The mathematics behind a “wormhole”

Let me tell you something that will sound ridiculous at first. Take a piece of paper. Draw a dot on the left and a dot on the right. The shortest path between them, if you are a little ant walking on the surface, is a straight line. Fine. Now fold the paper in half so the two dots touch each other. Suddenly there is a new path between them: you just go straight through, right where they touch. Zero distance. No new paper was added. The paper did not tear. The dots did not move. Nothing changed except the shape of the surface. And now something that was far away is right next to something else. That is the whole idea. Everything that follows, all the tensors and coordinate substitutions and Kruskal diagrams, is just making this precise in the language of general relativity.

Einstein and Rosen did not set out to find a wormhole in 1935. They were trying to solve a problem that bothered Einstein his whole life: what is a particle? In quantum mechanics, an electron is a point. A mathematical point, zero size, infinite density if you try to pack its mass into that zero volume. Einstein found this repulsive. He wanted to describe particles using nothing but the geometry of spacetime, with no singularities, no infinities, no special matter fields stuck in by hand. The idea was that maybe an electron is not a thing sitting in space. Maybe it is a shape of space. And the shape they found, falling right out of the same equations that describe black holes, was a tunnel connecting two separate sheets of the universe.¹The 1935 paper is titled “The Particle Problem in the General Theory of Relativity.” Einstein and Rosen were specifically trying to model elementary particles as singularity-free solutions to the coupled Einstein-Maxwell equations. The bridge was a side effect of that attempt. Their particle model ultimately failed because the construction requires the mass of the bridge to be zero, which does not match any known particle, and because the bridge is dynamically unstable. But the geometric structure they uncovered was real and has outlasted the original motivation by nearly a century.

To see how they found it, you need to start with the Schwarzschild metric, the exact solution to Einstein’s equations outside any static, spherically symmetric mass $M$. In the standard coordinates $(t, r, \theta, \phi)$, it looks like this: $$ds^2 = -\left(1 – \frac{r_S}{r}\right)c^2\, dt^2 + \left(1 – \frac{r_S}{r}\right)^{-1} dr^2 + r^2\, d\theta^2 + r^2\sin^2\theta\, d\phi^2,$$ where $r_S = 2GM/c^2$ is the Schwarzschild radius. Now, you might look at this and think: two things blow up. At $r = r_S$, the coefficient of $dr^2$ goes to infinity. And at $r = 0$, the curvature itself diverges. So the metric has two bad points. The obvious thing to assume is that these are both genuine physical singularities where the geometry breaks down. Einstein and Rosen questioned that assumption. What if the blow-up at $r = r_S$ is not a property of the geometry, but a property of the coordinates? What if you are using the wrong map?

Here is the test. A genuine singularity, like the one at $r = 0$, shows up in coordinate-independent quantities. The Kretschmer scalar $K = R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ diverges at $r = 0$ regardless of what coordinates you use. But at $r = r_S$, this scalar equals $K = 48G^2M^2/r_S^6 c^4$, which is perfectly finite. So the blow-up in the metric components at $r = r_S$ is not a physical singularity. It is a coordinate artifact, like the way polar coordinates go haywire at the origin even though nothing special happens to flat space there. Einstein and Rosen took this seriously and asked: what happens if we change coordinates to remove the fake singularity?²This distinction between coordinate singularities and true physical singularities is one of the central conceptual lessons of general relativity. The metric tensor’s components depend on the coordinate system; a divergence in a component can always be blamed on the coordinates. Physical singularities must show up in scalar quantities built from the Riemann tensor, quantities that are invariant under coordinate changes. At $r = r_S$ in Schwarzschild coordinates, all such scalars are finite. At $r = 0$, they all blow up. This asymmetry is what Einstein and Rosen were exploiting.

