The acceleration of gravity is indiscernable from acceleration due to changes in translational motion.In other words, you can't tell the difference between accelerations caused by different means just by examining the acceleration itself.
The logic might be called "six of one half dozen of the other" or more exactly, it involves the "identity of indiscernables" - compliments to Monseur Leibniz. I use it myself as a first line attack-tic.Of course, there's something more to say...........
For instance, if someone accelerates away from earth and turns around (always accelerating) then comes back and stops next to the person who stays on earth, when both get out to look at one another, "who stayed has a beard and who went has none".
Some people get excited about "turning around" and the fact that the earth's g-field vectors converge at the center of the earth. This is nitpicking. One can "blend" out these difficulties by appropriate sneaky accelerations and by reshaping the earth's gravitational charge, etc. We're looking for the bald, singular fact ... not for degrees of minutiae.
There are some hilarious attempts to show that this or that type of atom does not respond to gravity in the same way. Some 'elements' fall faster than others after all. This logically misguided idea would have been recognized by Galileo for what it is. Even in his primitive era, he knew enough to see the fundamentals involved. Namely, if you drop a bucket of nails, they will fall at the same rate as one nail because each 'one' nail is falling separately inside of the bucket anyway. So what difference would it make if you drop 'em one at a time?
The foregoing is a logical statement not a physical one. If mass came in some sort of flavors which behaved in different ways in gravitational fields, each flavor might just fall at different rates/paths. But there is only one flavor - vanilla (it go down \/ ). The idea that common objects are "neopolitan" is a 'preposterous postulate' running counter to the minimalist approach to phenomena which nature always displays.
General relativity is on even more solid logical ground than special relativity since it's core principle is simple and fully in keeping with the minimal philosophy of existence. Each time I approach relativity theory I come away with a greater admiration for Albert. (He might not even deserve it though ... for if one 'guesses' right all that logically follows will also be true ... and AE will look like a deep, deep thinker even if he never thought to that depth. But I think he actually did ; ) else I should have more to say on this page.