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8 min read

— Prof. River Song

This post contains spoilers for the recent episode of Doctor Who, ‘World Enough and Time’. (Although the events of the episode have been trailed in such detail that everything is obvious and predictable, and there are no surprises left in the episode.)

General relativity and gravitational time dilation in the Doctor Who episode ‘World Enough and Time’

Although Doctor Who is ostensibly a sci-fi programme about time travel, it seldom gives any insight into the nature of time itself. Time, rather, is a series of arbitrary rules which the writers make up or violate at will to create tension, resolve the plot or explain away contrivances. The most memorable comment on the nature of time was that ‘actually from a non-linear, non-subjective viewpoint, it’s more of a big ball of wibbly-wobbly… timey-wimey… stuff’. This is from ‘Blink’, written by Steven Moffat and starring David Tennant as the Tenth Doctor. The phrase has become somewhat of a running gag in the show because of its vagueness and its ability to explain away almost any situation. But last evening’s episode, ‘World Enough and Time’, also written by Moffat, perhaps addresses some of the licenses the show took previously with the notion of time and this flippant characterisation of it, and it offers a more accurate picture of time drawing from physics.

The Twelfth Doctor, played by Peter Capaldi nearing the end of his tenure, finds himself on board a strange spaceship on a colonising mission. It is four hundred miles long and carried a crew of fifty people with the purpose of setting up a colony on a new planet. Something went wrong and the ship veered close to a black hole. The ship began steering itself away slowly from the black hole, and it spent two days fighting its gravity. In the process, some members of the crew went down to the lower decks to find the engines and reverse its thrusters, only to disappear and never return. In their place, a group of grotesque, humanoid creatures which looked like hospital patients emerged from the lifts and began abducting all human members of the crew. Forty-nine members of the crew were taken, leaving only one humanoid figure who was the janitor. As the Doctor and his companions start investigating, they see that while the rest of the crew is presumed dead, the computers on the command deck show a population of several million life forms in the lower levels, possibly the invading creatures. However, the ship had no contact with any species outside of the crew so it was inconceivable that another race would have boarded the ship. The identity of those creatures, then, is a mystery.

The solution to this problem is in general relativity, specifically the gravitational dilation of time. The ship is several hundred miles long, and it is facing in the direction of the black hole. As the upper decks are closer to the centre than the lower decks, they experience a stronger gravitational influence. The Doctor explains that gravity slows down time. Because of this, time on the upper decks is slower by several orders of magnitude than time on the lower floors. When seconds or minutes pass on the command deck, days and months pass down below. These are obvious similarities to the film Interstellar. As two days had passed on the command deck, a thousand years had passed down below. The mysterious creatures did not board the ship from outside, but were descendants of the original crew, several generations later. As it would later transpire, these descendants converted themselves to Mondasian Cybermen after the intervening millennium saw their civilisation on the brink of collapse, and they were coming up to the command deck to capture any human survivors and turn them into cybermen.

This was a compelling set-up for what was a rather enjoyable episode, but the episode was let down by the usual Steven Moffat pitfalls (a vain obsession with its own hype and cleverness, glib breaking of the fourth wall, and contempt for its viewers). Moreover, everything in the episode was trailed with such detail (John Simms returning as the Master revealed in the news, Mondasian Cybermen making a return in the preview) it was obvious down to the very last twist what was going to happen. Still, it was nice to see Doctor Who trying to come to grips with an understanding of time which wasn’t just some vague joke about how time is weird and therefore allows all kinds of plot contrivances.

Despite its silliness, the suggestion that time was ‘wibbly-wobbly, timey-wimey’ was quite revealing. There are many results in physics and in the philosophy of time that seem to suggest that time is in fact not a linear progression from cause to effect. Time can dilate or contract depending on the relative velocity to that of light, and objects can travel into the future: this is already commonplace and unsurprising. There can be weird local events like causal loops and bootstrap paradoxes (which have been quite widely abused by Doctor Who screenwriters, my favourite being in the Comic Relief special ‘The Crash of Time’). There can be even weirder phenomena like backwards causation. Moreover, the distortion of space-time geodesics by bodies of high mass can bend the timeline of objects in a manner that they can loop back in upon themselves, thereby travelling back in time. Time is rather messy and ‘non-linear’.