The move they made was to introduce a new radial coordinate $u$, defined by the substitution $$r = u^2 + r_S.$$ That is it. One line. Let us think about what it does. In the original coordinate $r$, you start at the horizon $r = r_S$ and go out to infinity. It is a half-line, with a wall at the left end. The new coordinate $u = \sqrt{r – r_S}$ runs from zero to infinity for $r > r_S$. But we can let $u$ be negative too, since $r = u^2 + r_S$ gives the same $r$ for $u$ and $-u$. So now $u$ runs over the whole real line, from $-\infty$ through zero to $+\infty$, and both halves map to the same geometry. The half-line has been unfolded into a full line. The wall is gone. In these coordinates, the metric becomes $$ds^2 = -\frac{u^2}{u^2 + r_S}\, c^2\, dt^2 + 4(u^2 + r_S)\, du^2 + (u^2 + r_S)^2\, d\Omega^2$$ where $d\Omega^2 = d\theta^2 + \sin^2\theta\, d\phi^2$ is the angular part. At $u = 0$ the time component vanishes, yes, but every spatial component is completely regular. The infinity in $g_{11}$ is gone. And now the geometry has two ends: $u \to +\infty$ is one asymptotically flat universe, and $u \to -\infty$ is another.³The fact that $g_{00}$ vanishes at $u = 0$ while $g_{11}$ and the angular components remain finite signals that the surface $u = 0$ is a null surface, an event horizon. This is not a singularity; it is just a surface where the time coordinate becomes degenerate. Observers crossing this surface in free fall feel nothing special. The vanishing of $g_{00}$ here is physically equivalent to the fact that the escape velocity from the Schwarzschild radius equals the speed of light: time, as measured by a distant observer, appears to freeze. But the local physics is perfectly smooth.

Now let us understand what the throat actually is, geometrically. At every point in the Schwarzschild geometry, you can draw a two-sphere, the surface of constant $r$ and constant $t$. Its area is $4\pi r^2$. As you travel from one universe toward the bridge, $r = u^2 + r_S$ decreases as $u$ decreases toward zero. The area of the surrounding sphere shrinks. At $u = 0$, $r = r_S$, and the sphere has its minimum area $4\pi r_S^2$. Then, if you keep going into the other universe, $r$ grows again and the spheres expand. The throat is the moment of minimum sphere size, and the condition for it is just $dr/du = 2u = 0$, which is satisfied at $u = 0$. Nothing fancy. Just a minimum.⁴A wormhole throat is more precisely defined as a surface of minimal area in the sense of a trapped surface. The condition $dr/du = 0$ here is equivalent to saying the expansion of outgoing null geodesics, the quantity $\theta_+$, vanishes on the throat. For a traversable wormhole, you additionally need $d^2r/du^2 > 0$ at the throat, the flaring-out condition, which ensures the throat is a minimum rather than a maximum. The Einstein-Rosen bridge satisfies this too: $d^2r/du^2 = 2 > 0$ everywhere.

You can make this even more visual. Take a snapshot of the geometry at one instant of time, $t = \text{const}$, and look at the equatorial plane $\theta = \pi/2$. The spatial metric on that slice is $$dl^2 = \left(1 – \frac{r_S}{r}\right)^{-1} dr^2 + r^2\, d\phi^2.$$ This describes a curved two-dimensional surface. Now here is a beautiful trick: we can ask whether this surface, intrinsically curved as it is, can be embedded in ordinary flat three-dimensional space as the graph of some function $z = z(r)$. If we use cylindrical coordinates $(r, \phi, z)$ in that flat space, the metric on the surface $z = z(r)$ is $$dl^2 = \left(1 + \left(\frac{dz}{dr}\right)^2\right)dr^2 + r^2\, d\phi^2.$$ Matching this to the Schwarzschild spatial metric forces $$\frac{dz}{dr} = \sqrt{\frac{r_S}{r – r_S}},$$ and integrating gives $$z(r) = 2\sqrt{r_S(r – r_S)}.$$ This is a paraboloid opening outward, a funnel.⁵This surface is called a Flamm paraboloid, computed by Ludwig Flamm in 1916, a year after Schwarzschild found his metric. It is crucial to understand what this embedding is and is not. It is not a picture of what the Schwarzschild geometry looks like from outside. It is a mathematical representation of the intrinsic geometry of a spatial slice, drawn in flat space so our brains can process it. A being living on the two-dimensional Flamm surface would measure exactly the same distances and curvatures as an observer in the equatorial plane of the Schwarzschild spacetime. But the embedding space has no physical meaning; it is just a visualization aid. For the Einstein-Rosen bridge, you take two of these funnels, one for each universe, and glue them at their narrowest point, $r = r_S$. The result is the famous picture: two flat spaces connected by a tube.

