event horizon - What happens if you let a cable roll slip into a black hole?

1. 
   

   Does the cable roll spin faster the more cable goes into the black hole in reference of a observer standing next to it?

   
   


2. 
   

   Can gravity pull the cable that it exceed the speed of light inside a black hole in reference to the center of the black hole?

   
   

Answer

Whether it's a black hole or some other more ordinary mass pulling on your rope isn't actually that interesting. Let's think about a cable unrolling above Earth to start with.

What we have is a pulley with a rope hanging off one side. The weight of the rope exerts some force on the edge of the pulley, causing it to undergo angular acceleration (starts to spin). This releases some more rope, so the force increases a bit, acceleration increases, pulley spins faster and faster. Some assumptions:

• Let's assume that our spool of rope is very big, so that there's always a lot of rope left in it, and the change in the mass of coiled rope is unimportant.


• Let's also assume the bearings or whatever that allow the pulley to rotate are frictionless.


• Also, let's not complicate anything with relativistic speeds, yet.

Once the end of the rope hits the ground, the weight of the portion of the rope lying on the ground is no longer felt by the pulley. Since there is always a constant amount of rope between the pulley and the ground, the force is constant. The pulley continues to spin faster, but with constant acceleration. Still assuming speeds stay non-relativistic for now. Now for the first consideration with the black hole. Instead of "hitting the ground" the end of the rope now "hits the singularity". The weight of the portion of the rope absorbed by the singularity is no longer felt by the pulley. We're adding a bit of mass to the black hole, but let's assume that the mass of rope that we're feeding in is small compared to the mass of the black hole (similar to how you'd assume the mass of rope lying on the ground is much less than the mass of the Earth). This means the gravitational force exerted by the black hole doesn't increase. So we really have exactly the same scenario. Don't worry, it will get more complicated in a minute.

As the pulley keeps spinning faster, its edge (the part that's spinning fastest) grows to an appreciable fraction of the speed of light. The rope is coming out at the same speed that the edge of the pulley is moving, so it is also now moving relativistically. Now Newtonian intuition starts to break down. Another way to look at what has been going on is in terms of energy. The rope starts with some gravitational potential energy (from having a height above the Earth/black hole/whatever). As it falls it loses potential energy, which goes into the kinetic energy of the rope and rotating pulley. Newtonian considerations would tell you something along the lines of: $$-\Delta U = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$$ $v$ is the speed of the rope, and $\omega = v/r$ is the angular speed of the pulley (equal to the speed of the rope divided by the radius of the pulley). Anyway. This breaks down once speeds start getting relativistic. Energy is still conserved, but the change in speed obtained for a given energy input starts to decrease. You can put in more and more energy, and the speed will always keep increasing, but the increase for a given amount of energy gets smaller and smaller. The speed keeps increasing, approaching the speed of light, but never reaches it. You'll get to $.9c$, then $.99c$, then $.999c$, eventually $.9999999999999...999999c$, but never reach $1.0c$.

In practice, a lot of other things prevent you from getting to relativistic speeds in the first place. You might have a nearly frictionless pulley, but nearly isn't zero, and as speeds pick up the heat production from friction will destroy your equipment. Or air friction on the rope will increase with the speed until equilibrium is reached and the rope is falling at a constant speed. There is no air above a black hole of course... You need a lot of energy to get to relativistic speeds, which means to need to drop a LOT of rope. And the more rope you need to drop, the more needs to be coiled. But this increases the mass of the coil, which increases the energy needed to accelerate it. So more rope is needed. But this increases the mass of the coil... it's likely your coil will get so massive that it will collapse and form it's own black hole before you can get to relativistic speeds by unrolling it.

There are also worries about the tensile strength of the rope. You can't hang your spool arbitrarily high, at some point the weight of the rope below a given segment will be so much that the rope will break. This gets even worse around a black hole. As the rope approaches the singularity, the tidal force on the rope increases dramatically. Basically the BH is pulling harder on a nearby piece of rope than a more distant piece. As the singularity is approached, the tidal force tends to infinity, so even the most durable imaginable rope will be torn apart.

So far this has all been classical mechanics and special relativity. I won't even try to start bringing general relativity into this.

That's some ideas to get you thinking a bit. You can play with assumptions and adding in new considerations for different effects more or less ad infinitum. But the big take-away message is:

• Nothing with mass can move faster or equal to $c$, even if black holes are involved (massless particles like the photon can move at $c$, and in fact cannot move any faster or slower than this).


• The faster an object is moving, the more energy it takes to increase its speed by a fixed amount. This becomes important when speeds are a significant fraction of $c$. At much lower speeds the effect is safely ignored.