Shulchan Aruch (554:20) states, שלום שאלת אין באב בתשעה לחבירו, "One mustn't greet his friend on Tisha b'Av." But why not? The Beis HaMikdash was destroyed because of sinas chinam. It seems that Tisha b'Av is the ideal time to greet one's fellow man and increase love and peace.
To answer, we quote the following Gemara (Gittin 58.):
"There was an apprenticed carpenter whose eyes desired his boss's wife. Once, his boss, the carpenter, needed a loan. The apprentice said, 'Send your wife to me, and I will give her the money.'
The apprentice detained her in his home for three days. The carpenter came and asked, "Where is my wife?"
"I gave her the money and sent her back right away. But I heard that some youth defiled her on the way."
"What should I do?"
"I suggest you divorce her."
"But her kesubah is very large, and it will cost me a lot of money to divorce her."
"I'll lend you the money."
The carpenter divorced his wife, and the apprentice married her.
When the loan was due, the carpenter could not pay his debt, so the apprentice said, 'Work for me, and you will pay off your debt with your work.'
The apprentice and his new wife ate while the carpenter served them. As he poured them drinks, his tears fell into their cups. That is when the decree for the destruction of the Beis HaMikdash was sealed in heaven."
The Yaavatz (Reb Yaakov Emdin zt'l) notes that the apprentice didn't transgress an aveirah in this story. He married the carpenter's wife after she was divorced. Yet, because of his deeds, the Beis HaMikdash was destroyed. "From this, we learn that there are aveiros that aren't written explicitly, yet they are extremely grave and despised by Hashem. They are worse than the cardinal sins."
An example of this concept is the Gemara (Bava Metzia 30:), which states, "Yerushalayim was destroyed because they ruled by the laws of the Torah."
The Gemara asks, "Which laws should they follow, if not the laws of the Torah?"
The Gemara responds, "The [problem was they] followed the laws literally and never went beyond the letter of the law." They failed to consider the spirit of the law as well. They only focused on what is explicitly prohibited or permitted, and that isn't sufficient. A person can commit terrible aveiros, r'l, and it isn't an excuse that the aveirah isn't written explicitly in the Torah.
Now, let's return to our question. We want peace on Tisha b'Av, so why don't we greet people on Tisha b'Av?
The answer is that on Tisha b'Av, we seek true peace, not lip service. Throughout the year, people smile at their fellow man and greet one another, but they don't necessarily care about them in their hearts. On Tisha b'Av, we take a break from greeting our fellow man, reassess, and strive to reach a higher level. We want to go beyond the letter of the law, beyond what's commonly expected, and to develop genuine peace and love among Yidden.
