ENE-3: Quantum inequalities and the Alcubierre bill

Any L2 scheme at engineering scale; the bill is not merely large but larger than the universe.

Kind: constraint · Family: ENE (Energy) · Last reviewed: 2026-09-12

The result

Quantum inequalities: a quantum field can have negative energy density, but the magnitude times the duration squared is bounded, roughly by the Planck constant. The more negative, the shorter it lasts. Applied to the Alcubierre bubble, the wall must be thinner than a few hundred Planck lengths (about 10² v_b Planck lengths for their sampling choice), and for a 100 m bubble at v_b = 1 the total negative energy is about −6.2 × 10^62 kg, scaling with v_b: roughly ten orders of magnitude beyond the mass of the visible universe. A metre-thick wall would need about a quarter of a solar mass by the paper's order-of-magnitude constants, about half with exact ones, still unattainable

What it forbids or costs

Any L2 scheme at engineering scale; the bill is not merely large but larger than the universe

Escape hatches

Van Den Broeck's pocket geometry (ENE-4); the inequalities are derived for free fields and specific sampling functions, and interacting fields are less constrained

Who it bites

Physics proposals

• VIOLATES: Alcubierre 1994: the warp drive metric. Pfenning and Ford applied the quantum inequalities to this metric: the wall is at most about a hundred Planck lengths thick and a hundred-metre bubble needs $-6.2 \times 10^{62}\, v_s$ kg. Alcubierre invokes quantum field theory as the licence for negative energy, and the inequalities are the terms of that licence.
• VIOLATES: Everett and Roman 1997: the superluminal subway. The quantum inequality is applied in section 6: walls at most $\ell_P/\sigma^2 \approx 10^4\,\ell_P$ thick, and a total of order $10^{16}$ galactic masses per metre of tube. The paper is one of the entry's citations.
• VIOLATES: Krasnikov 1998: the tube laid on the way out. Krasnikov quotes Everett and Roman's inequality-based bill of $10^{32}$ galactic masses and does not dispute the calculation; his objection, developed in 2003, is that the flat-space inequality may not apply near an almost-formed time machine, which is an argument about the register entry's scope, not compliance with it.
• VIOLATES: Morris, Thorne and Yurtsever: the wormhole time machine (1988). Ford and Roman (1996) apply the quantum inequalities to the Morris–Thorne–Yurtsever wormhole parameters and find the exotic matter confined to a near-Planck band, which the construction does not accommodate.
• VIOLATES: Morris–Thorne traversable wormhole (1988). Ford and Roman (1996) apply the quantum inequalities to this class: the exotic matter must sit in a band only slightly wider than the Planck length, or the geometry's scales must differ enormously, both of which the paper's human-scale designs do not allow.
• VIOLATES: Natário 2002: warp drive with zero expansion. The energy density scales as $v_s^2 f'^2$, exactly as Alcubierre's, so the quantum-inequality bound on wall thickness applies unchanged; the paper offers no thicker wall and no smaller bill.
• VIOLATES: Visser's thin-shell wormholes (1989). The exotic matter is already a delta function, thinner than any quantum-inequality band, at a magnitude of a Planck mass per Planck length per edge; the inequalities' bound on magnitude times duration squared is exceeded by any static configuration of this kind.
• DODGES: Bobrick and Martire 2021: physical warp drives. Flattening by $\alpha_X$ cuts the Eulerian energy by $1/\alpha_X$ and lets the wall along $x$ approach Planck thickness so the Pfenning–Ford sampling inequality is met, at the price of an interior too thin for any observer; the paper admits the result still violates the averaged null condition and that such drives "more likely probe the limits of applicability of the quantum inequalities"
• DODGES: Lentz 2021: hyper-fast positive-energy solitons. Argues that "no such naturality conditions are known to restrict the stress-energy driving the positive energy solutions", so the Planck-thin-wall bound would not apply to a classically sourced bubble; once negative regions are present the bound returns and the paper has no answer to it
• DODGES: NASA Eagleworks warp field interferometer. Claims the bill falls "drastically" with thicker walls and further with bulk oscillation; the entry's quantum inequality is what forces the wall to be Planck-thin if the negative energy is quantum, and the documents assume classical negative energy in bulk without saying where it comes from
• DODGES: Van Den Broeck 1999: the pocket geometry. The wall bound $\Delta \leq 10^2 v_s L_P$ is accepted and the Ford and Roman inequality is verified in the neck with a ten-fold margin; the $-6.2 \times 10^{62}$ kg figure is evaded because it assumed the bubble radius equals the room inside, and the pocket breaks that assumption. The register lists this as ENE-3's escape hatch.
• DODGES: Warp drives in modified gravity. The quantum inequalities bound negative energy in quantum fields; if the source is a classical spin fluid or ordinary matter with positive density, they do not apply. The price moves to spin density and total mass, and no inequality on spin density is known.
• SATISFIES: The Scharnhorst effect. The static Casimir configuration is the standard case in which the quantum inequalities are respected; the effect asks for no more negative energy than that
• SILENT: Fell and Heisenberg 2021: positive energy from hidden geometric structure. Quantum inequalities are cited only for the Alcubierre bill; the paper assumes a classical source and, once its negative regions are admitted, never asks what supplies them or how thin the wall must be
• SILENT: Gao, Jafferis and Wall: traversable wormholes via a double-trace deformation (2017). The quantum inequalities are derived for free fields in flat space with timelike sampling; the paper does not address them, and whether they bound a one-loop AdS stress tensor under a boundary coupling is not settled in the literature.
• SILENT: Maldacena, Milekhin and Popov: traversable wormholes in four dimensions (2018), and the humanly traversable sequel (2021). The paper does not address the quantum inequalities; its negative energy is a static Casimir vacuum energy of a topologically non-trivial configuration rather than a transient negative pulse, and whether the flat-space inequalities apply to it is not settled in the literature.
• SILENT: The Mach effect thruster. If the negative-mass transient were real, the quantum inequalities would bound its magnitude and duration; the proposal treats it classically and never addresses them
• SILENT: White et al. 2021: the worldline-numerics Casimir pattern. Never computes the energy or wall thickness an Alcubierre bubble of the pillar's size would need at any speed, so the Pfenning–Ford bounds are never confronted; the estimate above shows they would not be met

