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Warp drives in modified gravity

Torsion, conformal gravity and other extensions move the warp bill from the matter side of the field equations to the geometry, and what each move costs.

Kind: physics · Loophole: L2 · Standing: K3 · Bill: B-neg, B-new, B-mass, B-boot, B-caus · Last reviewed: 2026-09-12

The claim

Every theorem that prices a warp drive in negative energy is a theorem about the source of general relativity: the Einstein equations turn a statement about geometry (light rays converge) into a statement about matter (energy density is positive), and the warp metrics violate the geometric statement. The claim of this family of proposals is that general relativity is the wrong theory to price the drive with. Gravity has been tested where torsion, higher-curvature terms and extra scalar fields are negligible; in a theory that agrees with Einstein's everywhere it has been tested and differs where curvature or spin density is extreme, the same Alcubierre metric can be sourced by matter that obeys the energy conditions, because the extra sector absorbs what in general relativity would have to be negative.

Two peer-reviewed papers make this claim with worked mathematics. DeBenedictis and Ilijic (2018) show that in Einstein-Cartan gravity, general relativity supplemented with torsion coupled to spin, the Alcubierre and Van Den Broeck metrics can be sourced by a Weyssenhoff spin fluid whose stress-energy satisfies the weak and null energy conditions everywhere, with the ship still in free fall since torsion vanishes outside matter. Varieschi and Burstein (2013) show that in Mannheim's conformal (Weyl) gravity, a fourth-order theory, the Alcubierre metric with a suitable shaping function has non-negative energy density for every Eulerian observer at speeds up to about $2.5c$ and needs no exotic matter. In its strongest form the claim is that the exotic-matter objection to warp drives is not a fact about spacetime but an artefact of second-order, torsion-free field equations, and that the objection therefore stands or falls with general relativity's completeness at Planck-scale wall densities, which nobody has tested.

What the proponents do not claim: that the horizon, the bootstrap or the closed timelike curves go away (DeBenedictis and Ilijic say explicitly they do not address them); that the required spin densities or coupling constants are attainable; that conformal gravity or Einstein-Cartan theory is established. They claim that the bill can be moved. This dossier grades where it moves to.

Origin and lineage

The idea is as old as the energy-condition objection. Alcubierre's 1994 letter says exotic matter is needed "just as it happens with wormholes"; the traversable wormhole literature then spent two decades showing that in extended theories of gravity the matter threading a throat can be ordinary while "the effective stress-energy tensor containing higher order curvature derivatives" carries the null energy condition violation, in f(R) gravity (Lobo and Oliveira 2009), in Brans-Dicke theory and in curvature-matter couplings [HIGH] S1. The general point that energy conditions in extended theories apply to an effective fluid mixing matter and geometry, and that it is "somewhat misleading" to read them as conditions on matter, is made by Capozziello, Lobo and Mimoso (2015) for scalar-tensor, f(R) and Gauss-Bonnet gravity [HIGH] S1.

For warp drives specifically the record is thin. Varieschi and Burstein, Loyola Marymount, "Conformal gravity and the Alcubierre warp drive metric", ISRN Astronomy and Astrophysics 2013, 482734, arXiv:1208.3706 [HIGH] S1. DeBenedictis (Simon Fraser) and Ilijic (Zagreb), "Energy condition respecting warp drives: the role of spin in Einstein-Cartan theory", Class. Quantum Grav. 35, 215001 (2018), arXiv:1807.09745 [HIGH] S1. A 2026 preprint on ScienceOpen recomputes the Einstein-Cartan thresholds in closed form and compares them with nuclear matter [LOW] S1. Santiago, Schuster and Visser (2022) supply the general objection (ENE-5). Searches of arXiv on 2026-09-12 for "warp drive" together with f(R), Brans-Dicke, scalar-tensor, Gauss-Bonnet, teleparallel or f(T) returned no paper applying a warp metric in any of those theories; the only "warp drive" and "modified gravity" hit is Santiago, Schuster and Visser's abstract. That absence is a finding: the f(R) and Brans-Dicke warp drives that get invoked in conversation do not exist in the literature, only their wormhole cousins do. Three adjacent items are recorded for completeness: Rodal (2025) on why a spatially varying gravitational coupling becomes a scalar-tensor theory constrained by the Solar System [LOW] S1; Jusufi and Lobo (2026) on a T-duality-regularised Alcubierre profile that keeps its exotic matter [LOW] S1; and Joseph (2022) on a Bohmian-gravity reformulation, on physics.gen-ph without a journal [LOW] S1. The register lists modified gravity as an escape hatch of ENE-2; this is the dossier that scores the hatch.

