No matter how many times he considered it, Jophiel shivered with awe. It was obviously an artefact, a made thing two light years in diameter. A ring around a supermassive black hole.

Stephen Baxter, Xeelee: Redemption (2018)

I’ve written about rotating space habitats in the past, and I’ve written about relativistic starships, so I guess it was almost inevitable I’d end up writing about the effect of relativity on space habitats that rotate really, really rapidly.

What inspired this post was my recent reading of Stephen Baxter’s novel Xeelee: Redemption. I’ve written about Baxter before—he specializes in huge vistas of space and time, exotic physics, and giant mysterious alien artefacts. This novel is part of his increasingly complicated Xeelee sequence, which I won’t even attempt to summarize for you. What intrigued me on this occasion was Baxter’s invocation of a relativistic ringworld, briefly described in the quotation above.

Ringworlds are science fiction’s big rotating space habitats, originally proposed by Larry Niven in his novel Ringworld (1970). Instead of spinning a structure a few tens of metres in diameter to produce centrifugal gravity, like the space station in the film 2001: A Space Odyssey, Niven imagined one that circled a star, with a radius comparable to Earth’s distance from the sun. Spin one of those so that it rotates once every nine days or so, and you have Earthlike centrifugal gravity on its inner, sun-facing surface.

If we stipulate that we want one Earth gravity (henceforth, 1g), then there are simple scaling laws to these things—the bigger they are, the longer it takes for them to rotate, but the faster the structure moves. The 11-metre diameter centrifuge in 2001: A Space Odyssey would have needed to rotate 13 times a minute, with a rim speed of 7m/s, to generate 1g.

Estimates vary for the “real” size of the space station in the same movie, but if we take the diameter of “300 yards” from Arthur C. Clarke’s novel, it would need to rotate once every 23.5 seconds, with a rim speed of 37m/s.

Niven’s Ringworld takes nine days to revolve, but has a rim speed of over a 1000 kilometres per second.

You get the picture. For any given level of centrifugal gravity, the rotation period and the rotation speed both vary with the square root of the radius.

So what Baxter noticed is that if you make a ringworld with a radius of one light-year, and rotate it with a rim speed equal to the speed of light, it will produce a radial acceleration of 1g.* In a sense, he pushed the ringworld concept to its extreme conclusion^†, since nothing can move faster than light. Indeed, nothing can move at the speed of light—so Baxter’s ring is just a hair slower. By my estimate, from figures given in the novel, the lowest “deck” of his complicated ringworld is moving at 99.999999999998% of light speed (that’s thirteen nines).

And this truly fabulous velocity is to a large extent the point. Clocks moving at close to the speed of light run slow, when checked by a stationary observer. This effect becomes more extreme with increasing velocity. The usual symbol for velocity when given as a fraction of the speed of light is β (beta), and from beta we can calculate the time dilation factor γ (gamma):

Here’s a graph of how gamma behaves with increasing beta—it hangs about very close to one for a long time, and then starts to rocket towards infinity as velocity approaches lightspeed (beta approaches one).

Plugging the mad velocity I derived above into this equation, we find that anyone inhabiting the lowest deck of Baxter’s giant alien ringworld would experience time dilation by a factor of five million—for every year spent in this extreme habitat, five million years would elapse in the outside world. This ability to “time travel into the far future” is a key plot element.

But there’s a problem. Quite a big one, actually.

The quantity gamma has wide relevance to relativistic transformations (even though I managed to write four posts about relativistic optics without mentioning it). As I’ve already said, it appears in the context of time dilation, but it is also the conversion factor for that other well-known relativistic transformation, length contraction. Objects moving at close to the speed of light are shortened (in the direction of travel) when measured by an observer at rest. A moving metre stick, aligned with its direction of flight, will measure only 1/γ metres to a stationary observer. Baxter also incorporates this into his story, telling us that the inhabitants of his relativistic ringworld measure its circumference to be much greater than what’s apparent to an outside observer.^‡

So far so good. But acceleration is also affected by gamma, for fairly obvious reasons. It’s measured in metres per second squared, and those metres and seconds are subject to length contraction and time dilation. An acceleration in the line of flight (for instance, a relativistic rocket boosting to even higher velocity) will take place using shorter metres and longer seconds, according to an unaccelerated observer nearby. So there is a transformation involving gamma cubed, between the moving and stationary reference frames, with the stationary observer always measuring lower acceleration than the moving observer. A rocket accelerating at a steady 1g (according to those aboard) will accelerate less and less as it approaches lightspeed, according to outside observers. The acceleration in the stationary reference frame decays steadily towards zero, the faster the rocket moves—which is why you can’t ever reach the speed of light simply by using a big rocket for a long time.

That’s not relevant to Baxter’s ringworld, which is spinning at constant speed. But the centripetal acceleration, experienced by those aboard the ringworld as “centrifugal gravity”, also undergoes a conversion between the moving and stationary reference frames. Because this acceleration is always transverse to the direction of movement of the ringworld “floor” at any given moment, it’s unaffected by length contraction, which only happens in the direction of movement. But things that occurs in one second of external time will occur in less than a second of time-dilated ringworld time—the ringworld inhabitants will experience an acceleration greater than that observed from outside, by a factor of gamma squared.

So the 1g centripetal acceleration required in order to keep something moving in a circle at close to lightspeed would be crushingly greater for anyone actually moving around that circle. In Baxter’s extreme case, with a gamma of five million, his “1g” habitat would experience 25 trillion gravities. Which is quite a lot.

To get the time-travel advantage of γ=5,000,000 without being catastrophically crushed to a monomolecular layer of goo, we need to make the relativistic ringworld a lot bigger. For a 1g internal environment, it needs to rotate to generate only one 25-trillionth of a gravity as measured by a stationary external observer. Keeping the floor velocity the same (to keep gamma the same), that means it has to be 25 trillion times bigger. Which is a radius of 25 trillion light-years, or 500 times the size of the observable Universe.

Even by Baxter’s standards, that would be … ambitious.

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* This neat correspondence between light-years, light speed and one Earth gravity is a remarkable coincidence, born of the fact that a year is approximately 30,000,000 seconds, light moves at approximately 300,000,000 metres per second, and the acceleration due to Earth’s gravity is about 10 metres per second squared. Divide light-speed by the length of Earth’s year, and you have Earth’s gravity; the units match. This correspondence was a significant plot element in T.J. Bass’s excellent novel Half Past Human (1971).

^† Baxter’s novel is full of plot homages to Niven’s original Ringworld, including a giant mountain with a surprise at the top.

^‡ As Baxter also notes, this mismatch between the radius and circumference of a rapidly rotating object generates a fruitful problem in relativity called the Ehrenfest Paradox.