The Thermodynamics of Transient Trapped Surfaces in the Geon Collapse

Abstract

It is shown that transient trapped surfaces form in a class of emerging globally hyperbolic spacetimes, within punctured Planck-scale neighbourhoods of the geon supported on intersecting singular supports whose intersection forms a characteristic core in a non-strongly causal setting. These neighbourhoods shrink towards the intersecting singular support in the distributional geometry. In particular, the trapped surfaces occur near the characteristic limit corresponding to the unstable equilibrium of the self-gravitating geon. They act as an effective classical barrier for descriptions formulated purely within smooth differential geometry. The area of these trapped-surface configurations, computed on Planck-referenced neighbourhoods, is shown to tend to zero both in the asymptotically flat limit of the emerging spacetime and in the geon limit. Thus, transient trapped surfaces evaporate in the sense that their area vanishes as classical and asymptotically flat spacetime emerges within the quantum foam framework. A state-counting generating function for the transient trapped surfaces is constructed from the coherent-state density operator. This generating function maps microscopic occupation-number sectors to macroscopic data and thereby allows a definition of Boltzmann entropy (not to be confused with the von Neumann entropy, which is zero for any pure coherent state). Since the coherent state is constructed to implement the correspondence principle, expectation values of the relevant quantised observables reproduce their classical values. In particular, the expectation value of the bosonic occupation-number operator serves as a microstate-counting variable in the coherent sector. The generating function takes the form of an exponential of this expectation value, leading to an entropy–area relation consistent with the Hawking–Bekenstein scaling.

1. Introduction

In Wheeler’s work on the concept of a geon, a self-gravitating electromagnetic entity, he states that “the geon completes the scheme of classical physics; one’s interest in following geons into quantum domain will depend upon one’s view of the relation between very small geons and elementary particles.” [1]. Here, we advocate Wheeler’s view in his description of geons, supplemented by Schwartz’s view on elementary particles, and hence the view that, while we can introduce a classical universe as a manifold with a metric field, we must equip it with a space of distributions and a vector subspace admitting a Hilbert space structure. This subspace should be continuously embedded in the space of distributions in order for elementary particles to be admitted in the universe [2]. This view leads to a route that allows for a precise, self-consistent, and comprehensive formulation of the geon, and of a well-defined sector of quantum gravity, thereby addressing at least part of the gap identified by Carlip in his recent review of quantum foam [3], namely that the concept of quantum fluctuations, and consequently the vacuum fluctuations of the metric in quantum gravity, remains vague and lacks a precise formulation. Moreover, we believe that the current framework should provide some clarification on the issue of reaching a consensus on the open question of the topology or type of foamy spacetime on which such fluctuations should be defined, which also remains open [3].

In the perspective introduced above, the geon is the limit of a smooth transition from differential geometry, realised along a sequence of globally hyperbolic spacetimes, to a distributional spacetime geometry at a characteristic limit where strong causality is broken, and hence global hyperbolicity breaks down [4]. This transition involves warping of the spacelike hypersurfaces, and of their timelike normal fields, towards a null-like self-gravitating distributional structure in an unstable equilibrium at the characteristic. These are properties shared with Wheeler’s electromagnetic geon [1], including that its unstable equilibrium resembles the balance act of a pencil at its tip. In fact, it is possible to argue that it will collapse under its own polarisation. This collapse causes a vacuum displacement, producing regions of large expansion and contraction around a vacuum shell that eventually fades. The dynamics described here is driven by locally positive and negative energy densities of equal magnitude, while the total energy remains zero, in the emerging globally hyperbolic and asymptotically flat spacetime.

As we shall see, this means that the hypotheses of a positive mass theorem (see, e.g., Huang and Lee [5] for a discussion of the positive mass theorem) cannot be applied uniformly across all geometric scales in this framework. The reason for this is that, although the energy conditions hold at intersecting singular supports whose intersection forms a characteristic core of the geon, in the distributional sense, one can rigorously show that neither the dominant, the weak, nor the strong energy condition holds locally on suitable open sets in a wide class of emerging globally hyperbolic spacetimes, even if the total energy is zero. That is, there exist spacetime elements in the sequence of emerging globally hyperbolic spacetimes where all three of these energy conditions fail locally while the total energy is zero.

It can further be shown that there exist Planck-scale neighbourhoods of the core of the geon where the trace of the extrinsic curvature, and hence the local Hubble parameter, undergoes large transitions of expansion and contraction (faster than any polynomial), thereby driving inflation in the emerging spacetime and effectively resolving the horizon problem. Similarly, transient trapped surfaces exist in punctured neighbourhoods of the geon core. More precisely, and as has been shown in [4] and will be elaborated on here, they occur in neighbourhoods whose scale shrinks towards the support, and hence in neighbourhoods of the associated characteristic limit. These trapped surfaces act as an effective classical barrier for descriptions formulated purely within smooth differential geometry, and hence shielding the self-gravitating geon in its unstable equilibrium in distribution geometry.

The narrative above has served to provide a coherent thread to support a conceptual understanding of quantum foam and the geon. In relation to this, it seems prudent to take the opportunity to point out that it provides scope for some adjustments to Wojciech’s famous Border Territory cartoon [6], simply since it does offer a smooth cross-over from quantum to classical, with a reinterpretation of Bohr’s somewhat harsh pose at the quantum-to-classical border. This follows immediately from its construction, since the correspondence principle is naturally part of the model and technically embedded by the admitted Gelfand triple. Thus, the geon framework, and especially the introduction of a well-defined smooth transition from distribution to differential geometry, demonstrates that any statement that the border between quantum theory and classical theory is ill-defined necessarily is not true. In fact, in the geon framework, Bohr is not the border watch, but actually the guide, facilitating a smooth cross-over, here via the decoherence of the collapsing geon into a coherent state and hence providing a smooth pathway to the emerging globally hyperbolic and asymptotically flat spacetime. Thus, contrary to Wojciech’s statement that there is no unified view of the universe as a whole, it is reasonable to advocate that Wheeler’s geon [1] in conjunction with Schwartz’s definition of an elementary particle [2] provides a pathway to a unified view of the universe.

In this work, we continue the inquiry into the geon framework via an excursion whose purpose is to learn more about the properties of trapped surfaces forming a barrier in the Planck-scale neighbourhood of the geon. Specifically, we briefly revisit the construction of the geon and its context in distributional geometry built from a renormalised Gaussian Colombeau-type model delta net that was introduced in [4]; see, e.g., Nigsch–Vickers [7] for general details of a model delta net construction. Within this framework, we model trapped surfaces, determine their surface area in Planck-referenced neighbourhoods where they arise, and assign a statistical entropy by counting quanta in the weakly interacting coherent sector, motivated in part, but strictly independent of [8,9]. We obtain an entropy–area relation for these configurations arising in the collapsing geon, and hence in the transition from the distributional geometric context to smooth differential geometry. Additionally, we show that the trapped surfaces evaporate geometrically in the sense that their area tends to zero in the asymptotically flat limit. The entropy is constrained by the same entropy–area relation.

In relation to entropy, within the geon framework, it should be observed that the coherent state is constructed so that the expectation values of the relevant quantised observables reproduce their classical values (in the sense of the correspondence principle). This provides a mechanism by which, within the geon framework, we can use the quantum field operators to construct the statistics for the microstates and connect them to a macrostate once we have defined the applicable state-counting generating function.

To summarise, and to place this work in relation to the “central dogma” formulation in the review of the entropy of Hawking radiation by Almheiri et al. [10], but here in the context of transient trapped surfaces, we show that transient trapped surfaces emerge in the aftermath of the geon collapse, that any such transient trapped surface behaves like a quantum system whose Boltzmann entropy is proportional to its surface area. That is, we show that within the Gaussian Quantum Foam framework, and hence within a self-consistent distributional sector of quantum gravity, the entropy can be determined from microstate counting in a coherent state, allowing us to compute the entropy and bridge the micro- and macroscopic perspectives via Boltzmann’s formula. This is then used to show that the area is indeed a macroscopic measure of these states. This effectively removes any need to introduce a “central dogma” described in [10] relating the microstate and macrostate perspectives, by here showing that the area of a transient trapped surface is a representation of the underlying quantum states. That is, within the context of a renormalised Gaussian Model Delta as a model for the geon and emerging asymptotic flat spacetime, then contrary to the central dogma, there is no need to introduce “a certain dose of belief” [10] since we derive the area and entropy from a gravity description that encompasses a fully quantum to classical evolution.

Before we proceed and hence at the onset of this discussion, we emphasise and reinforce, to avoid any confusion that may arise concerning the nature of entropy, that the entropy discussed here is not the von Neumann entropy of the pure coherent state, since it is trivially zero. It is a statistical entropy functional motivated by microstate counting in the weakly interacting many-quanta sector in the neighbourhood of the geon.

2. The Distribution Geometry of the Geon

In this section, we provide a precise definition of the geon and its context in the renormalised distributional geometry together with examples of how to work with the distributional geometry and to understand the geon as a field and its impact on the energy conditions in the emerging spacetime. We follow the construction of the geon framework in [4] in order to keep the current work reasonably self-consistent. The motivation behind the development of a distributional geometry can be traced to several considerations, but a central requirement is that it enables a smooth transition, in the sense of distributions, from globally hyperbolic spacetimes in classical differential geometry to a distributional geometric limit. In the present construction, on each fixed spatial hypersurface, each shift component has its singular support in a two-dimensional embedded surface of the hypersurface. Their common intersection at the spatial origin defines a distinguished core, which we identify as the core of the geon in the distribution geometry. This provides a regularised description of Planck-scale physics in which Wheeler’s self-gravitating but unstable geon may be understood as a distributional limit supported on intersecting embedded surfaces with the collapse driven by its own polarisation effects localised at the core, in a mathematically controlled setting. A related motivation is that the framework allows for the construction of a self-consistent and well-defined localised causal sector of quantum gravity.

We now proceed to the construction of distributional geometry, initiating the formalisation of the geon via a smooth transition from globally hyperbolic spacetimes—and consequently, classical differential geometry—to the framework of distribution geometry. At this point, readers familiar with the notion of a geon from Wheeler’s work [1] might be reluctant to accept a definition of a geon as a distribution, on the basis that it appears to diverge from Wheeler’s original conception of a self-gravitating electromagnetic entity representing a ‘fundamental body’ [1]. To address such a concern, we proactively here state that the geon here that arises from the transition indeed is a self-gravitating null structure in an inherent unstable equilibrium, and we also return to the work of Schwartz on the application of distribution theory to elementary particles [2]. In this framework, Schwartz defines the universe as a differentiable manifold M and equips it with the space of distributions ${\mathcal{D}}^{\prime }\left(M\right)$; this construction provides a rigorous functional-analytical basis that aligns with Hörmander’s Definition 6.3.3 on distributions in manifolds [11]. Furthermore, Schwartz characterised a scalar particle as a vector subspace of ${\mathcal{D}}^{\prime }\left(M\right)$ endowed with a continuously embedded Hilbert space structure. Thus, in this generalised Schwartzian sense, the geon is identified as the fundamental constituting particle, or body, of the universe. Within this distributional setting, a sequence of spacetimes ${\left\{(M^{\left(k\right)},g_{\mu \nu }^{\left(k\right)})\right\}}_{k\in \mathbb{N}}$ serve as the approximating nets that converge to the geon, thus conversely allowing for the emergence of classical geometry from a distributional limit.

Given this concise background, we now proceed and introduce a localisation and a restriction of the Schwartz space $\mathcal{S}\left({\mathbb{R}}^3\right)$ that will be an integral part of the precise definition of the geon.

Definition 1

(A Localised Restriction of Schwartz Space ${\mathcal{S}}_{\mathcal{G}}\left(U\right)$). Consider an open set $U\subset {\mathbb{R}}^3$. Theorem 1.4.1 in Hörmander [11] guarantees that for any open set U and any compact set $K\subset U$, there exists a compactly supported test function ${\varphi }_c\in C_0^{\infty }\left(U\right)$ with $0\le {\varphi }_c\le 1$, such that ${\varphi }_c=1$ in a neighbourhood of K. Such a function is called a cut-off function. By Lemma 7.1.8 in [11], the space of compactly supported test functions is dense in the Schwartz space. Using a cut-off function, we may therefore localise elements of the Schwartz space, and in particular the set of Gaussians $\mathcal{G}\left({\mathbb{R}}^3\right)\subset \mathcal{S}\left({\mathbb{R}}^3\right)$ to the Schwartz space on the open set U as follows:

$${\beta }_{{\varphi }_c}:={\varphi }_c\beta ,\beta \in \mathcal{G}\left({\mathbb{R}}^3\right)$$

This construction provides a rigorous definition of a localisation of elements in a restriction of Schwartz space to a space ${\mathcal{S}}_{\mathcal{G}}\left(U\right)$. To avoid having to deal with a burdensome notation, we write β for such elements with the understanding that each Gaussian is modulated by a compactly supported cut-off function.

This modulation resolves any concern as to whether a sequence of localised Gaussians used here converging to a distribution in agreement with Theorem 4.1.5 [11] can be used to represent a local distribution on a manifold in the sense of Definition 6.3.3 in [11] (distributions on a differentiable manifold). The set of all such local distributions then agrees with Schwartz definition [2] for distributions on a manifold.

We now have a sufficient requisite to state the definition of the geon.

Definition 2

(Geon). Let ${\kappa }^{\left(k\right)}:O_{{\kappa }^{\left(k\right)}}\subset M_{\left(k\right)}\to U_{{\kappa }^{\left(k\right)}}\subset {\mathbb{R}}^4$ be a local coordinate chart on any homotopic k-spacetime $(M_{\left(k\right)},g_{\mu \nu }^{\left(k\right)})$ in a sequence ${\left\{(M_{\left(k\right)},g_{\mu \nu }^{\left(k\right)})\right\}}_{k\in \mathbb{N}}$ of homotopic, globally hyperbolic spacetimes, and denote by ${\kappa }_t^{\left(k\right)}={\kappa }^{\left(k\right)}{|}_{{\mathrm{\Sigma }}_t^{\left(k\right)}}$ its restriction to a Cauchy surface ${\mathrm{\Sigma }}_t^{\left(k\right)}=\{p\in M_{\left(k\right)}:t_{\left(k\right)}\left(p\right)=t,t\in \mathbb{R}\}$.

Let the line element for any spacetime element in the sequence be given by

$$d{s}_{\left(k\right)}^2=-N_{\left(k\right)}^{2}d{t}^2+{\eta }_{ij}(d{x}^i+{\beta }_{\left(k\right)}^{i}dt)(d{x}^j+{\beta }_{\left(k\right)}^{j}dt),$$

(1)

in the local coordinate chart with coordinates $(t,x^1,x^2,x^3)$ (i.e., Cartesian coordinates $(t,x,y,z)$). Here, $N_{\left(k\right)}=k$ is the lapse function, and ${\eta }_{ij}$ is the induced Euclidean metric on ${\mathrm{\Sigma }}_t^{\left(k\right)}$. The vector ${\beta }_{\left(k\right)}$ is the shift vector field tangent to ${\mathrm{\Sigma }}_t^{\left(k\right)}$. In the chart ${\kappa }_t^{\left(k\right)}$, its components

$${\beta }_{\left(k\right)}:=({\beta }_{\left(k\right)}^1,{\beta }_{\left(k\right)}^2,{\beta }_{\left(k\right)}^3)$$

are functions on $U_{{\kappa }_t^{\left(k\right)}}$, and each belongs to the Schwartz space of localised Gaussians ${\mathcal{S}}_{\mathcal{G}}(U_{{\kappa }_t^{\left(k\right)}})$ in Definition 1 with

$${\beta }_{\left(k\right)}^i:={\beta }_{\left(k\right)}\left(x^i\right):={\displaystyle \frac{1}{\sqrt{4{\sigma }^2/k^2\pi }}}exp\left(-{\displaystyle \frac{{\left(x^i\right)}^2}{4{\sigma }^2/k^2}}\right),\sigma \ge 1.$$

(2)

Here, σ is a parameter controlling the range in which the effect of the shift vector can not be neglected. The distributional limit of the sequence ${\left\{(M_{\left(k\right)},g_{\mu \nu }^{\left(k\right)})\right\}}_{k\in \mathbb{N}}$, defined by the convergence ${\beta }_{\left(k\right)}^i\to {\tilde{\beta }}^i$ in the sense of distributions on $U_{{\kappa }_t^{\left(k\right)}}$ as $k\to \infty$ (i.e., each ${\beta }_{\left(k\right)}^i$ converges in ${\mathcal{S}}_{\mathcal{G}}^{\prime }\left(U_{{\kappa }_t}\right)$ to the Dirac measure ${\delta }_{x^i}$ in the coordinate $x^i$), determines a localised, distribution-valued spacetime element

$${(\tilde{M},{\tilde{g}}_{\mu \nu })}_{\mathcal{G}}:=\underset{k\to \infty }{lim}(M_{\left(k\right)},g_{\mu \nu }^{\left(k\right)}),$$

which is called a geon.

Next, we use the following definition to clarify the support structure of the geon:

Definition 3

(Singular Support Embedding and Geon Core). Let ${\beta }_{\left(k\right)}^i,i\in \{1,2,3\}$ denote the Gaussian ray components of the shift vector field as in Definition 2. Then, in the sense of distributions on a fixed Cauchy surface ${\mathrm{\Sigma }}_t$,

$${\tilde{\beta }}^i:=\underset{k\to \infty }{lim}{\beta }_{\left(k\right)}^i={\delta }_{{\mathrm{\Sigma }}_i},$$

where ${\delta }_{{\mathrm{\Sigma }}_i}$ denotes the Dirac surface measure supported on ${\mathrm{\Sigma }}_i=\{p\in {\mathrm{\Sigma }}_t:x^i\left(p\right)=0\}\subset {\mathrm{\Sigma }}_t$, i.e., ${\delta }_{{\mathrm{\Sigma }}_i}:=\delta \left(x^i\right)$ in the adapted local coordinates. Each ${\tilde{\beta }}^i$ therefore has singular support given by the two-dimensional embedded hypersurface

$$\mathrm{singsupp}\left({\tilde{\beta }}^i\right)={\mathrm{\Sigma }}_i:=\{x^i=0\}\subset {\mathrm{\Sigma }}_t.$$

The intersection of the singular support of each component

$$\mathcal{C}:={\displaystyle {\displaystyle \bigcap _{i=1}^3}}{\mathrm{\Sigma }}_i=\left\{(0,0,0)\right\}$$

is said to be the geon core.

