Nucleation and evolution of false vacuum bubbles in scalar-tensor gravity
In this presentation, we discuss the nucleation and subsequent evolution of false vacuum bubbles in the scalar-tensor gravity. First, we transform the scalar-tensor type theory of gravity to the standard Brans-Dicke type. Second, we transform the Brans-Dicke type theory from the Jordan frame to the Einstein frame. For a certain potential, a true vacuum bubble in the Einstein frame can be transformed to a false vacuum bubble in the Jordan frame by a conformal transformation. Thus, in the Jordan frame, the nucleation of a false vacuum bubble can be possible and its subsequent evolution can be described with the help of thin-wall approximation. False vacuum bubbles have physical importance: a set of false vacuum bubbles might generate a negative energy bath and it has further theoretical implications.
Center for Quantum Spacetime, Sogang University, Seoul 121-742, Republic of Korea Research Institute for Basic Science, Sogang University, Seoul 121-742, Republic of Korea \PACSesPACS: 98.80.Cq, 04.70.Dy, 04.62.+v, 11.27.+d
Recently, the application of modified gravity for general relativity has gained vivid interest. People are interested in the following questions: What is the motivation behind such a modified gravity theory? What are the observational consequences of the theory for black holes, cosmology or other areas? What is the theoretical implication of the modified gravity theory?
In this context, the scalar-tensor models are the most promising candidate, since they generally have quite clear theoretical motivations. In this paper, we will discuss the false vacuum bubbles in the scalar-tensor gravity. This issue is related to the following interesting and important problems:
Cosmology: Is the bubble universe via false vacuum bubbles viable with the scalar-tensor gravity?
General relativity: Is the Jordan frame equivalent to the Einstein frame, in the presence of false vacuum bubbles?
String theory: If a false vacuum bubble is allowed, what is its implication to holography?
Quantum gravity: If a false vacuum bubble or a bubble universe exists, does it cause the information loss problem?
Of course, these are too difficult questions, and hence it is difficult to give an ultimate answer to some of the questions; however, in this paper, we want to assert the fact that a false vacuum bubble in the scalar-tensor gravity is a good toy model to study these problems.
1.1 Scalar-tensor models
The prototype of a scalar-tensor model is the Brans-Dicke theory  given by the action:
Here, is a dimensionless coupling parameter, and the Einstein gravity is restored back in the limit. is the potential for the Brans-Dicke field .
Matching current observations with the Brans-Dicke theory without a potential requires to be greater than . However, the study for small parameters have theoretical interests for three distinct reasons. If a potential is involved with the Brans-Dicke field, then the dynamics of the Brans-Dicke field is restricted and hence one can obtain a viable model even though is smaller than . Via the chameleon mechanism, an effective potential can be derived when there is a local matter density around the solar system ; this can in turn make apparently look large for solar system experiments. Some string-inspired models allow various values for and hence the study for small universe has got some theoretical importance for understanding the nature of string theory . For example, dilaton gravity can be restored from string theory as the limit. In some cases, higher order corrections in string theory can make the dimensionless coupling to be field dependent: . The Randall-Sundrum model  allows the weak field limit of gravity on the brane to be . Moreover, apart from string theory, higher order curvature corrections can be described by scalar-tensor models; for example, in gravity, it is equivalent to the limit with a suitable potential .
1.2 Conformal transformations
Almost all scalar-tensor models can be transformed into the prototype Brans-Dicke theory via field redefinitions. We can further transform the model to the Einstein theory if . This is known to be a conformal transformation :
where is the metric in the Einstein frame and is the minimally coupled scalar field in the Einstein frame with the potential .
2 Nucleation of false vacuum bubbles
In order to describe the nucleation process, there is a well known procedure  which uses the symmetric metric in the Euclidean signature , and then obtains the bounce solution for the field and the scale factor . Unfortunately, it is well-known that a small false vacuum bubble
If this has to happen, we have to require some simple conditions on the potential : and , where subscripts and respectively denote the true and the false vacuum in the Jordan frame. For convenience, we define the effective force function as , choosing and , and we represent the conditions to be
Therefore, we require and we conclude that such false vacuum bubbles can form only in a de Sitter background with and . An explicit example of such a potential and a false vacuum bubble solution was reported in .
