Black Holes and Entropy

Beckenstein, J. (1973)

An analogy between thermal systems and black hole mechanics is established. The thermodynamic relation \(dE = TdS - PdV\) is analogous to \(dM = \Theta d\alpha + {\bf \Omega}\cdot d{\bf L} + \Phi dQ\). The latter two terms represent the work done on the black hole by an external agent that increases the black hole's angular momentum and charge by \(d{\bf L}\) and \(dQ\), respectively. The terms \({\bf \Omega}\cdot d{\bf L} + \Phi dQ\) are the analog of \(PdV\), leaving \(\Theta d\alpha\) the analog of \(TdS\), and in particular, motivates the association of black hole entropy with area.

Entropy is associated with information in accordance with Shannon's definition, so that a gain of information is associated with a decrease in entropy, \(\Delta I = - \Delta S\). Within the context of the second law of thermodynamics, "the entropy of a system not in thermal equilibrium increases because information about the internal configuration of the system is being lost during its evolution as a result of the washing out of the effects of the initial conditions" (p. 35-36).

Black holes are fully describable in terms of just three attributes: mass, charge, and angular momentum. The concept of entropy as applied to black holes refers to the equivalence class of black holes with the same mass, charge, and momentum; it does not refer to the thermal entropy internal to a particular black hole (p. 36). The black holes within a single equivalence class, while having the same external properties, differ in their internal configurations that are related to the properties of their previous stellar forms, and the details of their fateful collapses. These details are essentially initial data from the standpoint of the black hole system, and all of this information is lost when the body becomes a black hole. The black hole entropy applies to all possible black holes with the same mass, charge, and momentum, and so it quantifies the loss of information (the inaccessibility of data) regarding all possible initial configurations of these black holes.

The act of gaining information about the internal configuration (and, hence, the initial state) of a system reduces its entropy. It is not possible to reduce the entropy of a black hole because the event horizon acts as a "one-way membrane" preventing an external agent from acquiring any information about the black hole's internal configuration. (p. 36)

Beckenstein proposes that black hole entropy is some function of its (rationalized) area, \(S_{\rm bh} = f(\alpha)\). To determine the functional form, he considers the hypothetical case of a single elementary particle of minimum area \(\alpha \geq 2\hbar\) (the limit is imposed by the Compton wavelength being the minimal radius of the particle) free-falling into a black hole. Since the particle is elementary with no internal structure, the minimum amount of information that might be lost when it falls into the black hole is the answer to the question of whether the particle exists or not, giving it an information content of \(\ln 2\). Then, with \(\Delta S_{\rm bh, min} = \frac{\partial f}{\partial \alpha} \Delta \alpha_{\rm min}\), we find \(\frac{\partial f}{\partial \alpha}= \ln 2 \hbar^{-1}/2\). Upon integration we obtain the expression (Eq. 16)

\begin{equation} S_{\rm bh} = \frac{\ln 2}{2\hbar}\alpha. \end{equation}
The generalized second law of thermodynamics is proposed: \(\Delta S_{\rm bh} + \Delta S_c > 0\). The strict "greater than" (\(>\)) proposition is confusing given the discussion surrounding this expression on p. 39. Consider a body with with entropy \(S\) that falls into a black hole. When the object falls into the black hole, it is lost to the universe external to the black hole, and so the common entropy of the universe associated with this object decreases, \(\Delta S_c = -S\). This entropy can be taken to be in direct correspondence with the information content of the object, via Shannon's definition of information entropy -- it is information about the internal state of the object. Once it has fallen into the black hole, it is lost to our measuring devices forever, and so the entropy of the black hole should increase, \(\Delta S_{\rm bh} = -\Delta S_c\). This would give \(\Delta S_{\rm bh} + \Delta S_c = 0\), but it assumes that no information about the object was known before it disappeared into the black hole. If, instead, we had occasion to measure some aspect of this object before it crossed the event horizon, we would have reduced its entropy somewhat. But even though this information might have been known, it has now disappeared, along with the unknown details of the object's internal configuration, down the black hole. It has become part of the black hole's internal configuration which is inaccessible to us. So, we in general expect the entropy of the black hole to increase more than the common entropy decreases on account of the object being swallowed -- hence the \(>\) in the relation. My guess, however, is that if we were to consider as our object a vessel of gas at thermal equilibrium such that its entropy were maximized, the generalized second law would read \(\Delta S_{\rm bh} = -\Delta S_c\).

A clue to resolving the previous confusion comes from the statement that the black hole can be thought of as the maximum entropy state of a given form of matter. This argument is based on the observation that besides charge, mass, and momentum, all information about the collapsing body is lost when it becomes a black hole. If this is true, then we only expect \(\Delta S_{\rm bh} = -\Delta S_c\) when the object falling into the black hole is another black hole. But I need to think about that.

Quantum vacuum energy in general relativity

Ford, L.H. (1975)

Ford seeks to identify the physical zero point energy of the vacuum within the context of a restricted set of spacetimes. While the general problem relies on a covariant and cut-off independent renormalization and regularization of the stress-energy tensor, Ford appeals to analogy with the Casimir effect to gain insight into the problem of vacuum energy in the gravitational setting. The resulting program is much simpler than the full theory of renormalization and, in some cases, yields useful physical intuition.

First though, a comment of Hawking's mentioned in the paper that I find thought provoking: " unambiguous definition of particle number in a local inertial frame may only be given for modes of wavelength much less than \({\color{black}{l}}\) (the local radius of curvature). The uncertainty in particle number of modes of longer wavelength is associated with an energy density of the order \({\color{black}{l^{-4}}}\)" (p. 1) (italics mine). The referred to energy density is not necessarily that of the vacuum (they are particles, after all), and Ford mentions on the next page that the nature of the energy density is not known. While obviously relevant to early times when \({\color{black}{l }}\sim l_{\rm Planck}\), this seems relevant even today with \({\color{black}{l }}\sim H^{-1}\). What is the connection between the uncertainty in particle number for modes with \({\color{black}{}}\lambda \gtrsim H^{-1}\) and the cosmological constant?

