Tippett and Tsang's TARDIS: A Mathematical Model of a Time Machine

Can the laws of physics allow a real time machine? In 2017, two physicists published a paper describing, in the language of Einstein's general relativity, a "bubble" of curved space-time that could carry its passengers backwards and forwards in time. They nicknamed it the TARDIS, after the time-travelling police box in Doctor Who.

The short answer to the question everyone asks: no, it cannot be built. The model only works on paper, and it needs a kind of "exotic matter" with negative energy that nobody has ever found. But the paper is a serious and interesting piece of theoretical physics, and it shows why time travel keeps appearing in respectable science journals.

Who proposed it, and where was it published?

The authors are Ben Tippett, a mathematician and physicist at the University of British Columbia's Okanagan campus in Canada, and David Tsang, an astrophysicist who was then at the University of Maryland. Their paper, "Traversable acausal retrograde domains in spacetime", appeared in the peer-reviewed journal Classical and Quantum Gravity in March 2017. The initials of the title spell TARDIS; an earlier preprint version used "achronal" in place of "acausal."

The authors say their aim was to describe the geometry of something a layperson would recognise as a time machine: a box that moves back and forth through time and space as seen by an outside observer.

The physics behind it: curved space-time

To follow the idea, it helps to start with Einstein. General relativity treats the three dimensions of space and the one dimension of time as a single four-dimensional fabric, space-time. Mass and energy curve that fabric, and what we feel as gravity is simply objects following the curves.

That is why Earth orbits the Sun. In flat space-time a planet would travel in a straight line; near a massive star, the geometry is bent, and the planet's straightest possible path becomes a loop.

Crucially, it is not only space that bends. Time is part of the same fabric, so it can be stretched and warped as well. Clocks really do run slightly slower closer to Earth than in orbit, which is why GPS satellites must correct for relativity. Tippett and Tsang asked what would happen if time could be bent much more dramatically, into a closed loop.

How the TARDIS bubble would work

In physics jargon, a path through space-time that loops back to its own past is called a closed timelike curve. The TARDIS model builds one on purpose.

• Imagine a bubble of specially shaped space-time, with ordinary flat space-time outside it.
• The bubble moves along a large circular path through space and time.
• From the outside, it first moves slower than light, then faster than light to "turn around" in time, and finally slower again, completing the loop.
• Passengers inside feel only a steady, modest acceleration. Following that circle, their journey closes on itself, so they can arrive back before they left.

The diagram below, from the paper, shows the idea. Person A rides inside the bubble; person B watches from outside. The black arrows show the direction of time for each. For B, time runs forwards as usual. For A, the bubble's path means the arrow of time sweeps round, so the two observers experience the journey in completely different orders.

B. K. Tippett et. al.

Tippett put the core claim plainly to the Georgia Straight: "mathematically, it is possible."

The catch: exotic matter and other problems

The paper is honest about the obstacles, and they are large.

• Exotic matter. The walls of the bubble can only be made from material that violates the normal "energy conditions" of physics, effectively matter with negative energy density. In Tippett's words, "we need materials, which we call exotic matter, to bend space-time in these impossible ways." No such material has been found in usable form.
• Faster-than-light motion. To turn around in time, the bubble must move faster than light relative to the outside world. General relativity allows the geometry on paper, but nothing known in nature does it.
• Mathematical trouble spots. The authors note that their space-time is geodesically incomplete and contains naked singularities, points where the equations break down, and that it makes the surrounding space-time non-orientable. All of these are warning signs that the solution may not describe a physically realistic universe.

So the honest summary is that the TARDIS is a precise description of what a time machine would have to look like if general relativity were the whole story and exotic matter existed, not a blueprint.

How it compares with the Alcubierre warp drive

The model is often compared with the Alcubierre drive, proposed by physicist Miguel Alcubierre in 1994. That design is a warp drive, not a time machine: it would contract space in front of a ship and expand it behind, letting a "warp bubble" cross distances faster than light while the ship inside never locally breaks the light-speed limit.

The two ideas share a lot. Both use a bubble that separates an inner region from ordinary space-time, and both require negative energy. The key difference, as Tippett and Tsang point out, is that their bubble's path is closed in space-time, so it returns to its own past by design. Later physicists have shown that Alcubierre-style warp bubbles could also, in principle, be used to create time loops, which is one reason faster-than-light travel and time travel are so closely linked in relativity.

Will we ever travel back in time?

Most physicists doubt it. Stephen Hawking proposed a "chronology protection conjecture," the idea that the laws of physics, probably through quantum effects, will always prevent closed time loops from forming. It has never been proven, but it reflects a widespread intuition that nature forbids the paradoxes that backward time travel would cause, such as changing an event that led to your own trip.

What is certain is that time travel to the future is real and routine: astronauts on the International Space Station age very slightly less than people on the ground, a measurable effect of relativity. Travelling into the past is the hard part.

Common myths about this study

• "Physicists have built a time machine." No. They described a mathematical model. No device exists.
• "It proves time travel is possible." It shows that general relativity's equations permit such a geometry, if exotic matter exists. Whether reality allows it is an open question.
• "It is just science fiction." Also not quite. Closed timelike curves are a legitimate topic in gravitational physics, and studying them helps test where Einstein's theory needs to be combined with quantum mechanics.

Even if no one ever steps inside a TARDIS, papers like this one sharpen our understanding of space, time and causality, and that is reason enough for physicists to keep asking the question.

Sources: Tippett B.K. & Tsang D., Classical and Quantum Gravity 34 (2017) 095006; University of Bath research portal (paper record); arXiv preprint 1310.7985; Georgia Straight (April 2017); ScienceAlert; Wikipedia: Alcubierre drive. Diagram: B. K. Tippett et al.

Related:

• Stephen Hawking's Universe - The Story of Everything, Time Travel, Aliens