Rotations in Standard Notation

This is all now in tensor doc, so delete this doc in a few days from 4.12.12 Rotations in Standard Notation PhL 4.9.12 This seems all OK, and should go somewhere in tensor doc. [ The new home is in Section 7 (i) item 10. ] We know that for any F, we have RS = 1 and we have the various orthogonality relations for R. Suppose it happens that R is a rotation. How do you express that fact in standard notation? Answer: you first start in dev notation and you write RRT = RTR = 1 RT = R-1 = S which says (RRT)ab = (RTR)ab = δab or RanRTnb = RTanRnb = δab or RanRbn = RnaRnb = δab Converting this to standard notation we would write the two equations separately and say RanRbn = δab RnaRnb = δab These must BOTH be true if R is a real-orthogonal matrix in dev notation. There is no need for having the transpose T symbol appear in the standard notation here. If the first equation above were true, how would you derive the second? RanRbn = δab => Ran(S-1)bn = δab Now apply Sna on the right on both sides and sum on a, Ran(S-1)bn Sna = δab Sna Ranδba = δab Sna Rbn = Snb Note that this is NOT the usual relationship Rbn = Snb. But now we can use that to say Rbn = Snb = Rbn and this then is another way to state the rotation fact Rbn = Rbn Given this we can then say RanRbn = δab => Rbn Rbn = δab Comment: Suppose R is a rotation, as can be represented above in all these ways: [RTR = 1]DN Rbn Rbn = δab Rbn = Rbn Rbn = Snb RanRbn = δab This needs to be somewhere in tensor doc! Any one of these equations implies all the others! Suppose g is NOT g = 1. Then g' = RRg or g'ab = Raa'Rbb'ga'b' . In DN you would write g' = R g RT = R g R-1 but this does not say g' = g. Only if g=1 do you get g' = g.