Now on to Ford's program. He lays it out on p. 2: "The Casimir effect suggests that we look for examples where the presence of a gravitational field is a perturbation which shifts the vacuum energy by a finite amount from its Minkowski space value. If we can then deform a given manifold into Minkowski space, we can ask whether or not the deformation changes the vacuum energy density by a finite amount; if so, this difference is the physical vacuum energy density of the original manifold. More precisely, define a cutoff energy-momentum tensor whose vacuum expectation value is always finite. Vary some parameter which determines the strength of the gravitational field. If the vacuum expectation value of the cutoff energy-momentum tensor changes by a cutoff-independent quantity, this is the change in the physical energy-momentum tensor associated with the zero-point fluctuations of the field in question.'' The primary limitation is the requirement that a cutoff-independent energy-momentum tensor exists -- generally it does not. I wonder if the ability to define a cut-off independent energy-momentum tensor is related to our ability to isolate a unique vacuum. In the cases presented by Ford, a unique vacuum is available and is used to construct the vacuum expectation value of the cutoff-independent energy-momentum tensor that is to be compared with the Minkowski space version.

The first simple example that Ford gives is provided also, in a bit more detail, by Birrell and Davies pp. 91-93. The case considered is that of a locally flat geometry with nontrivial topology in the form of a compactified spatial dimension; in two-dimensions, the topology is \({\color{black}{S^1}}\times \mathbb{R}^1\) and the line element is Minkowski with \({\color{black}{x^\mu}}\) and \({\color{black}{x^\mu + L}}\) identified (where \({\color{black}{L}}\) is the length of the compactified dimension). The eigenfrequencies are \begin{equation} \label{eigen} {\color{black}{ \omega_n = \frac{2\pi n}{L}. }}\end{equation} The VEV of the cutoff energy-momentum tensor computed in the vacuum, \({\color{black}{|0_L}}\rangle\), is
To obtain the sum, notice that \({\color{black}{}}\sum \omega_n e^{-\alpha \omega_n} = -d/d\alpha \sum \left(e^{-2\pi\alpha/L}\right)^n = -d/da \frac{1}{1-e^{-2\pi \alpha /L}}\).

\begin{eqnarray} \langle 0_L| T_{00} | 0_L\rangle &=& 2 \sum_n \frac{1}{2}\omega_n e^{-\alpha \omega_n} \nonumber \\ \label{exp} &=& \frac{2\pi}{L}e^{-2\pi\alpha/L}\left(1-e^{-2\pi\alpha/L}\right)^{-2}. \end{eqnarray} Recall that in Minkowski space, the ultraviolet divergence of the vacuum is removed by normal ordering the field operators. For a general state \({\color{black}{|}}\psi\rangle\), this is equivalent to subtracting the VEV, \begin{equation}{\color{black}{ \langle \psi| : T_{\mu \nu} : |\psi \rangle = \langle \psi| T_{\mu \nu} |\psi \rangle - \langle 0 | T_{\mu \nu} | 0 \rangle. }}\end{equation} Since the compact space is still Minkowski, the state \({\color{black}{|0_L}}\rangle\) is in the Fock space spanned by the \({\color{black}{|}}\psi\rangle\), and so we can write, \begin{equation} \label{fin} {\color{black}{ \langle 0_L| : T_{00} : |0_L \rangle = \langle 0_L| T_{00} |0_L \rangle - \langle 0 | T_{00} | 0 \rangle, }}\end{equation} where the VEV \({\color{black}{}}\langle 0 | T_{00} | 0 \rangle = \lim_{L' \rightarrow \infty} \langle 0_{L'}| T_{00} |0_{L'} \rangle\). So, following Ford's program, the \({\color{black}{|0_L}}\rangle\) belong to a topologically nontrivial realization of Minkowski space. Interpreting the compactification as a perturbation, we increase \({\color{black}{L}}\) until we recover the "usual'' Minkowski space with topology \({\color{black}{}}\mathbb{R}^4\). The idea is to examine the shift in vacuum energy as we "decompactify'' the space: if it is finite, Ford says it's the physical vacuum energy, the energy associated with the normal-ordered expectation value \({\color{black}{}}\langle 0_L| : T_{00} : |0_L \rangle\). Taylor expanding the exponential functions in Eq. (\ref{exp}) gives \begin{equation}{\color{black}{ \langle 0_L| T_{00} |0_L \rangle = \frac{1}{2\pi \alpha^2} - \frac{\pi}{6L^2} + \mathcal{O}(\alpha^3). }}\end{equation} Taking \({\color{black}{L }}\rightarrow \infty\) gives \({\color{black}{}}\langle 0 | T_{00} | 0 \rangle = \frac{1}{2\pi \alpha^2}\). Performing the subtraction in Eq. (\ref{fin}) and removing the cutoff (\({\color{black}{}}\alpha \rightarrow 0\)) gives \({\color{black}{}}\langle 0_L| : T_{00} : |0_L \rangle = -\pi/6L^2\). The physical vacuum energy of the compactified space is negative, just as in the Casimir effect! One possible way to think about this result is that the compactified space is in some sense restricted -- it is bounded, much like the space in between the parallel conducting plates of the Casimir experiment. Therefore, the vacuum modes are forced into a discrete spectrum, Eq. (\ref{eigen}). There are now a countably infinite number of such modes (indexed by integer \({\color{black}{n}}\)), whereas in the topologically trivial Minkowski space the spectrum has an uncountably infinite number of modes varying continuously with wavenumber \({\color{black}{k}}\). In the Casimir effect, the vacuum outside the plates is in some sense "denser'' than that inside, resulting in a force that collapses the plates. As Ford has shown, this analogy carries over rather neatly to the gravitational case, with the vacuum density of usual Minkowski vacuum being greater than the compactified space.

