Being reborn as a god will have to have some kind of connection to the wizard Shazam and the Rock of Eternity, both of which provide the source for Black Adam's power. In the pre-Crisis origin, Black Adam's body is resurrected by way of Dr. Sivana's Reincarnation Machine, which is most likely what brings the antihero back in Black Adam. Both of these things are new to the Black Adam mythos, as the character was never a slave in the comics until the New 52 reboot, and while he had children, there's not a specific story that says his son sacrificed himself to save Adam.
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There is quite a bit of mystery in how Black Adam will adapt his comics origin, but it's clear that some liberties are being taken to work it into the DCEU, which will likely be a combination of his many comic book back stories over the years. Black Adam has the power of a god, but he doesn't have it because his body's revival doesn't tie it to the reincarnation machine. Black Adam has a number of retconned origins going back to his initial creation in the pages of Fawcett Comics in 1945 leading up to the modern era, which has refined and altered his past in multiple ways to better fit into the DC Universe.
Black Adam was created by Otto Binder and C. C. Beck and first appeared in The Marvel Family #1 in 1945, where he was portrayed as an ancient Egyptian named Teth-Adam who was chosen by the wizard Shazam to become his champion and successor. He still serves as a champion for the wizard Shazam, but his home is destroyed and his family is killed by the evil priest Ahk-Ton, forcing him to return to his home and kill the villain. The new trailer for Black Adam shows the anti-hero being brought back to life and becoming a god, but his journey to get there is complex. Black Adam's resurrection and god-like powers are sure to play a role in the Shazam connections, as the trailer is holding back certain details to keep some mystery before audiences see it. The ancient Egyptian version of Adam held his son's body and said, "My son sacrificed himself to save me."
Black Adam was released into the modern era after the tomb was excavated and the scarab was uncovered. Black Adam was buried within the tomb of Ramses II after a spell was cast to house his soul inside a powerful scarab. The power corrupted Teth-Adam after he became a ruler. DC's many eras include Pre-Crisis, New 52, and Rebirth. The trailer for Black Adam shows Johnson's body being placed into a machine filled with containment fluid, while revealing images from his life during ancient times. After 5,000 years of traveling back to Earth, Adam must face Billy Batson and the Marvel Family before being defeated. Black Adam kills the pharoah to rule Egypt. Black Adam was sent to a distant star by Shazam because he couldn't revocation the power. Black Adam's origin has been shuffled around a lot, creating a confusing timeline of his true roots.
Black Adam's origin is slightly different in the modern era. Black Adam takes on the powers of the Egyptian deities. The trailer for Black Adam has Johnson saying that he was a slave until he died.
Black Adam's origin is not the same as many legacy comic book characters. I was reborn as a god.