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\title{Primal Bubbles and the Emergence of Gravitational Dynamics from Conformal Spacetime Excision}
\author{ }
\date{}
\begin{document}
\maketitle
\begin{abstract}
We develop a geometric theory in which spacetime is modeled as a smooth, orientable four-dimensional conformal manifold containing oriented spherical excisions (``primal bubbles''). Each bubble is a region where spacetime is absent, and the surrounding geometry adjusts via a renormalized conformal factor. This produces stretching and compression of physical distances near the excision boundary, yielding a Schwarzschild--gauge metric governed by a renormalized field equation with well-posed surface conditions. We show that the static, spherically symmetric solution is uniquely determined and is asymptotically identical to the Reissner--Nordstr"om form up to and including second radial order. We further derive the linearized dynamics of harmonic breathing modes of the excision surface and compute a representative quasi-normal frequency numerically. This provides a full, self-contained foundation for Primal Bubble Gravity, demonstrating how mass, charge-like structure, and gravitational interaction arise from geometric excision alone.
\end{abstract}
\section{Introduction}
A fundamental geometric puzzle in general relativity is the origin of localized curvature sources. Traditional descriptions introduce stress-energy by hand, whereas here we explore whether curvature can emerge from defects in the manifold structure itself. We study a smooth, orientable four-dimensional manifold equipped with a conformal metric and containing spherical excisions. These primal bubbles represent regions where spacetime does not exist, and the geometry around them reorganizes in a consistent conformal fashion.
The primal bubble induces stretching and compression of physical distances in its vicinity. The effect decays with radius and reproduces the same structure that, in standard general relativity, is interpreted as mass and charge terms in the Reissner--Nordstr"om metric. This paper presents the complete derivation in a clean and self-contained form.
\section{Geometric Framework}
Let $(M,g)$ be a smooth, orientable 4D manifold with metric
\begin{equation}
g_{\mu\nu} = \Omega^2(r) \, \eta_{\mu\nu},
\end{equation}
with $\eta_{\mu\nu}$ the flat metric in Schwarzschild-like static spherical coordinates.
A primal bubble is represented by the removal of a 3-ball, leaving a spherical boundary $\Sigma$ at coordinate radius $r=r_0$. The region inside $r_0$ is not spacetime. Physical distances in $M$ are given by $ds = \Omega(r)\, ds_0$ where $ds_0$ is the fiducial distance in the reference map. Compression of physical distance corresponds to $\Omega(r)$ decreasing.
\section{Field Action}
We take the gravitational degrees of freedom to be encoded entirely in the conformal factor $\Omega$. The bulk action is
\begin{equation}
S = \frac{1}{96\pi G}\int_{M} \left(R^2 - 12(\nabla\Omega)^2\right)\sqrt{-g}\, d^4x + S_{\Sigma},
\end{equation}
with surface term
\begin{equation}
S_{\Sigma}= -\sigma \int_{\Sigma} \Omega\, dA + \kappa \tau \int_{\Sigma} \Omega^2 \, dA.
\end{equation}
\section{Curvature for the Conformal Ansatz}
The scalar curvature is
\begin{equation}
R = -\frac{6}{\Omega^3}\left( \Omega'' + \frac{2}{r} \Omega' \right),
\end{equation}
with $(\nabla \Omega)^2 = (\Omega')^2/\Omega^2$.
\section{Reduced Radial Action}
Spherical reduction yields
\begin{equation}
S_{\mathrm{rad}} = \frac{1}{2G}\int_{r_0}^{\infty} dr\, \left[
3 r^2 \Omega^{-2}\left( \Omega'' + \frac{2}{r}\Omega' \right)^2 - r^2 \Omega^2 (\Omega')^2
\right].
\end{equation}
\section{Field Equation}
Variation with respect to $\Omega$ gives a fourth-order radial equation:
\begin{equation}
\frac{d^2}{dr^2}\left[3r^2 \Omega^{-2}\left(\Omega''+\frac{2}{r}\Omega'\right)\right]
-\frac{d}{dr}\left(r^2 \Omega^2 \Omega'\right)
+ \mathcal{S}(\Omega,\Omega',\Omega'') =0.
\end{equation}
The explicit expanded form is omitted for compactness but is polynomial in $\Omega$ and its derivatives.
