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Maldacena, Milekhin and Popov: traversable wormholes in four dimensions (2018), and the humanly traversable sequel (2021)

A pair of near-extremal magnetic black holes joined by a throat that charged fermions hold open with Casimir energy; it is longer inside than the road outside.

Kind: physics · Loophole: L3 · Standing: K3 · Bill: B-neg, B-mass, B-new · Last reviewed: 2026-09-12

The claim

Maldacena, Milekhin and Popov, arXiv:1807.04726, published Class. Quantum Grav. 40, 155016 (2023), "present a wormhole solution in four dimensions. It is a solution of an Einstein Maxwell theory plus charged massless fermions. The fermions give rise to a negative Casimir-like energy, which makes the wormhole possible" [HIGH] S1. Two magnetically charged black holes of opposite charge, each near extremality, are joined at the bottom of their throats. In the magnetic field a massless charged fermion has a zero-energy lowest Landau level of degeneracy $q$, the integer flux, so one four-dimensional fermion becomes $q$ massless two-dimensional fermions running along the field lines. In the wormhole those field lines close into circles, a massless fermion on a circle has negative Casimir energy, and that negative energy, fed into the Einstein equations, is what an ordinary Einstein–Maxwell throat lacks to stay open. "We should emphasize that this wormhole does not require any exotic matter. In fact, the ordinary matter of the Standard Model is enough!" [HIGH] S1 (§9.1). The Standard Model embedding uses weak hypercharge and works if the whole configuration is smaller than the electroweak scale, so the 2018 wormholes are microscopic [HIGH] S1 (§8).

The paper is as clear about the limit as about the result: "It is a long wormhole that does not lead to causality violations in the ambient space" [HIGH] S1. "By 'long' we mean that it takes longer to go through the wormhole than through the ambient space" [HIGH] S1 (§1). Short wormholes, the ones that would be shortcuts, are forbidden by the achronal averaged null energy condition, which the authors take as a theorem of quantum field theory [HIGH] S1.

The sequel, Maldacena and Milekhin, "Humanly traversable wormholes", Phys. Rev. D 103, 066007 (2021), keeps the mechanism and swaps the matter: a dark sector consisting of a four-dimensional conformal field theory with a gauged U(1), realised as a Randall–Sundrum II model, whose many light degrees of freedom supply enough Casimir energy for a wormhole large enough that a person survives the tides. "Using them, one could travel in less than a second between distant points in our galaxy. A second for the observer that goes through the wormhole. It would be tens of thousands of years for somebody looking from the outside" [HIGH] S1. The authors call this "science fiction" in their own introduction, and the catalogue takes them at their word: the claim is consistency with known physics plus one previously proposed extension, not a route.

Origin and lineage

The mechanism is Gao, Jafferis and Wall's: negative averaged null energy from quantum fields whose boundary conditions connect the two ends, here generated by field exchange between two nearby black holes rather than by a coupling written into a boundary Hamiltonian, and the coupled-SYK picture of Maldacena and Qi. The geometry is the charge-without-charge wormhole of Wheeler's programme; the paper's first figure is a 1966 Wheeler drawing of a wormhole threaded by magnetic field [HIGH] S1. The throat is the AdS$_2 \times S^2$ near-horizon region of an extremal Reissner–Nordström hole, and the wormhole is a pair of such throats glued at the bottom. The relation to ER = EPR is explicit: "The wormhole solution can be viewed as a pair of entangled black holes" close to the thermofield double state [HIGH] S1. The Morris–Thorne programme supplies the traversability conditions the 2021 paper applies. The Randall–Sundrum ingredient of the sequel is the same brane-world physics the register discusses under MAN-1 and the catalogue scores under brane-bulk shortcuts.

The mechanism

Ingredients. Einstein gravity, a U(1) gauge field with coupling $g$, and massless Weyl fermions of charge one [HIGH] S1 (§2.1). An extremal magnetic black hole of integer flux $q$ has radius $r_e = \sqrt{\pi}\, q\, l_p / g$, mass $M_e = r_e / G$, and a near-horizon region that is AdS$_2 \times S^2$ with both radii $r_e$ [HIGH] S1 (§2.2; Maldacena and Milekhin eq. 2.5). On the sphere the magnetic field $B = q / 2 r_e^2$ puts a charged massless fermion into Landau levels; for fermions the orbital and dipole energies cancel exactly in the lowest level, which has zero energy on the sphere and degeneracy $q$, so "a four dimensional chiral fermion gives rise to $q$ massless two dimensional chiral fermions" moving along the radial and time directions [HIGH] S1 (§2.3). Each is localised on a magnetic field line.