Now here is where I have to tell you something disappointing, and I want you to understand exactly why it is true, not just that it is. The Einstein-Rosen bridge is not traversable. You cannot go through it. No matter how fast you go, no matter what you do, no signal can get from one universe to the other by passing through the throat. The reason has nothing to do with engineering. It is built into the causal structure of the spacetime itself, and the way to see it clearly is with a tool called Kruskal-Szekeres coordinates.

The Schwarzschild coordinates $(t, r)$ cover only part of the full spacetime. Think of it like a map of the world drawn only for the northern hemisphere: perfectly accurate for what it covers, but useless for telling you about the south. George Kruskal (and independently Martin Szekeres) in 1960 found the global map, a coordinate system that covers the entire maximal Schwarzschild spacetime at once. In these coordinates $(T, X)$, the metric is $$ds^2 = \frac{32G^3M^3}{rc^6}\, e^{-r/r_S}\left(-dT^2 + dX^2\right) + r^2\, d\Omega^2,$$ where $r$ is determined implicitly by $$\left(\frac{r}{r_S} – 1\right)e^{r/r_S} = X^2 – T^2.$$ The whole spacetime now fits on a finite square diagram. Region I ($X > |T|$) is our universe. Region III ($X < -|T|$) is the other universe. The future singularity is the hyperbola $T^2 - X^2 = 1$ with $T > 0$, a curve that arcs over both regions like a ceiling. The event horizons are the diagonal lines $T = \pm X$.⁶The Kruskal extension is provably unique: it is the unique maximal analytic extension of the Schwarzschild metric. Every geodesic in this spacetime either reaches a curvature singularity, goes to infinity, or can be extended indefinitely. There is no larger spacetime that contains it as a proper subset. This uniqueness is what makes the Kruskal diagram authoritative: it is not one way of extending Schwarzschild, it is the only way.

Look at the diagram. The bridge, the moment when regions I and III are connected through the throat, corresponds to the slice $T = 0$. On that slice, you can draw a horizontal line from one region to the other and the geometry is the Flamm paraboloid we computed. But now look at what sits above $T = 0$: the future singularity. Any worldline moving forward in time, which in the Kruskal diagram means moving upward, and trying to get from region I to region III, has to cross the singularity first. The fastest possible traveler, a photon moving at 45 degrees in the Kruskal diagram, starts in region I at $T = 0$, moves toward the throat, and hits the singularity at $r = 0$ before ever reaching region III. The bridge opens and closes in zero time. It is not that you need to go faster; it is that there is no time available at all.⁷This can be made quantitative. A radial null geodesic in Kruskal coordinates satisfies $dX/dT = \pm 1$. A photon starting at $(T_0, X_0)$ in region I and moving toward region III (decreasing $X$) follows $X(T) = X_0 – (T – T_0)$. It hits the future singularity when $T^2 – X^2 = 1$, i.e. when $T^2 – (X_0 – T + T_0)^2 = 1$. One can check that this always happens before $X$ reaches $-|T|$ (region III), for any starting point with $X_0 > 0$ and $T_0 \geq 0$. The bridge always pinches off before any signal can cross.

