1. ## Relativistic ringworlds

I've just finished Stephen Baxter's Xeelee: Redemption, which features more of Baxter's vast mega-structures and time-periods. In this one, he uses a ringworld a light-year in radius, spinning at just a whisker short of light-speed, so that its inhabitants experience 1g of centripetal acceleration, while undergoing massive time dilation (a gamma factor of 5 million). It's a simple application of the a=v²/r equation for centripetal acceleration, and it comes out the neat way it does because of the odd coincidence that lightspeed divided by the length of a year is very close to being one gravity of acceleration.
Trouble is, Baxter neglected the effect of relativity on acceleration. We're familiar with the fact that, as you accelerate towards lightspeed, 1g of acceleration in your moving frame translates into considerably less in the rest frame - there's a factor of gamma-cubed involved for linear motion. A similar thing happens for transverse acceleration, of the kind required to steer your starship in a circle - a marginally more benign gamma-squared conversion.

So Baxter's ring has a centripetal acceleration of 1g in the rest frame, but its inhabitants would be pulling 25 trillion g in the frame corotating with the ringworld. Oops. We're going to need a bigger ring.

We need a radius of gyration that produces just one 25-trillionth of a g at lightspeed, in the rest frame. That turns out to be 2.4e13 light-years - inconveniently large, even by Baxter's standards.

Shame. It was such a nice idea.

Grant Hutchison

2. Originally Posted by grant hutchison
We need a radius of gyration that produces just one 25-trillionth of a g at lightspeed, in the rest frame. That turns out to be 2.4e13 light-years - inconveniently large, even by Baxter's standards.
Or rotate slower? Presumably, not much slower because of the non-linearity of relativistic effects (I suppose even I could do the math, but I don't think I can be bothered...)

3. Originally Posted by Strange
Or rotate slower? Presumably, not much slower because of the non-linearity of relativistic effects (I suppose even I could do the math, but I don't think I can be bothered...)
Good job I'm here then.
A 1g ringworld with a radius of a lightyear has a floor speed of about 0.7c, with a gamma of only 1.4. Ten lightyears gets you 0.96c, gamma 3.6. One hundred lightyears gets you 0.995c, gamma 10. It's ... unwieldy. And your building material costs don't just scale with the radius, but with gamma as well, because of relativistic length contraction around the circumference. So the 100ly ring has a (corotating) circumference of more than 6000 lightyears.

Grant Hutchison

4. Worlds orbiting supermassive black holes seem to be a popular choice at the moment. Gargantua and its planetary system in Interstellar is one example; Sean Raymond has described another version, with a million planets in concentric rings, distorted by time dilation and compression.

Shame that Baxter messed this one up. Does he know yet?

5. Originally Posted by eburacum45
Shame that Baxter messed this one up. Does he know yet?
I don't know. But I haven't really looked. (His website has never been particularly chatty, so I doubt if he would post anything about it there.)

Grant Hutchison

6. Originally Posted by grant hutchison
but its inhabitants would be pulling 25 trillion g in the frame corotating with the ringworld.
I don't really know how much acceleration people can tolerate, but I suspect that might be a little bit too much.

7. Originally Posted by grant hutchison
Good job I'm here then.
Exactly what I was thinking !

A 1g ringworld with a radius of a lightyear has a floor speed of about 0.7c, with a gamma of only 1.4.
Was there a reason he couldn't have done that? For example, did the story rely on the gamma of 5 million?

8. Originally Posted by Strange
Was there a reason he couldn't have done that? For example, did the story rely on the gamma of 5 million?
Very much so - time travel to the far future was a key plot element.

Grant Hutchison

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Originally Posted by grant hutchison
And your building material costs don't just scale with the radius, but with gamma as well, because of relativistic length contraction around the circumference. So the 100ly ring has a (corotating) circumference of more than 6000 lightyears.

Wow. Kind of makes you wonder what would happen to such a structure if you then slowed it back down! What would happen to it as it tried to fit 6,000 light years worth of material into a ring thats (now) only 600 light years around?

