Compactly Supported Nonlinear Constraint Deformations and Linearized Curvature Growth in Ori’s Chronology Model

1. Introduction

The regular vacuum–dust time-machine construction of Ori (Sections III–VI [1]) offers a concrete initial-value setting for the study of chronology horizons. This 2007 construction builds on Ori’s earlier compact-vacuum-core time-machine model [2]. The existence and geometry of this background are established results, not part of the present claims. Dietz, Dirmeier and Scherfner later established the geodesic incompleteness of the pseudo-Schwarzschild spacetime and ruled out a geodesically complete extension (Section 2–3 [3]). These results do not establish the perturbative conclusion considered here, and we do not treat the geodesic incompleteness as a new result. Rieger subsequently placed the pseudo-Schwarzschild geometry in a Misner-type causal classification and established isocausality results under explicit covering and quotient conditions (Proposition 7.1 [4]). That work does not study axial Einstein perturbations of Ori’s matched vacuum–dust initial data. Linear growth near a chronology horizon is likewise insufficient to prove an obstruction to nonlinear chronology violation.

We ask a more focused question. Can a compactly supported axial perturbation be tangent to an exact finite-amplitude Einstein constraint family on Ori’s initial slice, while its linearized vacuum development exhibits growth of a gauge-invariant curvature observable in a region causally protected from the dust interface? The same initial family underlies both conclusions, one about the nonlinear constraints and the other about the linearized evolution.

Theorem 1. 

Let $\mu >0$, $r_h=2\mu$, $r_0=3\mu$ and ${\theta }_c=\log (2+\sqrt{3})$. On the angular factor ${\mathbb{H}}^2$, let ρ be the distance from a fixed origin o, and write $B_R\left(o\right)$ for the metric ball of radius R. Fix an Ori core with angular cutoff $\rho ={\theta }_0$, where ${\theta }_0>2{\theta }_c$, and numbers $0<a<b$ satisfying $b+2{\theta }_c<{\theta }_0$. Let $p\in C_c^{\infty }\left((a,b)\right)$ be a nonzero real function odd about $(a+b)/2$. For all sufficiently small amplitudes ε, the core family given in (18) and (19) satisfies both nonlinear vacuum Einstein initial constraints exactly and agrees with Ori’s original core data outside the annulus. With the dust envelope and exterior unchanged, these data extend to a global Einstein–dust initial-data family satisfying the full constraints.

Its nonzero axial first variation determines a linearized vacuum Einstein solution on the protected interior. For the Weyl Pontryagin first variation $\dot{\mathcal{P}}$ of this solution, there exist $c,\delta >0$ such that

$${∥\dot{\mathcal{P}}\left(r\right)∥}_{L^2(B_{{\theta }_0-{\theta }_c}\left(o\right),\mathrm{d}{V}_{{\mathbb{H}}^2})}\ge c\left|\log {\displaystyle \frac{r-r_h}{r_h}}\right|(0<r-r_h<\delta ).$$

(1)

The last assertion is about the first variation at $\epsilon =0$ and thebackgroundhorizon. It is not a statement of nonlinear curvature blow-up.

The proof combines the explicit constraint factorization in Section 4, the spectral transfer in Section 6, and the local curvature calculation and finite-propagation argument in Section 7. Exact initial twist and finite-amplitude curvature information are collected afterward. Section 9 proves the combined theorem.

The axial master equations and the associated metric reconstruction are due to Kodama and Ishibashi (Section 5.2 [5]). Negative-curvature topological black-hole perturbations have also been studied in a different, asymptotically anti-de Sitter stability setting [6]. That black-hole stability problem is distinct from the Cauchy-horizon initial-value problem in the periodic Ori vacuum core considered here. The use of Pontryagin density as an axial curvature diagnostic is familiar from the Chern–Simons and nonmodal perturbation literature [7,10,11,12]. The curvature interpretation of odd-parity gauge invariants on Schwarzschild backgrounds also predates the present calculation (Sections IV and VI [9]). Fernández Tío and Dotti also studied the divergence of proper-time derivatives of invariant curvature perturbations near the Reissner–Nordström Cauchy horizon (Section IV [12]). Their background and initial-value problem differ from the Ori construction studied here. Classical gluing techniques and their relation to Killing initial data are also prior art [13,14]. Carlotto and Schoen established localized Einstein-constraint data in an asymptotically flat conical setting [15]. Their setting is distinct from the present compactly supported annular deformation on Ori’s initial slice. Any independent contribution lies in the explicit nonlinear annular constraint family and the protected curvature estimate for its first variation, rather than in the established formalisms.