So the bridge forms, it is real, it connects two universes, and it immediately closes. Einstein and Rosen did not know this in 1935 because Kruskal coordinates were not invented until 1960. In their $u$ coordinate picture, the bridge looks static, a permanent geometric tunnel between two sheets of space. They thought it would sit there forever. But the full causal analysis, which only became possible with the Kruskal extension, revealed that the bridge is not a static structure. It is an event: a moment when two sheets of space touch and then separate, like two soap bubbles briefly kissing before pulling apart. The geometry is doing something in time, not just in space, and the old coordinates were hiding it.

What Einstein and Rosen were actually hoping for was even more striking. They wanted the bridge to model a particle. An electron, in their picture, would not be a point of matter; it would be a tube of space, and the two mouths of the tube would appear, from outside, like two opposite electric charges. Lines of electric flux would thread the throat, emerging from one mouth looking like positive charge and entering the other looking like negative charge, but there would be no actual charge anywhere, just electric field lines passing through a hole in space. The field equations governing the electric field outside the bridge would be the ordinary source-free Maxwell equations, with zero charge density at every point, and yet integrating the electric flux over a sphere around either mouth would give a nonzero answer. The topology of the bridge mimics a charge without there being any charge.⁸This idea, that topological features of spacetime can mimic sources of physical fields, was later developed by Wheeler into a program he called “geometrodynamics.” He coined the phrase “charge without charge” and “mass without mass” to describe how wormhole topology can reproduce the effects of particles using only vacuum geometry. The program ultimately failed as a theory of elementary particles because the relevant wormholes are dynamically unstable, have the wrong quantum numbers, and cannot reproduce the spin-1/2 nature of fermions. But the conceptual framework influenced decades of thinking about quantum gravity and the quantum structure of spacetime. It is a gorgeous idea. It did not work as a theory of matter, but it opened a door that has never been fully closed.

John Wheeler picked up this thread in the 1950s and ran with it. He coined the word “wormhole” in 1957. He imagined that at the Planck scale, $\ell_P \approx 10^{-35}$ m, spacetime itself might be foamy with microscopic wormholes constantly forming and closing, a quantum froth of topology. He called this spacetime foam. And more recently, a conjecture called ER = EPR, proposed by Maldacena and Susskind in 2013, has suggested that quantum entanglement between two particles is not just analogous to a wormhole connecting them but literally the same thing, in some precise geometric sense. Two entangled particles sit in the two exterior regions of the Kruskal diagram, connected by a bridge that neither can traverse.⁹The ER = EPR conjecture (Einstein-Rosen = Einstein-Podolsky-Rosen) proposes that any pair of maximally entangled quantum systems is connected by a non-traversable wormhole. The conjecture resolves certain paradoxes about black hole evaporation by identifying the internal correlations of Hawking radiation with geometric bridges in the bulk spacetime. It remains a conjecture, not a theorem, but it has organized a large body of work in quantum gravity and holography. The non-traversability of the Einstein-Rosen bridge is essential to the conjecture’s consistency: if the bridge were traversable, it would allow superluminal communication between entangled systems, violating relativistic causality. The bridge is real. The shortcut is not. But the connection, the fact that two regions of spacetime that are causally disconnected are nonetheless geometrically joined, is as real as any solution to Einstein’s equations. That is the legacy of a single line: $r = u^2 + r_S$.

