So I've realized that with all the posts I've done on Doctor Who, I never actually looked at the theory behind how the T.A.R.D.I.S. can actually travel through time. There have actually been studies into how a time-travelling space might work in this universe and the findings have led physicists to believe it is theoretically possible for a T.A.R.D.I.S. to exist and to function as it does on the show in our universe.
The research paper is called Traversable Achronal Retrograde Domains In Spacetime (see what they did there?) and it was written by a pair of physicists named Ben Tippet and David Tsang. Tippet and Tsang proposed a spacetime geometry in which retrograde time travel, travelling back and forth along one's own timeline, is possible. A spacetime geometry is when spacetime- the fabric of the universe (where everything has happened, happens, and will happen in the future)- is arranged in a certain pattern. This is a fairly complex topic; however to put it simply, space exists of three dimensions (X,Y,Z) and time creates a fourth dimension. There are many theories as to how this spacetime geometry might be structured- the most famous being the Euclidean and Minkowski space.
In order for retrograde time travel to be possible, spacetime geometry must be structured in a way such that the time dimension curves around and back in on itself. This spacetime structure- called closed timelike curve or CTC- would theoretically allow one to hop from their current space and time to another space and time (i.e. the idea of the "time votex"- basically a wormhole- that the T.A.R.D.I.S. flies through). Essentially the T.A.R.D.I.S. would create its own sort of bubble containing a closed timelike curve, which it would use to travel through time and space.
So that's essentially what the universe would have to look like for the T.A.R.D.I.S. to exist. Given the vastness of the universe and how long it has and will exist, there actually is a possible these conditions already exist somewhere and that a race like the Timelords already exists out there too.
So that's about it, thanks for taking the time to read this. If you're interesting in learn more you can find Traversal Achronal Retrograde Domains In Timespace by Tippet and Tsang or you can check The Blue Box White Paper, which takes a more basic approach to the concept; both of these papers should be online if you search for the title.