Traversable Achronal Retrograde Domains In Spacetime is a new paper by
Caltech/Memorial Gallifrey physicists Benjamin K. Tippett and David Tsang that attempts to describe the spacetime through which Doctor Who's Tardis travels; one that "goes forward and back in time, and left and right in space." It's a bit heavy going, so they've also published The Blue Box White Paper, a lay-friendly, 17 page summary for people with "no technical knowledge of Einstein's Theory of General Relativity." The discussion continues on Tippet's Tumblr.
A person travelling within the TARDIS would describe it as a room which is constantly accelerating
forwards. Furthermore, any events which occur inside of the TARDIS bubble must satisfy Novikov's self
consistency condition11 .
A person outside the bubble would describe an entierly different scene (see Fig. 10). Due to its closed
trajectory, there will be a time before the bubble (and its contents) exists. Abruptly, we would see a bubble
appear and split into two, the two boxes moving away from one another. Initially, they will be moving at
superluminal speeds, but they will decelerate to a stand-still. The pair of boxes will then begin accelerating
towards one another until they exceed the speed of light, merge and disappear. Mysteriously, the contents
of one bubble will be appear to evolve forwards in time, while the other will evolve backwards.
To illustrate how time is experienced inside and outside of the bubble, imagine that there are two people
in our spacetime: Amy, who is travelling inside the bubble; and Barbara, who has been left behind. Suppose
that the two women are holding large clocks, and that the walls of the bubble are transparent, so that the
two women can see one another (See Fig. 11).