Is a Man Permitted to Go on a Bus If He May Inadvertently Bump Into a Woman?

Question 1:  Is a man permitted to ride on a crowded bus or train if he knows that he may inadvertently bump into a woman? 

Answer 1: This question was asked to Rav Moshe Feinstein by a young man who was leaving Kollel to enter the workplace and was nervous about riding the bus for this very reason. Rav Moshe told him that touching a woman is only forbidden if it is done “derech chiba”, in a lewd or affectionate way, and not if it is done incidentally. Since bumping into someone by accident is not derech chiba, Rav Moshe told the man that he need not be concerned about it. 

Question 1:  Would that mean that it is permitted for a man to shake hands with a woman? 

Answer 2:  No. Rav Moshe felt strongly that it is prohibited for a man to shake hands with a woman because shaking hands is by definition considered derech chiba. 

Although he forbids it, in one teshuva he notes that some people do shake hands with members of the opposite gender, and he says they must be doing so under the assumption that shaking hands is not derech chiba, but he himself feels otherwise.