Next, we introduce the space of admissible test functions:

Definition 4

(Admissible test functions in the static Gaussian sector). Let $\mathcal{S}(U_{{\kappa }_t^{\left(k\right)}})$ denote the localised restricted spatial Schwartz space on the slice ${\mathrm{\Sigma }}_t$. A test function $\psi (t,x)=\chi \left(t\right)\phi \left(x\right)$ is said to be admissible if

$$\chi \in C_0^{\infty }\left((0,1)\right),{\int }_0^1\chi \left(t\right)dt=1,\phi \in \mathcal{S}(U_{{\kappa }_t^{\left(k\right)}}),$$

and the spatial factor φ satisfies, in a neighbourhood of $\{x^i=0\}\cap U_{{\kappa }_t^{\left(k\right)}}$,

$$\phi >0,\phi (\dots ,-x^i,\dots )=\phi (\dots ,x^i,\dots ),{\partial }_i^2\phi \le 0(\mathit{local}\mathit{concavity}\mathit{in}{x}^i).$$

Given these definitions, we provide a list of remarks to elaborate on some further details.

Remark 1

(Convergence in Distributions). That the limit in Definition 2 is well defined relies, in part, on the existence of approximating sequences converging in the sense of distributions (see, e.g., Theorem 4.1.5 in [11]), but more importantly on that, and as we will show, a renormalised distribution algebra can be constructed allowing for products of distributions.

Remark 2

(Scripting of the Sequence Index). The sequence index k for each spacetime element in the definition of the geon may appear as a subscript or a superscript on vectors, covectors, and tensors, without a difference in meaning. The index k is never summed over.

Remark 3

(The Notation ${\beta }_{\left(k\right)}^{x^i}$). In what follows, we will also use the following notion for the shift vector components in a spacetime element $(M_{\left(k\right)},g_{\mu \nu }^{\left(k\right)})$

$${\beta }_{\left(k\right)}^{x^i}:={\beta }_{\left(k\right)}\left(x^i\right).$$

Remark 4

(Lapse Selection $N_{\left(k\right)}=k$). The reason for the choice of lapse function, which scales linearly with the sequence index and is otherwise kept constant, is to guarantee distributional integrity of the framework in the transition to distribution geometry. It also safeguards the Lorentzian signature $(-,+,+,+)$ for any finite sequence index k, since from (2),

$${\beta }_i^{\left(k\right)}{\beta }_{\left(k\right)}^i={\displaystyle \frac{k^2}{4{\sigma }^2\pi }}\left(exp\left(-{\displaystyle \frac{k^2{x}^2}{2{\sigma }^2}}\right)+exp\left(-{\displaystyle \frac{k^2{y}^2}{2{\sigma }^2}}\right)+exp\left(-{\displaystyle \frac{k^2{z}^2}{2{\sigma }^2}}\right)\right)\le {\displaystyle \frac{3k^2}{4{\sigma }^2\pi }}<k^2,$$

where it has been used that $\sigma \ge 1$ by definition.

Remark 5

(The Bernal and Sánchez relation). It is natural to pose the question of how general the definition of the geon actually is. To provide some insight to this, we use the fact that Bernal and Sánchez have proved [12] that any globally hyperbolic spacetime, here translated into the context of any spacetime element $(M_{\left(k\right)},g_{\mu \nu }^{\left(k\right)})$ in the definition, admits a regular time function $t_{\left(k\right)}$ with a gradient $\nabla t_{\left(k\right)}$ that is timelike. The spacetime can homeomorphically be foliated by the time function $t_{\left(k\right)}$ and thus, it is isometric to the smooth product topology ${\mathrm{\Sigma }}^{\left(k\right)}\times \mathbb{R}$ where ${\mathrm{\Sigma }}^{\left(k\right)}$ is a spacelike Cauchy surface (level surface) and where each level surface of $t_{\left(k\right)}$ that is ${\mathrm{\Sigma }}_t^{\left(k\right)}=\{p\in M_{\left(k\right)}:t_{\left(k\right)}\left(p\right)=t\in \mathbb{R}\}$ is a spacelike Cauchy surface with a metric field on ${\mathrm{\Sigma }}^{\left(k\right)}\times \mathbb{R}$ such that the line element in Cartesian coordinates takes the form

$$d{s}_{\left(k\right)}^2=-N_{\left(k\right)}^{2}d{t}^2+h_{ij}^{\left(k\right)}d{x}^{i}d{x}^j.$$

Here, $N_{\left(k\right)}:{\mathrm{\Sigma }}^{\left(k\right)}\times \mathbb{R}\to (0,\infty )$ is the lapse function and the symmetric tensor field $h_{ij}^{\left(k\right)}$ is the induced Riemannian metric on the hypersurfaces. The lapse function is related to the time function by the square modulus of an eikonal equation

$${\parallel \nabla t_{\left(k\right)}\parallel }^2={\displaystyle \frac{1}{N_{\left(k\right)}^2}}.$$

It is straightforward to show that the metric field in Definition 2 can be cast in this form by introducing a relative-velocity gauge via the diffeomorphism ${\mathrm{\Phi }}_{\left(k\right)}$:

$${\kappa }_{\left(k\right)}\circ {\mathrm{\Phi }}_{\left(k\right)}^{-1}:{\mathrm{\Phi }}_{\left(k\right)}(O_{{\mathrm{\Phi }}_{\left(k\right)}}\cap O_{{\kappa }_{\left(k\right)}})⟶{\kappa }_{\left(k\right)}(O_{{\mathrm{\Phi }}_{\left(k\right)}}\cap O_{{\kappa }_{\left(k\right)}}),$$

where $O_{{\mathrm{\Phi }}_{\left(k\right)}}$ and $O_{{\kappa }_{\left(k\right)}}$ are open subsets in the manifold. Explicitly, take ${\mathrm{\Phi }}_{\left(k\right)}$ to be a foliation-preserving diffeomorphism

$${\mathrm{\Phi }}_{\left(k\right)}:(t,x^1,x^2,x^3)\mapsto (t^{\prime },x^{\prime 1},x^{\prime 2},x^{\prime 3}),$$

(3)

as a coordinate transformation representing a relative-velocity gauge towards orthonormal observers,

$$t^{\prime }=t,x^{\prime i}={\mathrm{\Phi }}_{\left(k\right)}^i:={\mathrm{\Phi }}_{\left(k\right)}^i(x,t),i=1,2,3,$$

(4)

with the flow equation

$${\partial }_t{\mathrm{\Phi }}_{\left(k\right)}^{-1i}(x^{\prime },t)=-{\beta }_{\left(k\right)}^i\left({\mathrm{\Phi }}_{\left(k\right)}^{-1}(x^{\prime },t)\right).$$

(5)

Here, we exclude the intersection of the embedded two-surfaces in Definition 3 that constitutes the singular support from any open set, as this ensures the existence of the diffeomorphism; otherwise, the flow Equation (5) is ill-defined on the intersection in the distributional limit. Using the coordinate transformation (4) and the flow Equation (5), it follows that

$$d{x}^i={\partial }_{x^{\prime j}}{\mathrm{\Phi }}_{\left(k\right)}^{-1i}d{x}^{\prime j}+{\partial }_t{\mathrm{\Phi }}_{\left(k\right)}^{-1i}dt={\partial }_{x^{\prime j}}{\mathrm{\Phi }}_{\left(k\right)}^{-1i}d{x}^{\prime j}-{\beta }_{\left(k\right)}^i\left({\mathrm{\Phi }}_{\left(k\right)}^{-1}(x^{\prime },t)\right)dt,$$

and hence

$${\partial }_{x^{\prime j}}{\mathrm{\Phi }}_{\left(k\right)}^{-1i}d{x}^{\prime j}=d{x}^i+{\beta }_{\left(k\right)}^{i}dt.$$

The line element (1) in Definition 2 then takes the form

$$d{s}_{\left(k\right)}^2=-N_{\left(k\right)}^{2}d{t}^2+{\eta }_{mn}{\partial }_{x^{\prime i}}{\mathrm{\Phi }}_{\left(k\right)}^{-1m}{\partial }_{x^{\prime j}}{\mathrm{\Phi }}_{\left(k\right)}^{-1n}d{x}^{\prime i}d{x}^{\prime j}.$$

Thus, the form here is the same as in the theorem of Bernal and Sánchez [12]. Therefore, we conclude that the metric field can be cast into the form admitted by any globally hyperbolic spacetime. We also notice that, in the diffeomorphism ${\mathrm{\Phi }}_{\left(k\right)}$, where the shift vector has been gauged to zero, the induced Riemannian metric $h_{ij}^{\prime \left(k\right)}$ is given by

$$h_{ij}^{\prime \left(k\right)}={\eta }_{mn}{\partial }_{x^{\prime i}}{\mathrm{\Phi }}_{\left(k\right)}^{-1m}{\partial }_{x^{\prime j}}{\mathrm{\Phi }}_{\left(k\right)}^{-1n}.$$

(6)

However, even if the shift vector has been gauged to zero, it can be shown that covariant quantities, such as the trace of the extrinsic curvature $K^{\prime \left(k\right)}$, still retain the effect of the shift vector [4]. To see this for the extrinsic curvature, use

$$K_{ij}^{\prime \left(k\right)}=-{\displaystyle \frac{1}{2N_{\left(k\right)}}}{\partial }_t{h}_{ij}^{\prime \left(k\right)}.$$

This leads to the following trace:

$$K^{\prime \left(k\right)}=h_{\left(k\right)}^{\prime ij}{K}_{ij}^{\prime \left(k\right)}=-{\displaystyle \frac{1}{2N_{\left(k\right)}}}{h}_{\left(k\right)}^{\prime ij}{\partial }_t{h}_{ij}^{\prime \left(k\right)}={\displaystyle \frac{1}{N_{\left(k\right)}}}{\partial }_i{\beta }_{\left(k\right)}^i,$$

(7)

and hence, the shift vector can be used to proxy the geon on the hypersurfaces. In passing, we remark that there exists a $k_0$ such that, if k satisfies

$$k>k_0\wedge {\ell }_{\left(k\right)}:={\displaystyle \frac{\sigma }{k}}<{\ell }_p,$$

with ${\ell }_p$ the Planck length, then there exist Planck-scale neighbourhoods, for example, the open disc ${\mathcal{D}}_{{\ell }_{\left(k\right)}}(0,0)$ (for any fixed z), in which the expansion and contraction, as well as their rates, separated by the vacuum shell of the singular support, are huge. On the other hand, these effects become negligible as asymptotically flat spacetime emerges. This follows from the fact that the shift vector components are elements of the localised and restricted Schwartz space. Thus, the extrinsic curvature fluctuations close to the geon cause an inflationary phase that can also effectively resolve the horizon problem as classical spacetime emerges.

We are now ready to proceed to the construction of the renormalised distribution algebra. To start, we return to Definition 2, which is based on the sequence ${\left\{(M_{\left(k\right)},g_{\mu \nu }^{\left(k\right)})\right\}}_{k\in \mathbb{N}}$ of globally hyperbolic and homotopic spacetimes converging in the sense of distributions. For each k, global hyperbolicity guarantees the existence of a smooth and regular global time function $t_{\left(k\right)}:M_{\left(k\right)}\to \mathbb{R}$, which, for all $t\in \mathbb{R}$, induces a foliation of the manifold by spacelike Cauchy surfaces ${\mathrm{\Sigma }}_t^{\left(k\right)}=\{p\in M_{\left(k\right)}:t_{\left(k\right)}\left(p\right)=t\}$; see [12] for a proof that global hyperbolicity is sufficient for the existence of such a foliation.

As in Definition 2, let ${\kappa }^{\left(k\right)}:O_{{\kappa }^{\left(k\right)}}\subset M_{\left(k\right)}\to U_{\kappa }^{\left(k\right)}\subset {\mathbb{R}}^4$ be a local coordinate chart on $M_{\left(k\right)}$, and denote by ${\kappa }_t^{\left(k\right)}={\kappa }^{\left(k\right)}{|}_{{\mathrm{\Sigma }}_t^{\left(k\right)}}$ its restriction to any fixed hypersurface.

On any such Cauchy surface, and for each spatial coordinate $x^i$ (with the index i suppressed when convenient), we introduce a Gaussian restriction of Schwartz space to allow for scaled Gaussian products. As described in Definition 1, we localise the Gaussian kernel by modulating it with a compactly supported cut-off function. This ensures that the restriction makes sense on the open sets and that, in the limit, the resulting objects are local distributions on the spacetime manifolds, in the sense of Definition 6.3.3 in [11].

Definition 5

(Scaled Gaussian Cauchy Surface Restriction of ${\mathcal{S}}_{\mathcal{G}}(U_{{\kappa }_t^{\left(k\right)}})$). 

$$\begin{array}{c}\hfill {\mathcal{S}}_{{\mathcal{G}}_g}(U_{{\kappa }_t^{\left(k\right)}}):=\{g_{\left(k\right)}^n(x,\alpha )={\displaystyle \frac{{\displaystyle \prod _{i=1}^n}{g}_{\left(k\right)}(x,{\alpha }_i)}{k^{n-1}}}|n\in \mathbb{N},g_{\left(k\right)}(x,{\alpha }_i)\in {\mathcal{S}}_{\mathcal{G}}(U_{{\kappa }_t^{\left(k\right)}}),\\ \hfill {\alpha }_i>0,{\int }_{U_{{\kappa }_t^{\left(k\right)}}}{g}_{\left(k\right)}(x,{\alpha }_i)dx={\displaystyle \frac{1}{\sqrt{{\alpha }_i}}}\},\end{array}$$

where $\alpha ={\displaystyle \sum _{i=1}^n}{\alpha }_i$ is the total scale.

Remark 6.

Throughout this work, $g_{\left(k\right)}$ or $g^{\left(k\right)}$ denotes the Gaussian generator sequence of the restricted space. This should not be confused with the metric field $g_{\mu \nu }^{\left(k\right)}$, which always has indices, except when taking the determinant.

To this space, we introduce a multiplication operation:

Definition 6

(Multiplication in ${\mathcal{S}}_{{\mathcal{G}}_g}(U_{{\kappa }_t^{\left(k\right)}})$). For any $g_{\left(k\right)}^m(x,\alpha ),g_{\left(k\right)}^n(x,\gamma )\in {\mathcal{S}}_{{\mathcal{G}}_g}(U_{{\kappa }_t^{\left(k\right)}})$, multiplication is defined by

$$\begin{array}{cc}\hfill g_{\left(k\right)}^m(x,\alpha )·g_{\left(k\right)}^n(x,\gamma )& :=g_{\left(k\right)}^{m+n}(x,\alpha +\gamma )\hfill \\ \hfill & :={\displaystyle \frac{{\displaystyle \prod _{i=1}^m}{g}_{\left(k\right)}(x,{\alpha }_i)·{\displaystyle \prod _{j=1}^n}{g}_{\left(k\right)}(x,{\gamma }_j)}{k^{m+n-1}}}\hfill \\ \hfill & ={\displaystyle \frac{{\displaystyle \prod _{i=1}^{m+n}}{g}_{\left(k\right)}(x,{\zeta }_i)}{k^{m+n-1}}}=g_{\left(k\right)}^{m+n}(x,\zeta )\in {\mathcal{S}}_{{\mathcal{G}}_g}(U_{{\kappa }_t^{\left(k\right)}}),\hfill \end{array}$$

where $\alpha ={\sum }_{i=1}^m{\alpha }_i$, $\gamma ={\sum }_{j=1}^n{\gamma }_j$, and $\zeta ={\sum }_{i=1}^{m+n}{\zeta }_i$, with

$${\zeta }_i=\left\{\begin{array}{cc}{\alpha }_i,\hfill & 1\le i\le m,\hfill \\ {\gamma }_{i-m},\hfill & m+1\le i\le m+n.\hfill \end{array}\right.$$

This multiplication is bilinear, commutative, associative, and closes ${\mathcal{S}}_{{\mathcal{G}}_g}(U_{{\kappa }_t^{\left(k\right)}})$; that is, ${\mathcal{S}}_{{\mathcal{G}}_g}(U_{{\kappa }_t^{\left(k\right)}})$ is a vector space and an algebra under multiplication.

In the setting of the geon, with the shift vector components given by (2), it is convenient to take $g_{\left(k\right)}(x,{\alpha }_i)$ as

$$g_{\left(k\right)}(x,{\alpha }_i):={\beta }_{\left(k\right)}\left(\sqrt{{\alpha }_i}x\right)={\displaystyle \frac{1}{\sqrt{4{\sigma }^2/k^2\pi }}}exp\left(-{\displaystyle \frac{{\alpha }_i{x}^2}{4{\sigma }^2/k^2}}\right),$$

(8)

where ${\alpha }_i>0$ and, as before, $\sigma \ge 1$ is a localisation parameter controlling the range where the effect of the shift vector cannot be neglected. As $k\to \infty$, the Gaussian (8) converges to a scaled Dirac measure:

$$\underset{k\to \infty }{lim}\langle g_{\left(k\right)}(x,{\alpha }_i),\varphi \left(x\right)\rangle ={\displaystyle \frac{1}{\sqrt{{\alpha }_i}}}\varphi \left(0\right),\forall \varphi \in \mathcal{S}(U_{{\kappa }_t^{\left(k\right)}}).$$

(9)

We now introduce the corresponding restricted distributional space, defined as follows.

Definition 7

(Scaled Gaussian Cauchy Surface Restriction of ${\mathcal{S}}^{\prime }(U_{{\kappa }_t^{\left(k\right)}})$). 

$${\mathcal{S}}_{{\mathcal{G}}_g}^{\prime }(U_{{\kappa }_t^{\left(k\right)}}):=\left\{\underset{k\to \infty }{lim}{g}_{\left(k\right)}^n(x,\alpha )|g_{\left(k\right)}^n(x,\alpha )\in {\mathcal{S}}_{{\mathcal{G}}_g}(U_{{\kappa }_t^{\left(k\right)}})\right\}\subset {\mathcal{S}}^{\prime }(U_{{\kappa }_t^{\left(k\right)}}).$$

Given the formal definitions of the space, multiplication of elements in the space, and its dual, we define the basic structure of the renormalised Gaussian distribution algebra. Before stating the definition, it should be clear from standard properties of Schwartz space (in particular that it is a vector space and an algebra under multiplication), together with Definitions 5 and 6, that any operation in the algebra, including differentiation, is carried out on the smooth and rapidly decreasing elements in ${\mathcal{S}}_{{\mathcal{G}}_g}(U_{{\kappa }_t^{\left(k\right)}})$ before the limit is taken.

