Can a woman claiming I am the father of her child just put my name on the birth certificate. At this point I believe the fact she is pregnant is false however if she is I am also unsure if it is mine... So again can she just add my name without my knowledge

No. She can put your name on child's BC if
1.You acknowledged your paternity in written.
2.There was judicial proceeding and you were found to be the father.

GV gave you the LEGAL answer. A BC is a legal document and the information on it has to be accurate. However, that might not stop the mother. If you had sex with this woman about the time she conceived, then she could believe you are the father and list you as such. You would then have to go to court to get your name off the certificate.

she could believe you are the father and list you as such. You would then have to go to court to get your name off the certificate.

After the last reform a mother cannot list whoever on the child's BC without signed paternity acknowledgment/ of course if there is no presumption applied/... It is valid at least for 46 states.

GV is correct. In most areas an Acknowledgment of Paternity must be signed by the "potential" father (FOB) before a name is placed on the BC.

I sit corrected.