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Fell and Heisenberg 2021: positive energy from hidden geometric structure

A Helmholtz split of the shift vector that turns Eulerian energy density into a sum of principal minors, and a superluminal example that is positive for those observers only.

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

The claim

Shaun Fell and Lavinia Heisenberg ask whether a compact superluminal soliton in plain general relativity can be generated by positive energy densities only, and answer that within one subclass of shift-vector spacetimes it can, at least as the Eulerian observers see it. Their tool is a decomposition. Split the shift vector into a gradient part and a divergence-free part. The Hamiltonian constraint then separates into a term built from the second-order principal minors of the Hessian of the potential, a cross term, a term that is never positive and a curl term. In the purely irrotational sector the energy density is just the sum of the three minors, which has a geometric reading: positivity is a statement about the convexity of the potential in each two-dimensional subspace. That reading makes it "a relatively simple matter" to write down configurations with positive semi-definite Eulerian energy, and explains why the classical drives fail: Alcubierre uses one component of the shift, so only the negative terms survive, and Natário's divergence-free shift is purely solenoidal, for which the energy is never positive.

They exhibit two superluminal configurations. The first is a piecewise potential with a central shift of magnitude 1.3 and positive energy, which they set aside because its corners are not physical. The second is a twice-differentiable potential built from Gaussians and error functions with a central shift of about 1.26, everywhere positive Eulerian energy, finite total energy of about $9.25 \times 10^{43}$ J, four orders of magnitude below the Sun's rest energy, and a Schwarzschild-like exterior that they credit, following Bobrick and Martire, for the absence of negative energy. The configuration is tuneable: moving energy from the front of the shell to the back raises the central shift without changing the total, which they offer as a way to accelerate. The paper also reports what the fuller check shows: the weak energy condition is violated in compact regions through the principal pressures, "no amount of modification to the configuration could get rid of these WEC-violating regions", and the strong and dominant conditions are violated too.

Origin and lineage

Fell and Heisenberg, ETH Zürich, arXiv:2104.06488, April 2021, version 4 matching the journal; Class. Quantum Grav. 38, 155020 (2021) [HIGH] S1. It generalises the potential ansatz of Lentz 2021 by dropping his hyperbolic constraint, adopts the "non-truncation" explanation of Bobrick and Martire 2021, and stands against Alcubierre 1994 and Natário 2002 as the cases its decomposition explains. Its authors also point to Einstein–Cartan constructions as a modified-gravity route, which belongs with warp drives in modified gravity. It is the third of the three 2021 claims in ENE-5 and is recorded in WRP-3 as having "proved positivity for Eulerian observers only".

The mechanism

With unit lapse, flat slices and a time-independent shift $\vec N$, the Hamiltonian constraint reads

$$16\pi\rho = 2\big(\partial_xN_x\partial_yN_y + \partial_xN_x\partial_zN_z + \partial_yN_y\partial_zN_z\big) - \tfrac12\big[(\partial_zN_y+\partial_yN_z)^2 + (\partial_yN_x+\partial_xN_y)^2 + (\partial_zN_x+\partial_xN_z)^2\big],$$

and the Alcubierre ansatz reduces it to $16\pi\rho = -\tfrac12[(\partial_yN_x)^2 + (\partial_zN_x)^2]$, so "any ansatz must contain at least two non-zero components of the shift vector field" to have a chance [HIGH] S1. The Helmholtz decomposition $\vec N = \vec\partial\phi + \vec\omega$, $\nabla\cdot\vec\omega = 0$, gives the line element

$$ds^2 = -dt^2 + \sum_i\big(dx^i + (\partial_i\phi + \omega_i)\,dt\big)^2$$

and, after what the paper calls tedious algebra,

$$16\pi\rho = 2\big(h_1 + h_2 + h_3\big) - 2\langle J, H\rangle_F - \langle J, J\rangle_F + \tfrac12|\nabla\times\vec\omega|^2,$$