Creation of Universes from Nothing

Vilenkin, A. (1982)

Vilenkin is looking within the context of closed universe models for an understanding of how inflation began. The de Sitter solution in a closed universe executes a bounce at \({\color{black}{t = 0}}\) when the universe has a minimal size \({\color{black}{a_{\rm min} = H^{-1}}}\). Vilenkin takes a page from Coleman and studies these dynamics with an eye towards the analogous process in particle mechanics: the bounce of the universe corresponds to a particle bouncing off a potential barrier at \({\color{black}{a=H^{-1}}}\) (with \({\color{black}{a(t)}}\) assuming the role of the position coordinate). The analogy in this case is a particle of mass \({\color{black}{m}}\) evolving under an inverted harmonic oscillator potential with a height \({\color{black}{2/m}}\). Taking the particle analogy further, we know that quantum mechanically, the particle can tunnel through this barrier rather than executing the classical bounce.

One immediate concern arises at this point: there's nothing stopping \({\color{black}{a(t)}}\) from taking on negative values in the particle analogy where it corresponds to the position of the particle, since quantum tunneling can take the particle from one side to the other. But what to make of \({\color{black}{a(t) < 0}}\) back in the physical picture? Changing the domain of the problem weakens the relevance of the analogy.

The tunneling amplitude is given by the bounce solution of the Euclidean equation of motion, obtained by taking \({\color{black}{t }}\rightarrow it\). The Euclidean version of the de Sitter solution is \begin{equation}{\color{black}{ a(t) = H^{-1}\cos(Ht) }}\end{equation} With this form of \({\color{black}{a(t)}}\), the closed FRW metric describes a four-sphere (the "well-known'' de Sitter instanton, although I could find no references to it beyond the Gibbons-Hawking paper, where they don't refer to it as such and it appears mainly calculational). In contrast to the problem of vacuum decay, in which the Euclidean problem possessed (asymptotic, as \({\color{black}{t }}\rightarrow \pm \infty\)) initial (and final) states separated by a trough, this Euclidean solution has no initial states and is in fact defined only for \({\color{black}{|t| < \pi/(2H)}}\). It does possess the bounce at \({\color{black}{t=0}}\), and so is effectively all trough. The key property of this instanton, then, is that it is compact -- it seems to describe a tunneling event from nothing, as Vilenkin claims.

Contrast this with decay problems. There we have a quantum system in a false vacuum with the ability to tunnel to a lower energy true vacuum. The amplitude for this tunneling event is given by the Euclidean bounce solution, that describes the analog classical trajectory of a particle evolving in \({\color{black}{-V}}\) -- from one state to the other and back again. So, there is a duality of sorts, between the physical quantum problem and the Euclidean classical one. Here, the physical problem is the evolution of the universe, evolving classically from some initial state in the infinite past, through a classical turnaround, to a state in the infinite future. But this is not part of the duality. Instead, we consider the quantum analog of this problem, that of a particle tunneling through a potential barrier. It is this analog problem that we then Euclideanize and solve for its bounce solution. So the duality here does not involve the original classical problem, but the analog quantum mechanical one, whereas in the decay problem we have a bonafide quantum system to start with. Adding the tunneling event to the otherwise classical physical picture of the evolving de Sitter universe is a gutsy move. But, while the de Sitter evolution is ostensibly classical, we do expect some quantum funny business to pop up at the bounce point. This is what's clever about Vilenkin's approach -- he's providing the quantum dynamics via analogy because there is no guidance from the purely quantum gravitational side.

To secure these ideas, Vilenkin draws analogy to the creation of particle/antiparticle pairs which helps ground some of the concept, but again the main difference is that pair creation is a fundamentally quantum mechanical phenomenon, whereas the bouncing of the de Sitter universe may or may not be. But again, Vilenkin's guess, and I would agree, is that it is: at the bounce, or creation event, or whatever, there are strong quantum effects and so the comparison with pair creation is probably apt.

But I don't understand his figure:

Fig. 1 Vilenkin's Figure 1.

\({\color{black}{a(t) > 0}}\), so what's with the portion of the horizontal axis to the left of \({\color{black}{a = 0}}\)? Are there two universes?

Having said that, Vilenkin argues that the de Sitter instanton solution exists, and hence as a stationary point of the Euclidean action, should give a "dominant contribution to the path integral of the theory''. If we grant that this path integral is relevant to the quantum cosmological dynamics affecting the universe's origins, then indeed the tunneling event can (and maybe probably should) be realized. This to me is really all he needs to say, but despite a few questions of relevance, the inspiration drawn from the particle analogy is creative and inspiring.

Infrared divergences in a class of Robertson-Walker universes

Ford, L.H. and Parker, Leonard (1977)

Among the infinite collection of possible vacuum states in curved spacetimes, "apparently natural" states of minimally coupled scalar fields are shown to result in infrared divergent quantities in Robertson-Walker spaces with powerlaw expansion (including de Sitter space in the limit that the exponent goes to infinity). The metrics sourced by the energy density and pressure of these states are shown to be inconsistent with Einstein's equations leading to the peculiar conclusion that these otherwise "sensible" states are not permitted under Einstein gravity. Though not explicitly discussed by Ford and Parker, this is the calculation revealing that the Bunch-Davies vacuum state does not exist for a massless, minimally coupled scalar field in de Sitter space.