From a microlocal perspective, and temporarily restricting attention to products whose distributional limits are of order at most one (in the usual sense that a distribution is bounded by a finite sum of suprema of derivatives of admissible test functions; see, Definition 2.1.1, [11]), all operations are performed at finite k on smooth representatives. Consequently, the wave front set is empty at finite k, whereas the wave front set of the limiting distributions may be non-empty. Nevertheless, because the operations are carried out at the level where the wave front set is empty, the resulting limit is microlocally well defined, even if the limiting distributions $u,v\in {\mathcal{S}}_{\mathcal{G}}^{\prime }$ have oppositely directed wave fronts meeting at some point x. That is, we may have sequences $u_{\left(k\right)}\to u$ and $v_{\left(k\right)}\to v$ as $k\to \infty$ such that, for some pair $(x,\xi )$, $(x,\xi )\in WF\left(u\right)$ and $(x,-\xi )\in WF\left(v\right)$. In this case, the usual product of distributions would not satisfy Hörmander’s wave front set criterion (Theorem 8.2.10 in [11]), but the product is nevertheless well defined in the renormalised Gaussian algebra by construction. In this way, we avoid invoking the wave front set criterion at the level where it would prohibit multiplication, while preserving the Leibniz rule at the level of smooth representatives.

However, whereas products involving only zero-order factors, or a zero-order and a first-order factor, are solely controlled by the scaling and hence require no additional considerations, renormalisation is required for products whose distributional limits are of second order. For this, it should be noted that renormalisation is defined on the total smooth n-fold product prior to taking the limit and not on each factor in the product. Indeed, for an n-fold product, the smooth representatives grow like $k^{n-1}$, so the naïve product diverges as $k\to \infty$. In the present construction, the explicit $1/k^{n-1}$ scaling of n-fold Gaussian products, together with subtraction of universal $k^2$-type terms for products yielding a second-order contribution, is essential: it cancels the growth, yielding a distribution supported on the common singular point set. This $k^2$-term represents the geometric bare mass monopole of the geon. The same line of argument can be continued to higher even-order products with higher-order renormalisation, but in this work, renormalisation to obtain well-defined second-order distributions is sufficient. We remark that renormalisation is not required for odd order products at least if the resulting odd order is less than or equal to three.

Given these somewhat informal clarifications, we are ready to state the definition of the renormalised algebra:

Definition 8

(Renormalised Differential Algebra for Quantum Foam, $({\mathcal{S}}_{{\mathcal{G}}_g}^{\prime }(U_{{\kappa }_t^{\left(k\right)}}),·,\partial )$). The Gaussian restriction of Schwartz space defined in 5, together with Definition 6 and its dual space defined in Definition 7, forms a restricted renormalised distribution algebra, denoted

$$({\mathcal{S}}_{{\mathcal{G}}_g}^{\prime }(U_{{\kappa }_t^{\left(k\right)}}),·,\partial ),$$

where linear operations, multiplication, formal differentiation and renormalisation are performed on smooth representatives in ${\mathcal{S}}_{{\mathcal{G}}_g}(U_{{\kappa }_t^{\left(k\right)}})$ prior to taking the distributional limit.

We now demonstrate that the geon is indeed a self-gravitating, massless entity. To begin, let us return to the definition of the geon, Definition 2, and consider the effects on the level surfaces ${\mathrm{\Sigma }}_t^{\left(k\right)}$ of the foliation by the regular global time function $t_{\left(k\right)}$ as the sequence index k increases. To understand these effects, we use the fact that the time function satisfies the eikonal partial differential equation (see Remark 5):

$${\displaystyle \frac{1}{\parallel \nabla t_{\left(k\right)}\parallel }}=N_{\left(k\right)}.$$

(10)

With $N_{\left(k\right)}=k$, this imposes the following constraint on the time function:

$$\parallel \nabla t_{\left(k\right)}\parallel ={\displaystyle \frac{1}{k}}.$$

(11)

The eikonal equation allows us to understand the hypersurface evolution towards the geon, and some of its properties, without actually solving it. Specifically, we note that as the index k increases, the timelike normal gradient one-form $\nabla t_{\left(k\right)}=\left({\partial }_{\mu }{t}_{\left(k\right)}\right)d{x}^{\mu }$ to each level surface, and hence to each spacelike Cauchy surface ${\mathrm{\Sigma }}_t^{\left(k\right)}=\{p\in M_{\left(k\right)}:t_{\left(k\right)}\left(p\right)=t\in \mathbb{R}\}$, is timelike for each finite k but converges to null in the limit $k\to \infty$. Similarly, the spacelike level surfaces converge to a characteristic (null) surface. An alternative view of this, with the chronological future $I^{+}$ and causal future $J^{+}$ understood relative to the level surface ${\mathrm{\Sigma }}_t^{\left(k\right)}$, is that for any fixed t and any $k>\ell$, the homotopy of spacetime elements together with $\parallel \nabla t_{\left(k\right)}\parallel =k^{-1}$ gives rise to a non-convex nesting

$$I^{+}\left({\mathrm{\Sigma }}_t^{(\ell )}\right)\subset I^{+}\left({\mathrm{\Sigma }}_t^{\left(k\right)}\right)\subset J^{+}\left({\mathrm{\Sigma }}_t^{\left(k\right)}\right).$$

(12)

Thus, as the index k tends to infinity, the level sets of the global time function become asymptotically characteristic (null), and there is a pile-up of almost characteristic surfaces at an initial limiting null hypersurface, say ${\mathrm{\Sigma }}_0$, where it can be shown that the null expansion vanishes [4]. It is this limiting null structure that constitutes the self-gravitating, massless geon, in agreement with Wheeler’s notion [1].

The non-convex nesting necessarily implies that strong causality is broken at the limit. This follows from the fact that there exist indices $\ell <k$ and corresponding spacetime elements $(M_{(\ell )},g_{\mu \nu }^{(\ell )})$ and $(M_{\left(k\right)},g_{\mu \nu }^{\left(k\right)})$ such that a timelike curve $\gamma \in I^{+}\left({\mathrm{\Sigma }}_t^{\left(k\right)}\right)$ can leave and re-enter the chronological future $I^{+}\left({\mathrm{\Sigma }}_t^{(\ell )}\right)$. This is a consequence of the non-convex nesting of chronological futures, in particular $I^{+}\left({\mathrm{\Sigma }}_t^{(\ell )}\right)\subset I^{+}\left({\mathrm{\Sigma }}_t^{\left(k\right)}\right)$. Hence, a regular global time function $t_{\left(k\right)}$ cannot be sustained in the limit $k\to \infty$. Consequently, the standard Hamiltonian formulation of time evolution—see e.g., Kiefer [13]—which relies on a stable spacelike foliation, loses its meaning at the intersection of the singular supports at the geon core. The physical mechanisms of the geon collapse and the subsequent emergence of the classical background are therefore captured not by a flow along an internal Hamiltonian time coordinate, but parametrically through the sequence index k. In this framework, the usual canonical “problem of time”—see e.g., Kiefer [13] for a discussion of the “problem of time”—is bypassed by replacing Hamiltonian evolution at the singular limit with the aforementioned parametric flow, while the breakdown of quantum field theory in these highly fluctuating singular regimes is handled through the introduction of the geon as a distribution and through distributional geometry, giving rise at the same time to a well-defined sector of quantum gravity.

A further remark can be made in relation to the “problem of time”, since this framework offers a concrete realisation of Wheeler’s assertion that “Time is not a primordial and precise concept; it must be secondary, derivative, and approximate” [14]. This is a consequence of the fact that a smooth and regular time function exists if and only if the spacetime is globally hyperbolic [12], and since global hyperbolicity in the present model arises only once the geon collapses, it follows that time itself emerges as a consequence of quantum geometric excitation, not as a background parameter, but as a response to vacuum displacement.

The detailed review here has aimed to show that it is possible to implement a self-consistent and precise framework of the geon, sharing features with Wheeler’s geon [1] and Schwartz’s view on elementary particles [2]. Additionally, the framework satisfies Cooperstock–Faraoni–Perry geon conditions [15]: Einstein’s field equations must be solved self-consistently; the solution must be regular; the spatial geometry should remain stable throughout the sequence; and the gravitational field should approach asymptotic flatness at infinity. These conditions are satisfied since each spacetime in the convergent sequence is a smooth, globally hyperbolic solution to the field equations, and the construction remains regular in a weak sense even at the core of the geon, now within distributional geometry. Moreover, the construction is stable throughout the sequence, and the smooth, regular time function foliates every element of the sequence. Furthermore, the construction itself guarantees asymptotic flatness. In conjunction with these structural properties, the model also reproduces the foamy characteristics Wheeler saw as intrinsic to the geon and the emerging spacetime [14], and which Jack Ng [16], in his phenomenological analysis, later emphasised: spacetime appears smooth at large scales but increasingly bubbly “and foamy” as one approaches the Planck scale. These are also features that appear in Carlip’s review [3].

We would expect all these features to be generic for any geon model based on the mechanism by which the Cauchy surfaces and their timelike normal fields warp towards a null limit. It should be noted, however, that the construction presented here is specific in that it is limited to a bosonic sector of quantum gravity. Extending the framework to fermionic degrees of freedom would likely require the development of an appropriate spinor formalism in distribution geometry, which lies beyond the scope of the present work.

Given the geon framework, we will now provide three examples, both to illustrate how the algebra is used and to offer some basic insight into the physics of the geon landscape. Of these examples, one involves a zero-order product for which no renormalisation is required, while the other two involve products that give rise to second-order terms that must be renormalised. The first of these latter examples concerns the wave operator on the level sets (Cauchy surfaces) of each spacetime element in the sequence converging to the geon. The last is introduced to give meaning to the energy conditions in the geon context.

2.1. Examples

2.1.1. Shift Element Multiplication in ${\mathcal{S}}_{{\mathcal{G}}_g}(U_{{\kappa }_t^{\left(k\right)}})$

In this first example, we start from the observation that, given (8), any product of n-scaled Gaussians in the restricted space of Definition 5 can be expressed as

$$g_{\left(k\right)}^n(x,\alpha )={\displaystyle \frac{1}{{\left(2\sigma \sqrt{\pi }\right)}^{n-1}}}{g}_{\left(k\right)}\left(x,{\alpha }_1+{\alpha }_2+\cdots +{\alpha }_n\right),$$

(13)

which converges in the distributional sense to a rescaled Dirac measure,

$$\underset{k\to \infty }{lim}\langle g_{\left(k\right)}^n(x,\alpha ),\varphi \left(x\right)\rangle ={\displaystyle \frac{1}{{\left(2\sigma \sqrt{\pi }\right)}^{n-1}\sqrt{{\alpha }_1+\cdots +{\alpha }_n}}}\varphi \left(0\right),$$

(14)

for all $\varphi \in \mathcal{S}(U_{{\kappa }_t^{\left(k\right)}})$.

For the multiplication operation introduced in Definition 6, we have

$$g_{\left(k\right)}^m(x,\alpha )g_{\left(k\right)}^n(x,\gamma )=g_{\left(k\right)}^{m+n}(x,\alpha +\gamma )={\displaystyle \frac{{\displaystyle \prod _{i=1}^m}{g}_{\left(k\right)}(x,{\alpha }_i)·{\displaystyle \prod _{j=1}^n}{g}_{\left(k\right)}(x,{\gamma }_j)}{k^{m+n-1}}}\in {\mathcal{S}}_{{\mathcal{G}}_g}(U_{{\kappa }_t^{\left(k\right)}}),$$

(15)

for any $g_{\left(k\right)}^m,g_{\left(k\right)}^n\in {\mathcal{S}}_{{\mathcal{G}}_g}(U_{{\kappa }_t^{\left(k\right)}})$. It then follows from (13) that the multiplication in Definition 6 can be written as

$$g_{\left(k\right)}^m(x,\alpha )g_{\left(k\right)}^n(x,\gamma )={\displaystyle \frac{1}{{\left(2\sigma \sqrt{\pi }\right)}^{m+n-1}}}{g}_{\left(k\right)}(x,\alpha +\gamma ),$$

(16)

and hence, in the sense of distributions, using (14), for any $\varphi \in \mathcal{S}(U_{{\kappa }_t^{\left(k\right)}})$,

$$\underset{k\to \infty }{lim}\langle g_{\left(k\right)}^m(x,\alpha )g_{\left(k\right)}^n(x,\gamma ),\varphi \left(x\right)\rangle ={\displaystyle \frac{1}{{\left(2\sigma \sqrt{\pi }\right)}^{m+n-1}\sqrt{\alpha +\gamma }}}\varphi \left(0\right),$$

(17)

where $\alpha ={\displaystyle \sum _{i=1}^m}{\alpha }_i$ and $\gamma ={\displaystyle \sum _{j=1}^n}{\gamma }_j$.

2.1.2. Geon Field Equation and Renormalisation on $U_{{\kappa }_t^{\left(k\right)}}\subset {\mathrm{\Sigma }}_t^{\left(k\right)}$

In this example, we show that it is possible to formulate a non-linear wave equation for the shift vector components on any level set even if the coefficients are not smooth. Specifically, we show that the source of the wave equation is driven by the second-order derivative ${\delta }_{{\mathrm{\Sigma }}_i}^{″}$ of the Dirac measure ${\delta }_{{\mathrm{\Sigma }}_i}$ with a positive coefficient.

Using the metric field for the geon—see line element in Definition 2—and the non-linear wave operator acting on ${\beta }_{\left(k\right)}^x$ on a level set, then we have

$$\begin{array}{cc}\hfill {\square }_{g^{\left(k\right)}}{\beta }_{\left(k\right)}^x& ={\displaystyle \frac{1}{\sqrt{|g^{\left(k\right)}|}}}{\partial }_{\mu }\left(\sqrt{|g^{\left(k\right)}|}{g}_{\left(k\right)}^{\mu \nu }{\partial }_{\nu }{\beta }_{\left(k\right)}^x\right)\hfill \\ \hfill & =-{\displaystyle \frac{1}{N_{\left(k\right)}^2}}(2{\beta }_{\left(k\right)}^x{\left({\partial }_x{\beta }_{\left(k\right)}^x\right)}^2+{\beta }_{\left(k\right)}^x{\partial }_x{\beta }_{\left(k\right)}^x{\partial }_y{\beta }_{\left(k\right)}^y\hfill \\ \hfill & +{\beta }_{\left(k\right)}^x{\partial }_x{\beta }_{\left(k\right)}^x{\partial }_z{\beta }_{\left(k\right)}^z-\left(N_{\left(k\right)}^2-{\left({\beta }_{\left(k\right)}^x\right)}^2\right){\partial }_x^2{\beta }_{\left(k\right)}^x).\hfill \end{array}$$

(18)

Remark 7

(Same form for ${\beta }_{\left(k\right)}^y$ and ${\beta }_{\left(k\right)}^z$). The wave equation has the same form in all spatial coordinates, so it is sufficient to only consider one of the components.

To analyse the singular structure of this equation for the lapse scaling $N_{\left(k\right)}=k$, and test functions in the admissible space in Definition 4, each term in (18) is evaluated in the distributional algebra $({\mathcal{S}}_{{\mathcal{G}}_g^{\prime }},·,\partial )$ defined in Definition 8. The cross-terms with ${\partial }_y{\beta }_{\left(k\right)}^y$ and ${\partial }_z{\beta }_{\left(k\right)}^z$ vanish when tested against an even, positive test function in the admissible space (Definition 4). The remaining non-vanishing terms in the wave Equation (18) are

$${\displaystyle \frac{2}{k^2}}{\beta }_{\left(k\right)}^x{\left({\partial }_x{\beta }_{\left(k\right)}^x\right)}^2,$$

(19)

and

$$\left(1-{\displaystyle \frac{{\left({\beta }_{\left(k\right)}^x\right)}^2}{k^2}}\right){\partial }_x^2{\beta }_{\left(k\right)}^x={\partial }_x^2{\beta }_{\left(k\right)}^x-{\displaystyle \frac{{\left({\beta }_{\left(k\right)}^x\right)}^2}{k^2}}{\partial }_x^2{\beta }_{\left(k\right)}^x.$$

(20)

Using the shift vector in Definition 2 and the standard second derivative of a Gaussian, we have, in a functional sense, that

$$\begin{array}{cc}\hfill \underset{k\to \infty }{lim}{\displaystyle \frac{2}{k^2}}{\beta }_{\left(k\right)}^x{\left({\partial }_x{\beta }_{\left(k\right)}^x\right)}^2& ={\displaystyle \frac{2}{36\pi {\sigma }^2}}\underset{k\to \infty }{lim}\left[{\displaystyle \frac{9k^4{x}^2}{4{\sigma }^4}}{\beta }_{\left(k\right)}^{\sqrt{3}x}-{\displaystyle \frac{3k^2}{2{\sigma }^2}}{\beta }_{\left(k\right)}^{\sqrt{3}x}+{\displaystyle \frac{3k^2}{2{\sigma }^2}}{\beta }_{\left(k\right)}^{\sqrt{3}x}\right]\hfill \\ & ={\displaystyle \frac{2}{36\pi {\sigma }^2}}\underset{k\to \infty }{lim}\left[{\partial }_x^2{\beta }_{\left(k\right)}^{\sqrt{3}x}+{\displaystyle \frac{3k^2}{2{\sigma }^2}}{\beta }_{\left(k\right)}^{\sqrt{3}x}\right]={\left[{\displaystyle \frac{\sqrt{3}}{54\pi {\sigma }^2}}{\delta }_{{\mathrm{\Sigma }}_x}^{\prime \prime }\right]}_r.\hfill \end{array}$$