The throat. Join two such throats through global AdS$_2$:

$$ds^2 = r_e^2\left[-(\rho^2 + 1)\,d\tau^2 + \frac{d\rho^2}{\rho^2 + 1} + d\theta^2 + \sin^2\theta\, d\phi^2\right],$$

with $\rho$ running from $-\infty$ to $+\infty$ and each end matched to the small-$(r - r_e)$ region of one black hole via $\tau = t/\ell$, $\rho = \ell (r - r_e)/r_e^2$ [HIGH] S1 (eqs 5.18–5.19). The time direction never shrinks, so there is no horizon and the configuration is at zero temperature. The free parameter $\ell$ sets the effective length of the throat as seen by a two-dimensional massless field, $L_{\rm throat} = \pi\ell$, which "is also the time that it takes to go through the wormhole, as measured in terms of the asymptotically flat spacetime coordinate $t$" [HIGH] S1 (eq. 5.20), and the energy gap $E_{\rm gap} \propto 1/\ell$.

The negative energy. Each lowest-Landau-level mode runs out of one throat along a field line in the flat region, into the other throat and back through the wormhole: a circle of length $L \sim \pi\ell + L_{\rm flat}$. For $\ell \gg d$, $L = \pi\ell$. Then $q$ two-dimensional fermions on a circle of length $L$ have Casimir energy $-q\,2\pi/12L$, corrected by the conformal anomaly of the curved AdS$_2$ strip, giving

$$E_{\rm wormhole} = +\frac{q\pi}{24 L} - \frac{q\,2\pi}{12 L} = -\frac{q}{8\ell},$$

equivalently a two-dimensional stress tensor with $\hat T_{tt} = \hat T_{xx} = -q/(8\pi\ell^2)\cdot 1/(4\pi r_e^2)$ [HIGH] S1 (eqs 5.24–5.27). The null-null component is negative along the null lines wrapping the cylinder; those lines are not achronal, so the achronal ANEC is untouched [HIGH] S1 (§1). All other modes are massive in the throat and only renormalise the radii.

Fixing the length. The classical energy of the configuration relative to two extremal black holes is that of two near-extremal holes at the temperature $T = 1/2\pi\ell$ that would give the same $t = 0$ slice, $r_e^3 / G\ell^2$. Adding the Casimir term and minimising,

$$E = \frac{r_e^3}{G\ell^2} - \frac{q}{8\ell} \quad\Longrightarrow\quad \ell = \frac{16\, r_e^3}{G\, q} = \frac{16\pi^{3/2} q^2 l_p}{g^3}, \qquad E_{\min} = -\frac{G q^2}{256\, r_e^3},$$

and a direct solution of the Einstein equations with the quantum stress tensor confirms the variational answer [HIGH] S1 (eqs 5.30–5.31, §5.3.2). The throat length $\pi\ell \propto q^2 l_p$ is much longer than the black holes' size $r_e \propto q\, l_p$: "The length, or depth, of the throat is very large, larger than $q^2 l_p$" [HIGH] S1 (§9.1).

Traversal times. "The proper time that it takes for an observer to go through the wormhole is of order the light crossing time of the black hole, or $r_e \sim q$. This is much smaller than the time it takes to go through the wormhole as seen from the outside, which is $\pi\ell \propto q^2$" [HIGH] S1 (§9.1). The ratio $\ell / r_e = \gamma$ is the boost factor a non-relativistic particle acquires at the centre of the throat; Maldacena and Milekhin call the wormhole "the ultimate roller coaster" and state that even without $d \ll \ell$ "the time through the wormhole is always longer than through the outside, $\pi\ell > d$" [HIGH] S1 (2021, §2.3, eq. 2.14).

Holding the mouths apart. Two opposite magnetic charges attract; the paper sets them orbiting with angular velocity below the gap $1/\ell$, so that the throat is not disrupted, and notes the orbit radiates gravitational and electromagnetic waves until the mouths merge, on a time scale parametrically longer than the traversal time [HIGH] S1 (§6). The configuration is "rather fragile. If enough energy is sent through the wormhole mouth, a pair of horizons will develop" [HIGH] S1 (§1).

The Standard Model. With weak hypercharge as the U(1), the three generations act like $N_f = 54$ charge-one flavours in the central charge $c = q\sum_i |q_i| / 2$, provided the mouth separation $d$ and $r_e$ are below the electroweak scale so the fermions are effectively massless and the Higgs vanishes in the throat [HIGH] S1 (§8). The wormholes are therefore sub-electroweak in size. The paper flags that at such distances the Standard Model may not be the whole story and, in one paragraph, proposes the RS2 dark sector that became the sequel [HIGH] S1 (§8).