Then there is the question of whether time is ‘non-subjective’: this is dubitable, and would depend on how one would define ‘subjective’. To begin with, the theories of relativity emerged out of a broader intellectual and philosophical context that situated the human subject — either the consciousness or the observer — within the aspect of nature being observed. Freudian psychoanalysis postulated consciousness and the subconscious as entities that could be studied. Saussurian linguistics was concerned with how language mediates one’s understanding of reality. Developments in quantum mechanics later on would further suggest that the properties of physical systems are dependent upon being observed. In the context of time and relativity, the fact that time is relative to an observer’s frame of reference situates the observer within the measurement of time, thereby making it ‘subjective’. Time dilates or contracts depending on the velocity of the body relative to the observer.

In contrast, one could also suggest that the observation of the dilation or contraction of time requires one to come outside of one’s subjectivity. The difference in the definition of ‘subjective’ is a difference in observation and experience. The experience of time within a frame of reference is constant, even if time itself dilates: a body travelling at a velocity near that of light will not be able to tell that time is dilating in its frame of reference. It is only from outside this frame of reference that the dilation of time can be observed. This needs an observer to think of time from outside their subjective experience of it, and is arguably ‘non-subjective’. I am not particularly convinced by this, and I would prefer to think of time as subjective. But I can allow for an interpretation of the experience of time that can service the Tenth Doctor’s characterisation of it.

With the Twelfth Doctor commenting on gravitational time dilation, however, there is a more basic problem: it is vaguely correct: gravity does slow down time, so to speak. But what the Doctor presents is an oversimplification. By claiming that ‘gravity slows down time’, and by describing a black hole as ‘superman gravity’ the Doctor characterises gravity as a force. One of the fundamental principles of Einsteinian relativity is that gravity is not a force, but a curvature of space-time caused by matter along which different bodies move. Gravitational attraction is the movement of a body along the geodesics of a bent space-time — a bend that causes the straight-line path of an object to correspondingly bend towards another body — rather than a force exerted by a body that pulls another object towards itself. So it is not so much gravity that bends time, as gravity is bent space-time, bent instead by matter (in this case the enormous and condensed mass of a black hole).

Unfortunately, none of this mattered very much for the episode: the relativistic time dilation had very little connection to the events of the plot, and it only served to imply the length of time Bill had been held in captivity, waiting in vain for the Doctor to rescue her. The Doctor never addressed or solved the problem to the plot that this caused. The story could have taken place without the time dilation premise and it would not have lost anything. It is both ironic and disappointing that the most interesting aspect of the episode was also tangential to the story.

Further Reading:

Einstein’s Geometric Gravity on the Einstein Online portal by the Max Planck Institute. You know this is a great resource when you see the names of so many physicists in the title. This web site should give thorough gloss on General Relativity, enough to make sense of Doctor Who at any rate.
Backwards Causation on the Stanford Encyclopaedia of Philosophy. Think of the SEP as high-brow Wikipedia. It’s a standard resource for reference or a quick introduction to topics in Philosophy, rigorously edited and peer-reviewed. The article on backwards causation should illuminate just how some strange and non-linear time can be.
The Young Centre of the Earth, an article by U.I. Uggerhøj, et al. in the European Journal of Physics. This is a further article to illustrate the differential in time at regions with different gravitation potentials, in particular refuting a calculation made by Richard Feynman about the relative ages of the core and the surface of the Earth.
General Relativity in the New Scientist Instant Expert series. Excellent beginner-friendly resource that I used to brush up on all this. Has great illustrations as well.

Acknowledgement: I’d like to thank Dr. Alasdair Richmond for teaching me everything I know about time travel, and for being perhaps the only Personal Tutor to surpass the Doctor himself.