In October of 1943, the Naval destroyer USS Eldridge was not in Philadelphia, an indisputable fact which makes the entire story seem even more implausible than it already is. However, I personally believe the naming of that ship as the one involved in the experiment may simply be an attempt at disinformation, designed to make the whole thing unbelievable. Not, as I say, that it was particularly believable to begin with.
The story goes that the Eldridge was undergoing a secret test at Philadelphia Navy Yard, using Einstein's Unified Field Theory to render the vessel invisible. However, there were unintended and quite unexpected side effects of this experiment. The ship teleported from Philadelphia to Norfolk, Virginia, and back. The entire ship and crew were immaterial during teleportation, and some crew members rematerialized fused within bulkheads and other objects. According to a possibly fake newspaper article supposedly from 1943, a copy of which appeared in William Moore and Charles Berlitz's 1979 book The Philadelphia Experiment, there was an altercation between two sailors in a Philadelphia bar who vanished into thin air. A photocopy of the supposed article had been sent anonymously to Moore and Berlitz.
The story of the Philadelphia Experiment was apparently originated by Carlos Allende, aka Carl Allen. In 1956, he sent several letters to Morris K. Jessup, author of The Case for the UFO. The second of these letters described the experiment, which Allende claimed to have witnessed from the merchant ship SS Andrew Furuseth. In 1957, Jessup was invited to the Office of Naval Research by Major Darrell Ritter, Special Projects Officer George Hoover, and Captain Sidney Sherby. They had received a copy of Jessup's book, with annotations by apparently three different writers, referred to by ONR as Mr. A., Mr. B., and Jemi. It later came to be believed that it was just one writer, Carlos Allende.
Jessup gave the ONR officers the letters Allende had sent him, and they sent them to Varo Manufacturing of Garland, Texas, along with the annotated copy of Jessup's book which they had received. Varo produced several copies of the book, with the annotations in red, and two of the letters as appendices. These copy of Jessup's book would be known as the Varo edition.