If you do not love Doctor Who, or love discussing the minutia of the program, you may wish to skip this post. You have been warned!
This post is largely to encapsulate two threads from Reddit: This one, and this one. While the idea that events from The Day of the Doctor might have influences the Doctor's abandoning Susan in The Dalek Invasion of Earth, speculating that the TARDIS telepathic fields may have something to do with how companions leave the program was notably less well received.
I for one prefer to look at the program from the standpoint that what is actually seen on the screen is true (within the context of the fictional universe). and then work from there what it means. I realize that some people prefer to work from what they want to believe is true, and then ignore contradictory data and/or complain about it being a mistake from the "obvious truth" of whatever they want to believe.
The Doctor and Susan
We know that thirteen Doctors (the 1st through the 12th plus the War Doctor) help to move Gallifrey to a pocket universe at the end of the Last Great Time War. We also know that Doctors earlier than 12 will forget the specifics of the time lock, because 12 is the senior Doctor present. When I talk about Time Lord memory, below, I will argue that 12 also does not recall what happened initially; only when that event occurs for his incarnation will he be able to recall it.
I am going to venture that the 1st Doctor became aware of the Time War during the end of The Dalek Invasion of Earth, and that he flew into his part of it shortly after (perhaps having put Barbara and Ian to sleep or out of the way in another part of the TARDIS), and before The Rescue. Thus, the 1st Doctor knows why he left Susan behind, but does not remember the specifics at the start of the next story.
The 1st Doctor then encounters a much older Susan in The Five Doctors. Although he is (again) doomed to forget the specifics of the adventure, we really do not know how long he travels with Susan between the end of this multi-Doctor story and the resumption of his original timeline. It should be noted that the Doctor encounters the older Susan after leaving her from Susan's point of view, but before leaving her from the Doctor's.
When he finally encounters the 12th Doctor in Twice Upon a Time, he no longer has any memory of his own later incarnations, which he has met at least twelve of.
Even so, I would posit that the Doctor leaves Susan both because (1) he has an intimation of the Time War, and (2) he knows that he will do so. He may have no direct memory of either when he leaves Susan, but he does have at least a partial intimation that guides his actions. Even his speech about returning may be prompted by a subconscious knowledge that he will encounter an older Susan - and that may be why the speech starts The Five Doctors!
Time, the TARDIS, and Parallel Universes
From a basic, four-dimensional standpoint, the universe is expanding. It started as a near-singularity, and will continue to expand until it reaches heat-death. At this point, from an outside vantage point, the universe will once again resemble a near-singularity, and any continued outward pressure will cause another big bang, resulting in a new universe.
The expansion of the universe, on the small scale, causes the phenomenon in quantum physics of particle-wave duality. The expansion is such that a particle spreads into a wave until acted on, where the energy from the interaction is enough to allow it to collapse back into a particle.
Although it is more complicated than that, this is largely why you remember the past but not the future, and why there is an arrow of time at all. Moving backwards requires not only the energy for the motion, but also the energy to deal with the expansion, and that requires more energy that you can obtain.
The TARDIS deals with this problem by side-stepping it. Rather than fight the expansion of the universe, the TARDIS slips into the Vortex, and then slips out at the desired time and place. Because the universe is expanding, though, the TARDIS has to compensate. Experienced subjectively from within, Planck's constant does not change. Objectively, from without, the constant is smaller in the denser past than it is in the expanded future. Eventually, the speed of light is also detectably affected; it was faster in the past during the early expansion and will slow to just over the Planck's constant just prior to the final heat death of the universe.
(This is not easy for the TARDIS to compensate for, which is why the Time Lords are forbidden to go too far forward or too far back. Also, the transduction barriers protecting Gallifrey probably break down at these extremes.)
BTW, that method of dealing with time, and the requirement to also match the "relative dimension" of the universe when the TARDIS materializes? It is part of the name, TARDIS. It is also seen to malfunction in Planet of the Giants.
Each of the potential variables where a particle may collapse from a wave creates a potential parallel universe. Most of these become "bubble universes" that collapse almost instantly. Some, like Pete's World, have more staying power. The wibbly-wobbly ball of timey-wimey stuff is a direct consequence of this. Free will is real in Doctor Who, and it selects which of those potential universes become the prime universe, which become full parallels, and which collapse. The past is no less changeable that the future. The expansion of the universe causes possibility to come into existence, and creates a 5th dimension, a kind of "temporal space" that Susan could not adequately describe to Ian.
