The one where Peri messes up a musical performance on stage, and then almost dies from rapidly ageing.
The TARDIS materializes aboard the U.S.S. Eldridge, from the infamous US Navy try for invisibility known as the Philadelphia Experiment. The Eldridge, and now the TARDIS, are trapped in a dimensional flux between Earth and an unknown micro-dimension. The only survivors of the crew are the bosun and a Professor Tessler; the rest have become time-looped automatons. The Doctor can pilot between Earth in 1943 and the Eldridge, but that's all.
The Doctor is able to trace the problem to an alternate dimension called Capron, whose ruler Osloo has been "mining power" from the wreck of the Eldridge and causing the problem. The Doctor conditions Peri to be able to survive in the alternate dimension, and the two of them head to Capron. The plan is to offer that world a different source of energy to replace the Eldridge, allowing the ship to return to normal space-time and freeing the TARDIS.
The Doctor manages to get himself captured in moments, and lands in a cell with Ezz, the rebellious son of the tyrannical Presidenta Osloo. Peri fares slightly better, passing herself off as an expected guest but not realizing that guest will be expected to sing opera for an audience of thousands. Osloo personally captures her, and when Peri asks her to shut down the power mines Osloo has her arrested.
With the help of a rebel named Yka, the Doctor and Ezz escape and rescue Peri. The four of them head back to the Eldridge. Peri becomes time-sick by leaving the TARDIS before her previous conditioning can be reversed; because she was conditioned for faster-moving time, she begins aging at an astounding rate. The Zero Room is able to cure her condition.
While Osloo fantasizes about conquering the Earth, the Doctor and Peri head to Washington D.C. to prevent the Philadelphia Experiment from ever taking place — the paradox will do less damage to the web of time than the Eldridge's current predicament. He tries leaving a note for Professor Tessler's younger self, but it ends up delivered to someone else and ignored.
The TARDIS returns to Capron, hoping that now that Ezz knows the true source of his mother's power he will be able to overthrow her. Unfortunately, they materialize right in front of Osloo and her guards, and she captures the TARDIS in seconds. Osloo has her troops hold Peri at gunpoint, forcing the Doctor to take her to the Eldridge as well. The Doctor, realizing with a sinking feeling that his attempt to change history has failed, is forced to do as she asks, giving the Caprons the same conditioning that he gave Peri earlier.
While Osloo tours the Eldridge, the Doctor gives the crew the bad news. Tessler and the bosun's destinies are caught up with the Eldridge's, and the ship is disintegrating rapidly. There's no way he can get Tessler back to his wife. Furious, Tessler throws in his lot with Osloo, offering her the plans to the Philadelphia Experiment and his assistance in her conquest of Earth. In exchange, Osloo will force the Doctor to violate causality and take Tessler home. The group returns to the TARDIS just in time as the Eldridge finally disintegrates around them.
The TARDIS returns to Capron. Osloo kills Tessler, then marches back out to marshal her technologically superior forces for conquest. Unfortunately for her, the Doctor didn't reverse the earlier conditioning, and both Osloo and the bosun age backwards to infancy. The Doctor leaves Ezz in charge and he and Peri depart, stopping only long enough to rig up the promised replacement power source.
Tropes:
- Achievements in Ignorance: Osloo manages to commandeer the TARDIS virtually effortlessly, never realizing that it's supposed to be difficult.
- Another Dimension: Capron is apparently located in a "micro-dimension," vastly smaller than the main Whoniverse.
- Artistic License – Physics: The Doctor claims that parallel dimensions are particle-sized and located within the universe that spawned them because there wouldn't be room to put them anywhere else.
- The Doctor plans to provide unlimited power for a million years to a micro-dimension using a single torch battery in his pocket, because a microverse uses only a "minute amount of energy."
- Bound and Gagged: Peri is briefly gagged when she first meets Tessler and the Bosun.
- Butt-Monkey: As usual, Peri. She manages to humiliate herself in front of most of a planet, (temporarily) ages to senescence in a matter of minutes, and falls through the deck of the Eldridge.
- Cold-Blooded Torture: The apparent specialty of Osloo's sidekick Bundth, though he never actually performs any.
- Cover Identity Anomaly: Despite not being able to pronounce the name of the person she's impersonating, Peri manages to get all the way to Osloo on sheer chutzpah.
- Forgot About His Powers: The Doctor is blackmailed by Osloo's troops holding guns on Peri, and Tessler is later shot by them. Only specially-designed weapons work in the TARDIS due to its state of temporal grace. Then again, other stories would show the temporal grace equipment is susceptible to failure when operating out of the normal universe. And is quite possibly broken.
- Glorious Leader: Presidenta Osloo frankly oozes palpable evil, yet she was elected along with a cadre of cooperative sycophants and is able to whip up a totalitarian dictatorship with ease.
- Hoist By Her Own Petard: Osloo is defeated when she refuses to listen to the Doctor’s warnings about how he needs to stabilise the dimensions after they travel between Earth and Capron, resulting in Osloo de-aging to infancy.
- Ignorant of Their Own Ignorance: While Osloo is dangerous, she tends to create her own ideas of situations based on what she likes to believe, such as assuming that the TARDIS is unique because that makes her ‘ownership’ of it all the more significant.
- It's All About Me: A good description of Osloo, as she defines everything by how it would benefit her and ignores anything anyone has to say to her that might contradict that belief.
- Kangaroo Court: Osloo's 'trial' for Ezz just consists of telling her counsellors that he's guilty of treason and then coming back to tell them what sentence she feels he should receive, never even bothering to tell them what he's actually meant to have done because she reasons that Ezz "knows" what he's done and her people just need to know he's guilty.
- La Résistance: One of the least effective examples, though they do manage an effective jailbreak.
- Large Ham: Osloo, dear Rassilon.
- O.O.C. Is Serious Business: For once, the Doctor is willing to take action that will change what he knows of history to save the Eldridge from its fate, although he does argue that its fate has already created a rift in time and his current actions will just undo that rift.
- Ominous Message from the Future: The Doctor tries to leave one to make its way to Professor Tessler's younger self, but arrives too late.
- Time Paradox: For once in the series, creating one is the lesser of two evils.
- You Have Outlived Your Usefulness: As soon as she finds out where the TARDIS is, Osloo condemns Mers to death. The arrival of the TARDIS interrupts them.