The evolution of a minimally coupled scalar, \({\color{black}{\phi_{; \mu}{}^{; \mu}}}\), under power law expansion, \({\color{black}{a(t) \propto t^c}}\), is considered. Under the usual mode decomposition, the mode function \({\color{black}{\psi_k}}\) satisfies \begin{equation}{\color{black}{ \frac{d^2\psi_k}{d\tau^2} + k^2 a^4 \psi_k = 0, }}\end{equation} with \({\color{black}{\tau = \int^t a^{-3}(t')dt'}}\). Note that this is not the conformal time, but is instead a redefinition of time that conveniently eliminates \({\color{black}{\psi_k'}}\) from the equation of motion. For power law expansion, we get \begin{equation} \label{modefp} {\color{black}{ \psi_k(\tau) = \sqrt{\tau}[c_1 H^{(1)}_\nu (z) + c_2 H^{(2)}_\nu (z)], }}\end{equation} with \({\color{black}{z = ka_0^2 \tau^b/|b|}}\) and \({\color{black}{b = (1-c)/(1-3c)}}\). Note that the \({\color{black}{c_i}}\) are the complex Fourier amplitudes, not the same \({\color{black}{c}}\) that appears in the power law expansion. With the mode function, we can calculate the energy density and pressure of the field in the small \({\color{black}{k}}\) limit; the intent is to identify under what conditions these quantities diverge for "sensible" states, defined as those for which \({\color{black}{|c_1 - c_2| \neq 0}}\) as \({\color{black}{k\rightarrow 0}}\). It is worthwhile to work through this calculation.

The small argument limit of the Hankle function, \({\color{black}{H^{(1)}_\nu (z) = J_\nu (z) + iY_\nu(z)}}\), is found by examining its series expansion and keeping only the lowest-order term. The Hankel expansion is built around the power series of the Bessel function, \begin{equation} \label{expfp} {\color{black}{ J_\nu (z) = \left(\frac{1}{2}z\right)^\nu \sum_{k=0}^\infty (-1)^k \frac{\left(\frac{1}{4}z^2\right)^k}{k! \Gamma(\nu + k + 1)}. }}\end{equation} It is clear that \({\color{black}{\lim_{z\rightarrow 0}J_\nu(z) = 0}}\), and so the only function contributing to \({\color{black}{H_\nu (z)}}\) in this limit is \({\color{black}{Y_\nu (z) \rightarrow -J_{-\nu}(z)/\sin \nu \pi}}\). Keeping only the \({\color{black}{k=0}}\) term we find \begin{equation}{\color{black}{ H^{(1)}_\nu (z) = -\frac{i}{\sin \nu \pi}\left(\frac{1}{2}z\right)^{-\nu}\frac{1}{\Gamma(1-\nu)}. }}\end{equation} Using a kinda awesome result called the Euler Reflection formula, \({\color{black}{\Gamma(1-x)\Gamma(x) = \pi/\sin\pi x}}\), the above limit becomes \begin{equation} \label{asymfp} {\color{black}{ H^{(1)}_\nu (z) = -\frac{i}{\pi}\left(\frac{1}{2}z\right)^{-\nu}\Gamma(\nu), }}\end{equation} which agrees with Eq. (3.23) of the paper. Now, in order to evaluate the energy density and pressure of \({\color{black}{\phi}}\), we will end up needing to take time derivatives of \({\color{black}{\psi_k(\tau)}}\). From Eqs. (\ref{modefp}) and (\ref{asymfp}), the time dependence enters as \({\color{black}{\tau^{-\nu b + 1/2}}}\), with \({\color{black}{\nu = |2b|^{-1}}}\). When \({\color{black}{b>0}}\), the exponent \({\color{black}{-\nu b + 1/2 = 0}}\) and the time dependence drops out of \({\color{black}{\psi_k}}\). In order to retain the time dependence, we cannot use the standard asymptotic form of \({\color{black}{H_\nu (z \rightarrow 0)}}\), but must instead keep next-order terms.

Going back to Eq. (\ref{expfp}), we now keep both \({\color{black}{k=0}}\) and \({\color{black}{k=1}}\) terms: \begin{equation}{\color{black}{ H^{(1)}_\nu (z) = -\frac{i}{\sin \nu \pi}\left(\frac{1}{2}z\right)^{-\nu}\left[\frac{1} {\Gamma(1-\nu)}- \frac{1}{4}z^2\frac{1}{\Gamma(2-\nu)}\right]. }}\end{equation} Before applying the awesome Euler formula, we rewrite \({\color{black}{\Gamma(2-\nu) = (1-\nu)\Gamma(1-\nu)}}\) giving Eq. (4.3) in the paper. With Eq. (\ref{modefp}) and \({\color{black}{d\psi_k/dt = a^{-3}d\psi_k/d\tau}}\), we determine \({\color{black}{\rho}}\) and \({\color{black}{p}}\) using Eqs. 4.2a and 4.2b from the paper with the lower limit of integration taken to be \({\color{black}{\epsilon}}\). For \({\color{black}{b>0}}\), after performing the integrations, only a single term remains (from the \({\color{black}{|\psi_k|^2}}\) term), \begin{equation} \label{rhopfp} {\color{black}{ \rho = -3p = |c_1-c_2|^2\frac{\Gamma^2(\nu)}{4\pi^4 a^2}(\nu a_0^2)^{-2\nu}(2\nu -5)^{-1}\epsilon^{5-2\nu}. }}\end{equation} There are other terms, but these are of larger powers of \({\color{black}{\epsilon}}\): \({\color{black}{\epsilon^{7-2\nu}}}\), \({\color{black}{\epsilon^{9-2\nu}}}\), etc. (the lowest power of \({\color{black}{\epsilon}}\) coming from the \({\color{black}{|\partial_0 \psi_k|^2}}\) term is \({\color{black}{\epsilon^{7-2\nu}}}\).) As \({\color{black}{\epsilon \rightarrow 0}}\), these terms die away faster than the \({\color{black}{\epsilon^{5-2\nu}}}\) term of Eq. (\ref{rhopfp}).