Here, the first term within the brackets in the equality before the renormalised result ${[·]}_r$ converges to a scaled second-order distributional derivative, while the second term represents the infinite bare mass contribution that we remove by renormalisation in the limit as $k\to \infty$, so that the result makes sense in distribution geometry. On the other hand, for finite values of the sequence index k, this second term represents a contribution, $m_{B+}^{\left(k\right)}$, to the observable mass of a geon:

$$m_{B+}^{\left(k\right)}={\displaystyle \frac{\sqrt{3}}{36\pi {\sigma }^4}}{k}^2.$$

(21)

Continuing with (20), we observe that it consists of a standard second-order derivative, converging to the second-order distributional derivative of the Dirac measure, and a term that can be worked out in a functional sense and in the algebra simply by computing the second derivative and rearranging the terms before the limit is taken. We then obtain (in a functional sense of smooth localised sequences converging in distributions)

$$\begin{array}{l}\underset{k\to \infty }{lim}{\displaystyle \frac{1}{k^2}}{\left({\beta }_{\left(k\right)}^x\right)}^2{\partial }_x^2{\beta }_{\left(k\right)}^x\hfill \\ =\underset{k\to \infty }{lim}{\displaystyle \frac{1}{{\left(2\sigma \pi \right)}^2}}\left[-{\displaystyle \frac{k^2}{2{\sigma }^2}}+{\displaystyle \frac{k^4{x}^2}{4{\sigma }^4}}\right]{\beta }_{\left(k\right)}^{\sqrt{3}x}\hfill \\ =\underset{k\to \infty }{lim}{\displaystyle \frac{1}{36\pi {\sigma }^2}}\left[{\partial }_x^2{\beta }_{\left(k\right)}^{\sqrt{3}x}-{\displaystyle \frac{3k^2}{{\sigma }^2}}{\beta }_{\left(k\right)}^{\sqrt{3}x}\right].\hfill \end{array}$$

(22)

The first term in this result converges to a second-order distributional derivative expression of the Dirac measure,

$${\displaystyle \frac{\sqrt{3}}{108\pi {\sigma }^2}}{\delta }_{{\mathrm{\Sigma }}_x}^{″},$$

(23)

while the remaining term represents part of the observable bare mass contribution, $m_{B-}^{\left(k\right)}$, in the field equation. Using the fact that the integral of a scaled Gaussian is the reciprocal of the scaling factor, we find

$$m_{B-}^{\left(k\right)}=-{\displaystyle \frac{\sqrt{3}}{36\pi {\sigma }^4}}{k}^2.$$

(24)

To have a sensible result in distribution geometry, this term must be renormalised in (22). Nevertheless, it should be noted that when including the bare mass contributions (21) and (24) to the field Equation (18) before the limit is taken, in each observable, emerging spacetime element $(M^{\left(k\right)},g_{\mu \nu }^{\left(k\right)})$, we have

$$m_B^{\left(k\right)}=m_{B+}^{\left(k\right)}+m_{B-}^{\left(k\right)}=0.$$

(25)

This is an expected result: in the emerging spacetime, each region of positive energy density is separated from a region of the same magnitude but negative energy density by a vacuum shell, and hence their net contribution must cancel. In conclusion, while each term in isolation must be renormalised to remain finite, the wave operator acting on the shift vector is finite for all finite k, and in distribution geometry, in the singular support ${\mathrm{\Sigma }}_i$, it is given by the second-order distributional derivative of the Dirac measure ${\delta }_{{\mathrm{\Sigma }}_i}$ with a bounded real-valued coefficient. To summarise, in a functional sense, we have

$$2{\beta }_{\left(k\right)}^x{\left({\partial }_x{\beta }_{\left(k\right)}^x\right)}^2-\left(1-{\displaystyle \frac{{\left({\beta }_{\left(k\right)}^x\right)}^2}{k^2}}\right){\partial }_x^2{\beta }_{\left(k\right)}^x\stackrel{{\mathcal{S}}_{{\mathcal{G}}_g}^{\prime }}{\to }\left({\displaystyle \frac{\sqrt{3}}{36\pi {\sigma }^2}}-1\right){\delta }_{{\mathrm{\Sigma }}_x}^{″}.$$

Substituting this into the wave Equation (18), we conclude that the field equation for the (unstable) Gaussian Quantum Foam element, or geon, on a Cauchy surface is

$$\underset{k\to \infty }{lim}{\square }_{g^{\left(k\right)}}{\beta }_{\left(k\right)}^x=\left(1-{\displaystyle \frac{\sqrt{3}}{36\pi {\sigma }^2}}\right){\delta }_{{\mathrm{\Sigma }}_x}^{″}.$$

(26)

Note that the coefficient

$$1-{\displaystyle \frac{\sqrt{3}}{36\pi {\sigma }^2}}>0,$$

since $\sigma \ge 1$. In conjunction, we have weakly ${\delta }_{{\mathrm{\Sigma }}_x}^{″}<0$ in the admissible space (Definition 4); it then follows that the geon is inherently unstable.

In the weak-field regime $\left|{\beta }_{\left(k\right)}\right|\ll N_{\left(k\right)}$ (and for slowly varying spatial dependence), the non-linear terms are negligible and the wave equation reduces to

$${\square }_{g^{\left(k\right)}}{\beta }_{\left(k\right)}^x={\partial }_x^2{\beta }_{\left(k\right)}^x+\mathcal{O}\left({\left({\beta }_{\left(k\right)}^x\right)}^2\right)=0.$$

2.1.3. Non-Uniformity of the Energy Conditions

In this example, taken from some unpublished material by the author of [17], it is shown how to use the algebra in a functional sense with explicit use of test functions to establish that the local energy conditions, specifically the local strong, dominant and weak energy conditions (see Kontou and Sanders [18] for details of these conditions in both classical and quantum contexts), hold at the core of the geon, but they do not hold uniformly across the sequence of globally hyperbolic spacetimes in Definition 2.

In particular, although the relevant inequalities are satisfied in the distributional limit at the singular support, for each finite spacetime element, there exist open subsets on which these conditions are violated. Moreover, a violation of the local weak energy condition, expressed as a negative projection of the stress–energy tensor along a future-directed timelike vector field, implies that the dominant energy condition fails as well. This follows since the dominant energy condition implies the weak energy condition; see, e.g., [18]. Also, while the energy conditions locally are violated across elements in the emerging asymptotic flat spacetime, the total energy as well as the momentum are actually zero on a total scale.

These results show that the formulation below of the positive mass theorem cannot be applied uniformly across geometric scales in this distribution-geometric framework, since its hypothesis, the dominant energy condition, fails on suitable open sets at finite scales. Here, we stress that the statement here is only in relation to the geon model and is not meant as an argument to falsify the statement on the positive mass theorem:

Theorem 1

(Positive mass theorem, see [5,19,20,21]). Let $(\mathrm{\Sigma },h,K)$ be a complete, asymptotically flat, three-dimensional initial data set satisfying the dominant energy condition. Then, the ADM energy–momentum $(E,P^1,P^2,P^3)$ satisfies

$$E\ge \left|P\right|,\left|P\right|:=\sqrt{{\left(P^1\right)}^2+{\left(P^2\right)}^2+{\left(P^3\right)}^2}.$$

Moreover, equality $E=\left|P\right|$ holds if and only if $(\mathrm{\Sigma },h)$ can be isometrically embedded into Minkowski spacetime with second fundamental form K.

In other words, while the dominant energy condition holds at the singular support in the distributional sense, it is not preserved uniformly across the sequence ${\left\{(M_{\left(k\right)},g_{\mu \nu }^{\left(k\right)})\right\}}_{k\in \mathbb{N}}$. Therefore, one cannot deduce a scale-uniform inequality of the form $E^{\left(k\right)}\ge \left|P^{\left(k\right)}\right|$ by appeal to the positive mass theorem above. Nevertheless, we shall see that even if the dominant energy condition is violated, we still have that $E^{\left(k\right)}\ge \left|P^{\left(k\right)}\right|$ holds, and specifically, we have $E^{\left(k\right)}=\left|P^{\left(k\right)}\right|=0$.

To see this, we first show that the dominant energy condition, and hence the weak energy condition, hold on the singular support of the geon but not in the emerging asymptotic flat spacetime, we then prove a theorem concerning the strong energy condition. To avoid any confusion, we remark that proving a statement about the strong energy condition does not imply that a corresponding statement about the weak energy condition can be made, since the strong energy condition does not imply the weak energy condition; see, e.g., [18].

Locally, a statement that the dominant energy condition is not violated is a statement that the local energy density should be greater than or equal to the momentum density. This can, geometrically, in the geon context and hence in relation to Definition 2, be formulated by the inequality

$${\rho }_{\left(k\right)}\ge \left|j_{\mu }^{\left(k\right)}{j}_{\left(k\right)}^{\mu }\right|,$$

(27)

where ${\rho }_{\left(k\right)}$ is the (projected geometrical) local energy density and $j_{\left(k\right)}^{\mu }$ is the (geometrical projected) momentum density. The local energy density, in the geon context, can be defined as the scalar projection of the stress–energy tensor along the worldline of a hypersurface-normal observer with four-velocity $u_{\left(k\right)}^{\mu }$ in the $3+1$ decomposition for any globally hyperbolic and homotopic spacetime element $(M_{\left(k\right)},g_{\mu \nu }^{\left(k\right)})$ in Definition 2. That is, using Einstein’s field equations,

$$R_{\mu \nu }^{\left(k\right)}-{\displaystyle \frac{1}{2}}{g}_{\mu \nu }^{\left(k\right)}{R}^{\left(k\right)}=8\pi T_{\mu \nu }^{\left(k\right)},$$

(28)

then, we have the energy density

$$\begin{array}{c}\hfill {\rho }^{\left(k\right)}:=T_{\mu \nu }^{\left(k\right)}{u}_{\left(k\right)}^{\mu }{u}_{\left(k\right)}^{\nu }={\displaystyle \frac{1}{8\pi }}\left(R_{\mu \nu }^{\left(k\right)}-{\displaystyle \frac{1}{2}}{g}_{\mu \nu }^{\left(k\right)}{R}^{\left(k\right)}\right)u_{\left(k\right)}^{\mu }{u}_{\left(k\right)}^{\nu }\\ \hfill ={\displaystyle \frac{1}{8\pi N_{\left(k\right)}^2}}\left({\partial }_x{\beta }_x^{\left(k\right)}{\partial }_y{\beta }_y^{\left(k\right)}+\left({\partial }_x{\beta }_x^{\left(k\right)}+{\partial }_y{\beta }_y^{\left(k\right)}\right){\partial }_z{\beta }_z^{\left(k\right)}\right).\end{array}$$

(29)

In these calculations, the SageMath [22] symbolic computation engine has been used to calculate the projection from the metric field given in the Definition 2 and the unique Levi-Civita connection.

To continue, $j_{\left(k\right)}^{\mu }$ is defined as the geometrical projected momentum density along the worldline of the hypersurface-normal observer with four-velocity $u_{\left(k\right)}^{\mu }$. That is,

$$j_{\mu }^{\left(k\right)}:=-h_{\mu }^{\left(k\right),\gamma }{T}_{\gamma \nu }^{\left(k\right)}{u}_{\left(k\right)}^{\nu }.$$

Here, it is straightforward to show that the projection with the four-velocity $u_{\left(k\right)}^{\nu }$ maps the stress–energy into the covector space as

$$T_{\mu \nu }^{\left(k\right)}{u}_{\left(k\right)}^{\nu }=(N^{\left(k\right)}{\rho }_{\left(k\right)},0,0,0),$$

and hence since

$$h_{\mu \nu }^{\left(k\right)}=g_{\mu \nu }^{\left(k\right)}+u_{\mu }^{\left(k\right)}{u}_{\nu }^{\left(k\right)},$$

and recalling that

$$u_{\mu }^{\left(k\right)}=(-N_{\left(k\right)},0,0,0),$$

then it immediately follows that

$$j_{\mu }^{\left(k\right)}=(0,0,0,0)\forall k.$$

(30)

We therefore have that the inequality (27) here reduces to

$${\rho }_{\left(k\right)}\ge 0.$$

Thus, and as logically expected since the dominant energy condition implies the weak energy condition (see, e.g., [18]), the dominant energy condition is violated at any point where the weak energy condition is violated. In particular, it here suffices to show that, for each finite spacetime element $(M_{\left(k\right)},g_{\mu \nu }^{\left(k\right)})$, there exists a neighbourhood containing points such that

$${\rho }_{\left(k\right)}<0,$$

so that the weak energy condition is violated there. Nevertheless, in distributions, at the geon limit and on the singular support, and within the admissible space of test functions in Definition 4, the dominant energy condition is not violated, since

$$\begin{array}{cc}\hfill & \underset{k\to \infty }{lim}{\int }_{U_{{\kappa }^{\left(k\right)}}}{\rho }^{\left(k\right)}\phi \sqrt{|g_{\left(k\right)}|}{d}^{4}x\hfill \\ \hfill & ={\displaystyle \frac{1}{8\pi }}\underset{k\to \infty }{lim}{\int }_{U_{{\kappa }_t^{\left(k\right)}}}{\displaystyle \frac{\left({\partial }_x{\beta }_x^{\left(k\right)}{\partial }_y{\beta }_y^{\left(k\right)}+\left({\partial }_x{\beta }_x^{\left(k\right)}+{\partial }_y{\beta }_y^{\left(k\right)}\right){\partial }_z{\beta }_z^{\left(k\right)}\right)\varphi }{k}}{d}^{3}x\hfill \\ \hfill & =0,\hfill \end{array}$$

(31)

where $\phi (t,\mathbf{x})$ is an admissible test function in the sense of Definition 4, i.e., $\phi (t,\mathbf{x})=\chi \left(t\right)\varphi \left(\mathbf{x}\right)$ with $\chi \in C_0^{\infty }\left((0,1)\right)$ satisfying ${\int }_0^1\chi \left(t\right)dt=1$, and $\varphi \in \mathcal{S}(U_{{\kappa }_t^{\left(k\right)}})$. In particular, for each $i\in \{1,2,3\}$, $\varphi$ is positive and even in the variable $x^i$, and locally concave in $x^i$ in a neighbourhood of ${\mathrm{\Sigma }}_i=\{x^i=0\}\cap U_{{\kappa }_t^{\left(k\right)}}$. This holds by the definition of the admissible test-function space Definition 4 and because the remaining scaling $k^{-1}$ forces weak convergence to zero, which may be seen by expanding around the singular support. To show that dominant energy condition is violated locally in the emerging spacetime, it is convenient to express (29) as follows:

$${\rho }^{\left(k\right)}={\displaystyle \frac{k^4}{32{\sigma }^4\pi N_{\left(k\right)}^2}}\left(xy{\beta }_{\left(k\right)}^x{\beta }_{\left(k\right)}^y+xz{\beta }_{\left(k\right)}^x{\beta }_{\left(k\right)}^z+yz{\beta }_{\left(k\right)}^y{\beta }_{\left(k\right)}^z\right).$$

(32)

As in the more general discussion of the operational meaning of the singularity theorems in the framework implementing Wheeler’s notion of the geon [4], it is then straightforward to conclude that for any finite k, the local projected energy density (32) takes both positive and negative values, separated by surfaces where the expression on the right-hand side vanishes. For example, it vanishes when $x=0\wedge z=0$ or $y=0\wedge z=0$, is positive in the first and third quadrants in the $x–y$ plane with $z=0$, and negative in the second and fourth. Thus, locally, the weak energy condition is not satisfied and consequently neither is the dominant energy condition.

In addition to the findings above, we may also note that the total energy in each spacetime element vanishes. This is an immediate consequence of the volume integral of ${\rho }^{\left(k\right)}$ over the support of the shift vector field:

$$E_{\left(k\right)}={\int }_{U_{{\kappa }_t^{\left(k\right)}}}{\rho }^{\left(k\right)}{N}_{\left(k\right)}{d}^{3}x={\int }_{U_{{\kappa }_t^{\left(k\right)}}}{\displaystyle \frac{\left({\partial }_x{\beta }_{\left(k\right)}^x{\partial }_y{\beta }_{\left(k\right)}^y+\left({\partial }_x{\beta }_{\left(k\right)}^x+{\partial }_y{\beta }_{\left(k\right)}^y\right){\partial }_z{\beta }_{\left(k\right)}^z\right)}{8\pi k}}{d}^{3}x=0,$$

since the integral is separable and, for each $i\in \{1,2,3\}$,

$${\int }_{U_{{\kappa }_{(t,x^i)}^{\left(k\right)}}}{\partial }_{x^i}{\beta }_{\left(k\right)}^{x^i}d{x}^i={\left[{\beta }_{\left(k\right)}^{x^i}\right]}_{\partial U_{{\kappa }_{(t,x^i)}^{\left(k\right)}}}=0,$$

where the boundary term vanishes because ${\beta }_{\left(k\right)}^{x^i}$ is localised by a compactly supported cut-off. Thus, and since the momentum density scalar is zero, we have $E_{\left(k\right)}=\left|P_{\left(k\right)}\right|=0$.

For the time being, we will leave out a precise formulation of these conclusions, specifically that while the dominant energy condition is valid in distributions, on the singular support, it fails locally on each spacetime element in the emerging and asymptotic flat spacetime. Instead, we proceed to show that even if the strong energy condition holds on the singular support, it is still violated in the emerging spacetime of the collapsing geon.

