Can he have his last name without being on birth certificate?

Georgia • Married💓 Poly 🤪 2 babies x
Okay so, i want my son to have his dad's last name but i dont want him to have same legal responsibility for my own reasons! So i wont be putting his name down, i hear he can sign responsibility over but.. I dont get it? 
So can i still have his last name for my little boy or? 

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