Fixed point in time

The Doctor appearing to die at Lake Silencio, on 22 April 2011, was a notable fixed point in time. (However, this is actually the Teselecta being attacked, not the Doctor) (DW: The Wedding of River Song)

Fixed points in time were moments in the space-time continuum at which events were set in stone and could never, ever be changed, no matter what. The Time Lords knew which points in time were "fixed", which the Ninth Doctor said was a maddening experience to have to go through. (DW: The Parting of the Ways) This was called 'the burden of the Time Lords' by the Tenth Doctor, who was trying to explain to Donna why he couldn't prevent Mount Vesuvius from erupting or save the people of Pompeii. (DW: The Fires of Pompeii)

Definition

Fixed points were events and/or individuals who had such long-standing impacts on the timeline that no one, not even Time Lords, dared interfere with their natural progression. The Doctor, free to interfere in alien invasions and save planets in most cases, could neither interfere nor interact with these fixed points, out of fear of damaging reality. Were a fixed point to be interfered with, time would often find a way to make the timeline continue with minimal changes.

For example when the Doctor saved Adelaide Brooke and two of her crew, Brooke committed suicide to preserve the timeline with minimal changes; the events that occured on Bowie Base One were revealed to the public by the surviving members of the crew with with Adelaide being hailed as a hero for her heroics in stopping the viral menace, which did not happen in the original timeline. (DW: The Waters of Mars)

Another example is that the Doctor was fated to die at Lake Silencio, meaning he HAD to die there. When Gantok prepared to kill him in revenge for being beaten at Live Chess, the Doctor was, much to his horror, saved from being shot by his non-destined killer by a trap hole that fed Gantok to the ravenous skulls of the Headless Monks. (DW:The Wedding of River Song)

However if one actually broke a fixed point in time, as when River Song refused to kill the Doctor, time would freeze and collapse; reality would "die". It should be noted that River is the only known person to change a fixed point to such a degree that the whole of reality was put in danger.

With a "still point in time", such as Lake Silencio, it was easier to create a fixed point in time.(DW: The Wedding of River Song) Where or when others are in time are currently unknown.

Notable fixed points

Implied to be a fixed point was the death of Pete Tyler on 7 November 1987. Rose Tyler saved his life and the paradox allowed the invasion of Reapers. The timeline returned to normal with the Reapers gone, when Pete realised what had happened and ran out in front of the car that should have killed him. The only thing changed in the new timeline was where he died, that the driver of the car which struck him stopped and someone (Rose) was with him when he died. (DW: Father's Day)

The destruction of Ockora in 2204 was a fixed point. When the Second Doctor warned the Selachian Supreme Leader, he realised they could have done something about it. The Supreme Leader and forty Selachian soldiers escaped the planet's destruction and boarded the Triumph, the ship which dropped the G-bomb which destroyed Ockora. They killed everyone on the ship, but the other G-bomb was accidentally set off, leading to the formation of another black hole. The Doctor considered the formation of another black hole an acceptably small change. (PDA: The Final Sanction)

Jack Harkness became a fixed point in time after his resurrection by Rose Tyler, which caused him to become immortal. (DW: The Parting of the Ways, Utopia) Jack temporarily lost his immortality due to Miracle Day, but Rex Matheson and he became immortal afterwards. (TW: The New World, The Blood Line)

Whether they both became fixed points is unknown.

The destruction of Pompeii by the Vesuvius volcano was a fixed point in time caused by the Doctor. As well as its eruption, the deaths of most of the citizens of Pompeii were fixed. The Doctor did save one family, despite his unwillingness to interfere further. (DW: The Fires of Pompeii)

The death of explorer Adelaide Brooke was one of the few times the Doctor intentionally interfered with a fixed point. His rationale was that, as the last surviving Time Lord, the Laws of Time were his to command. In the end, Brooke committed suicide, allowing the timeline to unfold with only minor changes. (DW: The Waters of Mars)