where $H$ is the Hessian of $\phi$, $J$ the Jacobian of $\vec\omega$, $h_1 = \partial_y^2\phi\,\partial_z^2\phi - (\partial_y\partial_z\phi)^2$ and cyclically the second-order principal minors of $H$, and $\langle\cdot,\cdot\rangle_F$ the Frobenius product [HIGH] S1. The momentum constraint becomes $16\pi\vec p = \nabla^2\vec\omega$, so an irrotational shift has zero Eulerian momentum. In the irrotational sector

$$8\pi\rho = h_1 + h_2 + h_3, \qquad \vec p = 0,$$

which is the "hidden geometric structure": positive energy needs the potential to be convex enough in the coordinate planes, and a purely solenoidal shift has $\rho = -\tfrac{1}{16\pi}K_{ij}K^{ij} \le 0$, which is Natário's case [HIGH] S1.

The smooth example is the potential of their equation (11), a combination of Gaussians in $r - (x^2+y^2+z^2)\Pi/m$ and $r + (x^2+y^2+z^2)\Pi/n$ with error functions, parameters $(\Pi, r, V, \sigma) = (1/4, 6, 10, 1)$, and spatial weights $m(x,y,z) > n(x,y,z)$ ahead and $m < n$ behind to tilt the gradient; with those choices the central shift magnitude is about 1.26, superluminal, the energy density is everywhere positive and peaked behind the centre, and the shift is radial and asymptotically Minkowskian outside [HIGH] S1. In SI units the peak Eulerian density is $\rho_{max} \approx 3.2 \times 10^{26}$ kg m$^{-3}$, "an astronomical energy density concentrated in a very small region" that "more than likely" forms a black hole, and the total energy is $E_{total} \approx 9.25 \times 10^{43}$ J against $1.78 \times 10^{47}$ J for the Sun [HIGH] S1. The authors could not solve the partial differential equation for the minors numerically and say a purpose-built solver would be needed [HIGH] S1.

The fuller check is in the paper itself. Analysing the Lorentz-invariant eigenvalues of the stress-energy, the Eulerian density is the first eigenvalue and is non-negative, but "the principle momenta constraints is where the violations occur, namely, $\rho + p_{p_i} < 0$ for some $i \in \{1, 2, 3\}$ in compact regions in the distribution" [HIGH] S1. The strong energy condition, in the irrotational Eulerian form ${}^4R_{\mu\nu}n^\mu n^\nu = -\langle H,H\rangle_F - \nabla\phi\cdot\nabla(\nabla^2\phi)$, is violated by both configurations, and the dominant condition fails because $|\rho| - |p_{p_i}| < 0$ in compact regions [HIGH] S1.

Santiago, Schuster and Visser add three things. The decomposition is the matrix identity $K^2 - \mathrm{tr}(K^2) = 2\,\mathrm{tr}(\mathrm{cof}\,K_{ij})$, "a curiosity which is unfortunately less useful than one might hope", because for any zero-vorticity drive the Eulerian density is a pure spatial divergence, $\rho = \tfrac{1}{16\pi}\partial_i(\Phi_{,i}\Phi_{,jj} - \Phi_{,j}\Phi_{,ij})$, which integrates to zero under fall-off, so positive somewhere forces negative elsewhere; and their null-condition theorem covers the whole class regardless [HIGH] S1. The first example's potential is only piecewise differentiable, so the metric is discontinuous and the Riemann tensor carries squares of delta functions, "mathematically and physically not viable" [HIGH] S1. Of the second they write that at its most general the potential is $F(r, \sigma(x,y,z)) + v_* z$, and that the specific example does what they call the equivalent of setting $\sigma \to 1$ and $v_* \to 0$, leaving a radial flow that "is simply not viable for describing a warp drive spacetime"; they note that radial-flow potentials of this type reproduce Painlevé–Gullstrand Schwarzschild and de Sitter, not warp drives [HIGH] S1. That reading of the second example disagrees with the paper's own description of it as axisymmetric with a level central shift, and no reply from Fell and Heisenberg on the point has been found [MED] S1.

