The embedding diagramof the Schwarzschild wormhole illustratedat 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 thatin reality the Schwarzschild wormhole is dynamic, and unstable.The tremendous gravity impels the wormhole both to elongatealong its length, and to shrink about its middle.Watch two white holes merge, form a wormhole,then fall apart into two black holes (52K GIF
movie);orsame movie, double-size on screen (same 52K GIF).
The yellow arrowsindicate the directionality of the horizons.A person (or signal) can pass through a horizononly 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.
Suppose, despite the objections,that our Universe were attached to another Universe through aSchwarzschild 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 thenormal view.For simplicity, I have supposed that the other Universe containsstars exactly like ours, so it’s a bit like looking through adistorted mirror.
Only after falling through the horizon of the black holeare we able to see the other Universethrough 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 experimentin 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 theother 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 Universethrough the screen formed by the outward Schwarzschild surface(white).I’ll fix it when I get the time.
In principle, a wormhole could be stabilized by threading its throatwith ‘exotic matter’.In the stable wormhole at left,the exotic matter forms a thinspherical shell(which appears in the diagram as a circle,since the embedding diagram is a 2-dimensional representationof 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 wormholelies 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 specifywhat the energy-momentum contentof 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 isM. 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.
