What is the ruling on sitting next to a woman on public transportation without any physical contact, whether she is old or young, and does the ruling differ if there is physical contact with an old woman, by analogy with the permissibility of shaking her hand?

It is permissible to sit next to an old woman as long as there is no physical contact. If contact occurs, there is no sin if no temptation arises, because the fear of temptation is absent with an old woman. However, if temptation is feared, then it is not permissible. As for a foreign woman who is not old, it is permissible to sit next to her if no physical contact occurs, but with contact, it is not permissible. There is no difference in forbidden touch whether there is a barrier or not, in order to block the means to temptation.