Alien Math · follow-up to PG-019
The wormhole bill
A doorway through space one metre wide costs about minus 1.1 Jupiters of energy. The minus sign is the whole problem: energy anyone can collect is positive, and no amount of it converts.
The bill
The wall, pinned
The record for PG-019 names a materials wall: the lens plate n(ρ) = 1 + (a/2ρ)² needs n → ∞ at the centre and n = 2 at the throat, and printed pillars stop at n = 1.34, so the plate only lenses for b ≥ 1.15a. The energy wall is the same Ellis metric built in real gravity, which that record never priced. Here it is priced; the numeric integral matches the closed form to 15 digits.
| The bill (for throat radius a = 1 m) | Value | Scales as |
|---|---|---|
| Ellis throat, total negative energy (−πc⁴a/2G) | −1.9×1044 J = −1.1 Jupiter masses | linear in a |
| Visser thin-shell (flat cut-and-paste), negative mass | −1.4 Jupiter masses | linear in a |
| Ford–Roman quantum inequality | negative energy must sit in a band "many orders of magnitude smaller than the throat" | tighter for bigger throats |
| Casimir energy density at 10 nm gap | −4.3×104 J/m³ | 1/d⁴ |
| Casimir volume needed at 10 nm | 4.4×1039 m³ ≈ 3 trillion Suns | |
| Positive rest energy of the plates (graphene) vs Casimir negative energy | 1.6×1014 : 1, net positive | |
| Sun's full output to collect 1.9×1044 J (positive) | 16 billion years |
The last two rows carry the argument. Energy you can harvest is positive; the plates that make Casimir negative energy outweigh it by 14 orders of magnitude. The budget is a sign constraint (the null energy condition), so Kardashev scale alone does not buy it.
Seven ways round
Routes to the budget, ranked
The routes that dodge the sign exist on paper. Each one trades the sign problem for a new one: unknown physics, a black hole of ten thousand Suns, or no faster-than-light at all.
Ranked by the chance the route ever yields a throat something can go through. "Gap" is an order-of-magnitude call. Figures marked ~ are from memory and were not re-read for this page.
| # | Route | What would need to be true | Gap today | Known no-go | Odds call |
|---|---|---|---|---|---|
| 1 | Find one, don't build one (primordial / relic wormholes) | Early-universe topology left throats around; we spot them by their lensing | Unknown existence; microlensing searches set upper limits only (~) | None in principle; needs exotic matter to have stayed put | Low, but the only route with zero new tech |
| 2 | Maldacena–Milekhin "humanly traversable" | Randall–Sundrum extra dimension + a dark U(1) + two extremal magnetic black holes joined; fermion Casimir energy pays the sign | Needs that dark sector (nobody has seen one) and black holes with r_e > 1.5×107 m, about 104 Suns | Outside clocks: tens of thousands of years, so no shortcut | Best on-paper physics; a 1 s ride for the traveller |
| 3 | Gao–Jafferis–Wall coupled throats (teleportation) | Couple two entangled black holes (or two qubit registers) so a signal crosses | Works now as a quantum circuit; carries qubits; people stay home | Never beats the classical channel it rides on | Certain for information, zero for travel |
| 4 | Modified gravity (Gauss–Bonnet, f(R), braneworld) | Extra curvature terms supply the NEC violation, ordinary matter stays positive | No sign of those terms at the needed strength; GW170817 fixed c_gw = c to ~10-15 | Kills many scalar–tensor versions | Low |
| 5 | Dirac-fermion + Maxwell wormholes (Blázquez-Salcedo, Knoll, Radu 2021) | Charged fermions alone hold the throat open | Needs charge-to-mass far beyond any known particle; throats microscopic (~) | Smoothness at the throat disputed (Bolokhov et al. 2021, ~) | Low |
| 6 | Engineered negative energy (Casimir, squeezed light) on a Visser thin shell | Make, hold and place −1.4 Jupiter masses per metre in a Planck-thin band | Net sign wrong by ~1014; band thickness set by Ford–Roman | Quantum inequalities; plate rest mass | Effectively nil |
| 7 | Kardashev brute force (Dyson swarm, galactic output) | Positive energy can be turned into negative energy | Energy available (16 Gyr of Sun, 55 days of a galaxy); conversion impossible | NEC; averaged NEC theorem (Graham–Olum) | Nil on its own; multiplies any route that solves the sign |
Pinch-zoom near home
Warp shells, priced
To pinch space around a ship locally, the cheapest solution anyone has published that obeys every energy condition is a shell of 2.4 Jupiter masses, 10–20 m across, that goes slower than light and cannot accelerate itself. Its average density is 1.5×1023 kg/m³, about 650,000× nuclear density, and its inner wall sits at 1.5× its own Schwarzschild radius (6.7 m). Faster than light still needs negative energy in every version checked.
