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The Philadelphia Experiment forms the basis for a claim that in 1943 the U.S. Navy attempted to use "speculative physics" to render a ship, the U.S.S. Eldridge, invisible. Allegedly, they were successful. There are some pretty detailed accounts of the whole thing. This last one includes a wave of rebuttals and attacks from both sides. A thorough analysis can be found here.

Some of the sources explain that the Navy denies it all. The link given, though, is dead, and there have been allegations that there's a conspiracy at work. On the other hand, quotations from the U.S. Navy on the subject exist. Quoting part of one,

Over the years, the Navy has received innumerable queries about the so-called "Philadelphia Experiment" or "Project" and the alleged role of the Office of Naval Research (ONR) in it. The majority of these inquiries are directed to the Office of Naval Research or to the Fourth Naval District in Philadelphia. The frequency of these queries predictably intensifies each time the experiment is mentioned by the popular press, often in a science fiction book.

The genesis of the Philadelphia Experiment myth dates back to 1955 with the publication of The Case for UFO's by the late Morris K. Jessup.

Some time after the publication of the book, Jessup received correspondence from a Carlos Miquel Allende, who gave his address as R.D. #1, Box 223, New Kensington, Pa. In his correspondence, Allende commented on Jessup's book and gave details of an alleged secret naval experiment conducted by the Navy in Philadelphia in 1943. During the experiment, according to Allende, a ship was rendered invisible and teleported to and from Norfolk in a few minutes, with some terrible after-effects for crew members. Supposedly, this incredible feat was accomplished by applying Einstein's "unified field" theory. Allende claimed that he had witnessed the experiment from another ship and that the incident was reported in a Philadelphia newspaper. The identity of the newspaper has never been established. Similarly, the identity of Allende is unknown, and no information exists on his present address.

. . .

Personnel at the Fourth Naval District believe that the questions surrounding the so-called "Philadelphia Experiment" arise from quite routine research which occurred during World War II at the Philadelphia Naval Shipyard. Until recently, it was believed that the foundation for the apocryphal stories arose from degaussing experiments which have the effect of making a ship undetectable or "invisible" to magnetic mines. Another likely genesis of the bizarre stories about levitation, teleportation and effects on human crew members might be attributed to experiments with the generating plant of a destroyer, the USS Timmerman. In the 1950's this ship was part of an experiment to test the effects of a small, high-frequency generator providing l,000 hz instead of the standard 400hz. The higher frequency generator produced corona discharges, and other well known phenomena associated with high frequency generators. None of the crew suffered effects from the experiment.

ONR has never conducted any investigations on invisibility, either in 1943 or at any other time (ONR was established in 1946.) In view of present scientific knowledge, ONR scientists do not believe that such an experiment could be possible except in the realm of science fiction.

The whole idea seems like a load of rubbish, especially since the principles the experiment would be based on are extremely unsound. I'm quite skeptical about the claim, but I'm not fully convinced that it was fake, because there are claims that the ship's logs (which purportedly put the ship somewhere completely different, far from Philadelphia) were falsified.

In short, there doesn't seem to be a neutral (i.e. not by the Navy or the original claimants) record of the actions of the U.S.S. Eldridge during this time period, which for me would be the clincher. At least, I can't find one.

Did the Philadelphia Experiment ever take place?

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    Is there reason to think that anyone outside the military would be privy to the actions of a warship? – Nate Eldredge Mar 26 '15 at 1:34
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    The electrical equipment reminds me of degaussing a ship to avoid magnetic mines. – ChrisW Mar 26 '15 at 9:00
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    As a nitpick, you can't use 'speculative physics' to render a ship invisible, or do anything else useful. You have to get the physics past the 'speculative' stage before you can do anything with it. – DJClayworth Mar 26 '15 at 17:00
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    @DJClayworth I'm only using the term because it was described as such. I don't believe that even the concepts that are being referred to as "speculative physics" actual ever existed. But I get what you mean; it hadn't occurred to me. – HDE 226868 Mar 26 '15 at 20:19
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    Both of the proposed explanations, degaussing (which was aimed to make the ship "invisible" to certain sensors) and superior power generation, could have have had significant tactical advantages and thus would be good reasons for secrecy at the height of WWII. So I think falsified records and other signs of secrecy are just as consistent with the much more mundane explanations provided. – TheBlackCat Mar 27 '15 at 12:10

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