My question involves paternity law for the State of: Georgia

My wife was pregnant when we separated and I was told that she had a miscarriage. I found out today that she had the baby in April. We have a mutual restraining order, so I cannot contact her. What rights do I have in requesting from the hospital if my name is on the birth certificate? She is with someone else now, and I do not know who's name is on certificate. I do not know childs name or sex.