Theorem 2

(Non-Uniformity of the Strong Energy Condition). Let ${\beta }_{\left(k\right)}^i$ be the Gaussian components of the shift vector field on a sequence of globally hyperbolic and homotopic spacetimes ${\left\{(M_{\left(k\right)},g_{\mu \nu }^{\left(k\right)})\right\}}_{k\in \mathbb{N}}$ in Definition 2. Given the projection of the Ricci curvature along the four-velocity $u_{\left(k\right)}^{\mu }$ of an Eulerian observer, i.e., the scalar $R_{\mu \nu }^{\left(k\right)}{u}_{\left(k\right)}^{\mu }{u}_{\left(k\right)}^{\nu }$, along timelike worldlines orthogonal to any Cauchy surface ${\mathrm{\Sigma }}_t^{\left(k\right)}$ by

$$\begin{array}{cc}\hfill & R_{\mu \nu }^{\left(k\right)}{u}_{\left(k\right)}^{\mu }{u}_{\left(k\right)}^{\nu }=\hfill \\ \hfill & -{\displaystyle \frac{1}{N_{\left(k\right)}^2}}\left({\left({\partial }_x{\beta }_{\left(k\right)}^x\right)}^2+{\beta }_{\left(k\right)}^x{\partial }_x^2{\beta }_{\left(k\right)}^x+{\left({\partial }_y{\beta }_{\left(k\right)}^y\right)}^2+{\beta }_{\left(k\right)}^y{\partial }_y^2{\beta }_{\left(k\right)}^y+{\left({\partial }_z{\beta }_{\left(k\right)}^z\right)}^2+{\beta }_{\left(k\right)}^z{\partial }_z^2{\beta }_{\left(k\right)}^z\right).\hfill \end{array}$$

(33)

Then, the following hold:

(i) 

(Non-negativity at the singular support) For all finite $k\in \mathbb{N}$, the scalar projection of the Ricci curvature is strictly greater than zero at the origin:

$$R_{\mu \nu }^{\left(k\right)}{u}_{\left(k\right)}^{\mu }{u}_{\left(k\right)}^{\nu }{|}_{\mathbf{x}=0}>0.$$

In the distributional limit $k\to \infty$, this scalar projection satisfies

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}{\int }_{U_{{\kappa }^{\left(k\right)}}}{R}_{\mu \nu }^{\left(k\right)}{u}_{\left(k\right)}^{\mu }{u}_{\left(k\right)}^{\nu }\phi \sqrt{|g_{\left(k\right)}|}{d}^{4}x\\ \hfill =\underset{k\to \infty }{lim}{\int }_{U_{{\kappa }_t^{\left(k\right)}}}{R}_{\mu \nu }^{\left(k\right)}{u}_{\left(k\right)}^{\mu }{u}_{\left(k\right)}^{\nu }\varphi N_{\left(k\right)}{d}^{3}x\\ \hfill =-{\displaystyle \frac{\sqrt{2}}{8\sqrt{\pi }\sigma }}{\displaystyle \sum _{i=1}^3}\langle {\delta }_{x^i}^{″},\varphi \rangle ,\end{array}$$

where $\phi (t,x)$ is an admissible test function in the sense of Definition 4, i.e., $\phi (t,x)=\chi \left(t\right)\varphi \left(x\right)$ with $\chi \in C_0^{\infty }\left((0,1)\right)$ satisfying ${\int }_0^1\chi \left(t\right)dt=1$, and $\varphi \in \mathcal{S}(U_{{\kappa }_t^{\left(k\right)}})$. In particular, for each $i\in \{1,2,3\}$, ϕ is positive and even in the variable $x^i$, and locally concave in $x^i$ in a neighbourhood of $\{x^i=0\}\cap U_{{\kappa }_t^{\left(k\right)}}$. This defines the distributional curvature content of geon core at its singular support $\mathcal{C}={\bigcap }_i{\mathrm{\Sigma }}_i,{\mathrm{\Sigma }}_i=\{x^i=0\}$; see Definition 3.

(ii) 

(Local sign structure for finite k) For any finite $k\in \mathbb{N}$, there exists a point $x_0^{\left(k\right)}\in U_{{\kappa }_t^{\left(k\right)}}^{\left(k\right)}\setminus \mathcal{C}$ such that

$$R_{\mu \nu }^{\left(k\right)}{u}_{\left(k\right)}^{\mu }{u}_{\left(k\right)}^{\nu }\left(x_0^{\left(k\right)}\right)=0.$$

Moreover, there exists an open ball $B_{{\kappa }_t^{\left(k\right)}}\subset U_{{\kappa }_t^{\left(k\right)}}^{\left(k\right)}\setminus \left\{0\right\}$, centred at $x_0^{\left(k\right)}$, such that within $B_{{\kappa }_t^{\left(k\right)}}$, there exist disjoint subregions $B_{{\kappa }_t^{\left(k\right)}}^{+}\subset B_{{\kappa }_t^{\left(k\right)}}$ and $B_{{\kappa }_t^{\left(k\right)}}^{-}\subset B_{{\kappa }_t^{\left(k\right)}}$ satisfying

$$R_{\mu \nu }^{\left(k\right)}{u}_{\left(k\right)}^{\mu }{u}_{\left(k\right)}^{\nu }>0in{B}_{{\kappa }_t^{\left(k\right)}}^{+},R_{\mu \nu }^{\left(k\right)}{u}_{\left(k\right)}^{\mu }{u}_{\left(k\right)}^{\nu }<0in{B}_{{\kappa }_t^{\left(k\right)}}^{-}.$$

That is, the projected scalar Ricci curvature changes sign on open sets away from the singular support, and hence the strong energy condition is violated locally in those regions.

Proof. 

Locally, and for each spatial coordinate $x^i$ (with $i=1,2,3$), we have from (33)

$$-{\displaystyle \frac{1}{k^2}}\left({\left({\partial }_{x^i}{\beta }_{\left(k\right)}^{x^i}\right)}^2+{\beta }_{\left(k\right)}^{x^i}{\partial }_{x^i}^2{\beta }_{\left(k\right)}^{x^i}\right)=-{\displaystyle \frac{1}{2k\sigma \sqrt{\pi }}}\left({\displaystyle \frac{k^4}{2{\sigma }^4}}{\left(x^i\right)}^2-{\displaystyle \frac{k^2}{2{\sigma }^2}}\right){\beta }_{\left(k\right)}^{\sqrt{2}{x}^i}.$$

Thus, we can write

$$R_{\mu \nu }^{\left(k\right)}{u}_{\left(k\right)}^{\mu }{u}_{\left(k\right)}^{\nu }={\displaystyle \sum _{i=1}^3}{p}_{\sigma }^{\left(k\right)}\left(x^i\right){\displaystyle \frac{{\beta }_{\left(k\right)}^{\sqrt{2}{x}^i}}{2k\sigma \sqrt{\pi }}},$$

(34)

where

$$p_{\sigma }^{\left(k\right)}\left(x^i\right)=-\left({\displaystyle \frac{k^4}{2{\sigma }^4}}{\left(x^i\right)}^2-{\displaystyle \frac{k^2}{2{\sigma }^2}}\right).$$

For any finite k, $R_{\mu \nu }^{\left(k\right)}{u}_{\left(k\right)}^{\mu }{u}_{\left(k\right)}^{\nu }>0$ at the origin since $p_{\sigma }^{\left(k\right)}\left(0\right)>0$ and ${\beta }_{\left(k\right)}\left(0\right)>0$.

The polynomial $p_{\sigma }^{\left(k\right)}$ has two real roots in each coordinate, located at

$$x_{\left(k\right)}^i=\pm {\displaystyle \frac{\sigma }{k}}.$$

Moreover,

$${\displaystyle \frac{d{p}_{\sigma }^{\left(k\right)}\left(x^i\right)}{d{x}^i}}=-{\displaystyle \frac{k^4}{{\sigma }^4}}{x}^i,$$

so $p_{\sigma }^{\left(k\right)}$ is strictly increasing for $x^i<0$ and strictly decreasing for $x^i>0$. Hence, $p_{\sigma }^{\left(k\right)}$ changes sign across each root.

To obtain a zero of the full expression, consider the diagonal line $\gamma \left(r\right)=(r,r,r)$. Along $\gamma$,

$$R_{\mu \nu }^{\left(k\right)}{u}_{\left(k\right)}^{\mu }{u}_{\left(k\right)}^{\nu }\left(\gamma \left(r\right)\right)=3p_{\sigma }^{\left(k\right)}\left(r\right){\displaystyle \frac{{\beta }_{\left(k\right)}^{\sqrt{2}r}}{2k\sigma \sqrt{\pi }}},$$

so it vanishes at $r=\pm \sigma /k$. By continuity, there is an open ball centred at $x_0^{\left(k\right)}=\gamma (\sigma /k)$ in which the expression attains both positive and negative values on disjoint open subregions, establishing the local sign-changing structure away from the core of the geon $\mathcal{C}$.

Finally, returning to (34) and observing that the polynomial factor is the factor from taking the second derivative of the Gaussian, we find

$$R_{\mu \nu }^{\left(k\right)}{u}_{\left(k\right)}^{\mu }{u}_{\left(k\right)}^{\nu }=-{\displaystyle \frac{1}{4k\sigma \sqrt{\pi }}}{\displaystyle \sum _{i=1}^3}{\partial }_{x^i}^2{\beta }_{\left(k\right)}^{\sqrt{2}{x}^i}.$$

Thus, we obtain in a functional meaning that

$$\begin{array}{cc}\hfill \underset{k\to \infty }{lim}{\int }_{U_{{\kappa }^{\left(k\right)}}}{R}_{\mu \nu }^{\left(k\right)}{u}_{\left(k\right)}^{\mu }{u}_{\left(k\right)}^{\nu }\phi \sqrt{|g_{\left(k\right)}|}{d}^{4}x& =\underset{k\to \infty }{lim}{\int }_{U_{{\kappa }_t^{\left(k\right)}}}{R}_{\mu \nu }^{\left(k\right)}{u}_{\left(k\right)}^{\mu }{u}_{\left(k\right)}^{\nu }\varphi N_{\left(k\right)}{d}^{3}x\hfill \\ \hfill & =-{\displaystyle \frac{1}{4\sigma \sqrt{\pi }}}\underset{k\to \infty }{lim}{\int }_{U_{{\kappa }_t^{\left(k\right)}}}{\displaystyle \sum _{i=1}^3}{\partial }_{x^i}^2{\beta }_{\left(k\right)}^{\sqrt{2}{x}^i}\varphi d^{3}x\hfill \\ \hfill & =-{\displaystyle \frac{\sqrt{2}}{8\sqrt{\pi }\sigma }}{\displaystyle \sum _{i=1}^3}\langle {\delta }_{x^i}^{″},\varphi \rangle \hfill \\ \hfill & =-{\displaystyle \frac{\sqrt{2}}{8\sqrt{\pi }\sigma }}{\displaystyle \sum _{i=1}^3}{\partial }_{x^i}^2\varphi {|}_{x^i=0}.\hfill \end{array}$$

By Definition 4, ${\partial }_{x^i}^2\varphi {|}_{x^i=0}\le 0$ for each $i=1,2,3$, and hence, the distributional strong energy condition holds at the geon core. □

Returning to Einstein’s field equations, (28),

$$R_{\mu \nu }^{\left(k\right)}-{\displaystyle \frac{1}{2}}{g}_{\mu \nu }^{\left(k\right)}{R}^{\left(k\right)}=8\pi T_{\mu \nu }^{\left(k\right)},$$

provides further insight into Theorem 2. First, by contracting with the metric field $g_{\mu \nu }^{\left(k\right)}$, i.e., taking the trace, it follows that for every finite value of the sequence index k, and hence for every globally hyperbolic spacetime element $(M_{\left(k\right)},g_{\mu \nu }^{\left(k\right)})$ in Definition 2,

$$T^{\left(k\right)}=-{\displaystyle \frac{1}{8\pi }}{R}^{\left(k\right)}.$$

Using $g_{\mu \nu }^{\left(k\right)}{u}_{\left(k\right)}^{\mu }{u}_{\left(k\right)}^{\nu }=-1$ and (29) allows us to write the strong energy condition in the renormalised Gaussian Model Delta Net framework, and in the $3+1$ context of the spacetimes in Definition 2, as

$$R_{\mu \nu }^{\left(k\right)}{u}_{\left(k\right)}^{\mu }{u}_{\left(k\right)}^{\nu }=8\pi \left(T_{\mu \nu }^{\left(k\right)}-{\displaystyle \frac{1}{2}}{g}_{\mu \nu }^{\left(k\right)}{T}^{\left(k\right)}\right)u_{\left(k\right)}^{\mu }{u}_{\left(k\right)}^{\nu }=8\pi {\rho }^{\left(k\right)}-{\displaystyle \frac{1}{2}}{R}^{\left(k\right)}.$$

(35)

Continuing to the trace $R^{\left(k\right)}$ of the Ricci tensor, it can be shown (using the SageMath symbolic computation engine, or by straightforward but somewhat tedious manual calculations involving the Levi-Civita connection and the metric field) that the Ricci scalar, as expected, is given by

$$\begin{array}{c}\hfill R^{\left(k\right)}={\displaystyle \frac{2}{N_{\left(k\right)}^2}}({\left({\partial }_x{\beta }_{\left(k\right)}^x\right)}^2+{\beta }_{\left(k\right)}^x{\partial }_x^2{\beta }_{\left(k\right)}^x+{\partial }_x{\beta }_{\left(k\right)}^x{\partial }_y{\beta }_{\left(k\right)}^y+{\left({\partial }_y{\beta }_{\left(k\right)}^y\right)}^2+{\beta }_{\left(k\right)}^y{\partial }_y^2{\beta }_{\left(k\right)}^y\\ \hfill +\left({\partial }_x{\beta }_{\left(k\right)}^x+{\partial }_y{\beta }_{\left(k\right)}^y\right){\partial }_z{\beta }_{\left(k\right)}^z+{\left({\partial }_z{\beta }_{\left(k\right)}^z\right)}^2+{\beta }_{\left(k\right)}^z{\partial }_z^2{\beta }_{\left(k\right)}^z)\\ \hfill ={\displaystyle \frac{2}{N_{\left(k\right)}^2}}({\left({\partial }_x{\beta }_{\left(k\right)}^x\right)}^2+{\beta }_{\left(k\right)}^x{\partial }_x^2{\beta }_{\left(k\right)}^x+{\left({\partial }_y{\beta }_{\left(k\right)}^y\right)}^2+{\beta }_{\left(k\right)}^y{\partial }_y^2{\beta }_{\left(k\right)}^y\\ \hfill +{\left({\partial }_z{\beta }_{\left(k\right)}^z\right)}^2+{\beta }_{\left(k\right)}^z{\partial }_z^2{\beta }_{\left(k\right)}^z+8\pi N_{\left(k\right)}^2{\rho }^{\left(k\right)}).\end{array}$$

(36)

Then, it follows in a functional sense that

$$\underset{k\to \infty }{lim}{\int }_{U_{{\kappa }_t^{\left(k\right)}}}{R}^{\left(k\right)}\varphi N_{\left(k\right)}{d}^{3}x={\displaystyle \frac{\sqrt{2}}{4\sqrt{\pi }\sigma }}{\displaystyle \sum _{i=1}^3}\langle {\delta }_{x^i}^{″},\varphi \rangle ={\displaystyle \frac{\sqrt{2}}{4\sqrt{\pi }\sigma }}{\displaystyle \sum _{i=1}^3}{\partial }_{x^i}^2\varphi {|}_{x^i=0}.$$

(37)

Here, the terms involving products of first-order derivatives in different direction, summing up to the energy density, vanish by the same line of reasoning used for the energy density in (31). Nevertheless, it should be noted that for finite k, the mixed first-derivative terms in ${\rho }^{\left(k\right)}$ coincide with the corresponding mixed terms in $\frac{1}{2}{R}^{\left(k\right)}$ with opposite sign, and hence, cancel identically in the strong energy condition (35). So, the mixed terms cancel exactly, leaving only the diagonal terms, in agreement with (33). In the distributional limit, the remaining diagonal contribution yields precisely the curvature content stated in Theorem 2.

Collecting all results allows us to state the main result:

Theorem 3

(Non-uniformity of the positive mass in the Geon). Although the dominant, weak, and strong energy conditions hold at the singular support in the distributional sense, for each finite k, they fail on suitable open subsets of $M_{\left(k\right)}$. In particular, the dominant energy condition is not satisfied uniformly across the Gaussian model delta net. Consequently, a positive mass statement cannot be invoked in a scale-uniform manner across the sequence, even if $E_{\left(k\right)}=\left|P_{\left(k\right)}\right|=0$ holds for each spacetime element $(M_{\left(k\right)},g_{\mu \nu }^{\left(k\right)})$.

All these examples have been used to illustrate how to extend the differential geometric context on a globally hyperbolic spacetime to a natural functional setting and how to do physics in distribution geometry. For further details, e.g., discussion on classical singularities and examples, we refer to [4].

Having established the geometry and the basic physics of the geon, we now proceed to the main theme of the work to show that when the geon collapses and is driven into oscillations, transient trapped surfaces emerge.

3. Transient Trapped Surfaces

In this section, we show that transient trapped surfaces exist in a neighbourhood of the singular support of the geon. More precisely, we show that transient trapped surfaces arise in quantum foam and in emerging globally hyperbolic, asymptotically flat spacetime elements in a neighbourhood punctured by the intersection of the singular supports of the geon, and hence in a neighbourhood of the characteristic hypersurfaces that form the limit of the self-gravitating but unstable geon.

The geon limit, Definition 2, in the distributional framework developed in [4] reviewed in the previous section, is the distributional limit at the null characteristic where strong causality is broken because the timelike normals to the spacelike hypersurfaces, as well as the hypersurfaces themselves, tend to null. This null aggregation, caused by the warping and the resulting non-convex causal embedding, can be used to argue that trapped surfaces will emerge, since it leads to a negative null expansion rate. Hence, a classical barrier will evolve, shielding the geon in its unstable equilibrium. Conversely, once the geon collapses under the back-reaction of its own polarisation and the hypersurfaces undergo large fluctuations of expansion and contraction, separated by a vacuum shell with regions of negative and positive energy density, the trapped surfaces will gradually “evaporate” and eventually disappear completely. Thus, the trapped surfaces are transient. The fluctuations of the hypersurfaces as they tend towards a spacelike and asymptotically flat geometry are the effect of the displacement of the vacuum caused by oscillations of the shift vector in a coherent state.

