Well, here's how the whole story got started...

An astronomer named Morris K. Jessup published a book called "The Case for the UFO" in 1955. In it, he stated his belief that the key to developing space travel could be found by researching Albert Einstein's Unified Field Theory. Soon after the book was published, Jessup began receiving letters from a man named either Carlos Miguel Allende or Carl M. Allen. Allende or Allen claimed that he had information about a secret Navy experiment at the Philadelphia Navy Yard that used Einstein's theories to try to turn a destroyer -- the U.S.S. Eldridge -- invisible.

According to Allende/Allen, the experiment, conducted sometime in October of 1943, not only made the ship invisible, it also teleported it to Norfolk, Virginia, turned the crew invisible and intangible, drove the crew insane, displaced some of them in time, and caused one of the crewmen to catch fire and burn for eighteen days straight. These were not in the mission parameters, so the Navy covered up the experiment and stuck the surviving crew members in various hospitals.

A year or two after Jessup started his correspondence with Allende/Allen, someone apparently mailed a copy of Jessup's book to the Office of Naval Research. The book had a number of UFO-related footnotes written inside the margins in three colors of ink (blue for "Mr. A", violet for "Mr. B", and aqua for "Jemi"). The ONR got in touch with Jessup and asked him about the book, and Jessup was able to identify the handwriting of Mr. A as Allende/Allen. The ONR had an undisclosed number of copies made of the annotated book -- what value they saw in the book is unclear. Unsurprisingly, the UFOlogist community took this as a sign that the annotated book contained Important Stuff, and Jessup began working on UFO research and the Philadelphia Experiment full-time. He was found dead of carbon monoxide poisoning in Florida in 1959; his death was ruled a suicide, which prompted the conspiracy theorists to nod sagely and nudge each other.

It makes a great story, but now for the (more than likely) truth. The Eldridge spent October 1943 in Bermuda and New York Harbor. The ship did visit Norfolk on November 1, but it got there by sailing, not by teleportation. Carlos Miguel Allende was a pseudonym of hoax artist Carl Meredith Allen of Springdale, Pennsylvania; he served in the Coast Guard during World War II and may have hatched the idea for the hoax after learning of a naval technique of degaussing the hulls of ships to make them invisible to magnetic mines. And finally, there is convincing proof that the annotated edition of Jessup's book was put together by a couple of rabid UFO fans who also worked for the ONR...

Primary research: Suppressed Transmission: The First Broadcast by Kenneth Hite, "On the Whole, I'd Rather Be Invisible: The Philadelphia Experiment", pp. 49-52.