What a wormhole solution is
A wormhole is a spacetime geometry with a throat: a region of minimal area joining two otherwise separate mouths, so that a path through the throat connects points that a path through the surrounding space would reach only by going the long way. Nothing about this description is exotic mathematics. It is a statement about the topology and curvature of a four-dimensional manifold, and general relativity has been in the business of such statements since 1915.
Established The first one appeared almost immediately. Flamm noticed in 1916 that the Schwarzschild solution, embedded in a higher-dimensional flat space for visualisation, has the shape of a funnel that can be continued through. Established Einstein and Rosen formalised the continuation in 1935, looking for a particle model rather than a means of transport; the object now called the Einstein–Rosen bridge is a feature of the maximally extended Schwarzschild geometry, joining two asymptotically flat regions. Established Wheeler named the structures wormholes in the 1950s and studied them as topological features of a quantum spacetime foam.
Established The Einstein–Rosen bridge is also completely useless for travel, and this was settled early. Fuller and Wheeler showed in 1962 that the Schwarzschild throat is dynamic: it opens, reaches maximum size and pinches off, and it does so fast enough that no causal signal can cross from one mouth to the other. Anything that enters hits the singularity. Established The bridge is a shortcut on a spatial diagram and not a shortcut in spacetime, which is the only sense that matters.
Morris and Thorne, and the method that produced them
Established The modern subject begins in 1988, with a paper written as a teaching exercise. Morris and Thorne asked what geometry a wormhole would need in order to be usable by a person, and set down explicit criteria: no event horizon, tidal accelerations survivable by a human body, a finite and reasonably short proper crossing time, and a throat that stays open.
Established Their method is the important part and is routinely misdescribed. They did not solve the field equations for a given matter distribution and discover a wormhole. They wrote down the metric they wanted — a static, spherically symmetric line element parameterised by a redshift function and a shape function — and then used the field equations in reverse to compute the stress-energy tensor required to hold it. Established This is entirely legitimate. The field equations are an equality, and reading them right-to-left is as valid as reading them left-to-right.
Established What it produces, though, is a bill rather than a blueprint. The stress-energy that comes back has a property that no observed matter has: at the throat, the tension exceeds the energy density, which is to say the null energy condition is violated. In the Morris–Thorne language, the throat must be threaded with exotic matter. Established This is not an artefact of their assumptions. Hochberg and Visser later characterised the throat locally, as a marginally anti-trapped surface, and showed that null energy condition violation at the throat follows generically, without the symmetry assumptions.
Established There is a structural theorem in the same direction. Topological censorship, proved by Friedman, Schleich and Witt in 1993, says that in a globally hyperbolic, asymptotically flat spacetime obeying the averaged null energy condition, every causal curve from past null infinity to future null infinity can be deformed to a curve in the asymptotic region. In plain terms: obey the averaged condition and any wormhole you have is not passable in a way that beats the outside route. Established The theorem converts “exotic matter is needed” from a feature of one construction into a structural requirement of the whole class.
The solution zoo, and what it does and does not show
Established There are now many traversable wormhole solutions. Ellis and Bronnikov independently found drainhole geometries in 1973, supported by a scalar field with the wrong sign of kinetic term — mathematically clean, physically a ghost. Established Visser constructed thin-shell wormholes in which the exotic matter is concentrated on a surface layer, which localises the problem without removing it. Established Rotating, charged, higher-dimensional and modified-gravity wormholes all exist on paper.
Established One result deserves particular care because it is regularly overstated. Visser, Kar and Dadhich showed in 2003 that the total quantity of exotic matter in a traversable wormhole can be made arbitrarily small, by choosing a geometry in which the violation is confined to an arbitrarily thin region. Established Read carefully, this is a statement about an integral, not about buildability: the density in the thin region rises as the region shrinks. Frontier Fewster and Roman replied in 2005 that once the quantum energy inequalities are applied, the parameter tuning required is extraordinary and the band thicknesses fall below any scale the semiclassical theory can describe. The small-total-exotic-matter result survives as mathematics and does not survive as an engineering route.
Frontier The most careful modern construction is Maldacena and Milekhin’s, which builds a traversable wormhole from near-extremal magnetically charged black holes held open by the Casimir energy of fermion fields, in a setting with an extra dimension. Frontier It is a genuine solution and the numbers are instructive: a mouth radius above about 1.5 × 107 m, roughly one second of proper time to cross, about 3 × 103 years of external time elapsed during that second, and an ambient temperature requirement below 10−26 eV. Established The authors state in their own paper that they have given no plausible mechanism by which such an object could form.
Established Note what the external time figure means. The crossing is traversable and it is not a shortcut: going through takes thousands of years of outside time to achieve one second of inside time, so the wormhole is slower than the vacuum. Frontier The same is true of the anti-de Sitter constructions of Gao, Jafferis and Wall, where a double-trace deformation opens the throat but the traversal can never beat a signal sent through the ambient space. Frontier In every controlled construction in the literature, the property that makes a wormhole interesting to a traveller is the property the construction does not have.
What has been looked for, and not found
Established No wormhole has been observed. Frontier This is a statement with content, because searches have been designed and run. A wormhole mouth would lens background light in a characteristic way, distinguishable in principle from an ordinary compact mass because the effective mass can be negative; Safonova, Torres and Romero worked out the microlensing signatures, and Takahashi and Asada used the SDSS Quasar Lens Search to place bounds on the cosmological density of Ellis wormholes and negative-mass compact objects. Frontier Both returned nulls, and the bounds are now tight enough that natural wormholes of the sizes that would be astrophysically interesting are constrained rather than merely unobserved.
Speculative A separate line of work concerns wormholes as a description of entanglement rather than of transport. The ER = EPR conjecture proposes that an entangled pair is connected by a non-traversable wormhole, which would make wormhole geometry a language for quantum correlations. Speculative It is an active and genuinely interesting research programme and it is not a proposal about travel; the wormholes in question are, by construction, not traversable.
Speculative In 2022 a group reported traversable wormhole dynamics on a quantum processor, and the reporting of it was the worst in this subject in a decade. Established What the experiment ran was 164 two-qubit gates on nine qubits, realising a heavily sparsified model whose holographic dual is a gravity toy; the paper itself states the equivalence holds in the semiclassical limit of an infinite number of qubits. Frontier A published comment argued that experiments implementing small commuting models of this kind lack the gravitational features being claimed, and in 2025 the original paper received a formal correction adding the phrase “averaged over all fermions” to the caption of the figure offered as evidence of thermalisation. Established No spacetime was bent in Burlingame.
What the solutions actually establish
Established The honest summary is narrower than the headline and more useful. General relativity permits traversable wormhole geometries; this is settled. Established Those geometries require stress-energy that violates the null energy condition at the throat; this is also settled, and it is structural rather than incidental. Established No known matter does that in bulk, no wormhole has been observed, and the best controlled constructions are slower than empty space and have no formation mechanism.
Handwave The claim that should never be made from this material is that the existence of the solutions makes wormholes a matter of engineering. A solution to the field equations is a conditional: if matter of this description existed in this configuration, then the geometry would follow. Established The whole weight of the question sits in the antecedent, which is what Part 2 is about.