| Metric | Speed | Energy needed | Catch |
|---|---|---|---|
| Alcubierre 1994 | Any, including > c | Negative; Pfenning–Ford wall ≤ ~100 v Planck lengths, total far above the visible universe's mass (~1010×, as usually quoted) | Crew can't signal the bubble's front; Hawking-like blueshift pile-up at the front wall (Finazzi–Liberati–Barceló 2009) |
| Van Den Broeck 1999 | > c | Negative, cut to a few Suns (~) | Same sign problem, Planck-scale geometry |
| Natário 2002 | > c | Negative, zero volume expansion | Shows the "squeeze ahead, stretch behind" picture is optional; the sign isn't |
| Lentz 2021 / Fell–Heisenberg 2021 | > c claimed | Positive for Eulerian observers | Santiago, Schuster & Visser: generic warp drives violate the NEC for some observer (~) |
| Bobrick & Martire 2021 | < c (positive), > c (negative) | Positive subluminal; negative cut 100× by shaping | Every warp drive is a shell that needs propulsion |
| Fuchs et al. 2024 | < c, constant | Positive: 4.49×1027 kg, R1 = 10 m, R2 = 20 m, shift β = 0.02 | No speed-up mechanism; a near-black-hole shell |
| Plain gravity (time pinch) | n/a | Any mass | Real today: GPS clocks gain 38 μs/day; it zooms time, not distance |
| Acoustic analog (PG-019 family) | > "c" for sound | A pump | Only sound sees the warp; this is what the lab can build |
In short: for slower-than-light pinch-zoom, Jupiter-class mass compressed to near-black-hole density plus a separate thruster. For faster than light, the same sign wall as the wormhole.
Proving Ground habit
Steelman, hole, attack
The strongest route is the one that needs no builder: if throats exist, they show up as Ellis lenses, and our plate already computes that lens exactly.
| Route | Steelman | Hole | Attack within reach |
|---|---|---|---|
| Find one | Nature makes what we can't; the Ellis lens has a unique light-curve shape (demagnification dip) | Any relic throat needs its own exotic support to survive 13.8 Gyr | Fit Ellis-lens curves to public microlensing data; reuse wormhole_lens.py's exact deflection |
| Maldacena–Milekhin | No exotic classical matter; Standard-Model-style fermions in a magnetic field give a 2D Casimir energy −c₂/(8ℓ) that pays the sign | Rests on a dark sector nobody has seen; 104-Sun magnetic black holes; outside clocks run 104 years | Compute the 2D Casimir energy on a ring and check the sign and the 1/ℓ law numerically |
| Gao–Jafferis–Wall | Shown inside AdS/CFT; already run as circuits (Jafferis et al. 2022) | The 2022 "wormhole" signal was disputed (Kobrin et al. 2023, ~); it is teleportation with geometry on top | Extend PG-021 on ibm_fez: a size-winding teleportation circuit with a pre-registered fidelity |
| Modified gravity | The NEC is a statement about matter; change the geometry side and matter can stay ordinary | Theories that do this tend to be ghosts or already ruled out by gravitational-wave speed | Literature-only for now; flag which survive GW170817 |
| Dirac wormholes | Ordinary-looking spin-½ fields, no phantom field | Charge-to-mass needed is unphysical; throat smoothness disputed | Reproduce their ODE system; check the throat matching condition we'd need |
| Casimir / squeezed vacuum | Negative energy density has been measured | Quantum inequalities: the more negative, the shorter and thinner; plates outweigh it 1014:1 | A Ford–Roman bound calculator: for a target throat, print the allowed band and duration |
| Kardashev | Energy is no limit: a galaxy makes 1.9×1044 J in 55 days | You cannot turn positive energy into negative energy | None: this is a multiplier |
On the bench
Runs within reach
A pilot result first. Its predictions were never locked, so it seeds a card and grades nothing. The lab ray-traced sound through an acoustic Alcubierre bubble, which is just a moving-fluid bubble, since the Alcubierre metric with c = 1 is exactly Unruh's acoustic metric. Below the speed of sound the forward ray's wavenumber grows only e0.7 to e2.3 over the run. Above it the ray freezes at a horizon on the front wall and blueshifts without limit, e100 to e240, at a rate within 5% of the wall's surface gravity. That is the Finazzi–Liberati–Barceló instability, reproduced in a bucket of moving water.