We formalise this narrative about the trapped surfaces in the following theorem, introduced in [4]:

Theorem 4

(Trapped Surfaces in the Geon Collapse). Let ${\beta }_{\left(k\right)}^i$ be the Gaussian components of the shift vector field on a sequence of globally hyperbolic and homotopic spacetimes ${\left\{(M_{\left(k\right)},g_{\mu \nu }^{\left(k\right)})\right\}}_{k\in \mathbb{N}}$ forming a geon according to Definition 2, with lapse $N_{\left(k\right)}=k$. Let $u_{\left(k\right)}^{\mu }$ denote the four-velocity of an Eulerian observer. For any closed spacelike two-surface $S_t^{\left(k\right)}\subset {\mathrm{\Sigma }}_t^{\left(k\right)}$, let $s_{\left(k\right)}^i$ be the outward unit normal to $S_t^{\left(k\right)}$ in ${\mathrm{\Sigma }}_t^{\left(k\right)}$, and let $q_{\mu \nu }^{\left(k\right)}=h_{\mu \nu }^{\left(k\right)}-s_{\mu }^{\left(k\right)}{s}_{\nu }^{\left(k\right)}$ be the induced metric on $S_t^{\left(k\right)}$. Define the local null expansion by

$${\theta }_{(\pm )}^{\left(k\right)}:=q_{\left(k\right)}^{\mu \nu }{\nabla }_{\mu }{\ell }_{\pm \nu }^{\left(k\right)}=\pm H^{\left(k\right)}-K_{\left(k\right)}+K_{ij}^{\left(k\right)}{s}_{\left(k\right)}^i{s}_{\left(k\right)}^j.$$

(38)

Here, with the null field ${\ell }_{\pm \left(k\right)}^{\mu }:=u_{\left(k\right)}^{\mu }\pm s_{\left(k\right)}^{\mu }$, and with $H^{\left(k\right)}=q_{\left(k\right)}^{ij}{\partial }_i{s}_{\left(k\right)j}$ representing the mean curvature.

Then, there exist a $k_0\in \mathbb{N}$ and a sequence $r_k>0$ with $r_k\to 0$ as $k\to \infty$ (for example, $r_k=min\{{\ell }_p,\sigma /k\}$) such that for all $k\ge k_0$, there exists an open subset

$$U_{r_k}\subset B_{r_k}\left(0\right)\setminus \mathcal{C}$$

with

$$\mathcal{C}={\displaystyle \bigcap _{i=1}^3}{\mathrm{\Sigma }}_i;$$

see Definition 3. For example,

$$U_{r_k}=B_{r_k}\left(0\right)\cap \{x^1<0,x^2<0,x^3<0\},$$

on which

$${\theta }_{(\pm )}^{\left(k\right)}\left(x\right)<0,\forall x\in U_{r_k}.$$

Proof. 

Using the line element (1) with flat induced spatial metric ${\eta }_{ij}$, the extrinsic curvature and its trace are

$$K_{ij}^{\left(k\right)}={\displaystyle \frac{1}{2N_{\left(k\right)}}}\left({\partial }_i{\beta }_j^{\left(k\right)}+{\partial }_j{\beta }_i^{\left(k\right)}\right),K^{\left(k\right)}={\displaystyle \frac{1}{N_{\left(k\right)}}}{\partial }_i{\beta }_{\left(k\right)}^i.$$

Substituting into (38) yields

$${\theta }_{(\pm )}^{\left(k\right)}=\pm H^{\left(k\right)}-{\displaystyle \frac{1}{N_{\left(k\right)}}}\left({\partial }_i{\beta }_{\left(k\right)}^i-s_{\left(k\right)}^i{s}_{\left(k\right)}^j{\partial }_i{\beta }_j^{\left(k\right)}\right)=\pm H^{\left(k\right)}-{\displaystyle \frac{1}{N_{\left(k\right)}}}{q}_{\left(k\right)}^{ij}{\partial }_i{\beta }_j^{\left(k\right)}.$$

Fix $r_k:=min\{{\ell }_p,\sigma /k\}$ and work on the punctured ball $B_{r_k}\left(0\right)\setminus \mathcal{C}$. For sufficiently large k, the domain scale is dictated by $r_k=\sigma /k$. The mean curvature $H^{\left(k\right)}$ of the embedded two-surface $S_t^{\left(k\right)}$ within this shrinking domain is bounded by the inverse of the radius, such that ${sup}_{x\in U_{r_k}}\left|H^{\left(k\right)}\left(x\right)\right|\le k/\sigma <\tilde{c}k$ for some constant $\tilde{c}>0$. From the explicit form of the Gaussian shift (see Definition 2 and (2)), and choosing for definiteness the half-ball

$$U_{r_k}:=B_{r_k}\left(0\right)\cap \{x^1<0,x^2<0,x^3<0\},$$

we have

$$q_{\left(k\right)}^{ij}{\partial }_i{\beta }_j^{\left(k\right)}>0$$

on $U_{r_k}$ for all sufficiently large k. Moreover, by the Gaussian scaling, $|{\partial }_i{\beta }_{\left(k\right)j}|$ grows like $k^3$ on $B_{r_k}\left(0\right)$ as $k\to \infty$. Hence, there exist constants $c>0$ and $k_0$ such that for all $k\ge k_0$,

$${\displaystyle \frac{1}{N_{\left(k\right)}}}{q}_{\left(k\right)}^{ij}{\partial }_i{\beta }_j^{\left(k\right)}\ge c{k}^2\mathrm{on}{U}_{r_k}.$$

Therefore, for $k\ge k_0$ and all $x\in U_{r_k}$,

$${\theta }_{(+)}^{\left(k\right)}\left(x\right)=H^{\left(k\right)}\left(x\right)-{\displaystyle \frac{1}{N_{\left(k\right)}}}{q}_{\left(k\right)}^{ij}{\partial }_i{\beta }_j^{\left(k\right)}\left(x\right)\le \tilde{c}k-c{k}^2<0,$$

and similarly,

$${\theta }_{(-)}^{\left(k\right)}\left(x\right)=-H^{\left(k\right)}\left(x\right)-{\displaystyle \frac{1}{N_{\left(k\right)}}}{q}_{\left(k\right)}^{ij}{\partial }_i{\beta }_j^{\left(k\right)}\left(x\right)\le \tilde{c}k-c{k}^2<0,$$

for all $x\in U_{r_k}$ and all sufficiently large k. □

Thus, we have established the existence of trapped surfaces in the neighbourhood of the geon, and we can draw some further conclusions. First, there exist finite sequence indices $k\ge k_0$ (finite resolution) for which trapped surfaces form in finite but Planck-scale neighbourhoods and act as an effective barrier for descriptions formulated solely within smooth differential geometry. If one attempts to probe length scales below the scale at which these trapped surfaces arise, then a classical (and, in the present setting, semiclassical) description ceases to be adequate, and one is forced into the distributional regime and into the unstable, self-gravitating geon itself. In this sense, these “primordial black hole” configurations obstruct access to sub-Planckian scales within the smooth geometric sector. This is consistent with the Dvali–Gomez discussion of a generalised uncertainty relation [23]. Nevertheless, by using the smooth and microlocally compatible extension into distributional geometry reviewed in Section 2, one can in principle approach the boundary of the unstable geon in the distributional setting and identify the mechanism that produces this barrier at the onset of time, when the geon collapses.

In the next section, we will venture deeper into the geometry of the transient trapped surfaces and calculate their surface area and show that it decreases as the asymptotically flat spacetime emerge.

4. Area of a Transient Trapped Surface

In this section, we prove a basic area proposition for transient trapped surfaces in quantum foam. The motivation is to understand whether the result is aligned with Hawking’s area theorem, stating that the area of a black hole will never decrease as the spacetime evolves, and with Bekenstein’s arguments that allow one to assign an entropy to a trapped surface; see, e.g., Witten’s review on black hole thermodynamics for details on both Hawking’s and Bekenstein’s work [24]. It should already at this stage be stated clearly that we do not anticipate Hawking’s original statement to hold in the setting of emerging spacetimes: Hawking’s area theorem applies to the area of an event horizon under suitable energy conditions (in particular, a null energy condition; see, e.g., Wald [25] for the theorem and its proof), whereas, in our framework, the transient trapped surfaces arise in regions where the standard energy conditions—including the null, weak, dominant, and strong energy conditions—fail locally on open sets [4]. Consequently, the hypotheses of Hawking’s theorem are not satisfied, and there is no obstruction to a decrease in the area associated with these transient trapped surfaces.

Rather, as classical spacetime emerges and tends towards asymptotic flatness, we expect the area of a transient trapped surface to decrease and vanish. In this sense, within the quantum foam framework there is a natural mechanism for ‘evaporation’ that is compatible with Hawking’s semi-classical result (1974) that black holes radiate and can evaporate [26]. Hawking nevertheless also noted explicitly that his calculation neglects the back-reaction of the emitted quanta on the metric and quantum fluctuations of the metric, and that these omissions might alter the picture [26]. In our framework, we address precisely this caveat by exhibiting a collapse/evaporation mechanism in which back-reaction does not obstruct the formation of transient trapped surfaces; rather, it is the back-reaction at the geon collapse that gives rise to them.

To prove the area proposition, we use only standard results from differential geometry by constructing the local area density ${\rho }_{A_{\left(k\right)}}$ for a trapped surface for any finite $k>k_0\in \mathbb{N}$ (where $k_0$ is the lower bound for the existence of a trapped surface). Concretely, to determine ${\rho }_{A_{\left(k\right)}}$, we parametrise the surface, identify the tangent vectors, and calculate the area spanned by them. Integrating over all such local area elements gives the total surface area.

To be specific to the current context, to determine the area of a trapped two-surface ${\mathcal{T}}^{\left(k\right)}$ in the emerging spacetime, we rely on the fact that in the quantum foam of the geon (Definition 2), the induced metric field on each Cauchy surface ${\mathrm{\Sigma }}_t^{\left(k\right)}$ is Euclidean, ${\eta }_{ij}$. Thus, we can use standard results for embedded surfaces in Euclidean space: the area is the integral of the norm of the wedge product of the tangent vectors (equivalently, the square root of the determinant of the first fundamental form); see, e.g., Thorpe [27].

In the proof of Theorem 4, we obtained trapped regions in punctured neighbourhoods shrinking towards the singular support as $k\to \infty$. Accordingly, we compute the area on a Planck-referenced neighbourhood whose scale is compatible with the Gaussian localisation. Define

$$r_k:=min\{{\ell }_p,\sigma /k\},$$

and fix a centre $p_0=(x_0,y_0)$ in the third quadrant, with

$$x_0<-r_k{y}_0<-r_k$$

so that for all k, the closed disc

$$D_k:=\overline{D(p_0,r_k)}=\{(x,y)\in {\mathbb{R}}^2:{(x-x_0)}^2+{(y-y_0)}^2\le r_k^2\}$$

lies entirely in $\{x<0,y<0\}$ and, in particular, does not intersect any of the singular support ${\mathrm{\Sigma }}_i=\{x^i=0\}$ defined in Definition 3. We parametrise the trapped surface ${\mathcal{T}}^{\left(k\right)}$ by a local Euclidean embedding on $D_k$. That is, we consider

$${\overrightarrow{\beta }}^{\left(k\right)}:D_k\subset {\mathbb{R}}^2\to {\mathrm{\Sigma }}_t^{\left(k\right)},(x,y)\mapsto {\overrightarrow{\beta }}^{\left(k\right)}(x,y).$$

Explicitly, in the Gaussian Quantum Foam framework of Definition 2, we may, since the Gaussian shift components are separable, without loss of generality, restrict the analysis to the slice $z=0$, and hence parametrise the two-surface ${\mathcal{T}}^{\left(k\right)}$ over $D_k$ by the embedding

$${\overrightarrow{\beta }}^{\left(k\right)}(x,y)={\ell }_p\left({\beta }_{\left(k\right)}^x\left(x\right),{\beta }_{\left(k\right)}^y\left(y\right),0\right).$$

Here, we have reinstated physical units; accordingly, the normalisation of the shift components ${\beta }_{\left(k\right)}^i$ carries the appropriate Planck-length dependence, although it is not displayed explicitly in the notation above. The corresponding local area density is

$$\begin{array}{cc}\hfill {\rho }_{A_{\left(k\right)}}(x,y)& :={\ell }_p^2∥{\partial }_x{\overrightarrow{\beta }}^{\left(k\right)}(x,y)\times {\partial }_y{\overrightarrow{\beta }}^{\left(k\right)}(x,y)∥\hfill \\ \hfill & ={\ell }_p^2\left|{\partial }_x{\beta }_{\left(k\right)}^x\left(x\right){\partial }_y{\beta }_{\left(k\right)}^y\left(y\right)\right|.\hfill \end{array}$$

(Here, the absolute value is forced by the norm. If, in addition, one chooses $D_k$ inside a region where ${\partial }_x{\beta }_{\left(k\right)}^x\ge 0$ and ${\partial }_y{\beta }_{\left(k\right)}^y\ge 0$ hold throughout $D_k$, then the absolute value may be dropped on $D_k$.)

For finite k, the area of the trapped-surface patch ${\mathcal{T}}^{\left(k\right)}$ over $D_k$ is

$$\begin{array}{cc}\hfill A_{\left(k\right)}& :={\int }_{D_k}{\rho }_{A_{\left(k\right)}}(x,y)dxdy\hfill \\ \hfill & ={\ell }_p^2{\int }_{D_k}\left|{\partial }_x{\beta }_{\left(k\right)}^x\left(x\right){\partial }_y{\beta }_{\left(k\right)}^y\left(y\right)\right|dxdy\hfill \\ \hfill & ={\alpha }_{\left(k\right)}{\ell }_p^2={\alpha }_{\left(k\right)}{\displaystyle \frac{\hslash G}{c^3}}={\alpha }_{\left(k\right)}{\displaystyle \frac{m_p^2{G}^2}{c^4}},\hfill \end{array}$$

where

$${\alpha }_{\left(k\right)}:={\displaystyle \frac{k^2{\ell }_p^2}{4\pi {\sigma }^2}}{C}_{\left(k\right)}.$$

Here, $C_{\left(k\right)}>0$ is dimensionless and encodes the remaining geometric contribution of the integral over $D_k$ after factoring out the explicit ${\ell }_p$-dependence. In particular, by asymptotic flatness ($k\to 1$) in Gaussian Quantum Foam, one has $C_{\left(k\right)}\to 0$; hence, $A_{\left(k\right)}\to 0$. Moreover, since $r_k\to 0$ as $k\to \infty$, the neighbourhood over which the trapped region is supported shrinks towards the singular support, and the corresponding area contribution vanishes in the geon limit as well.

We conclude that the surface area of the transient trapped surface tends to zero both in the asymptotically flat limit ($k\to 1$) and in the geon limit ($k\to \infty$). Thus, even if a transient trapped surface forms during collapse, it evaporates within the quantum foam framework. To summarise, we have the following proposition:

Proposition 1

(Evaporation of Transient Trapped Surfaces in Gaussian Quantum Foam). Let ${\left\{(M_{\left(k\right)},g_{\mu \nu }^{\left(k\right)})\right\}}_{k\in \mathbb{N}}$ be a Gaussian Quantum Foam sequence in the sense of Definition 2, with induced Euclidean metric ${\eta }_{ij}$ on each Cauchy surface ${\mathrm{\Sigma }}_t^{\left(k\right)}$, and let $k_0\in \mathbb{N}$ denote the lower bound for the existence of a trapped surface. For each $k>k_0$, let ${\mathcal{T}}^{\left(k\right)}\subset {\mathrm{\Sigma }}_t^{\left(k\right)}$ be a transient trapped two-surface arising in the punctured neighbourhoods of Theorem 4. Define $r_k:=min\{{\ell }_p,\sigma /k\}$ and let $D_k=\overline{D(p_0,r_k)}\subset \{x<0,y<0\}$ be a closed disc as above.

From Definition 2, the Gaussian shift components are separable in the sense that ${\beta }_{\left(k\right)}^x={\beta }_{\left(k\right)}^x\left(x\right)$, ${\beta }_{\left(k\right)}^y={\beta }_{\left(k\right)}^y\left(y\right)$, and ${\beta }_{\left(k\right)}^z={\beta }_{\left(k\right)}^z\left(z\right)$. In particular, we may, without loss of generality, restrict the analysis to the slice $z=0$, and hence parametrise the two-surface ${\mathcal{T}}^{\left(k\right)}$ over $D_k$ by the embedding

$${\overrightarrow{\beta }}^{\left(k\right)}:D_k\subset {\mathbb{R}}^2\to {\mathrm{\Sigma }}_t^{\left(k\right)},(x,y)\mapsto {\overrightarrow{\beta }}^{\left(k\right)}(x,y)={\ell }_p\left({\beta }_{\left(k\right)}^x\left(x\right),{\beta }_{\left(k\right)}^y\left(y\right),0\right),$$

where ${\beta }_{\left(k\right)}^x$ and ${\beta }_{\left(k\right)}^y$ are the Gaussian shift components of the foam. Then, the area $A_{\left(k\right)}$ satisfies

$$\begin{array}{cc}\hfill A_{\left(k\right)}& ={\ell }_p^2{\int }_{D_k}∥{\partial }_x{\overrightarrow{\beta }}^{\left(k\right)}(x,y)\times {\partial }_y{\overrightarrow{\beta }}^{\left(k\right)}(x,y)∥dxdy\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill & ={\ell }_p^2{\int }_{D_k}\left|{\partial }_x{\beta }_{\left(k\right)}^x\left(x\right){\partial }_y{\beta }_{\left(k\right)}^y\left(y\right)\right|dxdy\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill & ={\alpha }_{\left(k\right)}{\ell }_p^2,\hfill \end{array}$$

(41)

where ${\alpha }_{\left(k\right)}={\displaystyle \frac{k^2{\ell }_p^2}{4\pi {\sigma }^2}}{C}_{\left(k\right)}$ with $C_{\left(k\right)}>0$ and dimensionless. In particular, $A_{\left(k\right)}\to 0$ as $k\to 1$ (asymptotic flatness), and $A_{\left(k\right)}\to 0$ as $k\to \infty$ (geon limit, since $r_k\to 0$). Consequently, the area of transient trapped surfaces vanish in the asymptotically flat limit of Gaussian Quantum Foam.

5. Entropy of a Transient Trapped Surface

In the previous section, we formulated a basic proposition for the area of a transient trapped surface in quantum foam. We concluded that the claim in [4], namely that the trapped surfaces are transient, not only follows immediately from Theorem 4, but also from Proposition 1, where it was shown that the area of the trapped surface vanishes in the asymptotically flat limit.

In this section, we formulate and prove an entropy theorem by quantising the shift vector on the hyper surfaces and constructing a coherent state in which the expectation values of each diffeomorphism-invariant scalar observable (that can be expressed in the renormalised distribution algebra) coincide with their classical counterparts.