2. The Ori Initial Slice

On Ori’s vacuum core the inherited metric can be written as

$$g_{\mathrm{O}}=-f{\left(r\right)}^{-1}\mathrm{d}{r}^2+f\left(r\right)\mathrm{d}{\tau }^2+r^2{h}_{{\mathbb{H}}^2},f\left(r\right)=1-{\displaystyle \frac{2\mu }{r}},r>2\mu ,$$

(2)

where $\tau$ is a periodic coordinate, $\mu >0$, and $h_{{\mathbb{H}}^2}=\mathrm{d}{\rho }^2+{\sinh }^2\rho \mathrm{d}{\vartheta }^2$. Here $\rho$ is the radial coordinate denoted $\theta$ in [1], and $\rho ={\theta }_0$ is the core cutoff. The original core initial slice is $r=r_0=3\mu$ with $r_h=2\mu$ and $f_0=1/3$. The background induced metric and second fundamental form are ${\gamma }_0=f_0\mathrm{d}{\tau }^2+r_0^2{h}_{{\mathbb{H}}^2}$ and $K_0={\left(\sqrt{3}{r}_0\right)}^{-1}{\gamma }_0$. We use the sign convention $K=-\frac{1}{2}{\mathcal{L}}_n\gamma$. The global vacuum–dust matching and chronology geometry are due to Ori (Sections III–VI [1]).

For later reference, on a general r-slice the principal extrinsic curvatures in the convention used here are

$${K_{\tau }}^{\tau }={\displaystyle \frac{f^{\prime }}{2\sqrt{f}}},{K_A}^B={\displaystyle \frac{\sqrt{f}}{r}}{{\delta }_A}^B.$$

(3)

They coincide at $r_0=3\mu$, as used in the nonlinear constraint construction below.

3. An Anisotropic Axial Constraint Ansatz

Fix the inherited umbilical Ori slice described above. Thus

$${\gamma }_0=f_0\mathrm{d}{\tau }^2+\sigma ,f_0={\displaystyle \frac{1}{3}},\sigma =r_0^2{h}_{{\mathbb{H}}^2},r_0=3\mu ,k_0={\displaystyle \frac{1}{\sqrt{3}{r}_0}},$$

(4)

where $\tau$ is periodic. For a fixed nonzero $\varphi \in C_c^{\infty }\left({\mathbb{H}}^2\right)$ set

$$V={*}_{\sigma }\mathrm{d}\varphi ,T=\mathrm{d}\tau \otimes V+V\otimes \mathrm{d}\tau ,S={\left|T\right|}_{{\gamma }_0}^2=2f_0^{-1}{\left|V\right|}_{\sigma }^2.$$

(5)

The Hodge star on one-forms in dimension two is independent of a conformal rescaling. In particular $D^{\sigma A}{V}_A=0$. We write D and ${\Delta }_{\sigma }$ for the Levi-Civita derivative and scalar Laplacian of $\sigma$.

For a smooth real function w on ${\mathbb{H}}^2$, independent of $\tau$, and a real amplitude $\epsilon$, consider

$$\begin{array}{cc}\hfill \gamma \left[w\right]& =f_0{e}^{2w}\mathrm{d}{\tau }^2+e^{-2w}\sigma ,\hfill \\ \hfill K[w,\epsilon ]& =k_0\gamma \left[w\right]+\epsilon e^{-w}(\mathrm{d}\tau \otimes V+V\otimes \mathrm{d}\tau ).\hfill \end{array}$$

(6)

The circle and hyperbolic factors scale in opposite directions. The exponent in the trace-free part is chosen so that the exact momentum constraint holds without an approximation.

Proposition 3.1. 

For every smooth τ-independent w, the initial data (6) satisfy

$${tr}_{\gamma \left[w\right]}K[w,\epsilon ]=3k_0,{div}_{\gamma \left[w\right]}\left(K[w,\epsilon ]-3k_0\gamma \left[w\right]\right)=0.$$

(7)

Their vacuum Hamiltonian residual has the exact pointwise form

$$\mathcal{H}[w,\epsilon ]=c_0(1-e^{2w})-2e^{2w}{\left|Dw\right|}_{\sigma }^2-{\epsilon }^2{e}^{-2w}S,c_0={\displaystyle \frac{2}{r_0^2}}=6k_0^2.$$

(8)

Neither an elliptic inverse nor a truncation in ε is needed for these identities.

Proof. 

