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Natário 2002: warp drive with zero expansion

The general warp drive class, and a member of it that neither contracts nor expands space and still needs negative energy.

Kind: physics · Loophole: L2 · Standing: K4 · Bill: B-neg, B-mass, B-boot, B-caus · Last reviewed: 2026-09-12

The claim

The usual story of the Alcubierre drive, that space is contracted ahead of the bubble and expanded behind it and that the bubble moves in response, is not what makes it work. Natário's claim is that a warp drive is any globally hyperbolic spacetime built from flat Euclidean slices with unit lapse and a time-dependent shift vector field $X$ on Euclidean 3-space, and that the bubble moves because $X$ carries it: the region where $X$ vanishes (the interior) slides through the region where $X$ equals the ship's velocity (the exterior, in ship-fixed coordinates), and whether space ahead is contracted or not is a detail of the particular $X$. To prove it, he constructs a divergence-free $X$, so that the expansion of the Eulerian volume elements is identically zero everywhere, and shows the resulting spacetime carries a ship at arbitrary speed exactly as Alcubierre's does. Along the way he proves that every non-flat member of the class violates the weak or the strong energy condition, gives the general formula for the energy density on flat slices, and shows that any member with $v_s > 1$ has a horizon, a visibility horizon and infinite blueshift at the horizon, whatever the choice of $X$.

The claim is therefore two things: a definition and a counterexample. The definition has become the standard one; the register's WRP-1 entry adopts it and Santiago, Schuster and Visser's 2022 theorem is proved for it. The counterexample removes the heuristic without removing the bill. Natário does not claim the zero-expansion drive is cheaper, buildable or free of the pathologies of Alcubierre's; he says the opposite, that the pathologies are properties of the class.

Origin and lineage

José Natário, Instituto Superior Técnico, Lisbon, "Warp drive with zero expansion", Class. Quantum Grav. 19, 1157 (2002), arXiv:gr-qc/0110086 [HIGH] S1. It answers the heuristic reading of Alcubierre 1994 and generalises the geometry using the extrinsic-curvature language of Wald and the rate-of-strain tensor of fluid mechanics. It cites Visser, Bassett and Liberati and Gao and Wald for the expectation that all superluminal spacetimes violate energy conditions (ENE-2), Clark, Hiscock and Larson for the optics of the Alcubierre bubble and Low for the argument that horizons are unavoidable.

Downstream: Lobo and Visser (2004) analysed the Natário drive's energy conditions in linearised gravity; Bobrick and Martire (2021) built their classification on the same shift-vector form (WRP-2); Santiago, Schuster and Visser (2022) named the class "generic Natário warp drives" and proved NEC violation for it (ENE-5); the 2021 positive-energy claims of Lentz, Fell and Heisenberg and Bobrick and Martire all live inside it, as does the subluminal shell of Fuchs et al. (2024) (WRP-3). The modified gravity survey uses the class definition to say what modified gravity does and does not change.

The mechanism

Units $G = c = 1$, signature $(-,+,+,+)$. Definition. A warp drive spacetime is $\mathbb{R}^4$ with the line element

$$ds^2 = -dt^2 + \sum_{i=1}^{3}\bigl(dx^i - X^i\,dt\bigr)^2,$$

for three bounded smooth functions $X^i(t, x, y, z)$. The induced metric on the slices $t = \mathrm{const}$ is the flat Euclidean metric, the lapse is 1, and the whole geometry is encoded in the time-dependent vector field $X = X^i\partial_i$ on Euclidean 3-space [HIGH] S1. The future unit normal is $n = \partial_t + X$; the observers with this four-velocity are the Eulerian observers, and they are in free fall (the curves $\dot t = 1$, $\dot x^i = X^i$ solve the geodesic equations) [HIGH] S1. The extrinsic curvature of the slices is

$$K_{ij} = \tfrac{1}{2}\bigl(\partial_i X_j + \partial_j X_i\bigr),$$

which is the rate-of-strain tensor of fluid mechanics for the "flow" $X$, and its trace is the expansion of the Eulerian volume elements

$$\theta = K^i{}_i = \nabla \cdot X .$$

The spacetime is flat wherever $X$ is a Killing field of the Euclidean metric, in particular wherever $X$ is spatially constant [HIGH] S1.

