We further investigate spin orbit coupling in ballistic carbon nanotubes in relation to two recent experiments [L. Hueso, et, al., Nature 410 445 (2007) and F. Kuemmeth, et, al., Nature 425 448 (2008)]. Spin-orbit coupling in ballistic nanotubes is due to the tube's curvature and a transverse electric field. The semi-quantitative agreement between the measured spin-orbit splitting in [F. Kuemmeth, et, al., Nature 425 448 (2008)] and our previous theoretical estimate is improved by the inclusion of effects of a transverse electric field. This induces asymmetries in the spin-orbit coupling for electrons and holes, in agreement with experiments. Furthermore, spin-orbit induced spin precession is an alternative explanation of spin relaxation reported in [L. Hueso, et, al., Nature 410 445 (2007)].