Concretely, we assign entropy to the coherent displacement of the vacuum induced by the shift vector by using the universality of the occupation number as put forward by Dvali and Gomez [8] in their analysis of the fundamental properties of Planck-scale black holes of weakly coupled bosons, which they argue form a Bose–Einstein condensate marking the classical limit of information before the characteristic scale and hence the limit of the self-gravitating but unstable entity that we have defined as Wheeler’s geon in Definition 2. One may question whether the black holes in the Dvali–Gomez analysis truly are Bose–Einstein condensates. A more likely and precise statement is that they constitute the classical limit before the condensate represented by the self-gravitating geon in its unstable equilibrium.

Leaving this question aside, we do agree that the occupation number, or rather the expected value of it over all bosonic states, is universal in relation to black holes and, in this context, to transient trapped surfaces. The meaning of this is that we expect it to be a defining quantity for the thermodynamic quantities of a transient trapped surface, here the area and entropy. In fact, we will argue that the entropy is proportional to the expected value of the occupation number, since the state-counting generating function in a coherent state takes the form of an exponential of the expected value of the occupation-number operator. This is in agreement with the conclusions of Bashkirov and Sukhanov [9], where they identify the corresponding coherent-state generating function as the exponential of the expected value of the occupation number, as well as with the more general conclusion of Dvali and Gomez [8] for black hole entropy.

Remark 8.

It should be emphasised, at the beginning of this discussion, that the entropy discussed here is not the von Neumann entropy of the pure coherent state $|\alpha \rangle$, which will be introduced later in this section, since it is trivially zero. Rather, it is a thermodynamic/statistical entropy functional motivated by microstate counting in a weakly interacting many-quanta sector [8,9].

We begin with observations from the construction of the quantum field theory of the collapsing geon, which can be carried out because the definition of the geon and the associated algebra admit a Gelfand triple. The unstable field equation of the geon (see Section 2.1.2) demonstrates the onset of decoherence and motivates a description of the quantised spacetime geometry that remains as close as possible to classical physics, namely in terms of coherent states. The choice of coherent states is motivated by several considerations, introduced in [4]:

First, the correspondence principle encoded in the Gelfand triple structure dictates that the transition from the quantum to the classical regime should be smooth. Coherent states, being eigenstates of the annihilation operator with smooth eigenvalues, provide the closest quantum analogue of classical fields, in the sense that their expectation values evolve in a classically controlled manner, with fluctuations that are minimal in the usual uncertainty sense.

Secondly, a coherent state naturally describes a system in which the ground-state wave packet is displaced from the origin. In the context of the shift vector, this aligns with its role in encoding spatial displacements between hypersurfaces (level surfaces). Thus, the coherent state provides a natural quantum representation of the shift vector, ensuring that its expectation values reproduce the classical behaviour predicted by the correspondence principle. The role of coherent states in quantum mechanics and quantum optics was developed by Glauber [28].

Finally, Wheeler characterised fluctuations in spacetime geometry in terms of persistent vacuum fluctuations [29]. In the context of quantum foam, this suggests that the relevant displacement should be understood in terms of the shift vector on spatial hypersurfaces, with respect to a global time parameter. Accordingly, we must consider quantum states that naturally exhibit such displacement behaviour, expressed through the fluctuating shift vector field.

To quantise the shift vector field ${\beta }_{\left(k\right)}^i$ we follow the prescription in [4], that is, we use the Gelfand triple and the distribution algebra to introduce a well-defined Hilbert space across all finite values of the sequence index k. The fact that each shift vector component depends only on a single spatial coordinate implies that the one-particle Hilbert space factorises as a tensor product

$${\mathcal{H}}^{\left(k\right)}={\mathcal{H}}_1^{\left(k\right)}\otimes {\mathcal{H}}_2^{\left(k\right)}\otimes {\mathcal{H}}_3^{\left(k\right)},$$

where each ${\mathcal{H}}_i^{\left(k\right)}$ corresponds to the spatial direction generated by the basis element ${\partial }_i$. Each factor therefore represents an independent Gaussian ray spanning the Cauchy surface. This separable structure underlies the quantisation scheme and provides the framework for implementing a correspondence principle in the geon and quantum foam framework.

To construct a scalar quantum field theory, we establish the Hilbert space structure within a Gelfand triple.

Consider the sequence ${\left\{(M_{\left(k\right)},g_{\mu \nu }^{\left(k\right)})\right\}}_{k\in \mathbb{N}}$ of globally hyperbolic and homotopic spacetimes in Definition 2. By the already used theorem of Bernal and Sánchez [12], each spacetime admits a smooth global time function $t_{\left(k\right)}:M_{\left(k\right)}\to \mathbb{R}$ whose level sets

$${\mathrm{\Sigma }}_t^{\left(k\right)}=\{p\in M_{\left(k\right)}:t_{\left(k\right)}\left(p\right)=t\}$$

are spacelike Cauchy surfaces.

Fix a hypersurface ${\mathrm{\Sigma }}_t^{\left(k\right)}$ and a coordinate chart ${\kappa }_t^{\left(k\right)}:{\mathrm{\Sigma }}_t^{\left(k\right)}\to U_{{\kappa }_t^{\left(k\right)}}\subset {\mathbb{R}}^3$. We define the Hilbert space

$${\mathcal{H}}_t^{\left(k\right)}:=L^2\left(U_{{\kappa }_t^{\left(k\right)}},k^{-3}{d}^{3}x\right),$$

with inner product

$${\langle u_{\left(k\right)},v_{\left(k\right)}\rangle }_{{\mathcal{H}}_t^{\left(k\right)}}={\int }_{U_{{\kappa }_t^{\left(k\right)}}}{\displaystyle \frac{1}{k^3}}{u}_{\left(k\right)}^{\ast }\left(x\right)v_{\left(k\right)}\left(x\right)d^{3}x.$$

The localised Gaussian Schwartz space ${\mathcal{S}}_{\mathcal{G}}(U_{{\kappa }_t^{\left(k\right)}})$ is continuously and densely embedded in ${\mathcal{H}}_t^{\left(k\right)}$, and its dual ${\mathcal{S}}_{\mathcal{G}}^{\prime }(U_{{\kappa }_t^{\left(k\right)}})$ contains the distributional limits of the Gaussian sequence. Thus, we obtain a Gelfand triple

$${\mathcal{S}}_{\mathcal{G}}(U_{{\kappa }_t^{\left(k\right)}})\subset {\mathcal{H}}_t^{\left(k\right)}\subset {\mathcal{S}}_{\mathcal{G}}^{\prime }(U_{{\kappa }_t^{\left(k\right)}}).$$

Since each component ${\beta }_{\left(k\right)}^i$ depends only on $x^i$, the Hilbert space factorises as

$${\mathcal{H}}_t^{\left(k\right)}={\mathcal{H}}_1^{\left(k\right)}\otimes {\mathcal{H}}_2^{\left(k\right)}\otimes {\mathcal{H}}_3^{\left(k\right)},$$

where

$${\mathcal{H}}_i^{\left(k\right)}=L^2(\mathbb{R},k^{-1}d{x}^i),$$

with inner product

$${\langle u_{\left(k\right)}^i,v_{\left(k\right)}^i\rangle }_{{\mathcal{H}}_i^{\left(k\right)}}=\int {\displaystyle \frac{1}{k}}{u}_{\left(k\right)}^{i\ast }\left(x^i\right)v_{\left(k\right)}^i\left(x^i\right)d{x}^i.$$

Each one-dimensional Gaussian Schwartz space ${{\mathcal{S}}_{\mathcal{G}}^{\left(k\right)}}_i$ is densely embedded in ${\mathcal{H}}_i^{\left(k\right)}$, yielding the Gelfand triple

$${{\mathcal{S}}_{\mathcal{G}}^{\left(k\right)}}_i\subset {\mathcal{H}}_i^{\left(k\right)}\subset {{\mathcal{S}}_{\mathcal{G}}^{\prime \left(k\right)}}_i.$$

Remark 9.

Because the localised Gaussian Schwartz space is nuclear and densely embedded in the weighted Hilbert space, its dual naturally accommodates the distributional limits of the Gaussian delta-net. In this way, the quantum Hilbert-space structure and the distributional geometry coexist within a single rigged Hilbert space (Gelfand triple) framework adapted to the Gaussian sector.

We are now ready to proceed with the quantisation. Since each component ${\beta }_{\left(k\right)}^i$ is a function of its corresponding spatial coordinate $x^i$, $i\in \{1,2,3\}$, the quantisation proceeds independently in each spatial direction. Thus, each shift vector component is promoted to an operator-valued distribution ${\widehat{\beta }}_{\left(k\right)}^i$ and, in the Schrödinger picture, expanded in a local Gaussian wave-packet basis:

$${\widehat{\beta }}_{\left(k\right)}^i:={\widehat{\beta }}_{\left(k\right)}\left(x^i\right):={\int }_{\mathcal{F}(U_{{\kappa }_t^{\left(k\right)}}^i)}d{p}^i\left({\psi }_{\left(k\right)}(p^i,x^i){\widehat{a}}_{\left(k\right)}^i\left(p^i\right)+{\psi }_{\left(k\right)}^{\ast }(p^i,x^i){\widehat{a}}_{\left(k\right)}^{i†}\left(p^i\right)\right).$$

(42)

Here, the annihilation operator ${\widehat{a}}_{\left(k\right)}^i\left(p^i\right)$ and creation operator ${\widehat{a}}_{\left(k\right)}^{i†}\left(p^i\right)$ satisfy the bosonic commutation relations:

$$[{\widehat{a}}_{\left(k\right)}^i\left(p^i\right),{\widehat{a}}_{\left(k\right)}^{j†}\left(p^{\prime j}\right)]={\delta }^{ij}\delta (p^i-p^{\prime i}),$$

(43)

$$[{\widehat{a}}_{\left(k\right)}^i\left(p^i\right),{\widehat{a}}_{\left(k\right)}^j\left(p^{\prime j}\right)]=0,[{\widehat{a}}_{\left(k\right)}^{i†}\left(p^i\right),{\widehat{a}}_{\left(k\right)}^{j†}\left(p^{\prime j}\right)]=0.$$

(44)

The mode functions are chosen as Gaussian wave packets, defined via their momentum-space representation:

$${\psi }_{\left(k\right)}(p^i,x^i)={\left({\displaystyle \frac{{\sigma }^2}{\pi k^2}}\right)}^{1/4}exp\left(-{\displaystyle \frac{{\sigma }^2}{2k^2}}{\left(p^i\right)}^2\right)exp\left(i{p}^i{x}^i\right).$$

(45)

The (total) Fock space is constructed as a tensor product of three independent bosonic Fock spaces:

$$\mathcal{F}\left({\mathcal{H}}^{\left(k\right)}\right)=\mathcal{F}\left({\mathcal{H}}_1^{\left(k\right)}\right)\otimes \mathcal{F}\left({\mathcal{H}}_2^{\left(k\right)}\right)\otimes \mathcal{F}\left({\mathcal{H}}_3^{\left(k\right)}\right),$$

(46)

where each individual Fock space is built as

$$\mathcal{F}\left({\mathcal{H}}_i^{\left(k\right)}\right)=\mathbb{C}\oplus {\mathcal{H}}_i^{\left(k\right)}\oplus \left({\mathcal{H}}_i^{\left(k\right)}{\otimes }_s{\mathcal{H}}_i^{\left(k\right)}\right)\oplus \dots .$$

(47)

The vacuum state for the complete Fock space is given by

$$|0_{\left(k\right)}\rangle :=|0_{\left(k\right)}{\rangle }_1\otimes |0_{\left(k\right)}{\rangle }_2\otimes {|0_{\left(k\right)}\rangle }_3,$$

(48)

with vacuum states $|0_{\left(k\right)}{\rangle }_i$ satisfying

$${\widehat{a}}_{\left(k\right)}^i\left(p^i\right){|0_{\left(k\right)}\rangle }_i=0,\forall p^i.$$

(49)

One-particle states are given by

$$|1_{\left(k\right)}\left(p^i\right){\rangle }_i={\widehat{a}}_{\left(k\right)}^{i†}\left(p^i\right){|0_{\left(k\right)}\rangle }_i,$$

(50)

and multi-particle states follow as

$$|p_1^i,p_2^i,\dots ,p_n^i{\rangle }_i={\widehat{a}}_{\left(k\right)}^{i†}\left(p_1^i\right){\widehat{a}}_{\left(k\right)}^{i†}\left(p_2^i\right)\cdots {\widehat{a}}_{\left(k\right)}^{i†}\left(p_n^i\right){|0_{\left(k\right)}\rangle }_i.$$

(51)

A coherent state $|{\alpha }_{\left(k\right)}^i\rangle$ for each separate Fock space $\mathcal{F}\left({\mathcal{H}}_i^{\left(k\right)}\right)$ is defined as an eigenstate of the annihilation operator ${\widehat{a}}_{\left(k\right)}^i\left(p^i\right)$:

$${\widehat{a}}_{\left(k\right)}^i\left(p^i\right)|{\alpha }_{\left(k\right)}^i\rangle ={\alpha }_{\left(k\right)}^i\left(p^i\right)|{\alpha }_{\left(k\right)}^i\rangle ,\forall p^i.$$

(52)

Interpreted through the quantised shift vector (42), the eigenvalue ${\alpha }_{\left(k\right)}^i\left(p^i\right)$ encodes the displacement of the quantum field configuration away from the vacuum. The classical profile ${\beta }_{\left(k\right)}^i$ is realised quantum mechanically as the expectation value $\langle {\alpha }_{\left(k\right)}^i\left|{\widehat{\beta }}_{\left(k\right)}^i\right|{\alpha }_{\left(k\right)}^i\rangle$. To maintain this correspondence, the eigenvalue ${\alpha }_{\left(k\right)}^i\left(p^i\right)$ must be chosen accordingly. Within the framework of the Gelfand triple ${\mathcal{S}}_i^{\left(k\right)}\subset {\mathcal{H}}_i^{\left(k\right)}\subset {\mathcal{S}}_i^{\prime \left(k\right)}$, it is natural to choose ${\alpha }_{\left(k\right)}^i\left(p^i\right)$ as a Gaussian:

$${\alpha }_{\left(k\right)}^i\left(p^i\right)={\left({\displaystyle \frac{k^2}{256{\pi }^3{\sigma }^2}}\right)}^{1/4}exp\left(-{\displaystyle \frac{{\sigma }^2{\left(p^i\right)}^2}{2k^2}}\right).$$

(53)

Next, the complete coherent state for the shift vector is given by the tensor product

$$|{\alpha }_{\left(k\right)}\rangle :=|{\alpha }_{\left(k\right)}^1\rangle \otimes |{\alpha }_{\left(k\right)}^2\rangle \otimes |{\alpha }_{\left(k\right)}^3\rangle .$$

(54)

Given this construction, we are now ready to determine the mean bosonic occupation number in the emerging spacetime, and hence for finite values of the sequence index k. To avoid any notation problem, and in an attempt to simplify an already rich and cumbersome notation, let $a\in \{1,2,3\}$ denote a fixed spatial direction and define the phase-space number density operator by

$${\widehat{\mathcal{N}}}_{\left(k\right)}^a\left(p^a\right):={\widehat{a}}_{\left(k\right)}^{a†}\left(p^a\right){\widehat{a}}_{\left(k\right)}^a\left(p^a\right),$$

so that the corresponding (Planck-length-adjusted) number operator is

$${\widehat{N}}_{\left(k\right)}^a:={\ell }_p^2\int {\widehat{\mathcal{N}}}_{\left(k\right)}^a\left(p^a\right)d{p}^a.$$

Then, the expected number of quanta in the direction a in the coherent state $|{\alpha }_{\left(k\right)}\rangle$ is

$$\begin{array}{cc}\hfill \langle {\widehat{N}}_{\left(k\right)}^a\rangle & :={\ell }_p^2\int \langle {\alpha }_{\left(k\right)}|{\widehat{\mathcal{N}}}_{\left(k\right)}^a\left(p^a\right)|{\alpha }_{\left(k\right)}\rangle d{p}^a\hfill \end{array}$$

(55)

$$\begin{array}{cc}\hfill & ={\ell }_p^2\int \langle {\alpha }_{\left(k\right)}^a|{\widehat{\mathcal{N}}}_{\left(k\right)}^a\left(p^a\right)|{\alpha }_{\left(k\right)}^a\rangle d{p}^a\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill & ={\ell }_p^2\int {\left|{\alpha }_{\left(k\right)}^a\left(p^a\right)\right|}^{2}d{p}^a={\displaystyle \frac{k^2{\ell }_p^2}{16\pi {\sigma }^2}}.\hfill \end{array}$$

(57)

This follows since each coherent sector $|{\alpha }_{\left(k\right)}^a\rangle$, $a\in \{1,2,3\}$, is normalised, the expectation value of the number density in a coherent state is ${\left|{\alpha }_{\left(k\right)}^a\right|}^2$, and

$${\int }_{-\infty }^{\infty }exp\left(-{\displaystyle \frac{{\sigma }^2{\left(p^a\right)}^2}{k^2}}\right)d{p}^a={\displaystyle \frac{\sqrt{\pi }k}{\sigma }}.$$

To determine the entropy of a transient trapped surface, we must identify a state-counting functional that maps microscopic occupation-number sectors to macroscopic thermodynamic quantities. For a fixed direction a and sequence index k, the Fock space $\mathcal{F}\left({\mathcal{H}}_a^{\left(k\right)}\right)$ admits a mode decomposition over the momentum parameter $p^a$. We therefore consider the density matrix restricted to a single (arbitrary) mode $p^a$. For brevity, we temporarily suppress the explicit mode dependence in the state vectors and eigenvalues, writing $|{\alpha }_{\left(k\right)}^a\rangle \equiv |{\alpha }_{\left(k\right)}^a\left(p^a\right)\rangle$ and $|n_{\left(k\right)}^a\rangle \equiv |n_{\left(k\right)}^a\left(p^a\right)\rangle$. Generalising the approach of Glauber [28] and Bashkirov–Sukhanov [9], each diagonal element of the coherent density matrix ${\rho }_{nn}^{(k,a)}$ in this sector is given by

$${\rho }_{nn}^{(k,a)}:={\left|\langle n_{\left(k\right)}|{\alpha }_{\left(k\right)}^a\rangle \right|}^2={\displaystyle \frac{{\left|{\alpha }_{\left(k\right)}^a\right|}^{2n}}{n!}}{\left|\langle 0^a|{\alpha }_{\left(k\right)}^a\rangle \right|}^2.$$