For explicit calculations, we use the thin-wall approximation
Then, we can approximate the Euclidean action and obtain
where . The size of the thin-wall should then satisfy the following equation of motion
Therefore, we can smoothly connect from the bounce solution to the subsequent evolution at the surface, by choosing .
3 Subsequent dynamics of false vacuum bubbles
According to the discussion of the previous section, we can sure that the junction equation (Equation (8), ) is the thin-wall limit of the Euclidean thick-wall solution. Now, we will assume as we already observed from the Euclidean solution, although it is not necessarily true in general. Then we can rewrite the equation as
where we have fixed without loss of generality. Then we can rewrite the right hand side as , where is the effective tension. In this rewritten form, we can calculate the Lorentzian dynamics of the thin-wall, as we already calculated in the Einstein case. Since , the effective tension can be negative, and it is not difficult to check that the effective tension indeed becomes negative .
The fact that the tension becomes negative is important to decide the causal structure . A true vacuum bubble wall can expand to the asymptotic observer, but the same is impossible for a false vacuum bubble if the tension is positive . Therefore, we can interpret this in two ways: (1) Our solution is a false vacuum bubble in the Jordan frame, and it expands over the asymptotic observer via negative tension and (2) our solution is a true vacuum bubble in the Einstein frame, and it expands over the asymptotic observer via positive tension (the latter case is trivial). These two interpretations are of course consistent. Figure 1 explains this . In the Jordan frame, the size of the cosmological horizon of the inside is smaller than that of the outside; whereas in the Einstein frame, the size of the cosmological horizon of the inside is larger than that of the outside.
We can think of a limit such that the cosmological horizon of a false vacuum bubble in the Jordan frame is much smaller than that of the true vacuum . This might happen for the and limit and there is no reason to prohibit this combination to happen. Then, the formation of a black hole with mass in the Jordan frame can be allowed: . Dynamical formation of such a black hole is obtained to paste the solution from ‘a false vacuum bubble in a true vacuum (dS-dS)’ to ‘a black hole in a de Sitter space (Sch-dS)’(Figure 2). In the thin-wall limit, during the formation of a black hole, the corresponding solution in the Einstein frame is less clear; and hence, this should be discussed using the complete thick-wall solutions.
4 Lessons from double-null formalism
To describe the dynamics of the shell beyond the thin-wall approximation, we may need to employ numerical calculations. The authors and colleagues used the double-null formalism for this purpose . The double-null formalism is based on the double-null coordinates: . Then, we formulate the Einstein equations and matter field equations in this coordinates and solve them numerically
What we can infer from this numerical calculations for bubble dynamics are as follows:
If there is a violation of the null energy condition or if the negative energy is concentrated on the wall, then a false vacuum bubble can expand over the asymptotic region .
The former result is not a strange one, and we can explicitly show by numerical calculations. The latter is a trivial result, and we can explicitly demonstrate via numerical computations.
In the previous work, we approximated the semi-classical energy-momentum tensor using the -wave approximation: we use two-dimensional results for and divide by . There is indeed some well-developed justification in support of the use of the -wave approximation for the black hole cases
Figure 3 shows that a collapsing false vacuum bubble can emit negative energy flux along the out-going direction (Figure 12 in ). is the false vacuum region and is the true vacuum region. The wall between two regions is located around . Then the shell is formed towards an in-going null direction and will collapse toward the center. The blue region is where the null energy condition is violated, i.e., (the out-going part of the energy-momentum tensor) becomes negative, and hence the negative energy is emitted along the out-going null direction. Asymptotic region of is yellow, i.e., there are some part with positive out-going energy; however, we could see that the out-going energy can be controlled by initial conditions .
Now we want to make some comments on the simulation. In the previous calculation, we assumed a bubble in an almost flat background. Therefore, the choice for the asymptotic mass was always approximately zero. However, in many situations, the asymptotic mass can be different; then, the tendency for the emission of negative energy can be changed. However, it is a definite fact that there is a solution of Einstein gravity that allows a false vacuum bubble to emit negative energy, at least in the -wave approximation level.