For terrestrial Ian, the answer to the problem was simple and fixed. For Gallifreyan Susan, the answer to the problem was complex and variable.
The important point of this is that the skein of the current configuration of the prime universe hangs on certain events - fixed points - while everything else is in potential flux.
The TARDIS, too, understands this flux...better than the Doctor does if The Doctor's Wife is accepted at face value. The TARDIS understands the ramifications of the Doctor landing where they do. The TARDIS takes the Doctor where they need to go. The TARDIS also has an agenda, which it stole the Doctor to fulfil. The TARDIS also has access to the Doctor's most private thoughts (see The Time Monster). All of this has ramifications after the Time War.
The Transduction Barriers
Gallifrey is "oblique to time", largely due to the transduction barriers. These were created by Rassilon to set Gallifrey apart from the standard flow of time, protecting the past of the Time Lords from being tampered with.
Normally, it doesn't matter when a party approaches Gallifrey - it is always the present. To illustrate this better, imagine that you visited Gallifrey. Then you boarded your TARDIS, went 2 million years into the past, and took a starliner to visit Gallifrey again. Not only would it still be the "present" on Gallifrey, but the amount of time you spent in travelling is the same amount of time that has elapsed on the planet since your last visit. Simply put, the timeline for Gallifrey is independent of the timeline of the universe.
This is also true, in general, for those travelling in TARDISes. Thus, the 7th Doctor can know the Rani's age by knowing his own, and Romana can correct the 4th Doctor about his age. Other Time Lords - the Monk, the Rani, the Master - are always encountered sequentially, so that their timelines match up. This obliqueness to time is so strong that the 1st Doctor was able to visit the same event several times without encountering himself. Presumably, the Rani was able to avoid detection by the Doctor, although they had been at the same events, simply because their timelines were not in sync. Gallifreyan Standard Time (as it is sometimes called) has to sync up in both 4th-dimensional and 5th-dimensional time.
There are some exceptions to this, but they are rare: multi-Doctor stories and Clara managing to meet the child Doctor are the only ones that occur in the televised series.
The TARDIS operates in at least 7 dimensions, three of which are spatial and 4 of which are temporal. Standard operations only deal with 4 of these dimensions, but the time scoops used in the Death Zone (or to bring The Three Doctors together) operate using 5-dimensional engineering. The height of Gallifreyan technology, the Transduction Barrier around Gallifrey, using 6-dimensional engineering, which is why you might encounter a parallel like the Inferno world or Pete's World, but do not normally encounter parallel versions of the Time Lords or Gallifrey.
The Time Lord object to the experiments of Kartz and Reimer in The Two Doctors not merely because they object to others gaining time travel, but they object to others gaining time travel that can violate the transduction barriers. The Kartz-Reimer experiments clearly do this, allowing two versions of the same Time Lord to meet with potential repercussions that could destroy the universe.
Time Lord Memory
We know that the Doctor claims, on multiple occasions, that the human brain is small. However, the Time Lord brain appears to be the same size. It might be possible that part of the Time Lord's mind is augmented by the TARDIS....effectively, a Time Lord uses the telepathic circuits like a gigantic hard drive.
Another possibility is that a Time Lord's brain has more than four dimensions, and is therefore actually larger than the human brain. This possibility might explain how the Doctor (and other Time Lords) are able to perceive time differently than humans do. It also might explain why the Doctor apparently forgets most of what happens in multi-Doctor stories.
Your personal memories, proceeding one to another, follow a four-dimensional track. This would be true even if you were able to travel in time; your memories would still follow a line. So long as the evens occurred in the past of that four-dimensional track, you can remember them.
A Time Lord's mind has both a four-dimensional track (as does ours), and a five-dimensional track. If an event has occurred in the past of both of these tracks, the Time Lord can remember them relatively easily. It an event has occurred in the past of one of these tracks, but not the other, the differential makes it hard for the Time Lord to recall, even though they have lived through those events. This is similar, in a way, to how short-term memory and long-term memory work in humans.
Another result of this is that an effect can travel up the 5th-dimensional track without following the 4th-dimensional track. Proximity is important. This is why turning the 2nd Doctor into an androgum affected the 6th Doctor without changing every event of the Doctor's fourth-dimensional experience in-between.