(58)

That is, the probability for observing n quanta is described by a Poisson distribution. To determine $\langle 0^a|{\alpha }_{\left(k\right)}^a\rangle$, we use the fact that

$${\widehat{a}}_{\left(k\right)}^{a†}|n\rangle =\sqrt{n+1}|n+1\rangle$$

is equivalent to

$$\langle n|{\widehat{a}}_{\left(k\right)}^a=\sqrt{n+1}\langle n+1|.$$

This allows us to write

$$\langle n\left|{\widehat{a}}_{\left(k\right)}^a\right|{\alpha }_{\left(k\right)}^a\rangle =\sqrt{n+1}\langle n+1|{\alpha }_{\left(k\right)}^a\rangle ={\alpha }_{\left(k\right)}^a\langle n|{\alpha }_{\left(k\right)}^a\rangle .$$

Solving this recursion relation, we have

$$\langle n|{\alpha }_{\left(k\right)}^a\rangle ={\displaystyle \frac{{\left({\alpha }_{\left(k\right)}^a\right)}^n}{\sqrt{n!}}}\langle 0|{\alpha }_{\left(k\right)}^a\rangle .$$

From this requiring the coherent state to be normalised

$$\langle {\alpha }_{\left(k\right)}^a|{\alpha }_{\left(k\right)}^a\rangle ={\displaystyle \sum _{n=0}^{\infty }}\langle {\alpha }_{\left(k\right)}^a|n\rangle \langle n|{\alpha }_{\left(k\right)}^a\rangle ={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{\left(\right|{\alpha }_{\left(k\right)}^a{{|}^2)}^n}{n!}}{\left|\langle 0|{\alpha }_{\left(k\right)}^a\rangle \right|}^2=e^{|{\alpha }_{\left(k\right)}^a{|}^2}{\left|\langle 0|{\alpha }_{\left(k\right)}^a\rangle \right|}^2=1$$

we have

$$\langle 0|{\alpha }_{\left(k\right)}^a\rangle =e^{-{\displaystyle \frac{1}{2}}{\left|{\alpha }_{\left(k\right)}^a\right|}^2}.$$

Given this, and returning to the coherent density matrix with the elements along the diagonal given by (58) and taking the trace, we have

$$\mathrm{tr}\left({\rho }^{(k,a)}\right)={\displaystyle \sum _{n=0}^{\infty }}{\rho }_{nn}^{(k,a)}={\displaystyle \sum _{n=0}^{\infty }}{\left|\langle n_{\left(k\right)}|{\alpha }_{\left(k\right)}^a\rangle \right|}^2={\displaystyle \sum _{n=0}^{\infty }}{W}_n^{(k,a)}{e}^{-{\left|{\alpha }_{\left(k\right)}^a\right|}^2}=1.$$

Here, each term

$$W_n^{(k,a)}\left(p^a\right):={\displaystyle \frac{{\left|{\alpha }_{\left(k\right)}^a\left(p^a\right)\right|}^{2n}}{n!}}$$

is interpreted as the (unnormalised) state-counting weight associated with the n-quanta sector in the direction a at sequence index k, in the chosen mode $p^a$. This allows us to define a state-counting generating function for a single arbitrary mode $p^a$ by

$$Q_{\left(k\right)}^a\left(p^a\right):={\displaystyle \sum _{n=0}^{\infty }}{W}_n^{(k,a)}\left(p^a\right)={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{{\left|{\alpha }_{\left(k\right)}^a\left(p^a\right)\right|}^{2n}}{n!}}=exp\left(|{\alpha }_{\left(k\right)}^a\left(p^a\right){|}^2\right).$$

(59)

We emphasise that $Q_{\left(k\right)}^a\left(p^a\right)$ is not a canonical state-counting generating function (often denoted Z and depending on temperature); rather, it is an exponential generating function for the coherent-state occupation-number sectors.

Expanding over all modes, we then obtain the corresponding generating functional

$$Q_{\left(k\right)}^a:=exp\left(\int |{\alpha }_{\left(k\right)}^a\left(p^a\right){|}^{2}d{p}^a\right).$$

(60)

Thus, (60) motivates the directional entropy definition

$$S_{\left(k\right)}^a:=k_{\mathrm{B}}{\ell }_p^{2}ln{Q}_{\left(k\right)}^a=k_{\mathrm{B}}{\ell }_p^2\int {\left|{\alpha }_{\left(k\right)}^a\left(p^a\right)\right|}^{2}d{p}^a=k_{\mathrm{B}}\langle {\widehat{N}}_{\left(k\right)}^a\rangle .$$

(61)

In the present setting, using (55), this yields

$$S_{\left(k\right)}^a=k_{\mathrm{B}}\langle {\widehat{N}}_{\left(k\right)}^a\rangle =k_{\mathrm{B}}{\displaystyle \frac{k^2{\ell }_p^2}{16\pi {\sigma }^2}}.$$

The full entropy is the sum over the three independent sectors,

$$S_{\left(k\right)}:=S_{\left(k\right)}^1+S_{\left(k\right)}^2+S_{\left(k\right)}^3=k_{\mathrm{B}}{\displaystyle \frac{3k^2{\ell }_p^2}{16\pi {\sigma }^2}}.$$

Now, recall from Proposition 1 that the trapped-surface area satisfies

$$A_{\left(k\right)}={\alpha }_{\left(k\right)}{\ell }_p^2,{\alpha }_{\left(k\right)}={\displaystyle \frac{k^2{\ell }_p^2}{4\pi {\sigma }^2}}{C}_{\left(k\right)},$$

with $C_{\left(k\right)}>0$. Defining the effective length scale

$${\ell }_{\left(k\right)}^2:=C_{\left(k\right)}{\ell }_p^2,$$

we obtain the directional and total entropy–area relations

$$S_{\left(k\right)}^a=k_{\mathrm{B}}{\displaystyle \frac{A_{\left(k\right)}}{4{\ell }_{\left(k\right)}^2}},S_{\left(k\right)}=3k_{\mathrm{B}}{\displaystyle \frac{A_{\left(k\right)}}{4{\ell }_{\left(k\right)}^2}}.$$

To summarise, we have the following entropy theorem for transient trapped surfaces:

Theorem 5

(Weakly interacting entropy of a transient trapped surface). Fix $k<\infty$ and let $|{\alpha }_{\left(k\right)}\rangle$ be the coherent state defined by (52)–(54). For each $a\in \{1,2,3\}$, let ${\widehat{N}}_{\left(k\right)}^a$ denote the corresponding number operator and define the thermodynamic/statistical entropy by

$$S_{\left(k\right)}^a:=k_{\mathrm{B}}\langle {\alpha }_{\left(k\right)}|{\widehat{N}}_{\left(k\right)}^a|{\alpha }_{\left(k\right)}\rangle ,S_{\left(k\right)}:=S_{\left(k\right)}^1+S_{\left(k\right)}^2+S_{\left(k\right)}^3.$$

Then,

$$S_{\left(k\right)}^a=k_{\mathrm{B}}{\displaystyle \frac{k^2{\ell }_p^2}{16\pi {\sigma }^2}},S_{\left(k\right)}=k_{\mathrm{B}}{\displaystyle \frac{3k^2{\ell }_p^2}{16\pi {\sigma }^2}}.$$

Moreover, if the trapped-surface area is given by Proposition 1,

$$A_{\left(k\right)}={\alpha }_{\left(k\right)}{\ell }_p^2,{\alpha }_{\left(k\right)}={\displaystyle \frac{k^2{\ell }_p^2}{4\pi {\sigma }^2}}{C}_{\left(k\right)},$$

and we define

$${\ell }_{\left(k\right)}^2:=C_{\left(k\right)}{\ell }_p^2,$$

then

$$S_{\left(k\right)}^a=k_{\mathrm{B}}{\displaystyle \frac{A_{\left(k\right)}}{4{\ell }_{\left(k\right)}^2}},S_{\left(k\right)}=3k_{\mathrm{B}}{\displaystyle \frac{A_{\left(k\right)}}{4{\ell }_{\left(k\right)}^2}}.$$

In particular, in the asymptotically flat limit of Proposition 1, one has $A_{\left(k\right)}\to 0$, while the entropy satisfies the entropy–area relation above with ${\ell }_{\left(k\right)}^2=C_{\left(k\right)}{\ell }_p^2$.

6. Discussion

A purely classical universe can be represented by a pair consisting of a differentiable manifold and a metric field on the manifold. This pair represents a spacetime, and we typically require it to be globally hyperbolic so that it is stably causal and necessarily admits a regular time function, and hence can be foliated by the level surfaces of that time function. This construction gives rise to a wide class of physical spacetimes, but leaves unanswered the question of the origin of elementary particles. In fact, the pair does not, by itself, provide any mathematical structure with which to represent elementary particles. To accommodate elementary particles, the manifold must therefore be equipped with at least one additional structure.

Taking Schwartz’s view as an ansatz from his book Applications of Distributions to the Theory of Elementary Particles in Quantum Mechanics [2], or starting from Gårding and Wightman in their work on fields as operator-valued distributions in relativistic quantum theory [30] on Minkowski spacetime, which can be extended to (curved) globally hyperbolic spacetimes [31], then, to accommodate elementary particles, the manifold structure must admit, or be equipped with, a space of distributions. This would allow us to introduce elementary particles as a vector subspace, equipped with a continuously embedded Hilbert space structure. Another way of phrasing this is that the universe should admit a Gelfand triple so that elementary particles are admitted in a smooth sense and consequently incorporate a correspondence principle.

In this work, we have taken exactly that view, using Hörmander’s Definition 6.3.3 [11] with local distributions in coordinate patches as a basis in conjunction with a standard pullback and composition mechanism for intersecting coordinate patches.

The Gelfand triple allows us to trace this universe to its origin. This suggests that Wheeler’s geon, and hence a self-gravitating but unstable entity representing the notion of a ‘fundamental body’ [1], can be introduced as a distribution. Specifically, the geon can be made precise via a sequence of smooth spacetimes converging, in distributions, to a distributional entity. This can be motivated by basic distribution theory, since a distribution can be approximated by a sequence of test functions in conjunction with the development of a renormalised distribution algebra, based on a Gaussian model delta net, which was introduced in [4] and has been reviewed in this work.

Specifically, we have made the geon precise as a limiting structure in an unstable equilibrium, supported on intersecting singular supports of the shift vector components whose intersection forms a characteristic core in a non-strongly causal setting. As the sequence index of the elements in the net tends to infinity and converges to a distribution, the spacelike hypersurfaces as well as the timelike normal fields warp towards a limiting null surface. This is what causes the breach of strong causality at the characteristic core. This extreme distributional behaviour inherently destroys global hyperbolicity; the regular time function and the associated Hamiltonian generator lose their mathematical meaning. Consequently, the standard machinery of quantum field theory in curved spacetime collapses at this limit.

The physical mechanisms explored here—such as the ‘collapse’ of the unstable geon and the ‘evaporation’ of the transient trapped surfaces—are therefore not temporal processes tracked by an advancing Hamiltonian clock. Rather, these phenomena are rigorously captured parametrically through the sequence index. Therefore, the framework sidesteps the breakdown of the Hamiltonian, allowing for a topological transition from the highly warped distributional regime to the classical, asymptotically flat regime to act as an effective geometric emergence flow.

Furthermore, we have derived the field equation of the geon in this distributional setting, where its coefficients are neither smooth nor linear. The resulting inhomogeneous non-linear field equation is driven by the second-order distributional derivative of the Dirac measure and is as expected inherently unstable. Thus, as a null structure in a non-strongly causal setting, its polarisation effects physically necessitate this emergence flow; the unstable equilibrium cannot be sustained, driving the transition into the classical regime.

As has been further shown, in part in [4] and in a more comprehensive discussion in this discourse, the energy conditions hold at the origin, or the core of the geon, in the distributional sense, but fail on the emerging globally hyperbolic spacetime elements, even if the total energy, as well as the total momentum, is zero. Thus, classical spacetime necessarily emerges from these vacuum fluctuations, where the hypersurfaces undergo regions of expansion and contraction separated by a vacuum state shell. It has been shown here that these fluctuations are huge at the Planck scale, effectively providing the inflation mechanism for resolving the horizon problem.

In relation to this, it has been shown that while no trapped surfaces arise at the core of the geon [4], they do arise at the Planck scale in neighbourhoods punctured by the geon core. Here, we have shown that these trapped configurations are transient, computed their surface area on the Planck-referenced neighbourhoods where they arise, and proved that this area tends to zero in the asymptotically flat limit.

We have determined the entropy for the weakly interacting constituents of the trapped-surface configuration, defined via the expectation value of the occupation-number operator over all modes in a coherent state. In fact, we have argued that the construction of a coherent state implementing the correspondence principle, as enforced by the Gelfand triple admitted by the definition of the geon through a sequence of smooth spacetimes converging to a self-gravitating, null-like, but unstable condensate at the characteristic limit, allows us to use the expectation value of the occupation-number operator over all modes as a microstate-counting variable.

More explicitly, since the expectation values in the quantum foam of the geon in a coherent state coincide with the classical values, this allows us to use these expectation values to construct the required state-counting generating function from the number operator and thereby determine the entropy of a transient trapped surface. In this way, we obtain an entropy–area relation qualitatively consistent with Hawking’s area and evaporation picture [26] (see Witten for an introduction and modern review of black hole thermodynamics [24]), in the sense that the area (and hence the associated thermodynamic/statistical entropy) decreases and vanishes in the relevant limit. However, we emphasise that our trapped surfaces are transient and are not assumed to be event horizons, and the entropy employed here is not the von Neumann entropy that necessarily is zero in a pure coherent state. Moreover, the present framework is not a semi-classical approximation: the metric back-reaction and the accompanying vacuum fluctuations are intrinsic to the construction. In this sense, our results support the view that back-reaction does not prevent such configurations from forming; rather, the collapse of the geon is necessary for their appearance.

In addition, it is worthwhile to stress that we have had access to an explicit quantum description of the radiation sector produced in the aftermath of the collapse of the self-gravitating but unstable boson field (the geon) at the characteristic (null) limit. In particular, since the construction proceeds through a causal net of globally hyperbolic spacetime elements, quantum fields are well defined on each element, and one can track the occupation-number statistics in the coherent sector without invoking a semi-classical cut-off. At Planck-referenced scales, the resulting entropy-area relation is qualitatively aligned with the semi-classical black hole thermodynamic scaling.

A common conceptual gap in gravitational entropy formulas is that the geometric area term does not manifestly arise from tracing over an explicit microstate basis (cf. the “central dogma” discussion in [10]). In the present framework, the relevant area term is not postulated: for each finite spacetime element, the trapped-surface patch area is calculated directly from the induced geometry on a Cauchy surface in the Planck-referenced punctured neighbourhood where the configuration exists. The connection between this geometric area and the entropy is then made by the observation that, by construction, the coherent sector implements the correspondence principle, so that expectation values of the relevant quantised observables coincide with their classical values. This provides a mechanism by which the occupation-number expectation value over all modes can be used as a microstate-counting variable and, correspondingly, to introduce a state-counting generating function for these states. The resulting Boltzmann relation then yields an entropy that can be related directly to the geometric area.

Hawking emphasised that, although in classical general relativity, black holes can absorb but not emit, semi-classical quantum-field effects lead to particle creation and hence thermal radiation, and thus to a decrease in the black hole mass and eventual evaporation [32]. Hawking also noted that it is a reasonable approximation to treat an evaporating black hole as evolving through a sequence of approximately stationary solutions and to compute the emission rate on each, until the quasi-stationary approximation eventually breaks down [32].

In the present framework, it is plausible that the stress–energy supporting transient trapped configurations stems from the radiation sector produced during the geon collapse under its inherent vacuum polarisation. Since the trapped surfaces form in emerging globally hyperbolic spacetime elements, one can, in principle, analyse the associated radiation using quantum field theory on static curved spacetime on each element of the causal net. At the qualitative level, the present results support the view that metric back-reaction does not obstruct evaporation-like behaviour and that it is likely that this is a feature carried over to black hole configurations.

Moreover, in relation to Hawking’s perspective on the fate of a radiating, and hence evaporating, black hole—and in the context of the uncertainty arising from not having a complete theory of quantum gravity available—the following possible scenarios were presented in [33]:

• The evaporation produces a naked singularity of negative mass which persists.
• The evaporation slows down and stops, leaving a remnant black hole of about the Planck mass.
• The black hole disappears completely, but all the information about the black hole states and any other locally conserved quantities escapes to infinity.
• The black hole disappears completely, taking with it the information about the black hole states and any other quantities which are not coupled to long-range fields.

In relation to the transient trapped surfaces, we have seen that they are created in punctured neighbourhoods of the geon core and that their associated entropy vanishes both in the core and in the asymptotically flat limit. Thus, the transient trapped surface disappears completely and all the information about the states and any other locally conserved quantities escapes to infinity. In fact, the very construction proceeds through a causal net of spacetime elements, each of which is globally hyperbolic. Therefore, the argument that non-global hyperbolicity forces a fundamental loss of unitarity does not apply at any finite stage of the model. In particular, each element admits a global time function that foliates the spacetime into Cauchy surfaces. It is solely at the core of the geon that strong causality is violated and unitarity is lost. At the core, however, the null expansion identically vanishes, and no trapped surfaces exist.

The usual route to Hawking’s pure-to-mixed conclusion is absent: the collapse of the unstable geon triggers decoherence into a coherent-sector description, but the resulting quantum state is still taken to be pure (in particular, coherent), so that the von Neumann entropy of the global state remains zero on every globally hyperbolic spacetime element emerging from the geon collapse.

Thus, the entropy relevant for the trapped-surface configurations is not the von Neumann entropy, but a thermodynamic/statistical entropy defined from occupation-number statistics in the weakly interacting coherent sector. In this, one can recover an entropy–area relation while maintaining global purity and at the same time show that the area of a transient trapped surface is a representation of the underlying quantum states.

Accordingly, within the internal logic of the geon, there is no information paradox: the global state remains pure, while the macroscopic entropy is controlled by occupation-number statistics.