# Entropic interpretation of the Hawking-Moss bounce

###### Abstract

We revisit the derivation of the Hawking-Moss transition rate. Using the static coordinates, we show that the Euclidean action is entirely determined by the contribution of the entropy of de Sitter space which is proportional to the surface area of the horizon. This holographic feature is common to any static spacetime with a horizon on which the shift vector vanishes.

###### pacs:

98.80.Cq,98.80.Qc,04.60.-m^{†}

^{†}preprint: RESCEU-14/16

When the Hawking-Moss bounce was first discovered Hawking:1981fz , it was interpreted as describing quantum tunneling from a de Sitter universe as a whole to another de Sitter space with a larger (effective) cosmological constant. Since a transition to a state with larger energy density is counterintuitive, many people perceived it with surprise. Another counterintuitive aspect is that it depends only on the potential energy densities before and after the transition independent of the “distance” in the field space.

The latter point was challenged by Weinberg later Weinberg:2006pc , who proposed a “thermal” interpretation in the limit the difference of the vacuum energy densities is small so that geometrically both states are described by de Sitter space with the small Hubble parameter.

Nowadays the Hawking-Moss instanton is playing more and more important roles in various fields of physics—not only in the inflationary cosmology Sato:2015dga , where it was originally applied, but also in the landscape of string theory Susskind:2003kw where exponentially large number of possible vacuum states with different energy densities exist.

In this Letter, we revisit the interpretation of the Hawking-Moss instanton to provide a new picture, namely, the entropic interpretation without any restrictions unlike in Weinberg:2006pc . The importance of the gravitational entropy term in the vacuum transition rate has been stressed by Gregory et al. in a different context Gregory:2013hja , who incorporated the effect of black hole on the false vacuum decay and showed that a term proportional to the black hole horizon area must be taken into account.

Before embarking on the entropic interpretation, we review the derivation of the Hawking-Moss bounce solution Hawking:1981fz ; Weinberg:2006pc ; Vilenkin:1983xq . Assuming that the bounce solution has O symmetry, the Euclidean metric can be characterized by one parameter and its function ,

(1) |

where represents the line element of the unit three-sphere. The Euclidean action of the Einstein gravity and a canonical scalar field with a potential is then written as follows.

(2) |

where an over-dot represents differentiation with respect to . From (2), the field equations read,

(3) | |||

(4) |

The Hawking-Moss solution corresponds to a static scalar field configuration with which is realized at potential extrema with . Hence (4) reads

(5) |

and its solution is given as

(6) | |||

(7) |

where is a field value at a potential extremum. Substituting the solution (6) into the action (2), we find

(8) |

Hawking and Moss Hawking:1981fz originally considered the transition from a false vacuum state to the local potential maximum and identified the transition rate as

(9) |

where the prefactor may be estimated as on dimensional grounds.

In Weinberg:2006pc , Weinberg proposed a thermal interpretation assuming , when is given by

(10) | |||

Here is the potential energy increment in the horizon , and is the Hawking temperature of de Sitter space. He argues that the gravitational effect is negligible because the geometry does not change practically before and after the transition thanks to the assumption . As a result the formula based on (10) is identical to the case a horizon-sized domain receives thermal fluctuation at the Hawking temperature.

In the rest of this Letter, however, we show that the exponent is completely determined by the gravitational entropy of the system, as the bulk energy of the scalar field is fully canceled out by the negative gravitational energy due to the Hamiltonian constraint. It is therefore concluded that only the gravitational entropy affects the Hawking-Moss transition, and that it does not break the conservation of energy.

In order to prove the above statement, it is essential to describe the (Euclidean) de Sitter space with a static metric . Here, we start with a more general Arnowitt-Deser-Misner (ADM) decomposition Arnowitt:1959ah

(11) |

where is the lapse function, is the shift vector, and is the spatial metric. The Latin indices run from to . Applying the Wick rotation to introduce the Euclidean time , (11) reads

(12) |

with . Here and hereafter we put a tilde on quantities in the Euclidean space which is multiplied by some power of upon Wick rotation. Correspondence to the unrotated Lorentzian counterpart is also shown below. For example, the extrinsic curvature of the const. three-space is expressed as

(13) |

where denotes covariant derivative with respect to .