When \({\color{black}{\nu > 5/2}}\), \({\color{black}{\rho}}\) and \({\color{black}{p}}\) are infrared divergent if the \({\color{black}{c_i}}\) are regular (if \({\color{black}{c_2 - c_1 \neq 0}}\) as \({\color{black}{k\rightarrow 0}}\)) -- these are the states of interest. Now for the main result: \({\color{black}{\nu > 5/2}}\) corresponds to powerlaw expansion with \({\color{black}{1 < c \leq 2}}\), and from the Einstein equations we know that these powers give an equation of state, \({\color{black}{\alpha = p/\rho}}\), in the range \({\color{black}{-1/3 < \alpha \leq -2/3}}\). It's close, but the equation of state of Eq. (\ref{rhopfp}), \({\color{black}{\alpha = -1/3}}\), lies outside this range. The conclusion is that the metrics associated with the infrared divergent states are not consistent with the Einstein equations. As a specific, important example, the Bunch-Davies vacuum corresponds to \({\color{black}{c_1 = 0}}\), \({\color{black}{c_2 = 1}}\), which are regular in the \({\color{black}{k \rightarrow 0}}\) limit. While \({\color{black}{\rho}}\) and \({\color{black}{p}}\) are finite, the long wavelength contributions dominate the correlation function, \({\color{black}{\langle \phi^2 \rangle}}\), causing it to diverge for regular states like the Bunch-Davies vacuum. It is not strictly true that the associated metric is not consistent with Einstein gravity, since \({\color{black}{T_{\mu \nu}}}\) is finite. But, this result is often cited as indicating that the Bunch-Davies vacuum does not exist for the massless, minimally coupled scalar field.

Supercooled Phase Transitions in the Very Early Universe

Hawking, S. W. and Moss, I. G. (1982)

This is the well-cited paper associated with the Hawking-Moss instanton. The paper is short and terse: the instanton interpretation is actually not given, nor is the solution discussed in the Euclidean context at any length. The Hawking-Moss (HM) instanton is an idea that has grown considerably since its somewhat vague conception in this paper; several authors have worked to develop it and place it in the context of other phase transition solutions. I have found the paper by Jensen and Steinhardt (1984) to be a good reference in this regard; it gives a more elaborated and detailed account of the HM solution and its relation to the Coleman-De Luccia (CDL) and Coleman bubbles. It is remarkable, however, how Hawking and Moss advance at least the rough outline of such an insightful and foundational idea with so few words or explanation. (I'm actually not sure, though, which is more impressive: that feat or the efforts of cosmologists who have since worked to understand the relevance and proper context of the HM solution.)

Hawking and Moss consider inflation with a nearly-Coleman Weinberg-type potential, i.e. there is a tunable (though expected small) mass term. If the mass is larger: \({\color{black}{m^2 > 2H^2(1-6\xi)}}\), then we have the type of bubble nucleation described by Coleman and De Luccia, to wit a 1st order phase transition that proceeds via quantum tunneling of the field from the metastable false vacuum to the stable true vacuum. These bubbles then expand and coalesce leading to a very inhomogeneous universe. The main result of this paper seems to be the third paragraph, second column of pg. 2: if the mass is smaller, tending towards the nearly-CW-type potential, \({\color{black}{m^2 < 2H^2(1-6\xi)}}\), then things change abruptly and significantly. Decay via tunneling to the stable state is still a solution, but the action is larger (p. 181 J&S), suppressing the amplitude of this transition relative to the alternative: a tunneling event from the metastable state to the unstable state -- the field perched very precariously atop the small barrier. Since the potential barrier corresponds to the wall of the nucleated bubble, this tunneling event corresponds to a bubble with an infinitely thick wall and infinitesimally thin interior (p. 181 J&S). It is argued (p. 1) that this way of tunneling is a result of the effects of curvature and the finite horizon, that they ensure a phase transition that occurs simultaneously at all points in the space leaving behind a homogeneous space. It seems that the existence of a finite horizon is used to argue against the CDL-type bubble when \({\color{black}{m^2 < 2H^2}}\), since these bubbles would have a radius surpassing the radius of the de Sitter space, \({\color{black}{H^{-1}}}\). Jensen and Steinhardt also allude to the role of gravity: "for small enough mass the gravitational effects can cause the entire universe , not just a small bubble, to jump from the metastable to the unstable phase (p. 177)". Additionally they assert that "bubbles [evolve] from the Coleman-type bubble, through the Coleman-De Luccia type bubble to the Hawking-Moss solution as gravity is made stronger and stronger (p. 186)." I have yet to fully understand in what sense gravity is "strong" in the HM scenario.

It was not clear to me on several read-throughs why this transition necessarily occurs simultaneously everywhere, but the discussion of Jensen and Steinhardt elucidates this point. Moving from the physical tunneling problem to the Euclideanized version of a particle rolling in a trough, the CDL tunneling events correspond to the Coleman "bounce" solutions: the particle starts near the metastable part of \({\color{black}{V}}\) and rolls down through the minimum and back up the unstable side. Solutions exist for suitably chosen release points that do no over- or undershoot. The Hawking-Moss solution (i.e. instanton in the Euclidean world) corresponds to a release point at the minimum of \({\color{black}{-V}}\), \({\color{black}{\phi_1}}\) (the unstable local maximum of \({\color{black}{V}}\)). There are no dynamics (is there really even a tunneling event?): the field starts off and stays put. Without dynamics, this solution is more correctly an initial condition of the universe (or at least that part within the causal de Sitter horizon): all points start out at \({\color{black}{\phi_1}}\). In terms of the de Sitter instanton of radius \(H^{-1}\), it is clear that the classical universe emerges from the phase transition with a size \({\color{black}{H(\phi_1)^{-1}}}\).

