In 1916 Ludwig Flamm noticed that Schwarzschild's brand-new solution, sliced at one instant, is a curved surface with a "throat" — the picture every popular account now draws as a funnel. In 1935 Einstein and Rosen re-derived the same throat as a bridge between two identical sheets of space and hoped it might model a particle. What none of them could see until Kruskal's 1960 coordinates was that the bridge is not static: follow it forward in time and the two sheets are born apart, each capped by a singularity, rush together, fuse into a wormhole no wider than the horizon, and pinch off — all so quickly that a traveler entering one mouth is crushed at the singularity before reaching the other side. This entry computes that whole history from the metric and animates it. It is the moving, honest version of the figure Kip Thorne had drawn for Black Holes & Time Warps.
1 · The exact embedding interactive
Take one moment of Kruskal time and one equatorial plane. The intrinsic geometry of that 2-surface — how circumference grows with proper radial distance — is fixed by the metric, and we can draw it faithfully as a surface of revolution \(z(r)\) in ordinary 3D space, where the height \(z\) is chosen so that distance measured along the surface equals the true proper distance. At \(T=0\) this is precisely Flamm's paraboloid. Away from \(T=0\) the slope changes, but the recipe is the same closed form.
Slice time
The circumference of any horizontal circle here is a real, measurable \(2\pi r\); the proper distance you would walk between two circles is the real arc length on the surface. What is not real is the vertical axis — "hyperspace" is a rendering crutch. Two universes drawn as parallel sheets are not a fixed distance apart; only the bridge between them is geometry.
2 · The dynamics — a throat that is a falling geodesic interactive
Slide Kruskal time and watch the neck radius trace out a curve. That curve is not arbitrary: the throat's worldline is an ordinary radial free-fall geodesic. A test sphere released from rest at \(r=r_s\) falls to \(r=0\) along a cycloid, and the wormhole's neck does exactly the same thing — it is born from the past singularity, expands to a maximum radius of one Schwarzschild radius, and falls back into the future singularity. Integrating the metric along the neck gives its total lifetime as a clean closed form.
Scrub the history
3 · Why nothing crosses — the Kruskal picture interactive
The cleanest way to see the whole spacetime at once is the Kruskal–Szekeres diagram, where light always travels at 45°. Our universe is the right wedge; the "other" universe is the left wedge; the two are the two mouths of the bridge. A horizontal line is one instant of Kruskal time — one frame of the animation above. Launch a light ray from the right universe toward the left and follow it: it crosses the future horizon into the upper wedge and runs straight into the singularity. The two universes can send signals into the shared black-hole interior, but never to each other.
Which slice?
This is Fuller & Wheeler's 1962 result stated geometrically: the Schwarzschild wormhole is non-traversable. To hold a throat open you would need matter that violates the averaged null energy condition — "exotic" matter with negative energy density along the light ray. That is the entire content of the Morris–Thorne traversable-wormhole program (1988), and it is a different spacetime from this one.
4 · The honest gap to a "real" wormhole context
Everything above is vacuum general relativity — no matter anywhere, just the geometry of empty curved space. That is its strength (it is exact) and its limitation (it is not a machine anyone could use). A traversable wormhole, the kind in fiction, requires three things this one does not have: a throat that stays open, which demands a stress-energy with negative energy density threading it (Morris–Thorne 1988); tidal forces gentle enough for a traveler to survive; and stability against the pulse of radiation that any real traversal would send through. Known physics supplies negative energy densities only in small, quantum amounts (the Casimir effect, squeezed light), and quantum energy inequalities appear to forbid the macroscopic, long-lived version. The Einstein–Rosen bridge is therefore best understood as what the vacuum alone permits — a real feature of the maximally-extended solution — and simultaneously as a proof of why a usable wormhole must be built from something the vacuum does not contain.
5 · Verification on load
Rows marked live are recomputed by this page in JavaScript when it loads, by the same routines that draw the surface — the embedding by direct integration of the metric, the lifetime by quadrature along the neck worldline, the throat radius by inverting the Kruskal relation. If any status reads FAIL, the picture above should not be trusted.