The mechanism

Units $G = c = 1$ unless stated. The general move. Most extended theories can be rearranged as

$$G_{\mu\nu} = \kappa\,T^{\mathrm{eff}}_{\mu\nu} = \kappa\bigl(T^{\mathrm{matter}}_{\mu\nu} + T^{\mathrm{geom}}_{\mu\nu}\bigr),$$

where $T^{\mathrm{geom}}$ collects torsion-squared terms, higher derivatives of curvature or scalar-field stresses. The Alcubierre geometry fixes the left side. General relativity says the right side is $T^{\mathrm{matter}}$ and it is negative in the wall. An extended theory lets $T^{\mathrm{matter}}$ be positive provided $T^{\mathrm{geom}}$ is negative enough there. Santiago, Schuster and Visser state the limit of this: energy conditions are, through the field equations, "typically immediately converted into purely geometrical convergence conditions", the null convergence condition $R_{ab}\ell^a\ell^b \geq 0$ and the timelike convergence condition $R_{ab}V^aV^b \geq 0$, and "even in modified gravity, physically reasonable warp drives will still violate the purely geometrical null convergence condition and the timelike convergence condition which, in turn, will place very strong constraints on any modified-gravity warp drive" [HIGH] S1. The Raychaudhuri equation does not know which theory sourced the metric. Whatever theory one adopts, the Alcubierre wall makes null rays diverge where general relativity would need negative energy to make them do so, and the theorems of ENE-2 (Olum's, Gao and Wald's) are proved from that focusing, with the energy condition entering only as the general-relativistic name for it. Modified gravity therefore buys a positive matter sector at the cost of a geometric sector that carries the negative part, and the question for each theory is what that sector is made of and what it costs.

Einstein-Cartan. The connection carries torsion $T^\alpha{}_{\beta\gamma}$; varying the action with respect to the metric and the contorsion gives two equations, one relating the full Einstein tensor and the modified torsion (superpotential) $S^{\alpha\beta\gamma}$ to the stress-energy, the other setting $S^{\alpha\beta\gamma} = \kappa\,\tau^{\alpha\beta\gamma}$ with $\tau$ the spin tensor. Torsion therefore vanishes wherever spin vanishes: outside matter the theory is general relativity and test particles follow its geodesics [HIGH] S1. For a Weyssenhoff fluid, $\tau_{\alpha\beta}{}^{\gamma} = \tau_{\alpha\beta}u^\gamma$ with the Frenkel condition $\tau_{\alpha\beta}u^\beta = 0$, and after macroscopic averaging of unpolarised spins the quadratic terms survive, giving

$$G_{\mu\nu}\bigl[\{\}\bigr] - \kappa^2 s^2\bigl(-2u_\mu u_\nu - g_{\mu\nu}\bigr) = \kappa\,T_{\mu\nu}, \qquad s^2 := \tau_{\alpha\beta}\tau^{\alpha\beta},$$

with $G_{\mu\nu}[\{\}]$ the Einstein tensor of the Christoffel connection, that is, the ordinary Einstein tensor of the Alcubierre metric [HIGH] S1. The weak and null conditions on the matter then read

$$T_{\mu\nu}v^\mu v^\nu = \frac{1}{\kappa}\Bigl[G_{\mu\nu}\bigl[\{\}\bigr] - \kappa^2 s^2\bigl(-2u_\mu u_\nu - g_{\mu\nu}\bigr)\Bigr]v^\mu v^\nu \;\geq\; 0 .$$

