Arnold's Love Child -- The Birth Certificate

TMZ has obtained a copy of the birth certificate of the boy Arnold Schwarzenegger fathered ... and he's not listed as the father on the birth certificate.

Arnold Schwarzenegger Love Child
 
As TMZ reported, Mildred "Patty" Baena -- who worked as a housekeeper/assistant for Arnold and Maria -- is the mother.  She listed her then-husband, Rogelio Baena as the dad.

Baena used a different last name on the birth certificate -- Pena.

The boy was born on October 2, 1997, which now makes him 13. He was born at Saint Joseph's Hospital in Burbank, CA.