The humanly traversable version. A traveller feels tidal acceleration $a \sim ({\rm size})/r_e^2$ and must survive about $20g$ for a short time, which for a half-metre body gives

$$r_e > 1.5 \times 10^7\ \text{m} \approx 0.05\ \text{light-seconds},$$

so the mouths have the mass of intermediate-mass black holes, $M_e = r_e c^2 / G \approx 10^4$ solar masses [HIGH] S1 (eq. 3.26; the solar-mass conversion is arithmetic). In the RS2 model the two-dimensional central charge and the length are $c_2 = 4\pi R_5 r_e / (G_4 l_p \log(r_e / R_5))$ and $\ell = (4 r_e^2 / \pi R_5)\log(r_e / R_5)$; taking the minimal $r_e$ and the largest allowed AdS$_5$ radius $R_5 = 50\ \mu$m gives

$$r_e \sim 0.05\ \text{s}, \qquad c_2 \sim 7 \times 10^{72}, \qquad \ell \sim 3 \times 10^3\ \text{ly}, \qquad \gamma = \ell / r_e \sim 2 \times 10^{12}, \qquad E_{\rm bin} \sim -5 \times 10^9\ \text{kg}$$

[HIGH] S1 (eq. 3.27). The traversal takes a proper time $\pi r_e$, well under a second, and an outside time $\pi\ell \sim 10^4$ years. The dark sector must be colder than $1/\ell \sim 10^{-26}$ eV; any CMB photon that falls in is boosted by $\gamma$ and seen by the traveller boosted by $\gamma^2$, so "one would have to put the huge black hole inside a refrigerator"; the orbit lifetime for the smallest separation is about $10^{16}$ years; and "We have not given any plausible mechanism for their formation" [HIGH] S1 (§3.4, App. B, §4).

The headline number is $\pi\ell > d$: the wormhole is never shorter than the road.

What it costs

B-neg. The throat is held open by negative Casimir energy, a violation of the null energy condition along the field-line circles, with total energy $-q/8\ell$ in the 2018 solution and $-5 \times 10^9$ kg of binding energy in the human version. The averaged null energy condition is violated on non-achronal null curves only; the achronal version, which forbids shortcuts, is obeyed, and that is why the wormhole is long.

B-mass. Two near-extremal magnetically charged black holes with large integer flux $q \gg 1$, orbiting each other. The human version needs mouths of about $10^4$ solar masses each and a dark sector at $10^{-26}$ eV, colder than anything in the universe by many orders. No formation mechanism is offered in either paper, and the authors say so.

B-new. The 2018 solution can live in the Standard Model, at the cost of being smaller than $10^{-18}$ m. Anything larger needs new physics: a dark U(1) plus a conformal dark sector, realised as a Randall–Sundrum II brane world with an AdS$_5$ radius up to $50\ \mu$m. That is a specific, previously proposed extension, and it is not observed.

There is no B-caus: the wormhole cannot be made a time machine because it is longer than the outside (Maldacena and Milekhin, footnote a). There is no B-boot mark, but note that the construction does have the wormhole form of the bootstrap: the mouths are created together (the paper leaves how open) and separated at sublight speed, and the outside path is always faster than the inside.