The "senior Doctor" in the events that saved Gallifrey in The Day of the Doctor was the 12th Doctor. The War Doctor, 10th Doctor, and 11th Doctor therefore never know that Gallifrey was saved. This is also why we see the 12th Doctor initially not know where Gallifrey is; until the 5th dimensional differential is sorted out, he doesn't have clear memory of those events.
The War, 10th, and 11th Doctor then meet an unknown potential future Doctor played by Tom Baker (the Caretaker). If the Caretaker actually is a future Doctor, they will forget that event after it is finished.
This is also why Queen Elizabeth is so angry at the Doctor - his promise to return and marry her took place during the entanglement of 5th-dimensional memories, and he simply forgot that he had done so once it was over.
(The 10th and the 11th Doctors, by the way, offer perfect examples of how the post-Time War, pre-Restoration of Gallifrey Doctor views physical encounters very differently than any pre-Time War, or post-Restoration Doctor does. More on that to come.)
The Time War
One of the neat things about the Time War is that it probably started with the 4th Doctor's being interrogated by Davros and trying to alter Dalek evolution, but several other stories (both in later and earlier Classic Doctor Who) can be seen as portions of the Time War. This includes the idea that the Daleks would send a deceased Master back to Gallifrey in the TV Movie and multiple scenarios where Skaro and/or the Daleks appear to be defeated forever.
The Daleks not only have time travel, but they are one of the few species with time travel who are capable of being encountered out of sequence. I.e., Dalek time (unlike that of most of the universe) does not match Gallifreyan time. That might be a key to their surviving the Time War, but it also means that the Time War can be used to explain not only continuity differences between the Classic series and the New, but continuity differences in the Classic series itself. Once the Time War started (in 6th dimensional time) it had always been (in 4th and 5th dimensional time).
To both Time Lords and Daleks in the new series, the Time War is in their (relative, 5th dimensional) past. Only the fact that no Dalek in the new series believes the Time War is yet ongoing allows us to maintain with any confidence that the Daleks did not, in fact, gain a greater mastery of time than the Time Lords themselves.
TARDIS Reproduction, Time Lords, and Telepathic Circuits.
The TARDIS is designed for observation, and to allow its occupants to blend into societies without altering them. It is also designed to have six pilots. The basic argument here is that the TARDIS telepathic field suppresses romantic interest under normal circumstances. When that field is not operating in full force, it can cause over-compensation.
In the classic series, the "no hanky-panky in the TARDIS" rule meant that the Doctor's companions were not romantically attracted to him. In the new series, post-Time War, that early rule is out the window. Suddenly the Doctor is attracted to Rose, who is attracted to the Doctor. This continues with multiple companions until Gallifrey is restored, and the TARDIS no longer has a driving need to create new TARDISes. And it just stops. Completely.
The most obvious examples of this dynamic are in the TV Movie and in Human Nature/Family of Blood. In both cases, we see the Doctor severed from the telepathic circuits of the TARDIS, develop romantic feelings, and then see those romantic feelings simply disappear when the connection is restored.
Another clear example: Travelling with the 3rd and 4th Doctors, Sarah Jane Smith clearly saw the relationship as friendly but not romantic. Post-Time-War, when Sarah Jane Smith encounters the 10th Doctor, her memory is altered to the point where she believes that there really was a romantic attachment, and that she could not simply get on with her life. Both K-9 and Company and her appearance in The Five Doctors demonstrate that this was not true. She was definitely getting on with her life.
TARDISes are symbiotic with Time Lords. For most of the new series, the Doctor is not only the last Time Lord, but the TARDIS can be assumed to be the last TARDIS. No new Time Lords, no new TARDISes. The TARDIS has a vested interest in nudging the Doctor and his companions together.
(For that matter, it is relatively clear that the TARDIS telepathic circuits were used to push Donna Noble into creating the Metacrisis Doctor in Journey's End! The TARDIS also locked the doors, preventing Donna from leaving, in order to do so.)
Enter River Song
We can also talk about River Song, who is conceived in the TARDIS, becomes a Time Lord, and actually marries the Doctor.
