GIGO: Garbage In, Gigawatts Out?

[BurtJordaan]

It is time for us engineers to have some fun as again. :-))

It's a dream of many people: free energy, cars running on water, vacuum energy, over-unity machines, etc. Here's a dream that may (perhaps) one day come true. If we can find a friendly, rotating black hole within reachable distance and with a hospitable environment around its equator, we can perhaps colonize the space around it and have clean "free" energy for our colony!

Pictured below is a design based on Sir Roger Penrose's black hole energy extraction process. It shows a mega-city built on a rigid structure around the equator of a massive, spinning black hole. The city is at just the right distance from the hole so that the local gravity is at a comfortable 1g and that tidal forces and inertial frame dragging do not affect the structures adversely.


Credit: "Gravitation", Misner, Thorne, Wheeler (MTW), fig. 33.2

The city's garbage is processed at the top left by dumping it into suitable containers and pushing the containers at a carefully chosen angle towards the black hole. Due to frame dragging, the containers swing around the black hole in its ergosphere^[1], and so "steal" some of the black hole's angular momentum.

Just before the container starts to leave the ergosphere, the garbage is ejected towards the hole, to be "swallowed" by it. Due to black hole dynamics, the garbage instantly attains negative energy. The negative energy entering the black hole causes its rest mass to decrease. At the same time, the "lost" energy (or most of it) is added to the empty container as kinetic energy - the ultimate slingshot effect.

The result is that the empty container recoils outward at enormous velocity. The increased spacing of the small square blocks in the graphic indicate the relatively large outgoing speed. For a suitable ratio of garbage to container mass, the outgoing speed can be a large fraction of the local speed of light! This sort of thing is observed in nature when a star (one part of a binary system) is flung out of the center of a galaxy at a speed approaching the speed of light.

At the city's power station (top-center), the high-speed container is caught in a suitable water wheel-like contraption that dumps most of the empty container's huge kinetic energy into the rotation of a set of flywheels. These flywheels power generators that feed our city's electricity grid. The empty container is then reused for another load of garbage.

As stated in the authoritative MTW and mathematically shown by Penrose, if optimally executed, the net energy gain of every container's round trip approaches the total rest energy (E₀=mc²) of the garbage ejected, plus some fraction of the considerable angular momentum of the rotating black hole. This fantastic power station can produce around one gigawatt-hour of electric energy for every metric ton of garbage dumped into it - GIGO! What a pleasure!

Think about the benefits:

(a) Garbage completely out of sight and smell, hidden by an event horizon.

(b) Nearly 100% of the mass-energy of the garbage is turned into usable electricity.

(c) Some of the black hole's spin energy is thrown into the mix as well.

If this does not qualify for "over-unity", I don't know what will!

Oh, I can already hear the environmentalists of our city raising at least three issues:

(i) We are polluting the black hole.

(ii) We are "burning up" the black hole (if they can figure this one out!)

(iii) Why do citizens have to pay for garbage removal and then (again) for this "free" electricity?

--
Regards
Jorrie

Note [1] For the ergosphere, see http://en.wikipedia.org/wiki/Ergosphere

[Marshall]

Neat! :^D

I should be able to figure this out, but you may have the information more readily available. Do you happen to know if (under reasonable assumptions about the BH spin) there is any significant degree of time dilation (relative to distant observer) at the equatorial radius of the ergosphere? I kind of suspect not.

The explanation given here was definitely helpful. I had seen a brief account some years back but didn't quite understand or take the trouble to go through it. Negative energy is a hard idea to grasp. How can the gravitational energy given up by the infall be transferred to the kinetic energy of the empty container? I haven't given MTW a proper reading, just looked at different chapters at various times. Don't own a copy. So the above post's explanation was enlightening.

[Marshall]

Oh! maybe I see! Because of falling in all that distance, and maybe also because of frame dragging, the filled container achieves considerable speed. But it would normally lose much of its speed while coasting back up to the power station. However it does not because it ….
Nope. I still don't understand.

[Faradave]

The container would loose much of its energy on the return trip, but it returns without the garbage, having added the recoil of its loss.

I wonder if it could work with a star or other repository less massive than a black hole.

[BurtJordaan]

Do you happen to know if (under reasonable assumptions about the BH spin) there is any significant degree of time dilation (relative to distant observer) at the equatorial radius of the ergosphere? I kind of suspect not.

Marshall, I have 'engineerized' (SI units) some of MTWs equations for my Engineering Blog. Do not have the time right now to check the calcs, but here is the time dilation equation:


with

Symbol meanings:


The equatorial ergosphere starts at the normal Schwarzschild radius r_S = 2GM/c² and you can choose spin parameter anywhere up to half of r_S, which is the extreme Kerr rate.

Will be back later...

--
Regards
Jorrie

[BurtJordaan]

The container would loose much of its energy on the return trip, but it returns without the garbage, having added the recoil of its loss.

I wonder if it could work with a star or other repository less massive than a black hole.

With a star one would be limited to the amount of thrust you can generate by ejecting the garbage backwards at the right time, which will be tiny by comparison. The Penrose process with rotating BHs works by liberating at least some of the E=mc² energy of the garbage. More about that later.

--
Regards
Jorrie

[BurtJordaan]

The equatorial ergosphere starts at the normal Schwarzschild radius r_S = 2GM/c² and you can choose spin parameter anywhere up to half of r_S, which is the extreme Kerr rate.

Will be back later...

I made the calcs and for an extreme Kerr spin of a BH with 1.5 Solar masses, r_S=4.44 km. The time at the edge of the equatorial ergosphere is dilated to dtau/dt = 0.4, where dt is the incremental time of the distant observer. For spin factor a = 0.5GM/c², it comes down to 0.24 and for a = 0.1GM/c², it becomes 0.05.

The time dilation obviously becomes more sever as one enters the ergosphere and time 'stops' at the outer horizon with extreme spin.

Please check my calcs when you have time.

--
Regards
Jorrie

[BurtJordaan]

Oh! maybe I see! Because of falling in all that distance, and maybe also because of frame dragging, the filled container achieves considerable speed. But it would normally lose much of its speed while coasting back up to the power station. However it does not because it ….
Nope. I still don't understand.

Yes, frame dragging adds a little slingshot effect, but this is not what Sir Roger has been looking for. Also not just throwing reaction mass in the form of garbage overboard, which gives a tiny boost as well. He was greedy and wanted more, a lot more...

Similar to Hawking radiation, where the BH loses energy equal to the energy of a particle-antiparticle pair, one can theoretically approach the limit of adding delta_E = 2mc² to the container, where m is the mass of the garbage swallowed. For this an ideal arrangement would be needed (just, just above the ergosphere and perfect ejection), which may never be achieved.

The one Gigawatt per metric ton of garbage is apparently a reasonable compromise, but my guess is the process of converting that into electricity will also throw some away.

--
Regards
Jorrie

PS. I also used a borrowed copy of MTW, but I quoted this from it in my CR4 Blog:

"(Energy gain per trip) = (rest mass of garbage) + (amount, -ΔM, by which the hole's mass decreases)"