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Originally Posted by grant hutchison;2494964It's a simple application of the a=v[FONT=&
²/r equation for centripetal acceleration, and it comes out the neat way it does because of the odd coincidence that lightspeed divided by the length of a year is very close to being one gravity of acceleration. Trouble is, Baxter neglected the effect of relativity on acceleration.
That also puts the kibosh on the idea that we could travel to the stars in a reasonable time if we had unlimited propulsion power and could sustain acceleration of 1 g for a year or so. But in order to get the relativistic length-contraction advantage that brings the time down from centuries, we would run afoul of the effect you describe. So if the human body is only comfortable at 1 g, we gain no advantage from the relativistic regime, so we're talking centuries to get anywhere even in principle (never mind all the other issues)!

11. Originally Posted by Dave241
Wow. Kind of makes you wonder what would happen to such a structure if you then slowed it back down! What would happen to it as it tried to fit 6,000 light years worth of material into a ring thats (now) only 600 light years around?
Nothing good.
The reverse problem - spinning something up from a standing start to a relativistic rim speed - is the basis for something called the Ehrenfest "paradox" in relativity. (It's only a paradox if you assume a completely rigid object, which turns out to be incompatible with relativity.) Anyway, there's nothing to stop you having a rigid object that's already rotating at relativistic speeds - Baxter actually points out that his ringworld would have to be built already rotating, for this reason.

Grant Hutchison

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Originally Posted by grant hutchison
Nothing good.
A reference to the all-time best AI line from a movie? The AI in Interstellar was the only part I really liked.

13. Originally Posted by Ken G
A reference to the all-time best AI line from a movie? The AI in Interstellar was the only part I really liked.
I don't actually recall the line. I found TARS a somewhat slappable character, I'm afraid.

I don't know the origin of my own use of the phrase (I've just spent a few minutes trying to think where I might have acquired it) but I'm told I used it a lot at work (where a sort of weary pessimism was pretty much part of the person specification), so I guess it's not surprising it surfaces here, too.

Grant Hutchison

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The scene I liked was when TARS was asked what would happen if an attempt was made to dock with the malfunctioning space station, and TARS simply said, "nothing good." To me that was such a breath of fresh air over the obligatory "there would be a 99.4% chance of explosive decompression" or some typically Spock-like response.

I overstated the severity of the gamma^3 correction on interstellar travel, it hurts but with a proper acceleration of about a comfortable g for more than a few years t of proper time, the distance in light years covered in the rest frame is et/2 . So the exponential can certainly get you pretty far in a decade.

15. Originally Posted by Ken G
I overstated the severity of the gamma^3 correction on interstellar travel, it hurts but with a proper acceleration of about a comfortable g for more than a few years t of proper time, the distance in light years covered in the rest frame is et/2 . So the exponential can certainly get you pretty far in a decade.
Yes, rest-frame distance is a hyperbolic cosine function of on-board elapsed time, so it starts to look like a simple exponential fairly quickly.

Here's how it goes, at 1g with a mid-course turn-over so as to come to a halt at the destination.
relativistic journey.jpg
Fifty years gets you to the edge of the observable Universe.

Grant Hutchison

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Let's say one of these got built--and big enough so it is, what ((.99.

What happens to be if I try to run that one mile an hour to break lightspeed--what would I feel at one g?

Would it be like in anightmare--seeing the heat-death of the universe as I am slowly trying to run away from a plodding monster behind me and I feel as if in molasses?

There's your cosmic horror right there.

17. Originally Posted by publiusr
Let's say one of these got built--and big enough so it is, what ((.99.

What happens to be if I try to run that one mile an hour to break lightspeed--what would I feel at one g?

Would it be like in anightmare--seeing the heat-death of the universe as I am slowly trying to run away from a plodding monster behind me and I feel as if in molasses?

There's your cosmic horror right there.
No, nothing like that. You'd run that extra mile an hour (assuming you can run that slowly), and nothing much would change.

Grant Hutchison

18. Originally Posted by publiusr
Let's say one of these got built--and big enough so it is, what ((.99.

What happens to be if I try to run that one mile an hour to break lightspeed--what would I feel at one g?

Would it be like in anightmare--seeing the heat-death of the universe as I am slowly trying to run away from a plodding monster behind me and I feel as if in molasses?

There's your cosmic horror right there.
You'd feel like you were running in 1G. Subjectivity!

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