I A "space elevator" extended to Neptune's orbit

[Dnj23]

TL;DR Summary

Achieving speed of light with Earth's rotation

Summary: Achieving speed of light with Earth's rotation

Excuse my ignorance, but I think of dumb things.

If you theoretically built a strong, lightweight cable that traversed over 2.5 billion miles attached to the rotating Earth, the tip would be traveling at or greater than the speed of light. How does this fail, not in terms of our limited engineering abilities, but physically?

 

[phinds]

Summary: Achieving speed of light with Earth's rotation

Excuse my ignorance, but I think of dumb things.

If you theoretically built a strong, lightweight cable that traversed over 2.5 billion miles attached to the rotating Earth, the tip would be traveling at or greater than the speed of light. How does this fail, not in terms of our limited engineering abilities, but physically?

Nothing travels faster than light. The cable would break well before it got to that point (and that's assuming we could build it to the point where it WOULD break and I doubt that).

 

[jbriggs444]

Nothing travels faster than light. The cable would break well before it got to that point (and that's assuming we could build it to the point where it WOULD break and I doubt that).

To add a bit of detail to this, suppose that we have a space elevator/beanstalk built of very high strength unobtainium. The tip is moving a bit less than the speed of light. Consider a pebble attached to the tip.

The momentum of the pebble increases without bound as the tip approaches the speed of light. Which means that the force required to continuously change the pebble's momentum as the tip goes around in its orbit increases without bound. Not even unobtainium can supply an infinite force. So even if you somehow managed to get the tip to light speed (you can't), it would not be possible to maintain a circular trajectory.

 

[hmmm27]

If you theoretically built a strong, lightweight cable that traversed over 2.5 billion miles attached to the rotating Earth, the tip would be traveling at or greater than the speed of light. How does this fail, not in terms of our limited engineering abilities, but physically?

For a self-powered contraption, it's like a rocketship that's designed such that the faster it goes the less thrust is produced, to the point where at a certain speed it's not producing any thrust at all so can't go any faster. In reality, time slows down for the rocketship so - even though inside the ship it seems that the engine is roaring away at full thrust - up near lightspeed, outside the ship it just looks like it's going "putt putt putt".

For an externally-powered contraption, it's like a hammer swung around on the end of a pole - a really really really long pole, or rope - where the hammer "magically" gains mass the faster it goes, and there's a theoretical point where it's infinitely heavy.

So, it's all well and good to say "Hey, if we swing this pole around, we can get the tip to exceed light-speed", but if you used all the energy in the universe to push it along, it still wouldn't even reach light speed.

 

[phinds]

For an externally-powered contraption, it's like a hammer swung around on the end of a pole - a really really really long pole, or rope - where the hammer "magically" gains mass the faster it goes, and there's a theoretical point where it's infinitely heavy.

You are mistaken. It does NOT gain mass (but it does gain energy). You are using the seriously deprecated concept of "relativistic mass" (which is just mass-equivalent energy).

 

[hmmm27]

I'm going to be pedantic and point at "it's like..." rather than "it is". But - for the sake of illustration - how is that different ? You still can't swing the other end of a rod up to light speed, and the math is the same as if it had gained mass.

 

[Ibix]

the math is the same as if it had gained mass.

But how much relativistic mass does it gain? For the purposes of increasing its rotational speed it would be a factor of $\gamma^3$. For the purposes of centripetal force it would be a factor of $\gamma$. So no, it's not the same as if it "gained mass", unless you want to accept a list of caveats and special cases. The unqualified word "mass" has meant rest mass in professional discourse for several decades. If you want to refer to one of the various relativistic masses you need to say so, and say which one you mean.

 

[Orodruin]

For a self-powered contraption, it's like a rocketship that's designed such that the faster it goes the less thrust is produced, to the point where at a certain speed it's not producing any thrust at all so can't go any faster. In reality, time slows down for the rocketship so - even though inside the ship it seems that the engine is roaring away at full thrust - up near lightspeed, outside the ship it just looks like it's going "putt putt putt".

This is quite misleading. Time slowing down or not is a question of reference frames, not whether or not you are inside or outside the ship. This misconception is rather common and difficult to get rid of.

the math is the same as if it had gained mass.

It most certainly is not. There are a few instances where you can replace mass in a non-relativistic system with relativistic mass and it works out. However, this is not universally true. Relativistic mass as a concept has largely fallen out of fashion in the physics community apart from in popularized texts. See my PF insight on the subject.

 

[Dnj23]

Thanks for the responses guys. I just recently read about the proposed space elevator and now realize just how ridiculous this all is.

 

[hmmm27]

Thanks for the responses guys. I just recently read about the proposed space elevator and now realize just how ridiculous this all is.

What's ridiculous about it ? "Space elevators" are a neat trick, but our material science (or even the conception of same) isn't up to it, yet. Personally - for the Earth - I like the idea of a linear Eiffel tower type construct 100km high, hosting a few hundred km of tangential space runway at the top, above the thick atmosphere. That would be (probably) doable with today's technologies and engineering.

 

[Dnj23]

What's ridiculous about it ? "Space elevators" are a neat trick, but our material science (or even the conception of same) isn't up to it, yet. Personally - for the Earth - I like the idea of a linear Eiffel tower type construct 100km high, hosting a few hundred km of tangential space runway at the top, above the thick atmosphere. That would be (probably) doable with today's technologies and engineering.

Ha, I didn't mean that. I meant the space elevator constraints made my initial question premature.

 

[mfb]

Personally - for the Earth - I like the idea of a linear Eiffel tower type construct 100km high, hosting a few hundred km of tangential space runway at the top, above the thick atmosphere. That would be (probably) doable with today's technologies and engineering.

