Depends how long it is. If you're in intergalactic space and there's no nearby masses to cause trouble, minimum length is just minimum speed for the generator and gearbox to work times the Hubble constant. It already is quite workable at "just" a light-year.
The wire itself contains mass-energy (I think? I mean, you make the rules). If it's a decent-size (edit: as in non-microscopic) black hole it'll more than offset what's lost to Hawking radiation.
The basic idea could even work in practice, although material strength limitations mean the approach tends to be less about hooks and more about capturing the radiative heat of things colliding on their way in.
Then you drew a line segment between points (1,1) and (3,1). You can stretch the rubber until it’s significantly longer, but your line is always exactly 2 units long, even if the rubber stretches.
Ah, but then we couldn't see or experience gravity at all!
In differential geometry there's a very important distinction between coordinate distance and actual distance. The globe and GPS coordinates give a good example - one degree of longitude is throwing distance at South Pole Station, but ~111km at the equator, even though it's still one degree. On a curved surface additive coordinates will never describe actual distance exactly. In some cases, like a 2-spherical planet, they're even guaranteed to break down entirely somewhere (like the exact poles).
It was a blunder mentioning rubber - this isn't about bowling balls on a trampoline. I just meant that solid matter has a natural spacing between atoms, and if something continuously pulls it away from that - like expanding space or, I don't know, two conveyor belts going opposite ways - it's going to respond with a constant tension offsetting the effect. Or break.
The reason the planet orbits the star is that the star has warped space such that (given speed the planet is traveling) an orbit is a straight line.
And annoyingly, that's only possible in more dimensions than we can picture. All a 1+1 dimensional spacetime can do is expand or contract.
Gravity is mass warping spacetime.
Okay, nitpick
but if it's something relativistic space-like momentum can be just as important as energy. The matter half of the Einstein equation(s) treats every component of 4-momentum equally.
I'm not sure what's going on with your power calculation there. Edit: It should be the Hubble constant times length times whatever tension you let build up in the wire from the generator's resistance to movement. Planetary masses are only significant as your contraption gradually starts the endpoints moving.
Even mithril will have to break before it's moving faster than lightspeed. Otherwise, sure, fill your boots. If you set up in a gravitationally neutral area, it should start noticeably moving at around a light year of extension. You'll need way more to overcome the escape velocity of the Earth, sun and Milky Way, though.
Why not?
Wire doesn't actually provide a constraint either, we just tend to pretend it does when we're sure it won't break. At hypersonic speeds metal acts like a liquid, and at relativistic speeds matter particles do act like radiation.
Zoom in enough and it's all just waves. Presumably the mithril too, if it's at all compatible with relativity.
Matter doesn't stretch endlessly, though. Think about the individual mithril atoms, there's the same number but they have more space to cover as time goes on. The wire would gain some tension as space tries to expand it, and relaxing that tension could be used for power generation.
Or it would break. We can fudge something unnaturally strong but at some point the speed of sound in the material starts brushing against lightspeed and it just has to give.
The mithril wire would be put under tension by the expansion of space, the same as if it was laying on a stretching piece of rubber. Space would largely ignore it, because it's only sensitive to sizeable changes in the stress-energy tensor, and a segment of wire isn't that heavy.
Or just directly have the wire wrapped around an alternator making electricity.
No transponders makes you harder to target, but also increases the risk of an accedent, That's fine, though - it's not like somebody would deploy a sizeable force of military planes in a combat ready position, for an extended period, over a peacetime area, right? Right? /s
I feel like those situations are already life-threatening to some degree, so I'll take my chances.
I'd assume growth over the course of just exiting a car wouldn't be too significant.
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