow accuracy and yet adequate weight to produce rich tones.
I hope this will be helpful. That's the best way I can describe the concept.