Making gravity

What the game asks for

This is the field in sim.js, in the 0.65× Hollow, where the ground is 97.5 m from the centre.

• It points at the nearest ground, on all three shapes: the sphere, the Lens and Rama.
• It is 22 m/s² everywhere on the ground, 2.2 times Earth's, and the same at every latitude.
• It fades with height to nothing at the middle. A faint extra pull of up to 2% keeps pointing at the nearest ground, so nobody can hover in the middle for ever. Climbing to the middle costs 575 J per kilogram, as much as climbing 59 m on Earth.
• It is still, so nothing pushes you sideways when you run, and there are no weightless poles.
• The floating rocks have their own gravity wells, and near a rock the ground's pull fades.
• It can change. The world can morph into other shapes mid-match, and field events send bands of extra push across it at 24 m/s.

Four ways to make the pull

The Loom is the machine in the hull that makes and holds the fourth.

What the field is made of

In Einstein's theory, Gauss's law for gravity counts energy plus three times the pressure, a sum called the active mass (Ehlers and colleagues, 2005). Applied to our field, it says the air carries −3.1×10¹⁵ kg of active mass, densest 15–20 m above the ground, with 83% of it in the lowest 30 m. Negative active mass pushes things away from the middle, toward the ground.

The ground carries +3.1×10¹⁵ kg, so below it the two cancel. That is why the hull has no pull and the droids hold on to rails.

Is it possible?

Route 1: negative energy, ruled out at this size

• What it needs: −1.2×10⁹ kg/m³ at the peak, through a layer 20 m thick, held for ever.
• Quantum physics caps negative energy: the deeper the dip below zero, the shorter it can last. Averaged over a time t, the energy density can't go below −3ħ/(32π²c³t⁴) (Ford and Roman), which is about 4×10⁻⁷⁹ kg/m³ over one second. Our density could last about 10⁻²² s.
• The only lasting negative energy we know is the Casimir effect between metal plates, about −5×10⁻²¹ kg/m³ with the plates 1 µm apart, and only its force has been measured. To reach our density the plates would have to be 45 fm apart, the size of an atomic nucleus.
• By our estimate, light rays through the middle would also break the averaged null energy condition, which says the energy along a light ray's whole path can't add up to less than zero. It is proven for quantum fields in flat space (2016–17).

As with the Alcubierre warp drive, Einstein's equations allow it and quantum fields rule it out. No proof covers every field in curved space, so "ruled out" means by everything we know.

Route 2: tension, allowed but enormous

• Pressure is a source of gravity. Tension counts three times against the energy, so a medium whose tension equals its energy has negative active mass while its energy stays positive. This is how dark energy speeds up the universe's expansion (Riess and colleagues, 1998; Perlmutter and colleagues, 1999), and why domain walls repel (Ipser and Sikivie, 1984).
• The Loom holds a layer of false vacuum against the ground, empty space in a state with more energy than ordinary empty space. At the peak the layer holds 6×10⁸ kg/m³ of vacuum energy under equal tension: 10³⁵ times the universe's dark energy, and about 10⁸ times less than the difference between the vacuum inside a proton and outside it (in the bag model). Its energy is positive, so the quantum cap doesn't apply.
• It has to be held. A false vacuum decays (Coleman, 1977), and no still lump of a field holds itself together (Derrick, 1964). So the Loom holds the layer the way a trap holds an atom, and the Keeper rewrites any patch that falls out of place. Catching decays one by one can't work, because a decay bubble grows at nearly the speed of light.
• The tension needs a frame. In anything that stands still, tension must be matched by compression (von Laue's theorem), and the ground and the outer skin carry about half each. The frame's mass is the layer's energy divided by the frame's stiffness, so it must be the stiffest matter possible, with sound moving at nearly the speed of light (Zel'dovich, 1962), stiffer than even neutron-star cores (Bedaque and Steiner, 2015). The frame comes to about 1.6×10¹⁵ kg, a membrane tens of nanometres thick under the soil, and like the false vacuum it comes from the story.
• From outside, the world has weight. With the spare ground for morphs its energy comes to about 4.7×10¹⁵ kg, as much as an asteroid 15 km across. On the outer skin you'd feel about 32 m/s², more than on the meadow.

Building a world

The sphere's layer and frame hold at least 2.8×10³² J, the world's own mass times c², and with the spare ground for morphs (next section) a world holds 4.2×10³² J. No gain in efficiency can make that smaller.

The real limit is heat. Energy the Loom fails to store becomes heat, and it has to leave a world 195 m across before the world melts. So worlds are built in a yard with radiators about 100 km across, glowing at 2,000 K and giving off about 2×10¹⁶ W, and the build time depends on how little the Loom loses:

We take five centuries per world. The best transformers today lose about one part in a few hundred, so a Loom that loses 1 in 1.5 million comes from the story.

