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In Star Trek IV Spock takes the bird of prey back in time to retrieve two whales. He does this by slingshotting the ship around the sun.

It's my understanding that the Federation cannot travel in time (all other events have either been accidents or tailgating someone else's technology). As far as I'm aware Spock suddenly manages this feat and it's not duplicated again.

How did he suddenly know how to travel in time and why was this knowledge suddenly forgotten?

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Although the film doesn't make it immediately obvious, the reality is that Spock and the Enterprise crew have already travelled through time on at least two other occasions using the "Slingshot effect"; once by accident (in TOS : "Tomorrow is Yesterday" when the crew inadvertantly stray into the gravity well of a black star) and once on purpose (in TOS: "Assignment: Earth").

The slingshot sequence in Star Trek IV is a call-back to those episodes and certainly explains why Spock knows how to effect an accurate time jump.

As to why we never see this again, in-universe it's pretty clear that the maneuver is very unsafe and rarely attempted. On the first occasion the Enterprise is very nearly destroyed;

Scott : If I can't stop us soon enough, we may overshoot our time, and if I stop the engines suddenly the strain may tear us apart. Anyway we do it, it's a mighty rough ride.

Out of universe, the fact that the Enterprise very rarely engages in time travel is hand-waved away by the show writer (Ronald D Moore) who simply said that;

"I would assume that the precise calculations involved in using the slingshot method are something of a closely-guarded secret."

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    Indeed. It is not inconceivable that Spock, having discovered it in Tomorrow is Yesterday, is the only one who knows how to do it. – Joshua Jan 8 '15 at 22:59
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    Slingshot method > Save Romulus > New movie universe aborted. Just had that thought while reading this answer. – Kalamane Mar 19 '15 at 21:40
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    @Kalamane - per my answer here we have canonical proof that OldSpock is unwilling to alter the timelines again, regardless of NuSpock's begging.. – Valorum Mar 19 '15 at 21:46
  • @Joshua But Spock presumably used this method by design in "Assignment: Earth"? – jim Jun 27 '17 at 19:29
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The slingshot effect is actually a form of performing time travel. From Memory Alpha:

The slingshot effect, also known as the light-speed breakaway factor, was a method of time travel through the use of an artificially-created time warp. This maneuver was 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 would then break away from the stellar body, creating a whiplash effect which could transport the vessel through time. Performing this maneuver required extremely precise calculations to be made, such as availability of fuel components, acceleration, and mass of a vessel through a time continuum. (TOS: "Tomorrow is Yesterday"; Star Trek IV: The Voyage Home)

The only reason they performed time travel was so that they could prevent the destruction of Earth in the future. I would assume that time travel is most likely banned because doing so would cause alterations to the universe, for instance in the New Star Trek movies it caused a second divergent timeline to emerge and caused the original Spock to be stuck there.

Anyway Spock actually did most of the computations for the time jump from memory. From the Memory Alpha page for Star Trek IV: The Voyage Home:

On Earth, a faint transmission believed to be from Admiral Kirk is received and Cartwright orders it put through. Kirk advises Starfleet of their analysis of the probe's signal, tells them that Spock's theory is that only the extinct humpback whale can properly answer the probe and because of this, they are going to try time travel and they are computing their trajectory at the same time. At that moment, Kirk's signal degrades. Cartwright orders the transmission picked back up, but just then the windows behind him shatter and the wind and rain begin to blow into Starfleet Headquarters. At this point, all anyone in the command center can do is wait.

On the Bounty, Spock has completed his calculations and informs Kirk their time target is the late 20th century. Unfortunately he can't be more precise because of the limits of the equipment aboard the Bounty. Additionally he had to program some of the variables for his time travel computations from memory. When McCoy worriedly recites a line from Hamlet and Spock recognizes it, Kirk establishes his faith in Spock's memory and has the ship prepared for warp speed. Kirk orders Chekov to raise the shields and then tells Sulu to engage the Bounty's warp drive.

There's no reason to assume that the Federation CAN'T time travel, but are most likely not willing to do so in fear of changing events. The events in the movie is a rare occasion where they would be willing to allow time travel as an option.

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    If you’re going to quote chunks of Memory Alpha, you should acknowledge it and include a link in your answer. Downvote from me for lack of referencing. scifi.stackexchange.com/help/referencing – alexwlchan Apr 24 '14 at 21:57
  • Agree with @alexwlchan - Editing out the irrelevant bits would have made it a much better answer. – Valorum Apr 24 '14 at 22:15
  • Most people tend to like bigger quotes instead of small snidbits of information. – DoctorWho22 Apr 25 '14 at 2:11
  • Nobody wants to read a whole page of that. – Josh Schwarzzeskywalker Mar 31 '15 at 22:51
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Deep Space 9 had a time travel episode in Trials and Tribblations where they visited the TOS Trouble with Tribbles episode. It was an absolutely hilarious episode but they showed the Bureau of Temporal Anomalies interviewing Sisko and cautioning him about interfering with the timelines. So they can travel through time but it is frowned upon.

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