The Doctor also interfered with a fixed point in time shortly before or after the previous event, by saving the life of Emily Winter, a film actress in 1920s-era Hollywood. He was put on trial by the Shadow Proclamation for this. (IDW: Fugitive)

The Doctor's apparent death in his eleventh incarnation, was a fixed point in time arranged by The Silence. The Doctor was seemingly killed in Utah, at Lake Silencio, on 22 April 2011 at 5:02 pm. When River Song tried to prevent this, an alternate timeline was created where all of time occurred at the same time and it was always 22 April 2011, at 5:02 pm. The Doctor set things right by kissing River, shorting out the time differential between them and making events revert to the moment when she was supposed to kill him. It was later shown that the fixed point was actually not his death, and the Doctor who had "died" was actually the Teselecta - with the Doctor himself safely inside. (DW: The Impossible Astronaut, The Wedding of River Song)

The death of one of his companions, Adric, was tied to the extinction of the dinosaurs. When the Doctor's other companions urged him to use the TARDIS to rescue Adric, he refused. His explanation implied that this event was a fixed point.

Time in flux

The opposite of fixed points was time in flux. At these points time could change completely.

Flux points were relatively insignificant (on a universal scale) events that could be altered with relatively little to no consequence. The Doctor often meddled at these moments. (DW: The Christmas Invasion, QR: I am a Dalek) When the Tenth Doctor first met Martha Jones, he told her that "Crossing into established events is strictly forbidden ... except for cheap tricks." (DW: Smith and Jones)

Fluxing points

• 1969 was in flux, as discovered by the First Doctor when it was revealed that WOTAN was supposed to conquer Earth (PDA: The Time Travellers) instead of being defeated by the Doctor. (DW: The War Machines)
• Sarah Jane Smith was shown an alternative timeline where the Earth of 1980 was a barren wasteland by the Fourth Doctor, who explained 1911 was in flux because of the threat of Sutekh, noting that history could be altered on that kind of scale when dealing with a being of Sutekh's immense power. (DW: Pyramids of Mars)
• The Doctor explained to Rose Tyler when she said that he couldn't give dead human corpses to the Gelth, as she knew for a fact that dead bodies weren't walking around in 1869: time was in flux, changing every second and her "cosy little world can be rewritten like that". (DW: The Unquiet Dead)
• Although the Doctor told Rose that Harriet Jones would serve three successive terms as British Prime Minister, he himself would, in his next incarnation, cause her political downfall, leading to her losing the title of Prime Minister to Harold Saxon. (DW: The Sound of Drums) She would die the following year at the hands of the Daleks during their invasion of Earth. (DW: The Stolen Earth)
• When Edward VII, along with Balmoral Castle, vanished into thin air, the Doctor explained that with him gone, the whole future of the royal family was threatened and there would be no George V, George VI, Elizabeth II, Charles III and Camilla, William V and so on. (QR: Revenge of the Judoon)
• When the Doctor and Donna were chasing Reverend Golightly to save Agatha Christie, the Doctor explained that Agatha could die and most of the books she wrote would disappear. (DW: The Unicorn and the Wasp)
• After they saw Frank Openshaw exterminated by a Dalek, Rose Tyler and the Doctor meddled in history for him so that he met his wife Sandra years earlier than in the original timeline. (QR: I am a Dalek)
• In 2020, when a drilling operation in Cwmtaff disturbed a Silurian civilisation, the Eleventh Doctor told Amy Pond, Nasreen Chaudhry and Eldane that this encounter could lead to either a peaceful relationship or a devastating war. The Doctor called the event an opportunity. (DW: Cold Blood)
• When the Dalek Emperor and his flagship survived the end of the Time war in 199,909, they secretly altered the history of the Fourth Great and Bountiful Human Empire, controlling it via the Jagrafess. While the Ninth Doctor remembered it as the human race at its height, humanity's development was instead stunted. (DW: The Long Game, Bad Wolf)
• At one point, the Korven altered the history of Earth (which they had previously invaded in 2480), causing part of it to be controlled by the corrupt, totalitarian Department. (K9TV: The Eclipse of the Korven)