• The builders don't need all of the Sun. A yard draws about 2.7×10²² W while it charges, 1/14,000 of the Sun's output, so a partial swarm of collectors is enough. That is short of Kardashev Type II, which uses all of a star's power, and about 1.6 on Sagan's sliding scale.
• Fleets are built in parallel. With all of the Sun's output, about 14,000 yards could run at once and finish about 29 worlds a year.
• A failing world flashes. It releases its 4.2×10³² J as light, as bright as about a million Suns for one second. At Earth's distance that is 1.5×10⁹ J per square metre, and the side of Earth facing it would take about half the energy of the impact that killed the dinosaurs, so fleets orbit far from inhabited planets. That the energy comes out as light is a story choice.

Any shape

The ground's compression pulls like positive mass, and the air's tension pushes like negative mass, so together they make a gravitational double layer. Like a capacitor, a closed double layer of even strength pulls only between its two layers, whatever its shape (the solid-angle theorem of potential theory). That is exact only for a thin layer, and ours is 60 m thick, so the Loom corrects the difference. So the pull points at the nearest ground on a lens, a tube or a world with a waist.

The pull has a budget. By Gauss's law, pull times ground area is fixed by the charge, so during a match the Loom can only move pull around: making one patch heavier makes another lighter, and adding charge takes centuries.

Morphs

For drastic morphs every world keeps spare ground: half a sphere's worth of fully charged panels, each with its own layer and frame, folded against the outer skin into a closed shell that pulls nothing inside it. An unfolded panel brings its charge along, so the pull stays at 22 m/s² on any shape up to 1.5 times the sphere's ground area. The charge is only moved, so a morph makes no heat.

The Loom can also move charge between panels to make a shape heavier or lighter on purpose: pull = 22 m/s² × charge in use ÷ ground area, both counted in spheres, with at most 1.5 spheres of charge. So the sphere itself could pull up to 33 m/s².

Keeping the volume means the air isn't squeezed, and every shape here is under 1.5 spheres, so all of them can keep 22 m/s². The game's current Lens and Rama need 1.06 and 0.95.

• Where layers meet, their pulls add. In a narrow waist, opposite grounds pull against each other and leave a weightless bridge. At a lens's rim, loose things collect along the crease. In a lens thinner than 120 m, twice the layer's height, the two layers overlap across the whole middle.
• Everything in one place falls the same way. A gravity gun moves everyone and everything in its area, rounds included. Holograms are the one exception the story allows.

The world in layers

Rocks, gates, events

• Rocks. At 0.65× the big rock pulls with 20 m/s² near its surface, which takes about 1.6×10¹³ kg of positive active mass. As ordinary rock that would be denser than a white dwarf, so each rock is a store of energy and stiff matter that the Loom keeps. The Loom also thins the vacuum layer under each rock, which is why the ground's pull fades inside a rock's well.
• Field events. The ground alone can't make a push that peaks in mid-air, so a band of push in open sky carries its own sources: about 7.5×10⁸ kg/m³ of active mass for the starting sweep, positive ahead of the band and negative behind it. A sweep is a flying double layer, a small copy of the wall. The Loom takes its charge from the ground nearby, so the ground gets lighter first and warns players. At 24 m/s a sweep is about twelve million times slower than light, the fastest any change in the field can travel.
• Gates. Recent theory allows wormholes you can pass through, held open by a little quantum negative energy (Gao, Jafferis and Wall, 2017; Maldacena, Milekhin and Popov, 2018), and in all of them going through is never faster than going around. Ours meet that condition: light crosses the world in 0.65 µs, and a trip through a gate takes a whole frame. They still need negative energy in the throat, and the known designs are microscopic or need a hidden dark sector, so a gate a person can walk through comes from the story.
• Void swim and air dash. Arms push on almost nothing in air, so these moves are small fields that the Loom makes when it senses what you mean to do, as it already does for the skins.
• The Skylance. The wells pull its rounds three times harder than they pull bodies. That breaks the equivalence principle, which says everything falls alike and has been tested to one part in 10¹⁵. This is still open, and the explanation that fits the story is that the rounds are projections, like the guns, which the Loom bends on purpose.

Wheeler

Wheeler named or popularised most of this world's vocabulary: the geon, the wormhole, the black hole, and "spacetime tells matter how to move", which is the game's main mechanic in one line.

The builders believe his ideas. Wheeler drew the universe as a U that grows observers who look back at its beginning and, he argued, help make that beginning real. His 1989 "it from bit" says the same: "every it … derives its function, its meaning, its very existence … from the apparatus-elicited answers to yes or no questions, binary choices, bits."

Bits don't set the energy cost. Erasing a bit costs heat (Landauer, 1961), but charging a world takes the vacuum from one known state to another and erases nothing (Bennett, 1982), and when the Keeper resets a patch the bit costs about ten million times less than the energy the patch lost.

Decided and open

• decidedGates are wormholes that are never faster than light. Morphs may be drastic, and players should feel them.
• new hereThe frame, the Loom as a trap, five centuries per world, the double layer with spare ground, and field events that carry their own sources.
• new hereThe Lantern gives off the skin's sunlight, and the geon is dropped.
• dropped"It from bit" as the mechanism, the negative-temperature screen, and the earlier 490-year charge, which assumed negative energy.
• openWhether some shapes are heavier or lighter on purpose, and whether a failing world's flash is visible.
• openThe names: the Loom, the Keeper, the Skin and the Lantern. Also whether the ground mist should show players where gravity is, and the Skylance explanation above.