| Test | this page | analytic / published | status |
|---|---|---|---|
| T=0 embedding vs Flamm z=2√(r−r_s), r=3r_s (live ∫) | … | 2√2 = 2.828427 r_s | … |
| Max throat radius over all T (live) | … | 1.000000 r_s (at T=0) | … |
| Throat radius at T=0.5 (live invert) | … | 0.898172 r_s | … |
| Throat proper lifetime ∫√((4/ρ)e^−ρ)dT (live) | … | π = 3.141593 r_s/c | … |
| Neck worldline = radial geodesic (dτ/dT)² check | … | (4/ρ)e^−ρ · matches | … |
| Embedding-failure radius at T=1.1 (live) | … | 2 ln 1.1 = 0.190620 r_s | … |
| Pinch-off time: throat radius → 0 (live) | … | |T| = 1.000000 | … |
6 · Boundary of validity mandatory
Honesty tier of this picture
Model class: exact analytic embedding of a chosen foliation. The intrinsic geometry of every slice is exact vacuum Schwarzschild; the areal radii, proper distances, throat radius and lifetime are closed-form or machine-precision quadrature. The idealizations are about the drawing and the slicing, never the underlying spacetime.
Direct
The areal radius, proper distance along the surface, throat radius r≤r_s, and lifetime πr_s/c are exact and verified on load.
Reduced
The vertical "hyperspace" axis and the fixed gap between the two sheets are illustrative — an embedding needs an extra dimension that carries no physics.
Abstraction
Not used except at the very tip of the |T|>1 spikes, where the Euclidean embedding fails and the surface is dashed.
No honest picture
The interior of the singularity itself — r=0 is a boundary of the manifold, drawn only as a marker.
Geometry exact
Each slice is the true equatorial 2-geometry of maximally-extended Schwarzschild. The T=0 case is Flamm's paraboloid to 10⁻¹⁴; the lifetime matches the radial-geodesic cycloid to 10⁻¹².
The "time" is a slicing
The opening-and-closing is a statement about the Kruskal foliation. The exterior spacetime is static (Birkhoff); a different time slicing shows a different history of the same unchanging geometry. This is the standard, and standardly caveated, way to animate the bridge.
Embedding fails near the spike
For |T|>1 the slice cannot be drawn as a Euclidean surface of revolution below r = 2r_s·ln|T|; the intrinsic curvature there needs a hyperbolic embedding. We render that stub dashed and mark the singularity rather than faking a smooth tip (as hand drawings do).
Non-traversable, vacuum only
This is empty-space GR: no matter, no traveler back-reaction, no exotic stress-energy. It is not a usable wormhole and cannot be made one without energy-condition-violating matter (§4). Fuller–Wheeler 1962.
Schwarzschild, not Kerr
Real black holes spin, and the maximal extension of Kerr has a genuinely different (and richer, still non-traversable) interior with an inner horizon and rings. The clean cycloidal bridge is a special feature of the non-rotating case.
Classical only
The eternal two-universe extension is a mathematical maximal solution; a bridge formed by real collapse has only the future half, and quantum effects at the horizon are outside classical GR entirely.
Bottom line. The shape and its motion are the exact geometry of the vacuum Schwarzschild wormhole, foliated by Kruskal time and verified on load. The vertical axis is a drawing aid, the "dynamics" is a property of the slicing, the tip of each spike is honestly non-embeddable, and the whole object is non-traversable by construction. Spin, quantum horizons, and traversable (exotic-matter) wormholes are declared out of scope.
References
- Flamm 1916. Beiträge zur Einsteinschen Gravitationstheorie. Phys. Z. 17, 448. — the original paraboloid; the T=0 surface here.
- Einstein & Rosen 1935. The Particle Problem in the General Theory of Relativity. Phys. Rev. 48, 73. APS — the "bridge" between two sheets.
- Wheeler 1955. Geons. Phys. Rev. 97, 511. APS — coins the topological "wormhole" viewpoint.
- Kruskal 1960. Maximal Extension of Schwarzschild Metric. Phys. Rev. 119, 1743. APS — the coordinates that reveal the dynamics.
- Szekeres 1960. On the singularities of a Riemannian manifold. Publ. Math. Debrecen 7, 285. — independent maximal extension.
- Fuller & Wheeler 1962. Causality and Multiply Connected Space-Time. Phys. Rev. 128, 919. APS — the pinch-off; the bridge is non-traversable.
- Misner, Thorne & Wheeler 1973. Gravitation, §31.6 & Box 31.1. Freeman. — the throat's dynamical evolution and embedding sequence.
- Morris & Thorne 1988. Wormholes in spacetime and their use for interstellar travel. Am. J. Phys. 56, 395. AAPT — what a traversable wormhole would require (§4).
- Thorne 1994. Black Holes & Time Warps: Einstein's Outrageous Legacy, Fig. 14.3. Norton. — the hand-drawn sequence this entry computes.
- Visser 1995. Lorentzian Wormholes: From Einstein to Hawking. AIP Press. — the modern reference on wormhole geometry and energy conditions.