To be clear, this is a way to destroy the planet Earth, approximately following the rules here. This is much more difficult than killing all humans or wrecking the biosphere. The main problem is that the planet is just so damn big, and most practical methods require a huge energy investment.
My method requires the expenditure of minimal energy, and in fact, we can generate power using it. However, we will need a few advancements in material science (and a bit of patience).
We start with a Space Elevator
We just need the cable, so no need to worry about creating (or powering) a climber. The methodology is well-established, and we’re close in material-science technology to achieving this. A comparatively small energy-expenditure will have to be spent to get the cable in orbit, but we can make that back later.
Once built, the elevator cable will be exerting a considerable upward force on the base station of the elevator. This is because the center of mass of the cable is above geosynchronous orbit, so the centrifugal force is greater than the force of gravity.
Start Playing out the Cable
Now, we start building more cable at the base, and letting it out as we do so. Because the overall force is in the upward direction, the cable will easily lift more of itself. In fact, as we do this, the center of mass shifts further out, and the force increases, so we can keep increasing the thickness of the cable as we go.
(Note: At this point, we can add a power station at the base with turbines driven by the upward-moving cable. This can make up the energy we spent to get the cable into orbit to begin with.)
We keep increasing the thickness of the cable, and essentially start converting Earth’s mass into the cable. The ultimate goal is to unravel the whole planet into this increasingly-thick strand of elevator.
As the cable lengthens, its end will achieve escape velocity, and we can plan ahead and plant charges such that predetermined lengths get cut off and escape the planet’s gravity well. This will help reduce the forces on the cable.
So, we’re unraveling the Earth and throwing chunks of it away, but where is the energy for this coming from?
The Earth Supplies the Energy
The best thing about this plan is that we’re not using up our energy to do so. As the cable plays out, the Earth’s rotation is what’s supplying the power. The Earth is exerting an eastward force on the elevator at its base, accelerating the whole thing.
This means the cable exerts an equivalent westward force on the Earth, slowing down the planet’s rotation. As that happens, the distance of geosynchronous orbit gets greater, so we’ll need to let the cable get longer and longer before we blow off end pieces. For the purposes of this plan, we assume our material science advances to keep up with this demand.
But is there Enough Energy?
The natural next question is whether there’s enough energy in the rotation of the Earth to achieve our ultimate goal.
The rotation of the planet stores ~2.138×1029J of energy (source).
It takes 6.24×107J to accelerate one kilogram to escape velocity (source).
Dividing the first by the second gives us the energy to lift 3.42×1021 kg.
Unfortunately, Earth’s mass is ~5.97×1024 kg., so we can only lift ~0.06% of the Earth’s mass using this method, before our planet stops rotating.
Now, this is not a total loss. With the Earth not rotating, we’ll have year-long day and night cycles, which will surely wreck the biosphere and kill off the majority of living things. But this wasn’t our goal, so it’s still a bad outcome.
The Moon to the Rescue
Not all is lost. Before we stop rotating altogether, when the Earth rotates once a month, we will tidally-lock to the Moon. We should plan ahead such that the cable is on the opposite side of the Earth from the moon (at the L3 point) at the moment this occurs.
(Note: We will have to figure out a way to keep the cable from intersecting with the Moon in the time leading up to this, but let’s assume we can add some steering to accomplish that. If we do lose the first cable to the Moon, we can build a new one at the Earth-Moon L3 point. Alternatively, we can use the methods described below to get rid of the Moon as a preliminary step.)
At this point, we can start extending the cable without slowing the rotation of the Earth, because the Moon will be exerting a force on the Earth to maintain that tidal lock.
The Moon Supplies the Energy
Now we’re stealing energy from the Moon to power our destruction of the planet. This energy comes from the rotation of the Moon around the Earth, which means the Moon’s orbit will degrade, and it will get closer toward the Earth, reversing a billions-year-old trend.
Paradoxically, this will actually cause the Moon-Earth tidally-locked system to rotate faster. This increase in rotation speed will help us reduce the length of the elevator and get rid of the Earth’s mass more quickly.
Is THAT enough energy?
There’s no need to do math here. The most energy we could possibly get out of this process is the energy it would take to put the Moon in its orbit, which means we could raise a Moon-size chunk of the Earth to an orbit the distance of the Moon. We’re talking 1.25% of the Earth’s mass — better than the first approach, but not by much. Plus, we want to achieve escape velocity, not just Moon-distance orbit.
Worse, we’re bringing a huge mass (the Moon) toward the Earth, so really, we’re not gaining anything.
But let’s keep this up, until the Earth and Moon are spinning rapidly about one another, and then the Moon crashes into the Earth. Apocalyptic times for everyone.
It’s doubtful much will survive on the planet at this point, but once the dust settles, the Earth will still exist as a planet, albeit much-reshaped, and rapidly-rotating.
Don’t Give Up
We hunker down and ride this process out. Then we restart with our matter-ejection method until the planet slows its rotation again.
There’s one more object we can use to acquire tidal lock – the Sun. We should make sure that we end up with our elevator cable passing through the Sun-Earth L2 point. The Earth will be rotating at a leisurely once-a-year pace, and the cable will have to be about two million miles long at this point. (Center of mass at L2, which is a million miles away.)
And, we start again, playing out the cable longer and longer.
This time, we’re stealing energy from the Earth’s rotation around the Sun, so we can’t lose. Either we get rid of all the Earth’s mass, or we crash the planet into the Sun (or get ripped apart inside the Sun’s Roche Limit). Either way, the Earth ends up destroyed by all the definitions we care about. I’ll leave calculating which happens first as an exercise for the reader.
The main lesson of this story is – don’t give up. We could’ve despaired after the first method didn’t suffice, or the second. But by persevering, we were able to achieve our ultimate goal.