Of course, the salient feature of this solution is that it ensures a perfectly homoegenous universe -- the bubble is effectively the whole of the observable universe (really it's a bubble made of nothing but bubble wall!). Subsequent evolution will see the field rolling down to either the true vacuum or back to the metastable state; presumably different parts of the universe will subscribe to either fate. Jensen and Steinhardt point out that the result would be a universe at the end of the day not much different from one forged out of many nearly-simultaneous CDL bubble collisions: "we do not believe one could distinguish between an HM event and many independent tunneling fluctuations occurring approximately simultaneously in the universe" (p. 187).

Fate of the False Vacuum: Semiclassical Theory

Coleman, S. (1977)

This is Coleman's famous paper on phase transitions in field theory. The decay rate of the false vacuum is shown to depend on the Euclidean version of the action, where the nucleation of the bubble of new phase is associated with the "bounce'' solution of the Euclidean field theory. I haven't quite cracked this paper yet -- there are aspects of the bounce that beguile me. In what follows, I will sketch the main ideas of the paper and pause to point out the various sources of my confusion. The bubble nucleation rate per unit volume is \({\color{black}{\Gamma V = A\exp(-B/\hbar)[1+ \mathcal{O} (\hbar)]}}\). The exponent, the tunnelling amplitude, is known from elementary quantum mechanics to be \({\color{black}{B = 2\int_{q_0}^{\sigma} ds \sqrt{2V}}}\). The field tunnels to the true vacuum along the path that minimizes this quantity: \begin{equation} \label{var} {\color{black}{ \delta \int_{q_0}^{\sigma} ds \sqrt{2V} = 0. }}\end{equation} A parallel with the principle of least action is immediately evident: the field evolves so as to minimize the quantity \({\color{black}{\int ds \sqrt{2(E-V)}}}\). This is the same form as the tunneling amplitude, with \({\color{black}{E=0}}\) and \({\color{black}{V \rightarrow -V}}\). By this observation, the solutions of Eq. (\ref{var}) are the paths in configuration space traced out by the solutions of the equations \begin{equation} \label{imt} {\color{black}{ \frac{d^2 q}{d\tau ^2} = \frac{\partial V}{\partial q}. }}\end{equation} Meanwhile, the ordinary Euler-Lagrange equations are \begin{equation} \label{el} {\color{black}{ \frac{d^2 q}{dt^2} = -\frac{\partial V}{\partial q}. }}\end{equation} Eq. (\ref{imt}) is the imaginary time version of Eq. (\ref{el}), that is, \({\color{black}{\tau = it}}\). Because this substitution takes the Minkowski metric (\({\color{black}{ds^2 = -dt^2 + d{\bf x}^2}}\)) to the Euclidean (\({\color{black}{ds^2 = d\tau^2 + d{\bf x}^2}}\)), the action obtained by taking \({\color{black}{V \rightarrow -V}}\) (from which the modified equation of motion, Eq. (\ref{imt}) is derived) is called the Euclidean action. Because \({\color{black}{E=0}}\), when \({\color{black}{q = q_0, \sigma}}\), \({\color{black}{dq/d\tau = 0}}\). Taking \({\color{black}{q=q_0}}\) as \({\color{black}{\tau \rightarrow - \infty}}\) (\({\color{black}{q_0}}\) is reached only asymptotically), we have then \begin{equation}{\color{black}{ \int_{q_0}^\sigma ds \sqrt{2V} = \int_{-\infty}^0 d\tau L_E. }}\end{equation} So, the Euclidean action describes the motion of a particle in the potential \({\color{black}{-V}}\): beginning at the hump at \({\color{black}{q=q_0}}\) in the infinite past and rolling down through the valley to the point \({\color{black}{q=\sigma}}\) at \({\color{black}{\tau = 0}}\). This Euclidean picture is equivalent to the tunnelling event. The motion for \({\color{black}{\tau > 0}}\) is the time-reversal of the motion for negative \({\color{black}{\tau}}\): the particle reaches the point \({\color{black}{q = \sigma}}\), comes to rest, and rolls back to \({\color{black}{q=q_0}}\). This reversal of motion is what Coleman refers to as "the bounce''. Referring back to the expression for \({\color{black}{B}}\) (just before Eq. (\ref{var})), we see that the \({\color{black}{B}}\) is the total Euclidean action for the bounce, \begin{equation}{\color{black}{ B = \int_{-\infty}^{\infty} d\tau L_E. }}\end{equation}

Fig. 1 Coleman's Figure 1.

Moving to field theory, we replace \({\color{black}{q(\tau)}}\) by \({\color{black}{\phi(\tau,{\bf x})}}\) and the initial condition \({\color{black}{\lim_{\tau \rightarrow \pm \infty}q(\tau) = q_0}}\) becomes two conditions: \({\color{black}{\lim_{\tau \rightarrow \pm \infty}\phi(\tau, {\bf x}) = \phi_+}}\) and \({\color{black}{\lim_{|{\bf x}| \rightarrow \pm \infty}\phi(\tau, {\bf x}) = \phi_+}}\). The condition \({\color{black}{dq/d\tau|_0 = 0}}\) becomes \({\color{black}{d\phi(0,{\bf x})/d\tau = 0}}\). The bounce solution is O(4)-symmetric (this is not proven by Coleman; the O(3) symmetry is evident by the spherical symmetry of the bubble) so we can replace \({\color{black}{\tau}}\) and \({\color{black}{{\bf x}}}\) by \({\color{black}{\rho = \sqrt{\tau^2 + |{\bf x}|^2}}}\). With this change, the two separate initial conditions in terms of \({\color{black}{\tau}}\) and \({\color{black}{{\bf x}}}\) above become one condition: \({\color{black}{\lim_{\rho \rightarrow \infty} \phi(\rho) = \phi_+}}\).