Remembering that, in The Doctor's Wife, it is clear that the TARDIS remembers the future and takes the Doctor where he needs to go, but is also apparently not bound by predestination. The TARDIS knows that Ganger-Amy is a Ganger. The TARDIS knows that Ganger-Melody is a Ganger. The TARDIS knows that the Doctor and River will have romantic meetings and spend a great deal of time off-screen together. There is, in fact, no reason to assume that they do not have children at some point. Say, a 24-year long evening on Darillium? Or during the hundreds of years the 11th Doctor spends off-screen?
We have already talked about how the future is not predetermined for the TARDIS, but exists in potentialities. If you accept that, you can also understand that the TARDIS can select those potentialities to some degree. The creation of River Song is not just some random thing that happens; the TARDIS is a part of it. And the TARDIS then goes on to have a fantastic relationship with River, allowing her to borrow the TARDIS whenever she needs to, and making sure that the TARDIS is there to rescue her when required.
Remember, the TARDIS takes the Doctor where they need to go.
(We can also assume that the pre-Hartnell/Timeless Child used the same TARDIS, which is why it became stuck as a police box again as soon as was feasible. In this case, the 1st Doctor almost steals the wrong TARDIS, but Clara puts him back on track without really realizing why that particular TARDIS was so important.)
The TARDIS remembers the future, but most of the future exists in potentiality (no matter where you are on the timeline). The 8th Doctor says, "The universe hangs by such a delicate thread of coincidences, that it would be useless to meddle with it, unless like me you’re a Time Lord.” Even moreso than the Doctor, the TARDIS has the ability to meddle with the future.
We are to believe that the 7th Doctor can manipulate Ace all he wants, and manipulate Davros into destroying Skaro, but the TARDIS doesn't manipulate its passengers? Even when we watch events unfold, repeatedly, which are otherwise inexplicable?
More On Those Telepathic Circuits
We know that the TARDIS telepathic field can:
- Make you think you hear your native language.
- Make you think the speaker's lips sync to that language.
- Make you think written text is in that language.
- Make you not notice the above. (This is a plot point in Masque of Mandragora.)
- Affect the emotions of those it is in contact with. (This is a plot point in The Edge of Destruction, and includes feelings of paranoia strong enough to make Susan attack/threaten Barbara with scissors.)
We can further extrapolate from decades of the program that:
- The telepathic field includes perceptual filters that encourage people to ignore it when it materializes....this is discussed a bit in the first episode of Torchwood, where the effect is partially transferred to a paving stone due to the energy discharge in Boom Town as well as being transferred to Martha, the Doctor, and Jack in The Sound of Drums.
- The language translation may not always be perfect, due to the limitations of the people being spoken to. What sounds like bafflegab to us might actually make sense if we had the Doctor's knowledge. Similarly, when Susan said the 5th dimension was "space", that was the closest the TARDIS could translate to a concept that Ian would understand.
- This alteration of appearance may go even farther. Linx, in The Time Warrior, was well aware that the Doctor was not human, and responded to his description that the Doctor would look that way "to your eyes". The Doctor never suggests that a companion "Looks Time Lord" before the Time War. It may be that differences he is normally aware of are being "smoothed out" by the TARDIS telepathic circuits.
For that matter, the racism of the past is more honestly depicted in Rosa than in Thin Ice. it is a serious question how much the TARDIS telepathic circuits play with the perceptions of the TARDIS crew and the people who encounter them. History may indeed be a white wash, and we do see signs of this in other stories (The Girl in the Fireplace comes to mind), but it also may be that the TARDIS really likes Bill and wants her to feel comfortable.
ROSE: They all speak English.
DOCTOR: No, you just hear English. It's a gift of the Tardis. The telepathic field, gets inside your brain and translates.
ROSE: It's inside my brain?
DOCTOR: Well, in a good way.
ROSE: Your machine gets inside my head. It gets inside and it changes my mind, and you didn't even ask?
DOCTOR: I didn't think about it like that.
Perhaps you also didn't think about it like that either?
When a companion leaves the TARDIS, the connection to the telepathic field is reduced (in some cases, probably, eliminated). The companion, having had their romantic inclinations suppressed, may feel a sudden upsurge in those feelings, over-compensating by falling for potential mates in ways which would, barring this explanation, be inexplicable.
Likewise, following the Time War, the TARDIS telepathic circuits were used to encourage reproduction, and led to the successful creation of another Time Lord in the person of River Song. When Gallifrey was restored, this effect ended immediately.
Obviously, this is not "official canon". Yet.
Thanks for entertaining these rambling thoughts!