Chapter 9 Questions

If one takes the Schwarzschild metric and performs a somewhat complicated coordinate transform, one will discover that the Schwarzschild solution to Einstein's equations is more extensive than first thought. The new solution consists of two asymptotically flat spaces connected together at the event horizon. At first blush, it appears as if one can travel from one of these spaces into the other, thus this wormhole seems to provide a bridge between two universes. However: no timelike worldline can bridge the gap. The other universe is strictly elsewhere from our universe. The best you could hope for is to meet someone from the other universe as you were falling into the singularity. Once inside the horizon, both universes lie in your past light cone.

Now for the wormhole's stability: If we examine the wormhole solution in our modified coordinates, we find that the solution evolves as measured in the modified time coordinate. Basically, the Schwarzschild "throats" form in both universes, connect, and then pinch off into singularities. Viewed from outside, using "ordinary" time coordinates, the whole thing just sits there, but in the modified time coordinates, those which would be more appropriate for someone falling in, time is very finite and ends at the singularity.

Are wormholes real? They are a real solution, but that doesn't mean that they exist in the universe. They do not result from the collapse of a star. They just have to be built in ab initio. There are several known weird solutions to Einstein's equations that probably don't have much to do with the real world. They are worth studying though because they provide fascinating insights into properties of the theory general relativity.

If you want to know more about wormholes, may I recommend Kip Thorne's well-written book Einstein's Outrageous Legacy.

Technically, no, because the black hole solution is a vacuum solution that has an external region that goes to flat space outside of the hole. The universe is not a vacuum and there is no known (or required) external spacetime. However, the closed model (see Chapter 11) shares some features with the internal solution of a black hole, in that gravity is going to bring everything to a collapse and a singularity, and no worldline can escape that fate.

Black holes would be hard to detect, and a great deal of unseen mass could be stored inside black holes. However, based on how we think black holes must form (from collapsing cores of relatively rare supermassive stars) there can't be that many black holes in comparison to more normal stars. Hence we expect that the total mass in black holes is only a small fraction of the mass we can see, so that they don't constitute a major component of the unseen, or "missing" mass in the universe. (See Chapter 15)

They aren't actually moving faster than light. The simply appear to be moving faster than light. The jet is beamed toward us, nearly along our line of sight. It moves only slightly slower than light, so the light carrying the history of its motion tends to be bunched up when it gets to us. Imagine somebody flying from Alpha Centari at close to c while you watch through a telescope. By the time the light gets here telling you that they have left, they are about to arrive, so to you the journey speed appears to be superluminal. But by subtracting out the known light travel time, you can compute the true speed, and verify that it is less than c.

It is possible that tiny black holes could have formed and that they would be radiating away via the Hawking process by now. But none has ever been detected. Most cosmologist don't regard this as very likely.

Somebody once seriously suggested that a mini black hole could account for the Tunguska event in Siberia. Depending on the size of the black hole (let's assume it truly is a mini-hole with a Schwarzschild radius that is submicroscopic) it would pass straight through the Earth, possibly wreaking a little havoc at the entry and exit points due to its strong tidal forces. It would actually accrete very little mass in its passage.

Over the lifetime of the solar system (10 billion years or so) there is virtually no chance that we will encounter a black hole. (Or any other star, for that matter.) If the universe expands forever, the matter that presently makes up the Earth may one day end up in a black hole. Most things may very well end up in black holes in the incredibly huge distant future. Then in an even longer period of time those black holes will evaporate.

Copyright © 2005 John F. Hawley