Fictional drives

• SILENT: 2001: A Space Odyssey (1968): the Star Gate. No negative energy is admitted, so the quantum inequalities are not engaged.
• SILENT: Andromeda: the slipstream. No negative energy, so no inequality.
• SILENT: Battlestar Galactica: the FTL jump (2004 series). No mechanism is described to which the quantum inequalities could be applied; a ship-sized shortcut opened for an instant would be priced by them and the show says nothing.
• SILENT: Campbell's Islands of Space (1931): where the word came from. If the self-made hyperspace is read as a bubble, its wall is priced by the quantum inequalities; the novel has no wall thickness, no wall, only a coil.
• SILENT: Commonwealth: CST wormholes with trains through them (Hamilton 2004). A train-sized throat held open for centuries would be priced by the quantum inequalities; the work has no such calculation.
• SILENT: Contact: the Machine and the tunnel system (Sagan 1985, Zemeckis 1997). The quantum inequalities bound how much negative energy can exist for how long; a tunnel wide enough for a dodecahedron, open long enough for a day's transit, is priced by them and the work never mentions the price.
• SILENT: Cordwainer Smith: planoforming, Space² and Space³. No negative energy, so no inequality is engaged.
• SILENT: Cowboy Bebop (1998): the astral gates. No negative energy is admitted, so the quantum inequalities are not engaged.
• SILENT: Death's End: curvature propulsion (Liu Cixin, 2010). If the engine is a warp shell the quantum inequalities bound any negative energy in its wall; the novel does not say whether there is any.
• SILENT: Doctor Who: the TARDIS and the time vortex. No negative energy is invoked, so no quantum inequality is engaged.
• SILENT: Dune: the Guild navigators and folding space (Herbert 1965). The quantum inequalities would price a fold large enough for a heighliner; the work has no such calculation and no mechanism to apply it to.
• SILENT: Elite Dangerous: the frame shift drive (Frontier Developments, 2014). A ship-sized bubble is the case priced at universe-scale negative energy; never addressed.
• SILENT: EVE Online: the warp drive (CCP Games, 2003). A ship-sized wall of depleted vacuum is the case Pfenning and Ford priced; the lore gives no thickness or energy.
• SILENT: Event Horizon (1997): the gravity drive. No negative energy is admitted, so no quantum inequality is engaged; the drive's power budget is never shown.
• SILENT: Farscape (1999): wormholes, and starburst as a separate thing. No negative energy is admitted, so the quantum inequalities are not engaged.
• SILENT: FTL: Faster Than Light, the jump drive as a game mechanic. No negative energy, so no inequality.
• SILENT: Futurama: the Planet Express ship's dark matter engines (1999 onward). A ship-sized bubble is the case priced at universe-scale negative energy; never addressed.
• SILENT: Guardians of the Galaxy Vol. 2 (2017): jump points. No negative energy is admitted, so the quantum inequalities are not engaged.
• SILENT: Homeworld: hyperspace cores and the Far Jumpers. No negative energy, so no inequality.
• SILENT: Hyperion: the farcasters (Simmons 1989). A doorway held open for centuries would be priced by the quantum inequalities; the work has no such calculation.
• SILENT: Interstellar: the wormhole near Saturn (2014). The quantum inequality bound on negative energy density and duration is not addressed in the film; the wormhole is open for at least the decades between its appearance and the Endurance's passage.