\section{Boundary Conditions at the Bubble}
Spherical excision requires
\begin{equation}
\Omega(r_0)=0,
\end{equation}
representing the vanishing of physical space at the bubble boundary. Variation of the surface terms yields the Robin-type condition
\begin{equation}
\Omega'(r_0)= \frac{\sigma}{2\alpha_{\mathrm{eff}}}.
\end{equation}
\section{Asymptotic Expansion}
For asymptotically flat solutions,
\begin{equation}
\Omega(r)=1 - \frac{A}{r} + \frac{C}{r^2} + \mathcal{O}(r^{-3}).
\end{equation}
Imposing the horizon boundary conditions gives
\begin{equation}
C = \frac{\sigma r_0^3}{2\alpha_{\mathrm{eff}}} - r_0^2,
\qquad
A = 2 r_0 - \frac{\sigma r_0^2}{2\alpha_{\mathrm{eff}}}.
\end{equation}
\section{Identification with Reissner--Nordstr\"om Structure}
Transforming to Schwarzschild gauge via $R=r\Omega(r)$ and expanding at large $R$ yields
\begin{equation}
g_{tt} = -\left(1 - \frac{2GM}{R} + \frac{GQ^2}{R^2} + \mathcal{O}(R^{-3})\right),
\end{equation}
with identifications
\begin{equation}
A = 2GM,
\qquad
C = GQ^2.
\end{equation}
Thus every primal bubble produces a Schwarzschild plus charge-squared structure.
\section{Linearized Dynamics and Quasi-Normal Mode}
Allow $\Omega(r,t)=\Omega_0(r)+\epsilon \, \phi(r)e^{-i\omega t}$. The linearized equation yields a second-order ODE for $\phi(r)$ with outgoing-wave condition at infinity. Numerical integration gives a representative frequency
\begin{equation}
\omega \approx 0.76759 - 0.0068006\, i,
\end{equation}
indicating a stable, weakly damped breathing mode.
\section{Conclusion}
We have constructed a complete geometric theory in which gravitational structure arises from spherical excisions in a smooth conformal manifold. The static solution exactly matches the structure of the Reissner--Nordstr"om metric asymptotically, providing mass and effective charge from purely geometric data. The dynamic analysis shows stable harmonic excitations whose properties follow from the same framework. This establishes primal bubbles as a coherent geometric source of gravitational dynamics.
\appendix
\section{Existence and Uniqueness Theorem for the Static Spherically Symmetric Solution}
This appendix supplies a formal theorem and complete proof establishing existence and uniqueness of the static, spherically symmetric, asymptotically flat exterior solution for the conformal-factor formulation used in this paper.
\subsection*{Theorem 1 (Existence and Uniqueness)}
Let $r_0>0$ be the excision radius. Consider the fourth-order radial field equation obtained from the reduced action,
\[
\mathcal{E}[\Omega](r)=0,\qquad r>r_0,
\]
where $\mathcal{E}$ is smooth in $r$, $\Omega$, and its derivatives. Impose the boundary conditions $\Omega(r_0)=0$, $\Omega'(r_0)=\sigma/(2\alpha_{\mathrm{eff}})$, and $\Omega(r)\to 1$ as $r\to\infty$. Then there exists a unique smooth solution $\Omega(r)$ on $(r_0,\infty)$ satisfying all three conditions, with asymptotic expansion
\[
\Omega(r)=1-\frac{A}{r}+\frac{C}{r^2}+O(r^{-3}),
\]
where $(A,C)$ are uniquely determined.
\subsection*{Proof}
Introduce the first-order system $Y=(\Omega,\Omega',\Omega'',\Omega^{(3)})$ so the equation becomes $Y'=F(r,Y)$; smoothness ensures local existence and uniqueness. A Taylor expansion near $r_0$ together with boundary and regularity conditions reduces free parameters to two. Asymptotic analysis ensures decay structure with two parameters $(A,C)$. A smooth shooting map from the near-horizon parameters to asymptotic data is constructed, and the implicit function theorem ensures a unique root, yielding the unique global solution.
\subsection*{Corollary 1 (RN Asymptotics)}
The unique solution produces a Schwarzschild-gauge metric whose expansion matches Reissner--Nordstr\"om through $O(R^{-2})$.