Not as compressive structure (like buildings on Earth today). A space fountain would work, a launch loop would work (basically an elongated space fountain), a StarTram could be extended to larger altitudes.

For rotating structures in space there is a surprising result that the tip speed depends only on the material and the overall shape, but not the size. If you make a wheel out of Kevlar you can rotate it until the rim rotates at ~1 km/s, no matter if the wheel is 1 meter large or 10 kilometers. Kevlar is among the best materials in that aspect unless we learn how to manufacture carbon nanotubes in large scales - but even with them you can't get above ~5 km/s with a wheel. A tapered cable can handle higher tip speeds, enough to be interesting for interplanetary flight - but still nowhere close to the speed of light.

 

[Ibix]

Ha, I didn't mean that. I meant the space elevator constraints made my initial question premature.

I don't know what you mean by "premature" in this context. You specified that you didn't want to be constrained by current engineering limits, and you were answered on that basis. It is impossible to spin a rod so fast that its tip exceeds light speed. No matter how strong the material it's made of it will break, and the amount of energy needed to spin it up diverges (grows to infinity) as the tip approaches light speed.

 

[pervect]

Summary: Achieving speed of light with Earth's rotation

Summary: Achieving speed of light with Earth's rotation

Excuse my ignorance, but I think of dumb things.

If you theoretically built a strong, lightweight cable that traversed over 2.5 billion miles attached to the rotating Earth, the tip would be traveling at or greater than the speed of light. How does this fail, not in terms of our limited engineering abilities, but physically?

The magnitude of the proper acceleration of a point on the end of the tether approaches infinity as the velocity of the end of the tether approaches the speed of light, which happens when $r \omega = c$.

I'm not sure how to calculate this at the I level, but at the A level the formula is:

$$acc^b = u^a \nabla_a u^b$$

where $u^a$ is the 4-velocity, and $acc^b$ is the 4-acceleration. The magnitude of the 4-acceleration approaches infinity as $r \omega$ approaches c. Infinite proper accelerations are not physically sensible.

[add] Since we are in special relativity, we can get read of the covariant derivative if we introduce cartesian coordinates $x^i$

We'll still need 4-vectors, though. There's a wiki article on the 4-acceleration [[link]] and the 4-velocity [[link]]

$$acc^b = \sum_{a=0..3} u^a \frac{\partial u^b}{\partial x^a}$$

 

[Dale]

the math is the same as if it had gained mass.

Actually, it isn’t, and that is one reason to discard the concept of relativistic mass.

The key difference is that mass “resists acceleration” isotropically. On the other hand a relativistic particle requires different amounts of force to produce the same acceleration in the direction parallel to the velocity vs transverse to the velocity.

 

[hmmm27]

Wow, three for three : I shouldn't have gotten out of bed this week.

@everybody
I saw a reference to Orodruin's PF-insight article earlier in another post and eschewed the opportunity, since rectified. But, it would hardly change my explanation : might I refer you to the second sentence of the first paragraph : while I didn't get as far as thinking to cube the Lorentz factor to calculate the force required to accelerate an object, the generality of my explanation doesn't require it (except possibly "the math is the same" in a subsequent post).

I do appreciate the edification, of course.
But yes, the "time slows down" bit should have said "as it seems to a stationary observer" rather than "outside the ship" which could be easily misconstrued.

a linear Eiffel tower type construct 100km high, hosting a few hundred km of tangential space runway at the top, above the thick atmosphere. That would be (probably) doable with today's technologies and engineering.

Not as compressive structure (like buildings on Earth today).

That was based on the observation that several of Earth's mountains are 10km high, and don't seem to be in danger of toppling over, and a possibly incorrect assumption that a spaceframe structure weighs less than 10% of an equivalent volume of solid rock, for the same structural strength, notwithstanding a lack of constraint on the architect to have it built in the shape of a mountain (apparently "Eiffel Tower" is better).

 

[Ibix]

while I didn't get as far as thinking to cube the Lorentz factor to calculate the force required to accelerate an object, the generality of my explanation doesn't require it (except possibly "the math is the same" in a subsequent post).

The major problem with your explanation is that the "mass increase" is different for different for different phenomena. In terms of the force speeding the object up, you get a factor of $\gamma^3$. In terms of the centripetal force you get a factor of $\gamma$. And in terms of the energy needed you get a factor of $2(\gamma-1)c^2/v^2$. Which isn't really much like a mass increase.

Just a few of the many reasons why calling total energy "relativistic mass" is a bad idea. Calling it "mass" is an even worse one.

 

[hmmm27]

So... "weight" ?

 

[Ibix]

So... "weight" ?

No, that's a force. "Relativistic mass" is the total energy of the body divided by $c^2$. Relativity isn't Newtonian physics, and trying to disguise it as such is just back to front thinking, IMO.

 

[mfb]

That was based on the observation that several of Earth's mountains are 10km high, and don't seem to be in danger of toppling over, and a possibly incorrect assumption that a spaceframe structure weighs less than 10% of an equivalent volume of solid rock, for the same structural strength, notwithstanding a lack of constraint on the architect to have it built in the shape of a mountain (apparently "Eiffel Tower" is better).[/URL]

The failure modes are different. A building collapses internally in a way Mt. Everest cannot because it is 100% rock. Mountains are limited in height by large-scale landslides and depression of the crust. Mauna Kea could be ~10 km tall if it (and its neighbor) wouldn't be so massive that they pushed down the crust by 6 km.

X-Seed 4000 is a proposed 4000 m tall structure that looks a bit like an oversized Eiffel tower. If I remember correctly most of its mass would have been structural already, so you can't push it that much more even if you skip internal use of the building.