While Georgia law does not restrict your moving, you have put yourself in a position that gives Georgia jurisdiction over any custody case filed (assuming you both have been here for at least 6 months). If you move, the father can file for legitimation and custody, and you will be forced to come back here to defend the case. If he finds out you are leaving, and files the suit before you leave, the court may prevent you from removing the child from the state until the issue of custody is resolved by the court. That could take several months, or longer. You should retain an experienced custody attorney before saying anything or doing anything.
I am exclusively a family law attorney, practicing primarily in the metro Atlanta, Georgia trial courts. However, I handle appeals from anywhere in Georgia.
The first question is whether the father legitimated the child. You should find out if signing a birth certificate in California elevates him to the status as the legal father. A California family law attorney should let you know in a simple phone consultation. If the child was born in Georgia, simply signing a birth certificate would not give him status as the legal father.
If the father didn't legitimate the child under California or Georgia law, he has no status and no right to visitation, custody, or notice of a move. Paying child support does not give him the right to notice.
Regardless of the status, if you are the father, file a petition to legitimate and for visitation or custody as soon as possible. At the minimum, you should consider consulting with an attorney in an office visit. Any custody order should require the mother to give 30 days notice before a move. Once notice is given, the father can file for a petition to modify visitation or custody.
Paula J. McGill
The McGill Law Firm, LLC
300 Colonial Center Parkway
Decatur, GA 30030