The first term is the general-relativistic energy density, negative in the wall. The second is $\kappa s^2\,\chi_{\mu\nu}v^\mu v^\nu$ with $\chi_{\mu\nu} = 2u_\mu u_\nu + g_{\mu\nu}$, and for normalised isochronous timelike $u$, $v$ this equals $2(u_\mu v^\mu)^2 - 1 \geq 1 > 0$ by the reversed Cauchy-Schwarz inequality, with the null case following. So "the WEC may be made positive for sufficiently large $s^2$": that is the whole theorem, and it is exact [HIGH] S1. Two caveats the authors record: a perfect or anisotropic fluid is algebraically incompatible with the warp metric's Segre type, so a spinless auxiliary structure must be added to the matter, and $T_{\mu\nu}$ is otherwise "whatever is required in order to create the warp drive". The fluid four-velocity mimics the ship's but is not in free fall.

The numbers, minimising $s^2$ over the worst observer at each point for $v_s = 1.2$: the peak requirement is $\kappa^2 s^2 \approx 5\ \mathrm{m}^{-2}$, an angular momentum density $L \approx 3.6 \times 10^{34}\ \mathrm{kg\,m^{-1}\,s^{-1}}$, or about $3.5 \times 10^{68}\,\hbar$ per cubic metre, which for spin-$\hbar$ carriers is $3.5 \times 10^{68}$ particles per cubic metre; the accompanying Eulerian energy density peaks near $0.3\ \mathrm{m}^{-2}$, about $3.64 \times 10^{43}$ J per cubic metre or a mass density of $4 \times 10^{26}$ kg per cubic metre [HIGH] S1. The authors' aside: "As is often the case with exotic solutions in gravitational field theory, the devil is in the details." A monochromatic photon field carrying that spin would need a field of order $9 \times 10^{22}\sqrt{\omega}$ V/m and would lose U(1) gauge invariance by coupling to torsion; high-spin composites are unstable and too massive; fundamental high-spin fields are problematic to couple [HIGH] S1. For the Van Den Broeck pocket with $P = 3$ fm, $\tilde P = 1$ fm, $\alpha = 5$, the peak is $\kappa^2 s^2 \approx 150\ \mathrm{fm}^{-2}$, $L \approx 1.9 \times 10^{35}$ kg per femtometre per second, about $10^{84}$ spin-one particles per cubic metre, with a mass density of about $8 \times 10^{11}$ kg per cubic femtometre, in a much smaller volume [HIGH] S1. The 2026 preprint's closed-form threshold for $R = 10$ m, $\sigma = 100\ \mathrm{m}^{-1}$, $v_s = 0.5c$ is about $4.0 \times 10^{35}\ \mathrm{kg\,m^{-1}\,s^{-1}}$, which it puts at 25.7 orders of magnitude beyond fully polarised nuclear-saturation matter [LOW] S1.

Conformal gravity. The action is the square of the Weyl tensor, $I_W = -\alpha_g\int d^4x\,\sqrt{-g}\,C_{\lambda\mu\nu\kappa}C^{\lambda\mu\nu\kappa}$, invariant under local rescalings $g \to \Omega^2 g$, and the field equations are Bach's, $W_{\mu\nu} = T_{\mu\nu}/4\alpha_g$, with the Bach tensor

$$W_{\mu\nu} = -\tfrac{1}{6}g_{\mu\nu}R^{;\lambda}{}_{;\lambda} + \tfrac{2}{3}R_{;\mu;\nu} + R_{\mu\nu}{}^{;\lambda}{}_{;\lambda} - R_\mu{}^\lambda{}_{;\nu;\lambda} - R_\nu{}^\lambda{}_{;\mu;\lambda} + \tfrac{2}{3}R\,R_{\mu\nu} - 2R_\mu{}^\lambda R_{\lambda\nu} + \tfrac{1}{2}g_{\mu\nu}R_{\lambda\rho}R^{\lambda\rho} - \tfrac{1}{6}g_{\mu\nu}R^2,$$

