Characters in "Doctor Who" are always amazed by its unassuming appearance (it looks like a blue police box from the 1960s) and that it's much bigger on the inside than it is on the outside.
But what we really want to know is how does it work, and when can we have one?
Physicists Ben Tippett and Dave Tsang actually wrote a paper about how TARDIS time travel might be possible. Tippet and Tsang propose that the TARDIS moves as a bubble of space-time back and forth along a loop of time.
If you connect a bunch of those loops, then the TARDIS could move to any point in space and time, just like it does in the TV show.
The paper is called "Traversable Achronal Retrograde Domains in Spacetime." Yes, the acronym spells out "TARDIS," and yes, they named their proposed space-time bubble that on purpose.
Tsang and Tippet also claim they work at the Gallifrey Polytechnic Institute and Gallifrey Institute of Technology (Doctor Who is from a place called Gallifrey). But in real life they're physics professors at McGill University and the University of British Columbia respectively. Basically, they are awesome Doctor Who fanboys.
How would it work?
Einstein's theory of general relativity tells us that space and time are not separate - they're wrapped up in four dimensions. There are three dimensions of space (up-down, left-right, and forward-backward) and the dimension of time (future-past). Together they combine to create the fabric of space-time in which all the matter in the universe exists.
Massive objects, like stars and galaxies, stretch and curve this fabric into themselves. Physicists don't really know how space-time warps, but it's theoretically possible to fold one of those curves back on itself, creating what's called a closed time-like curve (CTC).
It's basically a loop:
The objects inside the loop in the graphic are called light cones. Light cones mark the boundaries of space-time that any one event (like the burst of light from a supernova explosion) can reach.
For example, imagine you're standing at the red dot in the diagram below. Time is on the y-axis (left) and space is on the x-axis (bottom):
The area enclosed by the white lines is everything you can see without traveling at the speed of light. If a star exploded 10 light-years away from you (orange dot), then it would take 10 years for light from that event to reach you.
The only way to get outside of a light cone is by traveling faster than the speed of light. Normally light cones are arranged in a straight line, because time moves in a straight line like the right side of the diagram below. But CTCs tip light cones, making it possible to travel backward and forward in time, like the left side of the diagram:
If the TARDIS/bubble of space-time entered into one of those loops, it would be possible to travel backward and forward through space and time.
It would look kind of like the following graphic from Tippet and Tsang's paper. (It's obviously the TARDIS and Amy Pond from seasons five, six, and seven.)
No world-saving space traveler would be very effective just traveling in a circle. So Tippet and Tsang outlined a mathematical formula to chop up different space-time curves and splice them back together; basically, a way to form tunnels that could transport you to any time and place.
The opening sequence of Doctor Who episodes shows the TARDIS moving through space-time in a motion that actually kind of mimics these proposed tunnels, minus the clouds and lightning:
In later episodes, where the Doctor travels into the future, the space-time vortex in the opening credits appears red instead of blue.
Fans speculate that's because light emitted by an object moving away from a viewer (perhaps into the future) shifts toward longer wavelengths. I.e. As the object moves away, the wavelengths of its light stretch out, making it appear more red:
In episodes where the Doctor goes to the past, the tunnel appears blue, since as the opposite happens: When and object moves toward us in space-time, the wavelength of light compresses toward the blue end of the spectrum.
When can I have a TARDIS?
You can't. At least until we figure out a way to create CTCs and invent a material that can repel gravity and somehow travel faster than the speed of light, then use it to build a blue telephone booth.
Time travel is tricky and full of risky paradoxes - like the grandfather paradox, where you accidentally stop your own birth. And again, physicists still aren't sure about how the geometry of space-time works. (We're also waiting for the scientific papers that explain why the TARDIS is bigger on the inside than it is on the outside and how it can instantaneously appear and disappear.)
The consensus seems to be that time travel into the future is possible, but time travel into the past is much trickier and may not be possible at all.
So for now maybe we should just follow the Doctor's lead and call it a "big ball of wibbly wobbly timey-wimey stuff" and just enjoy the show: