Wormholes: What the Mathematics Allows

The Einstein-Rosen bridge is real. It is an exact feature of a known solution to Einstein’s field equations, and nothing can travel through it — not a ship, not a signal, not light. Both halves of that sentence have been settled since 1962. Popular wormhole stories, including the one where a black hole holds a universe inside it, keep the first half and drop the second.

Second in a series on spacetime, light and information: CERN, wormholes, gravitational lensing, entanglement, and ER=EPR.

Proved, permitted, guessed

Wormhole physics mixes three kinds of statement, and almost every wormhole story that goes wrong goes wrong by treating one kind as another.

Theorem. A proof. Fix the assumptions and the conclusion follows; there is nothing left to argue about.
Solution. A metric — a specification of the distances and times in some spacetime — that satisfies the field equations. It shows that a geometry is self-consistent, and says nothing about whether anything in the universe would produce it.
Speculation. Coherent mathematics with no confirmed prediction and no observational support. Worth working on; not evidence for anything.

Experiments need a fourth label. Contested means competent people are still arguing about what was measured.

What Einstein and Rosen actually built

In 1935 Einstein and Nathan Rosen published “The Particle Problem in the General Theory of Relativity” in Physical Review. They were not designing a shortcut between stars; they were trying to build particles out of geometry alone.

Their move was a change of variable. Substituting u² = r − 2m produces a metric that covers two identical sheets of space, joined where u = 0. Each sheet is asymptotically flat: far from the centre, gravity fades away and space looks ordinary.

That junction is the bridge. Move inward on one sheet and the spheres you cross get smaller, as expected. Then the shrinking stops: their area reaches a minimum at r = 2GM/c² and starts growing again, outward now, into a second exterior region.

Nothing has been bored through anything: this is one geometry, satisfying one set of equations, that turns out to have two far-away regions in it.

What it means to extend a solution

The coordinates Schwarzschild originally used misbehave at r = 2GM/c²: terms in the metric blow up. The failure is in the coordinates, not the geometry.

Compare a map projection. Lines of longitude crowd together and meet at the north pole, where the projection stops being usable. Nothing has happened to the ice: redraw the region and the pole is an ordinary place.

Extending a solution is that repair, performed on a spacetime. You choose coordinates that stay finite where the old ones failed, then follow the geometry as far as it goes. The extension is maximal when there is nowhere left to go: every path either continues forever or ends at a genuine singularity, where curvature itself is infinite and no change of coordinates helps.

Kruskal-Szekeres coordinates do this for the Schwarzschild solution. They fit the whole extended solution into one diagram — two exteriors, a black-hole interior, a white-hole interior — arranged so that light always travels at exactly 45 degrees. That is what makes the diagram worth drawing: causal structure — what can reach what — becomes something you check with a ruler.

The analogy has a limit: the pole is known to be there in advance, while a maximal extension is a mathematical continuation, and whether the regions it produces exist anywhere is a separate question — the one the next two sections answer.

That funnel is the dashed line on the right-hand diagram — one slice through a whole spacetime, not a corridor. On the right light travels at 45 degrees, so the pink ray leaving exterior 1 runs into the singularity instead of reaching exterior 2. Geometry after Einstein & Rosen (1935), Phys. Rev. 48, 73.

The bridge closes faster than light can cross it

The bridge is not a structure that stands there being difficult to enter. It is not there long enough to enter.

The funnel picture is a single moment: one horizontal slice through the right-hand diagram above. Stack the slices in time order and the throat is no corridor standing still — it narrows and pinches shut, in a finite time.

Fuller and Wheeler settled the consequence in 1962, in Physical Review: “The throat of the wormhole pinches off in a finite time and traps the signal in a region of infinite curvature.” Send light in and it does not come out the far side: it ends on the singularity.

Carroll’s lecture notes put it geometrically: “the wormhole closes up too quickly for any timelike observer to cross it from one region into the next.” A timelike observer is anything that moves slower than light — you, a probe, a ship. Even the fastest traveller physics permits is too slow — and so is light.

On the Kruskal diagram the two exteriors sit in each other’s elsewhere. Physicists say they are spacelike separated: no path anything real could follow connects them, because getting from one to the other would mean outrunning light. In the everyday version, two events in different places at the same instant are spacelike separated: you cannot be at both, and no signal of yours gets from one to the other.

So crossing between the two exteriors is the same category of impossible as arriving before you set out — a fact about the causal structure of the spacetime, not about the state of our technology. Established.

An astrophysical black hole has no other side

The second exterior belongs to a black hole that was never made, and every black hole in the sky was made.

Two different spacetimes are in play, not two versions of one object.

The eternal black hole. This is what the maximal extension describes. It has no beginning; nothing formed it. Regions III and IV in the figure above — the second exterior and the white hole — belong to this object, and are artefacts of that idealisation.
A black hole made by a collapsing star. A different spacetime, with a different past. Its past is not a white hole. Its past is the star.

Carroll is explicit: “there is no need for a white hole, however, because the past of such a spacetime looks nothing like that of the full Schwarzschild solution.” That geometry has one exterior and one interior: no bridge, and no second flat region for a bridge to reach.

The wrong intuition is natural: the extended solution really does contain a bridge to a second flat region, and the picture of it circulates detached from the two words that make it harmless — eternal, and non-traversable.

Traversable wormholes, and the matter they would need

Morris and Thorne asked a better question in the American Journal of Physics in 1988. The usual direction is: specify the matter, solve for the geometry. They ran it backwards — write down the geometry you want, then read off the stress-energy it would take to hold that shape. What they wanted was modest by science-fiction standards:

a throat that stays open instead of pinching shut;
no horizon — no one-way surface anywhere along the route, so the trip can be made in both directions;
tidal forces small enough to survive, a tidal force being the difference in pull between a traveller’s head and feet.

What came back was a bill.

The throat material “must possess a radial tension τ₀ with the enormous magnitude” charted below. Tension is pressure with the sign reversed: a gas pushes outward on its container, a stretched cable pulls inward on itself. The throat has to pull hard to keep from closing.

Then the decisive clause: “this tension must exceed the material’s density of mass-energy, ρ₀c².” That is the inequality τ₀ > ρ₀c², and it is the whole barrier.

The numbers point somewhere unexpected. At a throat radius of 10 metres the required tension is about 5×10⁴¹ dyn/cm², and it drops as the inverse square of the radius. At about 3 kilometres it is down to roughly 10³⁷ dyn/cm² — the pressure at the core of a neutron star. Extreme, but a number nature actually produces.

So size is not the barrier. The inequality does not move: every known form of matter sits on the wrong side of it, at every scale.

Required throat tension falls as the inverse square of the throat radius: τ₀ ≈ (5×10⁴¹ dyn/cm²)(10 m / b₀)². At b₀ = 3 km it is down to the pressure in a neutron star core. What never moves is the inequality τ₀ > ρ₀c², in the panel at right. Morris & Thorne, Am. J. Phys. 56, 395 (1988).

What “exotic” actually means

Morris and Thorne named the required stuff exotic matter, and were plain about it: “No known material has this τ₀ > ρ₀c² property, and such material would violate all the ‘energy conditions’ that underlie some deeply cherished theorems in general relativity.”

An energy condition is an assumption about how matter behaves — the kind of statement that goes into a proof as a premise. Several of the best-known theorems in general relativity take one as input.

Which means exotic matter is not excluded by those proofs. It is what you get by denying an assumption the proofs lean on, and a theorem whose premise fails says nothing about the case.

That is why traversable wormholes stay a live solution rather than a closed question. The gap between the solution and anything buildable is not one that effort closes: it is a material nobody has produced at any macroscopic scale.

A universe inside every black hole — speculative

Nikodem Popławski’s black-hole cosmology proposes that the singularity is not real, but an artefact of leaving something out of the theory: torsion.

General relativity gives matter one way to affect geometry: mass-energy curves spacetime. Einstein-Cartan-Sciama-Kibble gravity adds a second. Torsion is a twisting of spacetime, and in this theory it is sourced by the spin of fermions — the electrons and quarks ordinary matter is made of — rather than by their mass.

At extreme density that torsion produces an effective repulsion. Collapse then does not end; it bounces. The interior becomes a new closed region that expands — a universe inside the hole (Astrophysical Journal 832, 96, 2016).

Label this speculative and mean it. Torsion has never been detected. The regime where it would matter lies far beyond any density anyone can reach or observe.

The scenario’s own dynamics frustrate testing. Particle production drives a brief inflationary phase in which the interior “expands and isotropizes to the current state” — it smooths itself out in every direction, erasing the fingerprint you would go looking for. Popławski himself adds that “this scenario is only approximate.”

It is a small programme with no observational support, and that is a location, not a dismissal. Two things would count as evidence:

an independent detection of torsion;
or a quantified relic of the bounce — a large-scale anisotropy, say — shown to be incompatible with standard inflationary cosmology.

Neither exists.

The wormhole that was not made on a quantum computer

In November 2022 Nature published “Traversable wormhole dynamics on a quantum processor” (612, 51–55). The experiment was nine qubits on Google’s Sycamore chip, running a sparsified Sachdev-Ye-Kitaev model: a much-studied model of interacting fermions, sparsified meaning most of its interaction terms had been stripped out. The stripped-down version was not written down from theory; it was found by a machine-learning search.

The claim, in the paper’s own words: the experiment uses holography to “probe the conjectured ER=EPR relation between entanglement and spacetime geometry.” Holography, here, is the practice of translating a quantum system’s mathematics into the language of a curved spacetime. Count the hedges in that quoted phrase: probe, conjectured, relation. Three in ten words.

A headline has room for none of them, and they were the first thing to go.

Three months later Kobrin, Schuster and Yao posted a comment. Their objections, in order:

the learned Hamiltonian — the rule governing how the system evolves in time — does not thermalise, meaning it never settles into the scrambled equilibrium a thermal system reaches;
the teleportation signal resembles SYK only for the operators used in training;
the observed “perfect size winding” — the technical signature the gravitational reading rests on — is generic to small fully-commuting models, the ones where the order of operations makes no difference, and does not persist at larger sizes or in non-commuting ones.

In April 2025 Nature published an Author Correction. The authors maintain that the comment agrees with them on the key points — the mechanism is size winding, the system thermalises at the time of teleportation — and that the objections concern counterfactual scenarios outside the protocol they ran.

Status: contested, and contested narrowly. The live question is whether nine qubits reproduce the features that make the gravitational reading meaningful. What nobody on either side contests: nothing in that laboratory bent spacetime.

What to take away

Three claims, three grades.

Theorem. The Einstein-Rosen bridge is real mathematics and useless for travel: it closes faster than light can cross it, and its second exterior belongs to an eternal black hole — an object collapse does not produce. Established.

Solution. Traversable geometries exist on paper, and need matter whose tension exceeds its own energy density. Nothing known behaves that way: a solution without a material, and the distance from there to engineering is not measured in decades.

Speculation. A universe inside a black hole rests on a modification of general relativity that nobody has detected, in a regime nobody can reach. It may yet be right. It is not evidence.

Three different kinds of sentence. Holding them apart is the whole skill.