Figure 4 is an example of the expanding and inflating wall (Figure 19 in ). In this case, is the true vacuum region and is the false vacuum region; hence, the wall is formed towards an out-going direction. This figure explicitly shows, when the null-energy condition on the wall is violated or the negative energy is concentrated on the wall, the existence of inflation, i.e., the existence of an anti-trapping horizon (white curve) is possible. Perhaps, collapsing false vacuum bubbles can be the origin of the negative energy; and collapsing false vacuum bubbles may induce an expanding false vacuum bubble .
In this paper, we discussed the nucleation of a false vacuum bubble in the scalar-tensor gravity, especially in the Jordan frame. The existence of such a false vacuum bubble is not contradictory, since we can match the solution with a true vacuum bubble in the Einstein frame . We can continuously connect the Euclidean solution  to the Lorentzian solution , and perhaps, the solution can allow a formation of the bubble universe that is separated from the asymptotic region (Figures 1 and 2).
There are some supplementary evidences for the existence of a bubble universe. The nucleated false vacuum bubbles can be used as sources of negative energy fluxes (Figure 3, ). If the negative energy is concentrated on a wall of a false vacuum bubble, then it will allow the formation of a bubble universe (Figure 4, ), even though the bubble initially satisfies the null energy condition.
As answers for the questions discussed in the introduction, we have the following comments:
The existence of a bubble universe separated from the asymptotic region seems to be possible. At least, in the Jordan frame, we have a definite solution (Figure 2). If one observer sees the bubble at with the condition , then we can choose a space-like hypersurface such that the observer is inside the cosmological horizon and there is the second asymptotic region inside the observer along the space-like hypersurface.
The next natural question to address is the meaning associated with the information loss problem. It is not sufficient to say something regarding the information loss problem using only the false vacuum bubble; for example, the entropy of the inside region is smaller than that of the outside (note that, the entropy should be calculated in the Einstein frame), and hence it is not entirely inconsistent with the unitarity. If we clearly separate the space-time by a black hole (Figure 2), then the loss of information will be clearer. However, still there remains a choice of black hole complementarity or holography . Also, one can ask whether we can apply the arguments of the fuzzball conjecture or not  in this context.
If we include the semi-classical effects, the outside of a false vacuum bubble will emit negative energy along the out-going direction. However, outside of a true vacuum bubble will not emit negative energy, since the background temperature of the false vacuum region is hotter than the true vacuum region. Therefore, a false vacuum bubble and a true vacuum bubble can be distinguished by semi-classical effects. Then, although our false vacuum bubble solution in the Jordan frame is classically equivalent to the true vacuum bubble in the Einstein frame, they can be distinguished by including semi-classical effects.
If there is an emission of negative energy, it can be concentrated on a wall of a false vacuum bubble, and it can in turn induce an expanding and inflating false vacuum bubble. Then, it can produce a supplementary evidence for the existence of a bubble universe that is separated from the asymptotic region.
Thus, the nucleation and evolution of false vacuum bubbles in the scalar-tensor gravity will give us a good toy model to investigate for the existence of a bubble universe and the information loss problem of black holes. The existence of a bubble universe becomes clearer, but the implication on the information loss problem and holography in string theory is not so clear. Emission of negative energy from a false vacuum bubble is related to the problem whether the Jordan frame is equivalent to the Einstein frame or not. It seems that they are not equivalent at the quantum level; this fact should be studied in detail and should be supported with more concrete calculations, beyond the -wave approximation.
Acknowledgements.The authors would like to thank Hongsu Kim, Wonwoo Lee, Jakob Hansen, Dong-il Hwang, and Young Jae Lee for discussions and encouragement. The authors especially thank Remo Ruffini and Hyung Won Lee for the hospitality we received during the 12th Italian-Korean Symposium on Relativistic Astrophysics. DY thanks Raju Roychowdhury and Chaitali Roychowdhury for the careful English revision. This work is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MEST) through the Center for Quantum Spacetime (CQUeST) of Sogang University with grant number 2005-0049409.
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