Here is where my confusion begins to set in. The Euclidean equation of motion of the field in terms of \({\color{black}{\rho}}\) is \begin{equation} \label{eomrho} {\color{black}{ \frac{d^2\phi}{d\rho^2} + \frac{3}{\rho}\frac{d\phi}{d\rho} = U'(\phi) }}\end{equation} which is the equation of motion of a particle evolving in potential \({\color{black}{-U(\phi)}}\) in time \({\color{black}{\rho}}\). It starts with \({\color{black}{d\phi/d\rho|_0 = 0}}\) at \({\color{black}{\phi(\rho = 0) = \phi_-}}\) and evolves to come to rest at \({\color{black}{\phi_+}}\) at \({\color{black}{\rho = \infty}}\). So at "time'' zero, we are already starting in the true vacuum and evolving (seemingly backward) to the true vacuum. Meanwhile, the physical picture in Minkowski space is of a field starting out in the false vacuum and evolving to the true vacuum. Of course, though, \({\color{black}{\rho}}\) is not actual time. Consider \({\color{black}{\tau = 0}}\). At \({\color{black}{\rho = 0}}\) (corresponding to \({\color{black}{{\bf x=0}}}\)), we have the field in the true vacuum and then the limit \({\color{black}{\rho \rightarrow \infty}}\) (corresponding to \({\color{black}{{\bf x} \rightarrow \infty}}\)) makes sense, since the field should be in the false vacuum far from the true vacuum. In fact, this should be true at all \({\color{black}{\tau}}\). But at \({\color{black}{{\bf x}=0}}\), this version of events presumes that the decay has occurred by \({\color{black}{\tau =0}}\) -- that the bubble is already formed. The way that I see to make sense of this is to interpret the evolution of \({\color{black}{\phi}}\) from \({\color{black}{\rho=0 \rightarrow \infty}}\) as the time-reversal of the nucleation process.

On page 33, Coleman describes the situation thusly: \({\color{black}{}}\phi(0)\) starts out very close to \({\color{black}{}}\phi_-\). The particle then stays close to \({\color{black}{}}\phi_-\) until a very large time \({\color{black}{}}\rho = R\) (it is explained earlier that the friction term \({\color{black}{}}\propto \phi'\) in Eq. (\ref{eomrho}) is diminished at large \({\color{black}{}}\rho\)). Near time \({\color{black}{R}}\), the particle moves quickly through the valley and slowly comes to rest at \({\color{black}{}}\phi_+\) at time infinity. Translating back to field theoretic language, the bounce looks like a bubble of radius \({\color{black}{R}}\) with true vacuum within separated by a thin wall from false vacuum without.

This essentially confirms my suspicion of what's going on from the 2nd to last paragraph. The field loiters around \({\color{black}{\phi = \phi_-}}\) for a long "time'' \({\color{black}{\rho}}\). In the field theoretical picture, the field has already tunnelled to the true vacuum by time \({\color{black}{\tau = 0}}\). The condition that it remains near \({\color{black}{\phi_-}}\) until \({\color{black}{\rho}}\) is large translates to the picture that the bubble, as a function of \({\color{black}{{\bf x}}}\), has a large physical radius \({\color{black}{\rho = R}}\). So at time \({\color{black}{\tau = 0}}\), a bubble is nucleated with radius \({\color{black}{R}}\). After this, it grows according to the hyperboloid \({\color{black}{R^2 = |{\bf x}|^2 - t^2}}\). The initial bubble radius depends on the energy difference between the false and true vacua: in the thin wall approximation (that this difference, \({\color{black}{\epsilon}}\), actually isn't very large), \({\color{black}{R \propto 1/\epsilon}}\). This makes sense because the tunnelling probability depends on this energy difference -- the greater the difference, the greater the tunnelling rate, and the smaller the radius.

A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems

Linde, A. (1982)

This is Linde's overview of his attempt to improve on the 'old' inflation of Guth's first-order phase transition by considering a roll down period after the initial bubble nucleation. He demonstrates solutions to the horizon, flatness, isotropy, and monopole problems.

Linde first reviews the problems with 'old' inflation, namely the unacceptably inhomogeneous universe left behind by bubble collisions and the exceedingly low critical temperature required to get sufficient inflation out of an SU(5)-type model. These problems relate to the sudden first-order transition to the true vacuum; Linde's idea is to delay the decay so that the bubble expands by a huge amount -- enough to encompass the whole of the present-day observable universe.

There are several important scales to sort out: \({\color{black}{\phi_0}}\) is the VEV, and it corresponds to the scale \({\color{black}{T_1}}\) (the equilibrium temperature); \({\color{black}{\phi_1 \approx \phi=0}}\) is the order of the field after tunnelling, and it corresponds to the scale \({\color{black}{T_c}}\), which is the critical temperature of the phase transition (I believe that the term "supercooling'' refers to the great difference between \({\color{black}{T_c \ll T_1}}\)); \({\color{black}{T_0}}\) is referred to as the temperature at which inflation begins.

