content_image/338753.jpg content_image/338754.jpg content_image/338755.jpg content_image/338756.jpg content_image/338757.jpg content_image/338758.jpg
and speculate on similar constructions in more general flux compactifications. 

## II. THE ORIGINAL BUBBLE OF NOTHING 

The bubble of nothing geometry introduced by Witten [1] is easily obtained from the five-dimensional Schwarzschild black hole, 

<img src='content_image/338753.jpg'>

It will be convenient for us to express this metric in terms of a new radial coordinate, r = $^{√}$ρ 2 − ℓ $^{2}$, as 

<img src='content_image/338754.jpg'>

We can now Wick rotate two of the coordinates via 

to yield 

<img src='content_image/338755.jpg'>

<img src='content_image/338756.jpg'>

This is the bubble of nothing metric written in a somewhat unfamiliar gauge. (See the discussion in [19].) In the limit r → 0, it becomes 

<img src='content_image/338757.jpg'>

which is devoid of a conical singularity if we impose periodicity with 0 ≤ y < 2 π . In this limit the r − slice degenerates onto a 2 + 1 dimensional de Sitter space of size ℓ , representing the induced metric on the bubble surface. 

In the limit of r ≫ ℓ , the metric asymptotes to 

<img src='content_image/338758.jpg'>

which represents the cartesian product of 4 d Rindler space and a circle of circumference 2 πℓ . The bubble of nothing geometry can therefore be regarded as a deformation of Rindler space (times a circle), whereby the horizon region near r → 0 is replaced with a smooth tip consisting of dS$_{3}$ × B$_{2}$ , where B$_{2}$ is a cigar-shaped disk. This is illustrated in Fig. 1 below. 

