The Slingshot effect, also known as light-speed breakaway factor, is a method of time travel which involves flying around a gravity well at extreme velocities, the break away speed sending the ship backwards or forwards in time. A successful slingshot effect requires extremely precise calculations.
The first, known example of this method was an accident when the USS Enterprise - trying to break free from the gravity of the planet Psi 2000 - was sent three days back in time. Later the same ship was sent much farther back in time; when it was dragged into the gravity well of a black star. After that the Enterprise purposefully used the technique to conduct historical research in the 1960s. (TOS episodes: "The Naked Time", "Tomorrow is Yesterday" & "Assignment: Earth")
The Enterprise's ability to perform the slingshot maneuver was found by Federation scientists to be unique to the Enterprise's engines. It was theorized that some combination of Spock's calculations with the environmental effects at Psi 2000 modified the engines to produce Chronitons. However, for reasons unknown, the technique could not be duplicated on other starships in experiments. (DTI novel: Forgotten History)After having encountered the USS Enterprise during its five-year mission, intelligent apes Kira, Cornelius, and Dr. Milo orbited a parallel Earth destroyed by a nuclear detonation and considered their options. On a tricorder, they read this database entry: "Slingshot Effect: The slingshot effect, also known as the light-speed breakaway maneuver, is a spaceflight maneuver in which a vessel closely approaches an astronomical body, using the body's gravitational field to provide additional speed. This maneuver is performed by traveling at an extremely high warp factor towards a massive body with a high gravitational attraction, such as a star. After allowing the gravitational pull to accelerate the vessel to even faster speeds, the vessel then breaks away from the stellar body, creating a whiplash effect that can propel a ship into a time warp, allowing the vessel to travel through time. The slingshot effect was used to return the USS Enterprise to the 23rd Century when the ship was accidentally sent to the year 1969 by near-collision with a black star. Performing this maneuver requires the correct completion of extremely precise calculations, such as available fuel components, acceleration, and a vessel's mass through a time continuum." (TOS comic: "The Primate Directive")
The Federation Department of Temporal Investigations timeship, Timeship One, was constructed using the Enterprise's discarded engines after that ship had been refitted in 2270. Timeship One's first mission would have had it use the slingshot maneuver to visit the future. However, a future version of that same timeship arrived from the future, having completed the maneuver with devastating results. As a result, the planned launch was cancelled. (DTI novel: Forgotten History)
The technique was used again 2286 to travel back in time to obtain a pair of humpback whales to satisfy a probe, which was devastating 23rd century Earth, looking for the then-extinct species. (TOS movie: Star Trek IV: The Voyage Home)
The slingshot effect was used in 2364 when the USS Enterprise-D was caught in a gravimetric wave. The effect allowed Enterprise to escape the wave and travel back in time to the source of the wave and destroy it. (TNG comic: "History Lesson")
Montgomery Scott reviewed the logs of the original Enterprise crew, particularly Spock's time travel equations from the Bounty and created another slingshot travel using the Bounty 2 around the Arhennius star to attempt to save James T. Kirk from his apparent death aboard the Enterprise-B. The Enterprise-D followed Scott into the past. (Star Trek novel: Engines of Destiny)
After the crew of the USS Defiant was thrown forwards in time to 2400, they were unable to slingshot around Bajor-B'hava'el as they had no frame of reference on how to do it there. It was later discovered they had travelled to the future by way of the Pah-wraiths' Red wormhole. In this timeline, Jean-Luc Picard's timeship USS Phoenix was designed to withstand the effects of using the slingshot effect to travel twenty-five thousand years into the past. (DS9 novels: The Fall of Terok Nor, The War of the Prophets, Inferno)