The model is based on the Coleman-Weinberg potential in the SU(5) GUT. When \({\color{black}{T \ll \phi_0}}\) the false vacuum at \({\color{black}{\phi =0}}\) has a very slight positive concavity, i.e. \({\color{black}{\phi}}\) has a very small positive mass. A small potential barrier at \({\color{black}{\phi \ll \phi_0}}\) makes the state classically metastable; however, it is unstable in the presence of thermal and/or quantum fluctuations. The inflationary epoch proceeds as follows: 1) temperature corrections stabilize \({\color{black}{\phi}}\) in the \({\color{black}{\phi=0}}\) symmetric state when \({\color{black}{T \gg \phi_0}}\) (where \({\color{black}{V(\phi_0) = 0}}\); 2) Eventually the universe comes to be dominated by \({\color{black}{V(0)}}\); 3) the phase transition proceeds through a strongly supercooled state \({\color{black}{\phi = 0}}\) at \({\color{black}{T=T_c}}\), where \({\color{black}{T_c \ll \phi_0}}\) (\({\color{black}{\phi_0 \sim \phi(T_1)}}\)), the symmetry breaking scale; \({\color{black}{T_c \ll T_1}}\)); 4) the field tunnels to \({\color{black}{\phi \sim 3\phi_1}}\), where \({\color{black}{\phi_1}}\) is very close to \({\color{black}{\phi=0}}\); 5) after bubble nucleation, \({\color{black}{\phi}}\) grows as \({\color{black}{\sim e^{mt}}}\). Linde states that \({\color{black}{\tau}}\) is "several times greater than \({\color{black}{m^{-1}}}\)", but it's really a factor of \({\color{black}{\ln \phi_0/\phi_i}}\), where \({\color{black}{\phi_i}}\) is the value of the field when inflation begins (which is probably \({\color{black}{\mathcal{O}(10)}}\)). Linde chooses \({\color{black}{\tau \sim 5m^{-1}}}\). The bubble at this point is still inflating, in contrast to the case of nucleation of true vacuum bubbles in 'old' inflation ; 6) the field spends most of its time near \({\color{black}{\phi_1}}\) in the bubble (Hubble friction) so that \({\color{black}{V(\phi) = V(\phi_1) \approx V(0)}}\) for a time \({\color{black}{\tau \sim 5m^{-1} \sim T_c^{-1}}}\) (the latter equality follows from \({\color{black}{-m^2 = -\frac{2}{15}V''(T_c)}}\).)

For SU(5), during this period \({\color{black}{\tau \sim T^{-1}_c}}\) of exponential expansion, \({\color{black}{H = 1.5\times 10^{10}}}\) GeV, and so with \({\color{black}{T_c \sim 2 \times 10^6}}\) GeV, the universe grows by a staggering amount: \({\color{black}{e^{H\tau} = e^{H/T_c} = e^{7500} = 10^{3260}}}\)! Given that the size of a typical bubble at nucleation is \({\color{black}{10^{-20}}}\) cm, the size of the bubble after inflation is \({\color{black}{10^{3240}}}\) cm \({\color{black}{\gg \ell = 10^{28}}}\) cm, the size of the present-day observable universe. So, indeed, the whole of the observable universe (plus much to spare) comes from the growth of a single bubble. At the end of the paper Linde shows that this amount of expansion is sufficient to resolve the classic problems of big bang cosmology. I won't go into these.

In the above computation, the amount of expansion of the universe is identified with the expansion of the bubble. It is not initially clear that this is appropriate, since to an observer outside the bubble, the bubble walls expand at just about the speed of light resulting in a bubble that grows to a size of only \({\color{black}{\sim 10^{28}}}\) cm. Linde is treating the spacetime inside the bubble just like any other inflationary spacetime: there is no mention of the effect of the bubble wall on the expanding space. In effect, observers inside the bubble evidently perceive an infinite inflating universe.

Birth of Inflationary Universes

Vilenkin, A. (1983)

This is a follow-on paper to Vilenkin's 1982 work "Creation of Universes from Nothing". It is mostly an elaboration on the idea proposed in that work: that the origin of the universe could have been a quantum tunnelling event. The added details resolve a few nits with the previous work, though nothing Earth shattering. First, and simply, the Euclidean time is given by \({\color{black}{\tau = it}}\), so that we have \({\color{black}{a(t) = H^{-1}\cosh (Ht)}}\) and \({\color{black}{a(\tau) = H^{-1}\cos (H\tau)}}\), the latter defining the Euclidean instanton (compare with Eqs. (4) and (5) in Vilenkin 1982, which are both in terms of \({\color{black}{t}}\), with \({\color{black}{t \rightarrow -it}}\)). This is useful in revealing that the instanton occurs in a different time domain than the subsequent classical evolution. Next, Vilenkin points out importantly that the action of the de Sitter instanton is smaller than all other compact gravitational instantons (a result found by Hawking) and so gives the dominant contribution to the tunneling amplitude (\({\color{black}{\Gamma \sim \exp (-S)}}\)). In light of this observation, Vilenkin's earlier statement that "...if the effective potential has several minima, most of the universes will nucleate at the maximum with the smallest value of \({\color{black}{\rho_v}}\)", where \({\color{black}{\rho_v}}\) is the effective potential energy of the field, is confusing. Since \({\color{black}{S = -3m^4_{\rm Pl}/8\rho_v}}\), these transitions have the largest action, and hence the smallest transition rate. Why then should most universes end up in such states?

Vilenkin also nicely discusses his result in relation to the work of Hawking and Moss (1982), who examine the same instanton solution but in the context of quantum tunneling from one point of the inflationary effective potential to another, rather than from nothing to a point of the effective potential as in Vilenkin's program. He disputes the Hawking-Moss interpretation on the grounds that the finite de Sitter horizon \({\color{black}{H^{-1}}}\) precludes the universe from undergoing a transition everywhere (as no causal process could effect); however, the Hawking-Moss analysis proposes no such thing. In fact, in their paper they use the same claim to argue against the consistency of Coleman-DeLuccia bubble formation on the grounds that such bubbles would have radii surpassing the causal horizon. The size of the Hawking-Moss "bubble" is the size of the de Sitter; this can be identified with the whole universe only if the universe is closed. My suspicion, then, is that Vilenkin is misplaced in his criticism of the Hawking-Moss phase transition.

The remainder of the paper considers the evolution of the inflationary universe after the tunnelling event. Vilenkin gives an interesting take on inflaton fluctuations as Brownian motion of the inflaton field and appears to anticipate eternal inflation. I haven't worked through this section.