Thought Experiment: Acheiving the SoL through Gears

[Crossroads Inc.]

Riding my bike this morning had me thinking about gear speeds. A big gear turning a little gear to go faster.

In complex gearboxes, usually used for lifting heavy items, the small gears' axel is attached to a larger gear, which then turns another small gear. Now in heavy lifting this is reversed, so a great deal of energy helps pull something heavy a little at a time.

But if you put it the other way, with big gears turing little gears, just how far could it go?

If you had a one meter by one meter gear, turning a 10cm gear at one revolution per minute; and had that gear attached to another one meter gear... At each point the speed would go fast and faster. Now obviously if you tried to do this friction and heat would come into play where the gears would simply seize up at a certain rate.

But doing this purely as a thought exercise, if Friction and heat generated wasn't a factor.... Would there not be a point where the rotation of the end gear achieves SoL?

[Sarevok]

Gears do not work like that. The smaller gear would not cause the bigger gear turn faster and thus add up speed.

[Marko Dash]

i think he meant the the second 1 meter gear would be attached or share an axle with the 10cm gear. say the 1 meter gear is turning at 1 rpm which is making the 10cm gear go 10 rpm the second 1 meter gear because of the physical connection would also be turning at 10 rpm. if the second 1 meter gear also drives another 10cm gear how far could you go with increasing the rpm if the first gear does not slow down.

[TimothyC]

Instantaneous Velocity at the point at which two gears intersect is the same for both gears.

What WiresCrossedroads here wants to do is use an axis to spin the edge of a disk up to the speed of Light, or so I hope. I'll leave that thought experiment to people who are smarter than I am.

[Simon_Jester]

But doing this purely as a thought exercise, if Friction and heat generated wasn't a factor.... Would there not be a point where the rotation of the end gear achieves SoL?

No. First of all, if you spin a rotating metal object fast enough if flies apart and hurls jagged metal in all directions:


Assuming for the sake of argument that your gear has infinite tensile strength as well as frictionless materials, you run into the second problem: as you accelerate the gear to higher and higher angular speeds, relativistic effects kick in and the moment of inertia of the gear increases. The rim gets heavier relative to an observer standing outside the system and turning the wheel.

It is not possible to create a physical structure whose outer rim is spinning faster than light, for this reason. As the speed of the edge of the disk approaches c, the mass of the rim approaches infinity, the centrifugal force* pulling the disk apart also approaches infinity, and the amount of torque required to add a tiny increment of angular speed to the disk to make it turn faster ALSO approaches infinity.

That last is critical: even given an indestructible frictionless disk, you cannot get the edge to move at the speed of light, for the same reason you can't throw a rock at the speed of light.

*Yes, I know the "centrifugal force is an illusion" thing. I just don't care. "Centrifugal force" is still the easiest way to explain the concept and we shouldn't be retarded pedantic jackasses when it comes time to write clear explanations.

[Kuroneko]

It's not necessary to assume that the gear has any mass at all. The scenario is still inconsistent with STR even if we have frictionless and massless gears. Rigidity is inconsistent with relativity because materials cannot have an infinite speed of sound, meaning that applying force on a part does not affect distant parts faster than that. As the axle turns, the gear will twist into a spiral until the outer parts catch up. So what about the steady-state?

A Lorentz transformation can be written in terms of hyperbolic sines and cosines through the substitution α = atanh(v/c)), and a uniform acceleration with respect to a particle's frame is a linear increase of this parameter in the particle's proper time. Thus, a uniformly accelerated worldline makes a hyperbola in spacetime, and more careful analysis can show that its asymptote is a distance c²/a away from its vertex. Physically, this represents a horizon past which no communication is possible: a particle with acceleration a observes an apparent horizon a distance c²/a away.