References and Footnotes

• 1
  The 1935 paper is titled “The Particle Problem in the General Theory of Relativity.” Einstein and Rosen were specifically trying to model elementary particles as singularity-free solutions to the coupled Einstein-Maxwell equations. The bridge was a side effect of that attempt. Their particle model ultimately failed because the construction requires the mass of the bridge to be zero, which does not match any known particle, and because the bridge is dynamically unstable. But the geometric structure they uncovered was real and has outlasted the original motivation by nearly a century.
• 2
  This distinction between coordinate singularities and true physical singularities is one of the central conceptual lessons of general relativity. The metric tensor’s components depend on the coordinate system; a divergence in a component can always be blamed on the coordinates. Physical singularities must show up in scalar quantities built from the Riemann tensor, quantities that are invariant under coordinate changes. At $r = r_S$ in Schwarzschild coordinates, all such scalars are finite. At $r = 0$, they all blow up. This asymmetry is what Einstein and Rosen were exploiting.
• 3
  The fact that $g_{00}$ vanishes at $u = 0$ while $g_{11}$ and the angular components remain finite signals that the surface $u = 0$ is a null surface, an event horizon. This is not a singularity; it is just a surface where the time coordinate becomes degenerate. Observers crossing this surface in free fall feel nothing special. The vanishing of $g_{00}$ here is physically equivalent to the fact that the escape velocity from the Schwarzschild radius equals the speed of light: time, as measured by a distant observer, appears to freeze. But the local physics is perfectly smooth.
• 4
  A wormhole throat is more precisely defined as a surface of minimal area in the sense of a trapped surface. The condition $dr/du = 0$ here is equivalent to saying the expansion of outgoing null geodesics, the quantity $\theta_+$, vanishes on the throat. For a traversable wormhole, you additionally need $d^2r/du^2 > 0$ at the throat, the flaring-out condition, which ensures the throat is a minimum rather than a maximum. The Einstein-Rosen bridge satisfies this too: $d^2r/du^2 = 2 > 0$ everywhere.
• 5
  This surface is called a Flamm paraboloid, computed by Ludwig Flamm in 1916, a year after Schwarzschild found his metric. It is crucial to understand what this embedding is and is not. It is not a picture of what the Schwarzschild geometry looks like from outside. It is a mathematical representation of the intrinsic geometry of a spatial slice, drawn in flat space so our brains can process it. A being living on the two-dimensional Flamm surface would measure exactly the same distances and curvatures as an observer in the equatorial plane of the Schwarzschild spacetime. But the embedding space has no physical meaning; it is just a visualization aid.
• 6
  The Kruskal extension is provably unique: it is the unique maximal analytic extension of the Schwarzschild metric. Every geodesic in this spacetime either reaches a curvature singularity, goes to infinity, or can be extended indefinitely. There is no larger spacetime that contains it as a proper subset. This uniqueness is what makes the Kruskal diagram authoritative: it is not one way of extending Schwarzschild, it is the only way.
• 7
  This can be made quantitative. A radial null geodesic in Kruskal coordinates satisfies $dX/dT = \pm 1$. A photon starting at $(T_0, X_0)$ in region I and moving toward region III (decreasing $X$) follows $X(T) = X_0 – (T – T_0)$. It hits the future singularity when $T^2 – X^2 = 1$, i.e. when $T^2 – (X_0 – T + T_0)^2 = 1$. One can check that this always happens before $X$ reaches $-|T|$ (region III), for any starting point with $X_0 > 0$ and $T_0 \geq 0$. The bridge always pinches off before any signal can cross.
• 8
  This idea, that topological features of spacetime can mimic sources of physical fields, was later developed by Wheeler into a program he called “geometrodynamics.” He coined the phrase “charge without charge” and “mass without mass” to describe how wormhole topology can reproduce the effects of particles using only vacuum geometry. The program ultimately failed as a theory of elementary particles because the relevant wormholes are dynamically unstable, have the wrong quantum numbers, and cannot reproduce the spin-1/2 nature of fermions. But the conceptual framework influenced decades of thinking about quantum gravity and the quantum structure of spacetime.
• 9
  The ER = EPR conjecture (Einstein-Rosen = Einstein-Podolsky-Rosen) proposes that any pair of maximally entangled quantum systems is connected by a non-traversable wormhole. The conjecture resolves certain paradoxes about black hole evaporation by identifying the internal correlations of Hawking radiation with geometric bridges in the bulk spacetime. It remains a conjecture, not a theorem, but it has organized a large body of work in quantum gravity and holography. The non-traversability of the Einstein-Rosen bridge is essential to the conjecture’s consistency: if the bridge were traversable, it would allow superluminal communication between entangled systems, violating relativistic causality.