Constraint scoring

ConstraintVerdictNote
CAU-1SATISFIESThe traversal takes longer as seen from outside than the outside path ($\pi\ell > d$), so no signal through the wormhole arrives before a light signal round it, and nothing is reversed in any frame.
CAU-2SATISFIESNo closed timelike curves form; the authors state the long wormhole cannot be converted into a time machine.
CAU-3SATISFIESThe boost-one-mouth conversion fails because the through-time already exceeds the around-time by construction; the register's assembly step cannot close a loop.
CAU-4N/ANo faster-than-light sector exists for which a preferred frame would need to be chosen.
CAU-5SATISFIESThe entangled-black-hole reading transmits nothing faster than light; the fermion exchange that generates the coupling is itself an ordinary field propagating through the outside.
ENE-1VIOLATESThe Casimir-like stress tensor of the lowest-Landau-level fermions has negative null-null components along the field-line circles; the null energy condition fails there, and this is the mechanism.
ENE-2SATISFIESNo time advance relative to the flat exterior: the wormhole is longer than the outside, exactly as the theorems require of a configuration whose negative energy sits only on non-achronal geodesics.
ENE-3SILENTThe 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.
ENE-4N/AThe pocket geometry is a warp-bubble device with no counterpart.
ENE-5N/ANot a Natário-class warp metric.
ENE-6SATISFIESThe negative energy is Casimir energy of massless two-dimensional fields on a circle, $-q/8\ell$; the construction lives entirely within the Casimir-scale negative energy the register says is the only kind anyone has, and pays for it with a throat of length $q^2 l_p$ and a large flux $q$.
ENE-7SATISFIESThe paper's own words: a long wormhole, one that takes longer to go through than the ambient space, and short wormholes are forbidden by the achronal ANEC, which the authors take as a theorem. The negative Casimir energy sits on null lines wrapping the field-line circles, which are not achronal, so the theorem's hypothesis holds and its conclusion, no shortcut, is met by design. The register lists this construction as the escape hatch.
CON-1N/ANo bubble wall or wall horizon; the wormhole has no horizon at all until overloaded.
CON-2SATISFIESThe mouths must be brought together, connected and then separated at sublight speed, and the outside path stays faster; the paper accepts that it offers no way to create the configuration, which is CON-2's demand accepted rather than evaded.
CON-3N/ANo Krasnikov-type pre-laid metric.
STA-1N/ANo superluminal bubble wall; the throat's own stability under small perturbations is analysed in the paper (§5.4) and is a different question.
STA-2VIOLATESThe throat is held open by negative energy, as STA-2 says every traversable throat must be; here it is Casimir energy of Standard Model or dark-sector fermions rather than exotic matter, and the amount scales with the flux and the length.
STA-3SATISFIESNo chronology horizon forms because no time machine can be built from a long wormhole.
HAZ-1N/ANo superluminal bubble sweeps up particles; the analogous hazard, infalling particles boosted by $\gamma$, is scored under HAZ-3.
HAZ-2N/ANo bubble-wall Hawking bath; the dark sector must be colder than $10^{-26}$ eV for the throat to exist at all, which is the reverse problem.
HAZ-3SATISFIESMaldacena and Milekhin impose a $20g$ tidal bound on a half-metre body and derive $r_e > 1.5 \times 10^7$ m; the throat is boost-invariant so the $\gamma \sim 10^{12}$ traveller feels nothing extra, though any stray photon is a $\gamma^2$-boosted hazard.
LOR-1SATISFIESThe traveller free-falls; the boost to $\gamma \sim 10^{12}$ at the centre is gravitational and the world line stays timelike.
LOR-2N/ANo tachyonic matter.
LOR-3N/ANo group- or phase-velocity claim.
LOR-4N/ANo Scharnhorst-type effect.
WRP-1N/ANot a warp drive.
WRP-2N/AThe shell classification does not apply.
WRP-3N/AThe positive-energy warp argument does not bear on a wormhole.
MAN-1SATISFIESThe human version uses a Randall–Sundrum II bulk exactly as MAN-1 describes such models, and the bulk supplies degrees of freedom, not a shortcut: the wormhole remains longer than the brane path, so no apparent brane causality violation arises.

Status of the argument

2018: the paper appears on arXiv in July; it is published in Classical and Quantum Gravity in 2023 and has about 210 citations on INSPIRE as of 2026-09-12 [HIGH] S2.

2019: Fu, Grado-White and Marolf construct asymptotically flat traversable wormholes held apart by a cosmic string, with short transit times in a different sense (the transit is short relative to the wormhole's own scale, not the outside) [HIGH] S2 (abstract).

2020–2021: Maldacena and Milekhin, "Humanly traversable wormholes", supply the RS2 dark-sector version with the numbers above [HIGH] S1. Blázquez-Salcedo, Knoll and Radu report traversable wormholes in Einstein–Dirac–Maxwell theory "without needing any form of exotic matter", with classical spinor fields [HIGH] S1 (abstract); Bolokhov, Bronnikov, Krasnikov and Skvortsova note that the classical Dirac fields there are themselves carrying the exotic properties [HIGH] S1 (abstract); Konoplya and Zhidenko show the solutions require a non-smooth membrane at the throat with the fermionic charge density changing sign, and construct alternatives [HIGH] S1 (abstract, PRL 128, 091104, 2022). That thread is classical and is a cousin, not a descendant.

No peer-reviewed challenge to the Maldacena–Milekhin–Popov solution exists as of 2026-09-12. The open questions are the authors' own: no formation mechanism, no route from two separate black holes to the connected state, and a dark sector that must be colder than the universe.

Sources