fourth order in the metric [HIGH] S1. The stress-energy is read off as $T_{\mu\nu} = 4\alpha_g W_{\mu\nu}$ for the Alcubierre metric with two shaping functions: Alcubierre's tanh profile and a polynomial "Hartle shaping function" with parameter $m = 4$. In general relativity both give Alcubierre's negative $T^{00}$. In conformal gravity with $\alpha_g = 1$ the tanh profile gives a $T^{00}$ that is "in part positive and in part negative, thus still violating the WEC", while the Hartle profile gives a $T^{00}$ that is "completely non-negative" for $v_s = 1$, $R = 1$; scanning $v_s$ from $0.25c$ to $3c$, the density stays non-negative up to $v_s \approx 2.5c$ and develops negative lobes beyond [HIGH] S1. The strong condition is satisfied, the dominant condition is violated in the case examined. Integrating over a bubble of $R = 100$ m at $v_s = c$ gives $E = 1.86 \times 10^{10}\,\alpha_g$ erg; with the only published value of the coupling, Mannheim's $\alpha_g = 3.29 \times 10^{82}$ erg s, this is $E = 6.12 \times 10^{92}$ erg $= 3.42 \times 10^{38}$ solar masses of ordinary matter, which the authors note might be reduced by a factor of $10^{68}$ to $10^{80}$ if the microscopic coupling differs from the macroscopic one [HIGH] S1. The theory itself is contested: Flanagan argues its conformal invariance is an artefact of the variables and that in the weak-field, slow-motion limit it disagrees with general relativity and is ruled out by Solar System observations [MED] S1, a conclusion Mannheim disputes in the literature the paper cites.

What the mechanism shows in one line. In both theories the Alcubierre metric is unchanged, so every constraint that is about the metric (horizon, bootstrap, closed timelike curves, hazards, the null convergence condition) is unchanged, and the negative quantity that general relativity calls energy density is renamed: spin-squared torsion stress in one theory, Bach-tensor curvature stress in the other.

What it costs

B-neg. The geometric null convergence condition is violated regardless of theory, so the effective source is negative in the wall in every theory; the matter is positive only because a geometric term has been given the sign general relativity assigned to the matter. In the specific models the conditions are not even fully rescued: conformal gravity violates the dominant condition in the case computed, and the Einstein-Cartan spin fluid needs an auxiliary spinless structure whose stresses are "whatever is required" (ENE-1, ENE-5).

B-new. Torsion coupled to spin, or a fourth-order conformally invariant gravitational action, neither of which is in general relativity or has been observed; in the Einstein-Cartan case the carrier of $10^{68}\,\hbar$ per cubic metre has no candidate the authors find satisfactory, and a photon carrier would break U(1) gauge invariance.

B-mass. Spin densities of $10^{68}\,\hbar$ per cubic metre at mass densities of $4 \times 10^{26}$ kg per cubic metre for a $v_s = 1.2$ Alcubierre bubble, $10^{84}$ spin-one particles per cubic metre for the pocket; or $3.4 \times 10^{38}$ solar masses of ordinary matter in conformal gravity at the published coupling. These are positive but not payable.

B-boot. The horizon is a property of the metric and DeBenedictis and Ilijic acknowledge it remains; the wall matter's leading edge is spacelike for $v_s > 1$ whatever sources it (CON-1, CON-2).

B-caus. Everett's two-bubble loop uses only the metric (CAU-3); nothing in either theory picks a frame.

Constraint scoring

Every entry in the register, one row each, verdict from the physics vocabulary in GRADING.md, and a note that says why.