The determinant and inverse blocks of $\gamma \left[w\right]$ are

$$\sqrt{\det \gamma \left[w\right]}=\sqrt{f_0}{e}^{-w}\sqrt{\det \sigma },{\gamma }^{\tau \tau }=f_0^{-1}{e}^{-2w},{\gamma }^{AB}=e^{2w}{\sigma }^{AB}.$$

(9)

Put $Q=\epsilon e^{-w}T$. It is off-diagonal, so its trace vanishes identically. The momentum constraint reduces to ${div}_{\gamma \left[w\right]}Q=0$. For a symmetric tensor,

$${\nabla }_j{Q^j}_i={\displaystyle \frac{1}{\sqrt{\det \gamma }}}{\partial }_j\left(\sqrt{\det \gamma }{Q^j}_i\right)-{\displaystyle \frac{1}{2}}{Q}^{jk}{\partial }_i{\gamma }_{jk}.$$

(10)

The last term is zero because Q is off-diagonal and $\gamma \left[w\right]$ is block diagonal and circle-independent. For $i=\tau$, the remaining expression is the divergence of $\sqrt{\det \gamma }{Q^A}_{\tau }=\epsilon \sqrt{f_0}\sqrt{\det \sigma }{V}^A$, which vanishes since $D^{\sigma A}{V}_A=0$. For $i=A$, only the $j=\tau$ term survives and its circle derivative vanishes. This proves the momentum identity without assuming a special radial form of $\varphi$ or w.

Regard $\gamma \left[w\right]$ as a one-dimensional warped product over $({\mathbb{H}}^2,e^{-2w}\sigma )$ with fiber radius $\sqrt{f_0}{e}^w$. The two-dimensional conformal curvature formula and the warped-product formula give

$$R\left(\gamma \left[w\right]\right)=e^{2w}\left(R\left(\sigma \right)-{2\left|Dw\right|}_{\sigma }^2\right),R\left(\sigma \right)=-{\displaystyle \frac{2}{r_0^2}}.$$

(11)

The ${\Delta }_{\sigma }w$ terms cancel exactly. This cancellation is not a linearization artifact. The squared norm of the second fundamental form is

$${\left|K[w,\epsilon ]\right|}_{\gamma \left[w\right]}^2=3k_0^2+{\epsilon }^2{e}^{-2w}S.$$

(12)

Substitution into $R\left(\gamma \right)+{(trK)}^2-{\left|K\right|}^2$ proves (8). □

4. An Exact Compactly Supported Nonlinear Constraint Family in the Axial Sector

The exact Hamiltonian identity above admits a quadratic reduction. We use it to construct nonzero axial first-order data as tangents to smooth, compactly supported families satisfying the full nonlinear constraints exactly. The construction changes the axial seed at order ${\epsilon }^2$. It therefore does not solve the reduced equation with a freely prescribed, amplitude-independent $\varphi \in C_c^{\infty }\left({\mathbb{H}}^2\right)$.

4.1. A Quadratic Reduction of the Hamiltonian Constraint

Put $Y=e^{2w}>0$ in the exact Hamiltonian identity (8). Multiplication by Y gives

$$Y\mathcal{H}=c_{0}Y(1-Y)-{\displaystyle \frac{1}{2}}{\left|DY\right|}_{\sigma }^2-{\epsilon }^{2}S,c_0={\displaystyle \frac{2}{r_0^2}}.$$

(13)

This is an identity, rather than an approximation. Any exact solution with $S\ge 0$ and $Y>0$ obeys

$$0<Y\le 1,{\epsilon }^{2}S\le c_{0}Y(1-Y)\le {\displaystyle \frac{c_0}{4}}.$$

(14)

For a fixed nonnegative source, every exact solution must therefore satisfy the necessary amplitude bound ${\epsilon }^2{\parallel S\parallel }_{\infty }\le c_0/4$. This is neither an obstruction at arbitrarily small amplitude nor an existence theorem. The squared gradient in (13) also prevents a direct application of an elliptic inverse function theorem.

We next exploit the freedom to select a smooth family of axial potentials with a prescribed nonzero tangent at $\epsilon =0$. This choice gives exact constraint data with a prescribed linearization. It differs from holding the full finite-amplitude axial source fixed.

4.2. A Smooth Annular Seed with an Exact Nonlinear Completion

Write the hyperbolic-plane metric in polar coordinates as $h_{{\mathbb{H}}^2}=\mathrm{d}{\rho }^2+{\sinh }^2\rho \mathrm{d}{\vartheta }^2$. The inherited Ori initial data are

$${\gamma }_0=f_0\mathrm{d}{\tau }^2+r_0^2(\mathrm{d}{\rho }^2+{\sinh }^2\rho \mathrm{d}{\vartheta }^2),K_0=k_0{\gamma }_0,$$

(15)

where $r_0=3\mu$, $f_0=1/3$ and $k_0=1/\left(\sqrt{3}{r}_0\right)$. These data and their matching to Ori’s exterior are inherited from (Section III.B and Sections IV–VI [1]).

