If I'm not on the birth certificate but paying child support, do I have rights to my child?

In the state of Georgia I have been paying child support for a year and a half via a court order. I have made all payments on time. I am however not on the birth certificate. We have an arbitrary agreement on visitation, I get her Friday through Sunday and she has her Sunday evening through Friday evening. We also have a written agreement that was signed by both of us 2 years ago that I would have her Thursday through Sunday evening. She texted me today that she has gone to Ohio with my child (3yrs) with her boyfriend for 2 weeks. I am scared for my child's safety and well being. Do I have rights since I am paying child support to make any legal action for this? Can I call the authorities or a federal agency to get her to bring my child back? There is no telling when she will be back, she may never come back. What rights do I have in this situation? Thank you for your input.-Joseph