What it costs

B-neg: the paper's own eigenvalue analysis finds the weak, strong and dominant energy conditions violated in compact regions, and Santiago, Schuster and Visser find the null condition violated for the class. B-mass: $9.25 \times 10^{43}$ J is about $10^{27}$ kg, about 170 Earth masses, at a peak density the authors expect to collapse into a black hole; the paper's hope that the geometric reading lets this be reduced is a research direction, not a number. B-caus: a superluminal central shift with no preferred frame; the paper's causality section relies on global hyperbolicity and on horizons preventing acceleration through $c$, which is to say it relies on the drive not being reachable. No new field: the ansatz is general relativity with a classical source left unspecified.

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-1VIOLATESA central observer at shift 1.26 relative to distant observers, with no frame singled out, is a signal into some observer's past by the entry; the paper's causality section addresses closed curves within one spacetime, not the two-frame argument
CAU-2DODGESSection 3.3 argues that global hyperbolicity is a precondition of the construction, so the slices are Cauchy surfaces and closed timelike curves cannot form, and that horizons prevent manipulating the distribution past light speed; both are assumptions about a single drive, which is the entry's escape hatch rather than an answer
CAU-3VIOLATESEverett's two-bubble loop drops exactly the global hyperbolicity the paper leans on, and the Shoshany–Snodgrass gluing of two non-unit-lapse drives into a closed timelike geodesic reaches this class; the paper does not consider two devices
CAU-4SILENTNo preferred frame is chosen or discussed
CAU-5N/ANo entanglement signalling
ENE-1VIOLATESBy the paper's own account the weak, strong and dominant conditions fail in compact regions; only the Eulerian density is positive
ENE-2VIOLATESSuperluminal by construction; calls Olum's result a "claimed proof" and offers no way round it; Santiago, Schuster and Visser show the null condition fails for the class
ENE-3SILENTQuantum 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
ENE-4SILENTVan Den Broeck's pocket is cited as prior art and not used
ENE-5VIOLATESNamed in the entry: positivity established for the co-moving Eulerian observers only, which the paper's own eigenvalue analysis confirms
ENE-6SILENTNo source is named for the negative pressures the paper admits it needs
ENE-7N/AA warp bubble of trivial topology; no handle, so nothing for Friedman, Schleich and Witt to deform. The superluminal energy bill is charged under ENE-2, of which ENE-7 is the wormhole form.
CON-1SATISFIESConcedes that horizons form and "prevent the configurations from transporting an inertial observer from the subluminal regime to the superluminal regime", which is the Everett–Roman conclusion, and leaves open "whether the central vehicle can actually manipulate the energy density"
CON-2SILENTSketches acceleration by redistributing shell energy front to back and never asks who does the redistributing ahead of the bubble or how the shell is laid
CON-3N/ANot a tube
STA-1SILENTSays the spacetime "might still possess some of the other pathologies ... such as the formation of horizons" and goes no further
STA-2N/ANo wormhole throat
STA-3N/ANo time machine assembled
HAZ-1SILENTSwept-up matter never discussed
HAZ-2SILENTCrew flux never discussed
HAZ-3N/ANo throat
LOR-1SATISFIESMotion is geometric; the central observer is inertial and nothing is pushed through $c$
LOR-2N/ANo tachyons
LOR-3N/ANo signal-velocity claim
LOR-4N/ANot Scharnhorst light or a particle measurement
WRP-1SATISFIESUnit lapse, flat slices, shift vector: the zero-vorticity subclass of the Natário class, as Santiago, Schuster and Visser label it
WRP-2VIOLATESClaims a superluminal configuration from positive energy; the entry says superluminal needs exotic matter, and the paper's own pressures confirm it
WRP-3VIOLATESRecorded in the entry as positivity for Eulerian observers only; not a surviving superluminal positive-energy drive
MAN-1N/AModified gravity is mentioned as an alternative route; no extra dimensions in the mechanism

Status of the argument

Sources