Bubbles of “Nothing”

Note: Technical details and discussion are written with small print and can be skipped in a first reading

What is a bubble of “nothing”? The answer clearly depends on which area of reality you are thinking about. I’ll illustrate this with a few examples.

In physics, a “bubble of nothing” refers to an effect discovered by Witten twenty years ago. Witten showed that the standard so-called Kaluza-Klein vacuum, Minkowski spacetime cross a circle M x S^1, is unstable to nucleating a “bubble of nothing”. It is of course important to have some criteria for determining whether a ground state of the form M x S^1 is reasonable as a unification of gauge fields with general relativity.

First of all, one should impose that this vacuum should be stable at the classical and the semiclassical level. The Kaluza-Klein vacuum is classically stable but unstable against semiclassical decay. Now, even if a state is stable against small oscillations, it may be unstable at the semiclassical level. This can occur if the state is separated by only a finite barrier from a more stable state. It will then be unstable against decay by semiclassical barrier penetration. To look for a semiclassical instability of a vacuum state, one looks for a ”bounce” solution of the classical euclidean field equations.

How is this applied to the Kaluza-Klein vacuum? First you analytically continue the Kaluza-Klein vacuum to euclidean space, i.e. to

ds^2 = dx^2 + dy^2 + dz^2 + dt^2 + dphi^2,

where phi is a periodic variable running from 0 to 2pi R, so that dphi is the line element of the circle S^1. Equivalently,

ds^2 = dr^2 + r^2 dTheta^2 + dphi^2.

A solution to the Einstein equations with the same asymptotic behavior is

ds^2 = dr^2/(1-k/r^2) + r^2 dTheta^2 + (1-k/r^2) dphi^2.

This is actually the five-dimensional Schwarzschild solution (but should not be interpreted as a black hole). Regularity at the point where r^2 = k, requires that we set k = R^2, where R is the radius of the circle. The metric of the resulting space – continued to Minkowski space – is

ds^2 = dr^2/(1-R^2/r^2) + (1-R^2/r^2) dphi^2 – r^2dpsi^2 + cosh psi^2dOmega^2.

This space is nonsingular and geodesically complete (which roughly means that there are no possible light-rays that suddenly “end”, like there are in spacetimes with black holes); it is the space that the Kaluza-Klein vacuum decays into.

What happens in the decay of the Kaluza-Klein vacuum is that a hole spontaneously forms in space. As a function of time, t, the boundary of the hole is at x^2 = R^2 + t^2. After a very brief time, this hole – or bubble of “nothing” – is expanding to infinity at the speed of light. So, why is it called a “bubble of nothing”? This is because the Kaluza-Klein vacuum decays into nothing, or more precisely into a space which is bounded by a bubble of nothing – space does not exist “inside” this bubble – which is expanding to infinity and pushing to infinity anything it may meet!

In pseudoscience the examples of “bubbles of nothing” are abundant. I’ll just mention two examples. One is the crackpot book The Final Theory by Mark McCutcheon; another one seem to be the research carried out at The Quality of Life Research Center in Copenhagen, directed by holistic physician Søren Ventegodt.

In politics, there are plenty of examples of “bubbles of nothing”. Recently, there are the examples of various reactions to Israel’s extreme aggression against the palestinians and the people of Lebanon (the massacre in Qana being just one very recent example)- such as those by president Bush, Rice and many others (however strange, my friends over at Cosmic Variance have had nothing to say about this conflict). Another one is Bush’ veto against research in stem cells. (In an upcoming post I’ll explain why research in stem cells is so important).

In sports, a recent example of a “bubble of nothing” was apparently Floyd Landis’ “victory” in the 2006 Tour de France. But to be fair this doping “scandal” might in itself be a “bubble of nothing”. This is, however, not very likely (as recent findings showed that some of the testosterone in his body had come from an external source) and Floyd might end up being considered by historians as the most naive and stupid “winner” of the Tour de France.

The list of “bubbles of nothing” is endless. Other examples, anyone?

Update: Floyd Landis’ B sample was positive; welcome in the historybooks and adieu to the title as the winner of the 2006 Tour de France…;-)