The energy theorem. Since the slices are flat, the Hamiltonian constraint gives the energy density seen by Eulerian observers as

$$\rho = T_{ab}n^a n^b = \frac{1}{16\pi}\bigl(\theta^2 - K_{ij}K^{ij}\bigr).$$

If the strong energy condition holds and $\theta \neq 0$ somewhere, the Raychaudhuri equation makes $\theta$ blow up in finite time, which is impossible for bounded $\nabla\cdot X$; so the strong condition forces $\theta \equiv 0$, and then $\rho = -K_{ij}K^{ij}/16\pi \leq 0$ with equality only where $K_{ij} = 0$. Hence a warp drive spacetime that violates neither the strong nor the weak energy condition is flat [HIGH] S1. This is Theorem 1.7 and it is the cleanest short proof in the literature that the class pays in negative energy.

Alcubierre as a special case. Taking $X = v_s f(r_s)\,\partial_x$ gives

$$\theta = v_s f'(r_s)\frac{x - x_s}{r_s}, \qquad \rho = -\frac{v_s^2\,[f'(r_s)]^2}{32\pi}\,\frac{y^2 + z^2}{r_s^2},$$

reproducing Alcubierre's expansion and energy density [HIGH] S1. Replacing $x$ by $\xi = x - x_s(t)$ shifts $X$ by $-v_s$ so that the interior stands still and the exterior flows past at $-v_s$; a field $X$ that vanishes near the origin and equals $-v_s(t)\,\partial_x$ far away is said to generate a warp bubble with velocity $v_s$.

The zero-expansion drive. In spherical coordinates with the $x$ axis as polar axis, use the Euclidean identification of vectors with 2-forms. Since $\partial_x \sim d\bigl(\tfrac{1}{2}r^2\sin^2\theta\,d\varphi\bigr)$, the field

$$X \sim -v_s(t)\,d\bigl(f(r)\,r^2\sin^2\theta\,d\varphi\bigr) \;=\; -2 v_s f\cos\theta\; e_r \;+\; v_s\,(2f + r f')\sin\theta\; e_\theta,$$

with $f(r) = \tfrac{1}{2}$ for large $r$ and $f(r) = 0$ for small $r$, is the exterior derivative of a 2-form and therefore divergence-free, and it generates a warp bubble with velocity $v_s\,\partial_x$ [HIGH] S1. The rate-of-strain components are

$$K_{rr} = -2v_s f'\cos\theta, \quad K_{\theta\theta} = K_{\varphi\varphi} = v_s f'\cos\theta, \quad K_{r\theta} = v_s\sin\theta\Bigl(f' + \frac{r}{2}f''\Bigr), \quad K_{r\varphi} = K_{\theta\varphi} = 0,$$

so $\theta = K_{rr} + K_{\theta\theta} + K_{\varphi\varphi} = 0$ identically, and the energy density is

$$\rho = -\frac{1}{16\pi}K_{ij}K^{ij} = -\frac{v_s^2}{8\pi}\Bigl[\,3 (f')^2\cos^2\theta + \Bigl(f' + \frac{r}{2}f''\Bigr)^2\sin^2\theta\,\Bigr] \;\leq\; 0 .$$

Ahead of the bubble ($\cos\theta > 0$) the wall is compressed radially ($K_{rr} < 0$ where $f' > 0$) and this is exactly balanced by expansion in the two perpendicular directions: volume is preserved, nothing is "contracted in front and expanded behind", and the energy density is negative everywhere the wall has a gradient [HIGH] S1. That is the counterexample.