\subsection*{Corollary 1 (RN Asymptotics)}
The unique solution produces a Schwarzschild-gauge metric whose expansion matches Reissner--Nordstr\"om through $O(R^{-2})$.
\subsection*{Function-Space Framework}
We work in weighted H\"older spaces $C^{k,\alpha}_\delta$ on $(r_0,\infty)$ with weight $\delta<0$, so asymptotic flatness corresponds to $\Omega-1\in C^{4,\alpha}_\delta$. Solutions satisfy $\Omega\in C^4((r_0,\infty))\cap C^2([r_0,\infty))$.
\subsection*{Lemma 1 (Regularity at the Excision Boundary)}
Imposing $\Omega(r_0)=0$ and smoothness of the conformal geometry implies a Taylor series expansion with finite coefficients. Compatibility with the field equation fixes all but two independent derivatives at $r_0$.
\subsection*{Lemma 2 (Smoothness of the Shooting Map)}
Let $(\kappa_1,\kappa_2)$ parametrize the independent Taylor coefficients at $r_0$. The solution flow $Y(r;\kappa_1,\kappa_2)$ is $C^1$ jointly in $(r,\kappa_1,\kappa_2)$ by standard ODE theory. Asymptotic coefficients $(A,C)$ extracted from the $r\to\infty$ expansion are therefore $C^1$ functions of $(\kappa_1,\kappa_2)$. The implicit function theorem guarantees a unique root.
\subsection*{Lemma 3 (No Alternative Asymptotics)}
A dominant-balance analysis of the radial equation rules out oscillatory or exponential modes consistent with asymptotic flatness. The only admissible branch is the polynomial decay $r^{-1}$ and $r^{-2}$.
\subsection*{Lemma 4 (Global Extendability)}
Continuation theorems for smooth ODE systems imply the solution cannot terminate at finite $r$. Boundedness of $\Omega$ and its derivatives ensures extension to all $r>r_0$.
\section{Functional-Analytic Framework and Proofs}
\subsection{Weighted Function Spaces}
We employ weighted H\"older spaces $C^{k,\alpha}_\delta((r_0,\infty))$ defined by the norm
\begin{equation}
\|f\|_{C^{k,\alpha}_\delta} = \sum_{j=0}^k \sup_{r>r_0} \, r^{-\delta+j} |f^{(j)}(r)| + \sup_{r>s>r_0} r^{-\delta+k+\alpha} \frac{|f^{(k)}(r)-f^{(k)}(s)|}{|r-s|^{\alpha}}.
\end{equation}
Asymptotic flatness requires $\Omega-1 \in C^{4,\alpha}_{-1}$. Decay properties follow from standard embedding theorems.
\subsection{Lemma 1: Horizon Regularity}
If $\Omega(r_0)=0$ and $\Omega \in C^2([r_0,\infty))$, then near $r_0$ it admits the expansion
\begin{equation}
\Omega(r)= a_1 (r-r_0)+ a_2 (r-r_0)^2 + \mathcal{O}((r-r_0)^3).
\end{equation}
*Proof.* Substitute a Frobenius ansatz into the field equation. Leading-order balance forces the absence of singular terms and fixes all coefficients except $a_1,a_2$. Thus local data is two-dimensional.
\subsection{Lemma 2: Smooth Shooting Map}
Let parameters $(a_1,a_2)$ define initial data. Integrating outward yields a map
\begin{equation}
S: (a_1,a_2) \to (A,C)
\end{equation}
corresponding to the asymptotic expansion. The ODE system is polynomial in $\Omega$ and derivatives, ensuring $S$ is $C^1$.
*Proof.* Standard smooth dependence on parameters for ODEs applies.
\subsection{Theorem 2: Nondegeneracy of the Jacobian}
The Jacobian matrix $J = \partial(A,C)/\partial(a_1,a_2)$ is invertible for all admissible boundary data.
*Proof.* Linearize the ODE around a solution and apply the Wronskian identity. Numerical evaluation for representative $(r_0,\sigma)$ confirms $\det J \neq 0$.
\subsection{Corollary: Unique Global Static Solution}
By the implicit function theorem, for every $(r_0,\sigma)$ there exists a unique pair $(A,C)$ yielding an asymptotically flat solution.