ConstraintVerdictNote
CAU-1VIOLATESThe metrics are Alcubierre's and Van Den Broeck's unchanged; effective superluminal transport between flat-space points with no preferred frame is exactly the signal CAU-1 prices, and changing the field equations changes nothing about the causal structure.
CAU-2SILENTNeither paper mentions chronology protection; whether semiclassical back reaction in a theory with torsion or fourth-order gravity would forbid the two-bubble loop is unexamined.
CAU-3VIOLATESEverett's two-bubble construction is a statement about the metric and a boost, and the metric is unchanged.
CAU-4SILENTNo preferred frame is proposed in either theory; torsion is localised in matter and conformal gravity is generally covariant.
CAU-5N/ANo quantum signalling.
ENE-1DODGESThe matter stress-energy is made to satisfy the weak and null conditions (Einstein-Cartan) or the weak and strong conditions (conformal gravity) by assigning the negative part to torsion or Bach-tensor terms; Capozziello, Lobo and Mimoso show this is what energy conditions mean in extended theories. The register's conditions are on matter, and matter now obeys them; the effective source does not.
ENE-2DODGESENE-2 lists modified gravity as an escape hatch and this is the hatch. But Olum's and Gao and Wald's theorems are proved from the focusing of null rays, with the energy condition entering as its general-relativistic name, and Santiago, Schuster and Visser note the null convergence condition is violated in any theory; what is dodged is the price tag, not the focusing.
ENE-3DODGESThe 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.
ENE-4SATISFIESDeBenedictis and Ilijic carry the Van Den Broeck pocket as their second case and find ENE-4's trade reproduced in spin: a higher peak density (about $10^{84}$ spin-one particles per cubic metre) in a far smaller volume.
ENE-5DODGESThe theorem is proved in general relativity for the Natário class. The same paper's abstract says the geometric convergence conditions are violated even in modified gravity, so the escape is narrow: a modified-gravity warp drive must accept a geometric sector that focuses like negative energy.
ENE-6N/ANo negative energy is invoked; the point of the class is to source the metric with positive matter or spin, so the Casimir floor does not bear.
ENE-7N/AWarp metrics of trivial topology, whichever theory sources them; no handle, so the censorship theorem does not bear. The escape the class attempts is from ENE-2 and is scored there; ENE-7 is ENE-2 for wormholes.
CON-1VIOLATESThe horizon at the wall for $v_s > 1$ is a property of the metric; DeBenedictis and Ilijic state that warp drive spacetimes "contain an effective horizon" and that they did not address it. Torsion vanishes outside matter, so nothing in the ship's causal reach changes.
CON-2VIOLATESThe wall matter's leading edge is spacelike for $v_s > 1$ regardless of what sources it (Krasnikov's argument is about where the stress-energy is, not what it is), so the drive must be laid along the route in advance in these theories too.
CON-3N/ANo tube geometry is examined in any extended theory.
STA-1SILENTThe horizon is unchanged and Hiscock's divergence uses only the metric and a quantum field on it, so it is expected to carry over, but no renormalised stress-energy has been computed for a warp bubble in Einstein-Cartan or conformal gravity.
STA-2N/ANo throat.
STA-3SILENTThe two-bubble chronology horizon is neither constructed nor analysed in these theories.
HAZ-1VIOLATESMcMonigal, Lewis and O'Byrne's sweeping-up and burst are kinematics of test particles on the Alcubierre metric, and in Einstein-Cartan theory test particles outside matter follow the same geodesics; the hazard is inherited unchanged.
HAZ-2SILENTThe Hawking flux temperature is set by the wall thickness; with a classical spin-fluid source the ENE-3 wall bound need not hold, so the wall could be thick and the temperature low, but no calculation exists.
HAZ-3N/ANot a throat.
LOR-1SATISFIESThe ship is on a timelike geodesic of the unchanged metric, and in Einstein-Cartan theory the torsion, localised in matter, does not affect it.
LOR-2VIOLATESThe spacelike leading edge of the wall matter for $v_s > 1$ is a statement about the metric; in Einstein-Cartan theory that matter carries the spin, so it is the spin fluid itself that would have to move tachyonically on a first flight.
LOR-3N/ANo medium.
LOR-4N/ANot relevant.
WRP-1SATISFIESThe metrics studied are Alcubierre's and Van Den Broeck's; the class definition is unchanged by the choice of field equations.
WRP-2DODGESBobrick and Martire's classification of superluminal shells as exotic is a general-relativistic statement; in Einstein-Cartan theory the shell is ordinary matter with spin. The classification is evaded by changing the theory, which the entry did not contemplate.
WRP-3DODGESWRP-3's dated score is implicitly within general relativity. DeBenedictis and Ilijic (2018) is a peer-reviewed superluminal ($v_s = 1.2$) warp drive whose matter satisfies the weak and null conditions, in Einstein-Cartan theory; it evades the score by changing the theory and pays in $10^{68}\,\hbar$ per cubic metre of spin. The register entry should say so.
MAN-1N/AFour dimensions, no bulk in either theory.

Status of the argument

Sources