Wormhole Physics: The Einstein-Rosen Bridge Part 1/2

 A wormhole is a hypothetical celestial object which is a shortcut to a distant part of the universe. It can be used as a pathway for interstellar travel and even as a possible time machine. Wormholes are not some creation of science fiction. Like black holes, wormholes are also solutions of Einstein's field equations. However, unlike black holes, there is no known physical mechanism of how or reason why they can be created or they must exist. The only way, I know of, that a wormhole can be created naturally is by a black hole. But such a wormhole is not traversable. If you attempt to cross it, you die. It is like jumping into a black hole. In [2], the authors claim that traversable wormholes may be built by a highly advanced civilization.

The first paper on wormholes with serious calculations was the 1935 paper by Einstein and Rosen [1]. Einstein and Rosen used the term "bridge". The word "wormhole" was coined by John Archibald Wheeler much later. Einstein and Rosen didn't study the bridge by means of interstellar travel but to build a geometrical model of an elementary particle. Einstein and Rosen considered two types of bridges: neutral and quasicharged. In this note, we discuss the neutral bridge.

Let us consider the ordinary Schwarzschild metric with $c=G=1$
$$ds^2=-\left(1-\frac{2M}{r}\right)dt^2+\frac{dr^2}{1-\frac{2M}{r}}+r^2d\Omega^2$$
where $d\Omega^2=d\theta^2+\sin^2\theta d\phi^2$. Clearly, $r=0$ is a singularity. It is not a physical singularity but a coordinate singularity. What this means is that one can remove the singularity by a suitable coordinate transformation. Einstein and Rosen introduced the coordinate transformation $u^2=r-2M$ and this results in the Einstein-Rosen form
$$
ds^2=-\frac{u^2}{u^2+2M}dt^2+4(u^2+2M)du^2+(u^2+2M)^2d\Omega^2 \tag{1}
$$
where $-\infty<u<\infty$. The coordinate change removes the region $0\leq r<2M$ containing the singularity and twice covers the asymptotically flat region, $2M\leq r<\infty$. The region near $u=0$ is interpreted as a bridge connecting the asymptotically flat region $u=\infty$ with asymptotically flat region near $u=-\infty$. Such an interpretation makes sense. If we consider a spherical surface defined by taking $u=\mbox{constant}$, the area of this surface is $A(u)=4\pi(u^2+2M)^2$. The minimum area is $A(0)=16\pi M^2$. The narrowest part of this geometry with the minimum area is defined to be the throat and the region nearby is called the bridge. Note that $u=0$ ($r=2M$) corresponds to the event horizon, i.e. the throat is the event horizon. For the creation of a wormhole, the existence of a physical singularity (curvature singularity) surrounded by a horizon is essential. A physical singularity which is not surrounded by a horizon is called a naked singularity. No neutral Einstein-Rosen bridge construction is possible without a horizon. The neutral Einstein-Rosen bridge (1) is also called the Schwarzschild wormhole. The Schwarzschild wormhole is actually a black hole so it is not traversable. This rules out the possibility of using naturally formed wormholes for interstellar travel.

References:

1. Albert Einstein and N. Rosen, The particle problem in general theory of relativity, Phys. Rev., 48:73-77, 1935
2. Michael S. Morri, Kip S. Thorne, and Ulvi Yurtsever, Wormholes, Time Machines, and Weak Energy Condition, Physical Review Letters, Volume 61, Number 13, pp. 1446-1449, 26 September 1988