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I speak North American English. When I pronounce "R", the fleshy part under my chin inside my jawbone tends to move up. When I stress the "R", it really moves up and back. I think the back of my throat is constricting, or the root of my tongue is making my vocal tract smaller.
The sound is called retroflex approximant, which means that it produces a stream of air not narrow enough to be classified as a fricative consonant("sh","v") but also not wide enough to be classified as a vowel. The retroflexion means that the tongue is curling backwards while producing the sound.
I am not sure what is the muscle you mean.
Approaching the matter as a phonologist, I could ask what assimilations does "r" participate in. When "r" follows "t" or "d" in NA English, it becomes an r-colored palato-alveolar fricative, as in "treat" or "dream". In Hawaiian English, "r" causes preceding "s" to become palato-alveolar "sh", as in "street". This suggests that "r" is palato-alveolar, which I take to be, essentially, alveolar, but with the blade of the tongue raised and the tip of the tongue lowered.
I don't know what effect there is on the root of the tongue for an "r", but I'd expect the root to be advanced (rather than retracted, as you report), since "r" has some affinity with palatals, and they cause enlargement of the pharynx.
