The Schwarzschild metric admits negative square root as well as positive square root solutions for the geometry.
White Holes and Wormholes
By Andrew Hamilton, 2001
The complete Schwarzschild geometry consists of a black hole, a white hole, and two Universes connected at their horizons by a wormhole.
The negative square root solution inside the horizon represents a white hole. A white hole is a black hole running backwards in time. Just as black holes swallow things irretrievably, so also do white holes spit them out.
White holes cannot exist, since they violate the second law of thermodynamics.
General Relativity is time symmetric. It does not know about the second law of thermodynamics, and it does not know about which way cause and effect go. But we do.
The negative square root solution outside the horizon represents another Universe. The wormhole joining the two separate Universes is known as the Einstein-Rosen bridge.
Do Schwarzschild wormholes really exist?
Schwarzschild wormholes certainly exist as exact solutions of Einstein’s equations.
- When a realistic star collapses to a black hole, it does not produce a wormhole (see Collapse to a Black Hole);
- The complete Schwarzschild geometry includes a white hole, which violates the second law of thermodynamics (see above);
- Even if a Schwarzschild wormhole were somehow formed, it would be unstable and fly apart (see Instability of the Schwarzschild wormhole below).
Kruskal-Szekeres spacetime diagram of the wormhole
The Kruskal-Szekeres coordinate system is arranged so that the worldlines of radially infalling (yellow) and outgoing (ochre) light rays lie at 45o.
The white hole is the region at the bottom of the diagram, bounded by the two red antihorizons. The black hole is the region at the top of the diagram, bounded by the two pink-red horizons. Both white and black holes have singularities at their centres, the cyan lines. The regions at left and right outside the horizons are the two Universes. The two Universes are joined by a wormhole, the region of spacetime between the white hole and black hole singularities.
As long as the inhabitants of the two Universes remain outside the horizons, they cannot meet or communicate with each other. However, the inhabitants can meet after falling into the black hole. Having met, they also soon meet the singularity.
Instability of the Schwarzschild wormhole
The embedding diagram of the Schwarzschild wormhole illustrated at the top of the page seems to show a static wormhole. However, this is an illusion of the Schwarzschild coordinate system, which is ill-behaved at the horizon.
The Kruskal spacetime diagram reveals that in reality the Schwarzschild wormhole is dynamic, and unstable. The tremendous gravity impels the wormhole both to elongate along its length, and to shrink about its middle.
The yellow arrows indicate the directionality of the horizons. A person (or signal) can pass through a horizon only in the direction of the arrow, not the other way.
There is a certain arbitrariness to the shapes of these embedding diagrams – the spatial geometry at a given `time’ depends on what you decide to label as time, how you slice spacetime into hypersurfaces of constant time. The inset shows the slicing for the embedding diagrams adopted here, drawn on the Kruskal spacetime diagram.
Stabilizing a wormhole with exotic matter
In principle, a wormhole could be stabilized by threading its throat with `exotic matter’. In the stable wormhole at left, the exotic matter forms a thin spherical shell (which appears in the diagram as a circle, since the embedding diagram is a 2-dimensional representation of the 3-dimensional spatial geometry of the wormhole).
The shell of exotic matter has negative mass and positive surface pressure. The negative mass ensures that the throat of the wormhole lies outside the horizon, so that travellers can pass through it, while the positive surface pressure prevents the wormhole from collapsing.
In general relativity, one is free to specify whatever geometry one cares to imagine for spacetime; but then Einstein’s equations specify what the energy-momentum content of matter in that spacetime must be in order to produce that geometry. Generically, wormholes require negative mass exotic matter at their throats, in order to be traversible.
While the notion of negative mass is certainly bizarre, the vacuum fluctuations near a black hole are exotic, so perhaps exotic matter is not utterly impossible.A good reference is M. S. Morris & K. S. Thorne (1988), “Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity”, American Journal of Physics, 56, 395-412.
Unfortunately it is impossible for a traveller to pass through the wormhole from one Universe into the other. A traveller can pass through a horizon only in one direction, indicated by the yellow arrows. First, the traveller must wait until the two white holes have merged, and their horizons met. The traveller may then enter through one horizon. But having entered, the traveller cannot exit, either through that horizon or through the horizon on the other side. The fate of the traveller who ventures in is to die at the singularity which forms from the collapse of the wormhole.
The traveller can however see light signals from the other Universe.
The trapped region between the two horizons is the Schwarzschild bubble encountered on the trip into the black hole.
A glimpse through the wormhole
Suppose, despite the objections, that our Universe were attached to another Universe through a Schwarzschild wormhole. What would we see?
Here is a glimpse through the wormhole at the other Universe, visible through the Schwarzschild surface still ahead and below us. We are at 0.35 Schwarzschild radii from the central singularity. Compare this to the normal view. For simplicity, I have supposed that the other Universe contains stars exactly like ours, so it’s a bit like looking through a distorted mirror.
Only after falling through the horizon of the black hole are we able to see the other Universe through the throat of the wormhole. We are never able to enter the other Universe, and the penalty for seeing it is death at the singularity.
It would be foolhardy to attempt this fatal experiment in the hopes of glimpsing another Universe. As seen in the next section, when a realistic star collapses to form a black hole, it does not produce a wormhole.
29 May 1998 update. Oops, there’s yet another set of grid lines missing from this picture, and in the movie below. Through the mouth (pink) of the wormhole, we should be able to see the surface of the black hole as seen in the other Universe, curved into our view by the gravity of the black hole, in the same way that we can see the surface (red) of the black hole in our own Universe through the screen formed by the outward Schwarzschild surface (white). I’ll fix it when I get the time.
See also: https://en.wikipedia.org/wiki/Wormhole