This method of time travel is the most crude. While traveling at warp speed a starship can enter a curved trajectory around a star or other massive stellar body with a large enough gravitational field. By specifically maneuvering the ship and the trajectory in precise, detail the vessel can actually accelerate beyond the infinite velocity of Warp 10 and achieve time travel.
While this method will work using any ship that has a sufficiently powerful warp drive is also the most inaccurate. There is no way to know exactly when the ship will emerge around the other side of the sun.
Potential Dangers
Time travel in this method can disrupt the systems of an entire starship potentially leaving the ship unable to break free from the gravitational pull of the stellar body in question. There is also a great potential for severe damage leaving a vessel unable to return. Some previous attempts have seen that dilithium crystals of poor quality and quickly degrade and break down after a slingshot is made, leaving a vessel without power. Shields, sensors, and weapons have also been known to go offline leaving the vessel defenseless and unable to maneuver. Instances where a vessel has found itself within the atmosphere of a planet without sufficient power to ascend have also occurred.
Notable Uses
In the 23rd century the USS Enterprise used this method to travel backward through time twice - once by accident where the ship was forced nearly into Earth's atmosphere the second time on purpose for historical research. During the same century Kirk and crew also used this method to travel back through time using a stolen Klingon vessel and abduct two whales from the 20th century so they could respond to a devastating alien probe destroying Earth in the 23rd century.
Each time, the crew and ship were disrupted, in the latter case the Klingon vessel was almost unable to return home because the dilithium crystals were nearly degraded beyond repair by the process.
Starfleet
Starfleet command has been known to use this method. However, the exact field equations developed originally by Captain Spock to return the Enterprise home are a closely guarded secret. The only other organization known to possess these complex formulas is Solas Tempus who has agreed with Starfleet that this information should remain a secret. They have also agreed not to use these formulas for a slingshot effect due to the volatile and risky nature of the process.
Solas Tempus
Solas Tempus, under General Order from the admiralty, has banned the use of this method without special clearance or in the special case of certain emergency situations.