Horizons and blueshifts. For a stationary bubble, $\langle\partial_t,\partial_t\rangle = -1 + X^2$, so the spacetime is stationary only if $|v_s| < 1$. For $v_s > 1$, null geodesics satisfy $|d\mathbf{x}/dt - X| = 1$: a flash outside the bubble is a sphere expanding at unit speed while being carried along $X$ at speed $v_s$, so events inside the bubble cannot influence events far ahead. Assuming cylindrical symmetry there is a point on the forward axis where $\|X\| = 1$, and the surface through it making angle $\alpha$ with $X$, $\sin\alpha = 1/\|X\|$, is a horizon; far from the bubble this is the Mach cone $\sin\alpha = 1/v_s$. The interior is causally disconnected from part of its own wall, which Natário notes is unavoidable (Low 1999). A visibility horizon bounds what the crew can see. Light energy measured by Eulerian observers obeys $E\,(1 + X\cdot n) = E_0$, so the observer at the centre sees a blueshift of $1 + v_s$ dead ahead, none at $90^\circ$, and infinite blueshift for rays approaching the horizon; light sent backwards is redshifted by $1 + v_s$ [HIGH] S1. These results are independent of the choice of $X$ and reproduce the thin-wall limit of Clark, Hiscock and Larson's numerical optics for the Alcubierre case.

Headline number. The paper gives no total energy. The quantity that fixes the bill is the same as Alcubierre's: $\rho \propto -v_s^2 (f')^2$, so the density scales as the inverse square of the wall thickness and the quantum inequalities of ENE-3 apply in the same way; Lobo and Visser's linearised analysis of both the Alcubierre and Natário drives finds that the negative energy in the warp field must be a considerable fraction of the ship's own mass-energy even at arbitrarily low speeds [HIGH] S1. Santiago, Schuster and Visser show the zero-expansion drive violates the null and strong conditions as well as the weak, and that the whole class violates the NEC [HIGH] S1.

What it costs

B-neg. Theorem 1.7: any non-flat member of the class violates the weak or the strong energy condition, and for the zero-expansion example the Eulerian energy density is negative throughout the wall in closed form (ENE-1, ENE-5).

B-mass. The density has the same $v_s^2 f'^2$ structure as Alcubierre's, so the quantum-inequality wall bound and the inverse-thickness scaling of the total energy carry over; the linearised analysis of Lobo and Visser puts the negative energy at a considerable fraction of the ship's mass even at low speed [HIGH] S1. No total has been computed for the zero-expansion metric specifically, which is why the mark rests on the class and not on a number.

B-boot. Natário proves the horizon for any $X$ with $v_s > 1$: the crew are cut off from part of the wall and from everything ahead, so the bubble cannot be created or controlled from inside (CON-1); the bootstrap of CON-2 follows, since the metric ahead must already be in place.

B-caus. Everett's two-bubble construction needs only effective superluminal transport between flat regions and a Lorentz boost of the second bubble's exterior; nothing in it depends on the expansion profile, so it applies to the class (CAU-3) [MED] S1.