| Bubble speed (× sound) | Horizon? | ln(blueshift) over t = 20 | Fitted rate | Predicted κ |
|---|---|---|---|---|
| 0.5 | No | 0.7 | n/a | n/a |
| 0.9 | No | 2.3 | n/a | n/a |
| 1.5 | Yes, front wall | 102 | 5.1 | 5.3 |
| 2.0 | Yes | 152 | 7.6 | 8.0 |
| 5.0 | Yes | 243 | 12.1 | 12.8 |
Proposed card PG-028 "Pinch-zoom budget", predictions written before any run:
- Labyrinth plate reaches the throat (the real wall in the record). Coiled-channel unit cells give n = 2 at ρ = a/2. Prediction: a full-wave sim of a coiled-cell plate shows the capture shadow of radius a within 10%, below the cell's homogenisation limit. Fails if the shadow smears beyond 20%.
- Acoustic warp bubble, full wave equation. Prediction: front-wall blueshift rate matches κ within 5% for speeds 1.5–5; adding Bogoliubov dispersion caps the pile-up at k ≈ 1/ξ. Fails if the cap is absent or the rate is off by more than 10%.
- Ellis-lens finder. Injection and recovery on synthetic microlensing curves. Prediction: Ellis versus point-mass lens told apart at Δχ² > 25 for impact u₀ < 1 at 1% photometry. Then point it at public survey light curves.
- Ford–Roman calculator. Prediction: for a 1 m throat the allowed negative-energy band comes out under 10-20 m. Fails if it comes out macroscopic.
- Maldacena–Milekhin ring check. Prediction: free fermions on a ring give Casimir energy −c/(12·circumference)-type scaling (their −c₂/8ℓ), sign negative, 1/ℓ within 1%.
- Printed pillar plate (already built, ungraded): exit angles follow 2K(a/b) − π for b ≥ 1.15a.
Items 1 and 2 are the ones that move the sim forward; 3 is the only one that could ever find a real throat.
Sources
Sources
Opened for this page: Fuchs et al. 2024, constant-velocity physical warp drive · Maldacena & Milekhin, humanly traversable wormholes · Pfenning & Ford, the unphysical nature of warp drive · Ford & Roman 1996, QFT constrains traversable wormholes · Bobrick & Martire 2021, introducing physical warp drives.
From memory, with no re-read (the ~ marks above): Alcubierre 1994; Van Den Broeck 1999; Natário 2002; Lentz 2021; Fell & Heisenberg 2021; Santiago, Schuster & Visser 2022; Blázquez-Salcedo, Knoll & Radu 2021 and Bolokhov et al. 2021; Gao, Jafferis & Wall 2017; Jafferis et al. 2022 and Kobrin et al. 2023; Graham & Olum 2007; Finazzi, Liberati & Barceló 2009; Ellis-lens microlensing limits.
The lab's own numbers come from two short scripts, one for the bill and one for the ray trace.
This follows the wormhole lens in Every loophole sends a bill. The older interactive room is the wormhole room, and the card it grew from sits on the Proving Ground board.