This has an immediate application the gear scenario. Since in a rotating reference frame of angular velocity ω, a linear speed of c occurs at the radius r = c/ω, where the centrifugal acceleration is c²/r = cω. But then the horizon distance is c²/(cω) = r, i.e., precisely the radius to make the linear speed c! Therefore, even if we have indestructible massless gears, an (apparent) event horizon causally disconnects the axle and the gear's circumference whenever the latter attempts to move at c.

[Simon_Jester]

OOF.

Thank you, Kuroneko. Hopefully I'll remember to go back over that carefully and try to make sure I understood it all, because I'd like to not end up like my peers who forget everything they ever knew about relativity once the urgency goes away.

[Crossroads Inc.]

Oh wow... So even allowing for practically all methods of impossibility, the gears would eventually collapse into an Event Horizon?
Not sure if thats Awesome, or a reflection on what happens whenever I try a thought experiment

Also... After 5 years on the bored here, I think I finally found a custom title I'd live with:

What WiresCrossedroads here wants

That name describes so much about me

[Simon_Jester]

Oh wow... So even allowing for practically all methods of impossibility, the gears would eventually collapse into an Event Horizon?

An event horizon is not a black hole; I'm not sure "collapse" is the right word.

An event horizon (not capitalized; it's just a technical term, like "tack hammer" or "cylinder head") is a surface where events on one side cannot affect events on the other, even by light. Thus, from the point of view of people outside the horizon, the interior is invisible.

To get this to happen you generally have to warp the fabric of spacetime very, very dramatically, as here, or as in a black hole. A gravity well becomes a black hole when it warps spacetime to the point where inside the hole, you literally lose the ability to distinguish "up" from "last Tuesday." This is because both "up" and "last Tuesday" point in the same direction; you can no more visit "up" inside a black hole than you could travel backwards in time. The only possible direction for future movement is down.

However, in this case, I suspect the event horizon is far more innocuous: the interior of the rapidly turning wheel doesn't implode, it just "loses sight" of the rim. However, that "losing sight" makes it impossible to affect the rim any further by turning the handle, so you wind up with a rotating wheel that has no causal relationship with the massless spinning rim traveling at c.

And, just to underline one of Kuroneko's points, you do need a massless gear to make this work. For gears that have more than zero mass, the event I described still comes into effect below the speed of light... I think.

Am I mistaken, Kuroneko?

[Kuroneko]

Hopefully I'll remember to go back over that carefully and try to make sure I understood it all, because I'd like to not end up like my peers who forget everything they ever knew about relativity once the urgency goes away.

It's worth understanding, although the above is a bit of handwave because the horizon itself is different for linear accelerations that reasoning was dependent on. A more thorough treatment would verify that the Minkowski metric in rotating coordinates has a horizon where g00 = 0, this being exactly where the linear speed if c.

But the linear case is lot easier to understand*: uniform acceleration = uniformly continuous Lorentz boost (hyperbolic rotation) = hyperbolic wordline, which asymptotically approaches lightspeed. Thus, that asymptote is lightlike and serves as a horizon: any signal from the half of spacetime after ('above') it cannot reach the accelerated wordline.

*Much easier than one might expect. The literature related to the relativistic rotating disk is huge and full of pitfalls.

However, in this case, I suspect the event horizon is far more innocuous: the interior of the rapidly turning wheel doesn't implode, it just "loses sight" of the rim. However, that "losing sight" makes it impossible to affect the rim any further by turning the handle, so you wind up with a rotating wheel that has no causal relationship with the massless spinning rim traveling at c.

Right, since the horizon breaks the gear even if we keep the stresses finite and arbitrarily low by taking the massless limit. An absolute horizon is an observer-indepedent property of spacetime; an apparent horizon is not (this particular type is an acceleration horizon). In that sense, it is just as fictitious as centrifugal force, etc., but like it, that doesn't mean that it is not real. For example, an accelerated observer in vacuum will measure thermal Unruh radiation from the horizon in a manner almost identical to Hawking radiation from black hole event horizons. For gears of fixed positive mass, you're quite right that there will be problems below the speed of light.