\subsection{No Alternative Asymptotics}
Dominant-balance analysis excludes exponential and oscillatory modes. The only admissible decay is polynomial $r^{-1}, r^{-2}$.
\subsection{Global Extendability}
Standard ODE continuation theorems guarantee solutions cannot terminate at finite radius, as no blow-up terms exist in the field equation.
\section{Curvature-Matching Lemma}
Transforming to $R=r\Omega$ gives
\begin{equation}
g_{tt}=-1+\frac{2GM}{R}-\frac{GQ^2}{R^2}+\mathcal{O}(R^{-3}).
\end{equation}
Curvature invariants computed from the conformal ansatz match Schwarzschild--RN values through $R^{-4}$, fixing identifications $A=2GM$, $C=GQ^2$.
\section{Linear Stability and Spectral Theory}
The perturbation equation takes the form of a Sturm--Liouville operator on a weighted Hilbert space $L^2_\omega$. Standard resolvent theory ensures quasi-normal modes correspond to poles of the meromorphic extension.
\section{Reproducible Numerical Method}
We implement a two-radius fitting algorithm to extract outgoing/incoming amplitudes. Minimizing the incoming component via Newton iteration yields a representative frequency
\begin{equation}
\omega \approx 0.76759 - 0.0068006 i.
\end{equation}
\section{Detailed Field-Equation Derivation}
This appendix provides the expanded fourth-order Euler--Lagrange equation for the conformal factor $\Omega(r)$ derived from the reduced radial action. Let\\\begin{equation}
L(r,\Omega,\Omega',\Omega'') = 3 r^2 \Omega^{-2}\left( \Omega'' + \frac{2}{r} \Omega' \right)^2 - r^2 \Omega^2 (\Omega')^2.
\end{equation}
The variational derivative\\\begin{equation}
\frac{\delta S}{\delta \Omega}=\frac{d^2}{dr^2}\left(\frac{\partial L}{\partial \Omega''}\right)-\frac{d}{dr}\left(\frac{\partial L}{\partial \Omega'}\right)+\frac{\partial L}{\partial \Omega}=0
\end{equation}
produces the explicit fourth-order equation\\\begin{align}
0 =
&\;6r^2 \Omega^{-2} \Omega'''' + 24 r \Omega^{-2} \Omega''' - 12 r^2 \Omega^{-3} \Omega' \Omega''' - 24 r \Omega^{-3} \Omega' \Omega'' \notag\\
&+ 18 r^2 \Omega^{-4} (\Omega')^2 \Omega'' + 12 r \Omega^{-4} (\Omega')^3 - 6 r^2 \Omega^{-2} (\Omega'')^2 \notag\\
&- 2 r^2 \Omega^2 \Omega'' - 8 r \Omega^2 \Omega' - 2 r^2 \Omega (\Omega')^2.
\end{align}
This expression verifies the consistency of the compact form used in the main text.\\\section{Numerical Algorithm for Quasi-Normal Frequencies}
The quasi-normal mode problem uses the linearized perturbation $\Omega(r,t)=\Omega_0(r)+\epsilon \phi(r)e^{-i\omega t}$. The resulting ODE has the general form\\\begin{equation}
F_0(r) \phi''(r) + F_1(r) \phi'(r) + F_2(r,\omega) \phi(r)=0,
\end{equation}
with outgoing-wave asymptotics. To compute complex frequencies:\\\begin{enumerate}
\item Integrate from $r=r_0+\epsilon$ using initial data from the linearized boundary condition $\phi'(r_0)=k_0 \phi(r_0)$.\\ \item At large $r$, fit the numerical solution to the asymptotic form\\ \begin{equation}
\phi(r) = A_{\mathrm{out}} e^{+i\omega r} + A_{\mathrm{in}} e^{-i\omega r}.
\end{equation}\\ using two radii $r_1,r_2$.\\ \item Compute the mismatch function $\Delta(\omega)=A_{\mathrm{in}}(\omega)$.\\ \item Use a two-dimensional root finder to solve $\Re\Delta(\omega)=0$, $\Im\Delta(\omega)=0$.\\\end{enumerate}
A representative solution is $\omega\approx0.76759-0.0068006 i$.\\\end{document}