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-1VIOLATESAny member of the class with $v_s > 1$ carries the interior between flat-space points faster than light through that space; nothing in the definition breaks Lorentz invariance or singles out the exterior frame, so a boosted second bubble sends the ship into its own past.
CAU-2SILENTChronology protection is not discussed; whether Hawking's back reaction forbids the two-bubble loop for this class is unsettled and uncomputed.
CAU-3VIOLATESEverett's two-bubble loop uses only superluminal transport between flat regions plus a boost of the second bubble's exterior, which every member of the class provides; the profile of $X$ plays no role.
CAU-4SILENTNo preferred frame is proposed and general covariance is kept; the constraint asks what picks the frame that avoids the time machine, and the paper does not say.
CAU-5N/ANo quantum signalling is involved.
ENE-1VIOLATESTheorem 1.7 proves every non-flat warp drive spacetime violates the weak or the strong condition, and the zero-expansion example has $\rho \leq 0$ everywhere; Santiago, Schuster and Visser add the null condition.
ENE-2VIOLATESNatário cites Visser, Bassett and Liberati and Gao and Wald as the reason all superluminal spacetimes are expected to violate energy conditions, and his theorem is the class-wide confirmation for warp drives.
ENE-3VIOLATESThe 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.
ENE-4SILENTThe pocket trick (a conformal factor on the slices) is outside the class as defined, since it makes the slices non-flat; Natário does not use it and does not say whether a zero-expansion pocket exists.
ENE-5VIOLATESENE-5 is a theorem about this class by name: for generic Natário warp drives the NEC is violated for any nonzero velocity, and the zero-expansion drive is treated as its own case in the proof.
ENE-6SILENTThe paper does not invoke a source at all; the only negative energy ever made (Casimir) is tens of orders of magnitude short of any wall density, and the gap is not addressed.
ENE-7N/ATrivial topology by definition of the warp-drive class; no handle for Friedman, Schleich and Witt to deform. The energy-condition violation Natário's theorem proves is the ENE-2 verdict, of which ENE-7 is the wormhole version.
CON-1VIOLATESNatário derives the horizon at $\X\= 1$ and the causal disconnection of the interior from part of the wall for any $X$ with $v_s > 1$, calling it unavoidable; this is CON-1 proved for the class.
CON-2VIOLATESGiven the horizon, the field $X$ ahead of a superluminal bubble is outside the crew's future light cone and cannot be created from the ship; Krasnikov's spacelike leading edge argument is about the location of the wall matter, which is the same for any profile.
CON-3N/AThe Krasnikov tube is not a warp bubble; it has non-flat slices and a static structure, and the paper does not use it.
STA-1SILENTNatário exhibits the horizon and the infinite blueshift there, which are the ingredients of Hiscock's divergence, but no semiclassical stress-energy has been computed for the zero-expansion metric; Hiscock's and Finazzi, Liberati and Barceló's calculations are for Alcubierre profiles.
STA-2N/ANo wormhole throat.
STA-3SILENTThe chronology horizon would appear only in the two-bubble time machine, which is not constructed here and whose semiclassical fate is unsettled.
HAZ-1SILENTThe sweeping-up of particles by a superluminal wall was computed for the Alcubierre profile; the mechanism (a wall moving faster than light with a horizon ahead) is present in every member of the class, but the burst has not been computed for the zero-expansion drive.
HAZ-2SILENTThe horizon is present and its infinite blueshift is Natário's own result, so a Hawking flux to the centre is expected; the temperature calculation exists only for Alcubierre-type walls.
HAZ-3N/AWormhole tidal bound; not a throat geometry.
LOR-1SATISFIESThe Eulerian observers, including the ship, are in free fall inside their light cones; the exterior flows past. Nothing is pushed to $c$.
LOR-2VIOLATESFor $v_s > 1$ the wall matter ahead of the ship lies outside its light cone and moves on spacelike world lines in the exterior frame, so creating the bubble on demand needs tachyonic matter; the class inherits Krasnikov's objection to Alcubierre.
LOR-3N/ANo wave propagation in a medium.
LOR-4N/ANot relevant to a geometric drive.
WRP-1SATISFIESThis paper is the class definition WRP-1 cites: unit lapse, flat slices, a shift vector field carrying the ship, and the demonstration that expansion is incidental.
WRP-2SATISFIESBobrick and Martire's classification is built on the same shift-vector form; the zero-expansion drive is a superluminal shell of exotic material moving inertially and the paper says so.
WRP-3SATISFIESNo positive-energy claim is made; Theorem 1.7 is a proof that none can be made inside the class in general relativity, consistent with the absence WRP-3 records.
MAN-1N/AFour dimensions, trivial topology.

Status of the argument

Sources