#### The Particle Problem in the General Theory of Relativity

*In honor of the World Year of Physics, which commemorates Einstein’s “miraculous year” in 1905, we’re presenting papers from the Physical Review archive related to Einstein’s accomplishments.*

In the last decades of his life, Albert Einstein tried endlessly to unify electromagnetism with his own theory of gravity, general relativity. These efforts are mostly now regarded as quixotic, but a short proposal written in 1935 with a colleague has survived in unlikely fashion as the source of science-fiction ideas for speeding across the universe by means of “wormholes” through spacetime. From the modern perspective, the paper also illustrates how general relativity posed mathematical and conceptual difficulties that foxed even its creator.

Einstein and Nathan Rosen, both at the Institute for Advanced Study in Princeton, wanted to rid physics of singularities–points where mathematical quantities become infinite or otherwise ill-defined–such as the concept of a particle that has all its mass concentrated into an infinitely small geometrical point. In general relativity, a point mass curves spacetime around it in a way that was calculated by Karl Schwarzschild in 1916 [1]. The Schwarzschild solution has mathematical singularities both at zero and at the so-called Schwarzschild radius.

Reinterpretation of the Schwarzschild solution avoids these singularities, Einstein and Rosen argued in their 1935 *Phys. Rev.* paper. They imagined a path tracing radially inward. Instead of trying to cross the imaginary spherical shell at the singular radius and proceeding down to the center, Einstein and Rosen showed how to match the path onto another track that emerges outward again–but into a separate section of spacetime. Imagine funnel shapes pulled out of two adjacent rubber sheets and connected at their necks, providing a continuous, tube-shaped path from one surface to the other. This construction makes a smooth connection or bridge between two distinct pieces of spacetime.

Viewed from afar, either part of this solution represents the gravitational effect of a mass because spacetime is strongly curved, but no physical body is present. Einstein and Rosen added an electromagnetic field to their solution, so that it could also represent a charged body. They hoped their construction would offer a starting point for a unified theory of gravity and electromagnetism based purely on fields, avoiding point particles and the singularities that came with them.

Not until 1939 was the modern idea of a black hole broached (see Focus Landmarks story), and only later were the subtleties of the Schwarzschild solution fully understood. The singular radius that Einstein and Rosen worked hard to avoid became the black hole’s event horizon. Although it is a one-way surface–light can pass across it going inward, but cannot come out–all physical quantities remain well defined at the event horizon. No true singularities arise there.

Further theoretical work showed that the Einstein-Rosen “wormhole” is not, contrary to outward appearances, a stable structure. For an observer trying to pass through, the wormhole opens up and closes too quickly for even a photon to get through [2]. Later work suggested that exotic forms of energy threaded through a wormhole might keep it open [3] but it remains unclear whether such arrangements are physically feasible.

Although a 1916 paper by Ludwig Flamm from the University of Vienna [4] is sometimes cited as giving the first hint of a wormhole, “you definitely need hindsight to detect it,” says Matt Visser of Victoria University in Wellington, New Zealand. Einstein and Rosen were the first to take the idea seriously and to try to accomplish some physics with it, he adds. The original goal may have faded, but the Einstein-Rosen bridge still pops up occasionally as a handy solution to the pesky problem of intergalactic travel.

David Lindley is a freelance science writer in Alexandria, Virginia.

### References

- K. Schwarzschild, “On the Gravitational Field of a Point Mass in Einstein’s Theory,” Proceedings of the Prussian Academy of Sciences, 424 (1916).
- R. W. Fuller and J. A. Wheeler, “Causality and Multiply-Connected Space-Time,” Phys. Rev. 128, 919 (1962).
- M. S. Morris et al., “Wormholes, Time Machines, and the Weak Energy Condition,” Phys. Rev. Lett. 61, 1446 (1988).
- L. Flamm, “Comments on Einstein’s Theory of Gravity,” Physikalische Zeitschrift 17, 448 (1916).