## The equation of “Nothing”

## On ‘Nothing’

**Adam R. Brown** and **Alex Dahlen**

**Abstract**

Nothing-the absence of spacetime-can be either an endpoint of tunneling, as in the bubble of nothing, or a starting point for tunneling, as in the quantum creation of a universe. We argue that these two tunnelings can be treated within a unified framework, and that, in both cases, nothing should be thought of as the limit of anti-de Sitter space in which the curvature length approaches zero. To study nothing, we study decays in models with perturbatively stabilized extra dimensions, which admit not just bubbles of nothing-topology-changing transitions in which the extra dimensions pinch off and a hole forms in spacetime-but also a whole family of topology-preserving transitions that nonetheless smoothly hollow out and approach the bubble of nothing in one limit. The bubble solutions that are close to this limit, bubbles of next-to- nothing, give us a controlled setting in which to understand nothing. Armed with this understanding, we are able to embed proposed mechanisms for the reverse process, tunneling from nothing to something, within the relatively secure foundation of the Coleman-De Luccia formalism and show that the **Hawking-Turok instanton** does not mediate the quantum creation of a universe.

**1 Introduction**

`Nothing’ first made an appearance in modern physics with the work of Witten [1], who showed that spacetimes with compact extra dimensions can be unstable to decay – the compact extra dimensions can pinch off to form a bubble of nothing containing not only no matter and no felds but also no space and no time.

If the universe can tunnel to nothing, it is natural to ask whether it can tunnel from nothing -the quantum creation of a universe.

Several authors have addressed this question [2-6], but unfortunately the situation remains

somewhat murky.

At least part of this murkiness stems from the ambiguity of what is meant by `nothing’.

In this paper, we address this ambiguity.

Our first step is to ask what the decay of Kaluza-Klein spacetime tells us about nothing.

Unfortunately the bubble of nothing itself turns out to be somewhat enigmatic on this

question.

However, in Sec. 2, we show that in models with perturbatively stabilized extra dimensions, bubbles of nothing are not all we have to work with.

The bubble of nothing is not an isolated solution, instead these models admit whole families of possible decays which remove different amounts of the stabilizing potential.

Only the decay that removes all the stabilizing potential, the bubble of nothing, changes the topology of spacetime; the other tunneling solutions in the family are topology-preserving but they nonetheless smoothly approach the bubble of nothing in the limit in which all the stabilizing potential is removed.

Though we learn little from the bubble of nothing itself, we learn a great deal from the sequence of bubbles that approach it, the bubbles of next-to-nothing.

Two things happen to the bubble interior as the bubble of nothing is approached: the

extra dimensions shrink to zero size and smoothly pinch off; and simultaneously the effective potential becomes ever more negative.

From the perspective of the lower-dimension Einstein frame, therefore, the interior becomes more and more negatively curved, so that

`Nothing’ should be thought of as the limit of anti-de Sitter space in which the curvature length goes to zero.

As it goes deeper and deeper into anti-de Sitter space, the interior of the bubble empties out until nothing remains.

In Sec. 3 we turn to the quantum creation of the universe. Quantum transitions in dynamical spacetimes are described by the Coleman-De Luccia formalism [7].

This formalism is well-studied, well-understood, and is the most credible part of semiclassical quantum gravity.

While it is always used in the context of transitions to nothing, it is not always used for transitions from nothing, perhaps because of the ambiguity of how to treat nothing.

With our new understanding of nothing, however, we are able to embed otherwise poorly moored questions of the quantum creation of the universe within this secure foundation.

The Coleman-De Luccia formalism requires that the same instanton that describes tunneling to nothing should also describe tunneling from nothing.

Within the context of this unified framework, we should think of tunneling from something to nothing as down-tunneling, and tunneling from nothing to something as up-tunneling.

The `nothing’s in the two processes are therefore the same – they are both the limit of anti-de Sitter space as the curvature length goes to zero.

But, since up-tunneling from anti-de Sitter space is impossible, the quantum creation of a universe using a bubble of nothing instanton – the Hawking-Turok process [5]|is forbidden…..

Read more: http://arxiv.org

## Leave a Reply