The Euclidean Einstein action is expressed as

(14) |

Here is the Euclidean momentum conjugate to , is an induced metric on the boundary surface S with , and is the unit normal vector on the boundary surface S where we assume vanishes. and are the gravitational Hamiltonian and the momentum for the dynamics of the foliation . They are given by

(15) |

where denotes the three-curvature on the hypersurface and represents the trace of the extrinsic curvature .

The matter Euclidean action, on the other hand, is expressed as

(16) |

where

(17) |

is the momentum conjugate to , and

(18) |

are the matter part of the Hamiltonian and momentum, respectively.

Classical Euclidean solutions are found by taking variation of the total Euclidean action . From variation with respect to and , we find the Hamiltonian and the momentum constraints,

(19) |

Therefore for a static configuration with and , the total Euclidean action is simply given by the surface terms as

(20) |

For the particular case of de Sitter space, the static metric is given by

(21) |

where is the Hubble parameter, denotes radial coordinate and is the metric on the unit two sphere. In the following, we impose the periodic boundary condition on the Euclidean time with a period .

Introducing a foliation in the spacetime fixed at constant Euclidean time , which takes the value in the range , we can easily decompose the Euclidean de Sitter metric with

(22) | ||||

(23) | ||||

(24) |

where is the unit normal vector on the hypersurface . This manifold generally has a conical singularity at where . This implies that the curvature is divergent on the de Sitter horizon, although the horizon is not a physical singularity.

As we see below, the conical singularity can be avoided by a specific choice of , namely the inverse Hawking temperature . It should be noted, however, that the manifold still collapses to a single point on the horizon and, as shown below, this plays an important role in deriving the entropy term from the Euclidean action.

In the following, therefore, we regularize the collapsing part of the manifold by first restricting the integration to the region and then setting the regularization parameter to zero after the calculation of the Euclidean action (Fig. 1).

In this case the action (20) has only the first term, where the surface is located at with its normal vector given by

(25) |

Note that, since this surface is not a real boundary of the theory, being introduced just for the sake of regularization, one should not apply the Gibbons-Hawking boundary terms GH ; GHb here. The length of circumference of the manifold is and the relation

(26) |

should be satisfied to ensure the absence of the conical singularity. This is the reason we must identify with the inverse Hawking temperature .

Hence, the nonvanishing term in (20) is calculated as

(27) |

where is the area of the horizon given as

(28) |

Thus, for the case giving a static de Sitter space with potential energy density , the action of the Hawking-Moss instanton

(29) |

is entirely given by the contribution of the de Sitter entropy GH . This is primarily because in the static configuration the bulk term of the action vanishes due to the Hamiltonian constraint.

In this sense, one can extend the thermal interpretation of the Hawking-Moss solution more rigorously to argue that is indeed proportional to the thermodynamical probability of the state , , where is the free energy. Here, since , simply reads . In other words, the probability is just proportional to the number of internal states associated with the de Sitter space with the energy density . Thus the smallness of the transition rate (9) to a state with a higher potential energy density is not due to the largeness of the energy—in fact, the total energy is always zero—, but because of the smallness of the number of microscopic states there.

It is also interesting to note that the Hawking-Moss transition apparently violates the second law of thermodynamics, as it is a transition to a state with smaller entropy. However, since it is governed by a single order parameter , the transition itself is not a macroscopic process but a mere microscopic process analogous to the Brownian motion. Indeed the stochastic approach of inflation can also reproduce the desired probability distribution Linde:1991sk .

In conclusion, by using a static coordinate system, we have shown that the Hawking-Moss instanton is entirely given by the entropy of the de Sitter spacetime proportional to the horizon area. In this sense, this solution is holographic 'tHooft:1993gx . As is clear from the above analysis, this feature is common to any static spacetime with a horizon on which the shift vector vanishes . In our derivation, the Hamiltonian constraint, which asserts that the sum of the material energy and gravitational energy vanishes, plays an important role. Hence previous considerations based only on the energy of matter part Weinberg:2006pc ; HenryTye:2008xu cannot grasp the essential feature of the problem.

Acknowledgements. We are grateful to Masafumi Fukuma and Yuho Sakatani for valuable communications. This work was partially supported by JSPS Grant-in-Aid for Scientific Research 15H02082 (J.Y.), Grant-in-Aid for Scientific Research on Innovative Areas No. 15H05888 (J.Y.), and a research program of the Advanced Leading Graduate Course for Photon Science (ALPS) at the University of Tokyo (N.O.).

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