• SILENT: Mass Effect: the mass relays (BioWare 2007 onward). A corridor thousands of light years long, open for millennia, would be priced by the quantum inequalities; the work has no such calculation.
• SILENT: No Man's Sky: the hyperdrive, warp cells and range. No negative energy, so no inequality.
• SILENT: Outer Wilds (2019): Nomai warp cores. No negative energy is admitted, so the quantum inequalities are not engaged.
• SILENT: Space Battleship Yamato: warp by wave motion engine. No negative energy, so no inequality.
• SILENT: Star Citizen: quantum drive and jump points. Negative energy is admitted for the bubble and never quantified; the quantum inequalities are not engaged.
• SILENT: Star Trek: warp drive (1966 onward). Bubble scale is ship scale, which is the case Pfenning and Ford priced at universe-scale negative energy; never addressed.
• SILENT: Starfield (2023): the grav drive. No negative energy is admitted, so the quantum inequalities are not engaged.
• SILENT: Stargate: the gate network (1994 film, SG-1 1997 onward). A disc several metres across held open for thirty-eight minutes is priced by the quantum inequalities; the show gives a duration and never a reason for it.
• SILENT: Stargate: the ships' hyperdrives. No negative energy, so no inequality. The ZPM is described as drawing on vacuum energy in a pocket of subspace-time, which is the closest the series comes, and it never says the sign.
• SILENT: Starman Jones: the congruencies (Heinlein 1953). No mechanism is described to which the quantum inequalities could be applied.
• SILENT: Stellaris: warp, hyperlane and wormhole (Paradox Development Studio, 2016). Warp bubbles at ship scale are the case priced at universe-scale negative energy; never addressed.
• SILENT: Super Dimension Fortress Macross (1982): the space fold. No negative energy is admitted, so the quantum inequalities are not engaged.
• SILENT: The Alderson drive (Niven and Pournelle 1974). No mechanism is described to which the quantum inequalities could be applied.
• SILENT: The Expanse: the ring gates and the slow zone (Corey 2013 onward). A ring a thousand kilometres across, open for years, is priced by the quantum inequalities; the work has no such calculation.
• SILENT: The Forever War: collapsar jumps (Haldeman 1974). The quantum inequalities would price a throat three kilometres across held open forever; the book predates them and never touches the question.
• SILENT: The Hitchhiker's Guide to the Galaxy: the Infinite Improbability Drive and bistromathics. No negative energy is invoked, so no quantum inequality bounds it.
• SILENT: The Orville: quantum drive (2017 onward). A ship-sized bubble is the case Pfenning and Ford priced at universe-scale negative energy; the show's dysonium tanks are never sized against it.
• SILENT: Thor (2011): the Bifrost. No negative energy is admitted, so the quantum inequalities are not engaged.
• SILENT: Wing Commander (1990) and the 1999 film: jump points. No negative energy is admitted, so the quantum inequalities are not engaged.
• SILENT: Xenocide: travel through the Outside. No negative energy, so no inequality.

Sources

• Ford and Roman, Phys. Rev. D 51, 4277 (1995), arXiv:gr-qc/9410043; Ford and Roman, Phys. Rev. D 55, 2082 (1997), arXiv:gr-qc/9607003; Pfenning and Ford, Class. Quantum Grav. 14, 1743 (1997), arXiv:gr-qc/9702026 ✓