Fix $0<a<b<{\theta }_0$ and put ${\rho }_{*}=(a+b)/2$. Choose a real, nonzero function $p\in C_c^{\infty }\left((a,b)\right)$ with reflection parity

$$p(2{\rho }_{*}-\rho )=-p\left(\rho \right),\rho \in \mathbb{R}.$$

(16)

For example, any nonzero smooth odd bump about ${\rho }_{*}$ supported strictly inside $(a,b)$ has the required parity. Define

$$Q\left(\rho \right)=p{\left(\rho \right)}^2+p^{\prime }{\left(\rho \right)}^2,0<\left|\epsilon \right|<{\epsilon }_{*},{\epsilon }_{*}^2{\parallel Q\parallel }_{\infty }<1,$$

(17)

where ${\epsilon }_{*}>0$ is any fixed sufficiently small number. The construction also extends smoothly to $\epsilon =0$. Set

$$\begin{array}{cccc}\hfill Y_{\epsilon }\left(\rho \right)& =1-{\epsilon }^{2}p{\left(\rho \right)}^2,\hfill & \hfill w_{\epsilon }\left(\rho \right)& ={\displaystyle \frac{1}{2}}\log Y_{\epsilon }\left(\rho \right),\hfill \\ \hfill {\varphi }_{\epsilon }\left(\rho \right)& =\sqrt{f_0}{\int }_0^{\rho }p\left(s\right)\sqrt{1-{\epsilon }^{2}Q\left(s\right)}\mathrm{d}s,\hfill & \hfill V_{\epsilon }& ={*}_{\sigma }\mathrm{d}{\varphi }_{\epsilon }.\hfill \end{array}$$

(18)

Here $\sigma =r_0^2{h}_{{\mathbb{H}}^2}$. Let $T_{\epsilon }=\mathrm{d}\tau \otimes V_{\epsilon }+V_{\epsilon }\otimes \mathrm{d}\tau$. We use the anisotropic ansatz with this smoothly varying potential,

$$\begin{array}{cc}\hfill {\gamma }_{\epsilon }& =f_0{Y}_{\epsilon }\mathrm{d}{\tau }^2+Y_{\epsilon }^{-1}\sigma ,\hfill \\ \hfill K_{\epsilon }& =k_0{\gamma }_{\epsilon }+\epsilon Y_{\epsilon }^{-1/2}{T}_{\epsilon }.\hfill \end{array}$$

(19)

Proposition 4.1. 

The family (19) is smooth in space and in ε for $\left|\epsilon \right|<{\epsilon }_{*}$. For every such ε it satisfies bothexactvacuum Einstein initial-value constraints,

$$R\left({\gamma }_{\epsilon }\right)+{\left({tr}_{{\gamma }_{\epsilon }}{K}_{\epsilon }\right)}^2-{\left|K_{\epsilon }\right|}_{{\gamma }_{\epsilon }}^2=0,{div}_{{\gamma }_{\epsilon }}(K_{\epsilon }-{tr}_{{\gamma }_{\epsilon }}{K}_{\epsilon }{\gamma }_{\epsilon })=0.$$

(20)

The data agree identically with $({\gamma }_0,K_0)$ outside $S^1\times \{a<\rho <b\}$. Their first variation is

$${\left.{\partial }_{\epsilon }{\gamma }_{\epsilon }\right|}_0=0,{\left.{\partial }_{\epsilon }{K}_{\epsilon }\right|}_0=\mathrm{d}\tau \otimes \left({*}_{\sigma }\mathrm{d}{\varphi }_0\right)+\left({*}_{\sigma }\mathrm{d}{\varphi }_0\right)\otimes \mathrm{d}\tau \ne 0,$$

(21)

where

$${\varphi }_0\left(\rho \right)=\sqrt{f_0}{\int }_0^{\rho }p\left(s\right)\mathrm{d}s\in C_c^{\infty }\left({\mathbb{H}}^2\right)\setminus \left\{0\right\}.$$

(22)

Thus this particular nonzero, compactly supported axial linearized Einstein-constraint datum is tangent to an exact, nonlinear and compactly supported vacuum-constraint family.

Proof. 

Since p is odd about ${\rho }_{*}$ while $p^2$ and $p^{\prime 2}$ are even, the integrand $p\left(s\right)\sqrt{1-{\epsilon }^{2}Q\left(s\right)}$ is odd about ${\rho }_{*}$. Its integral over $(a,b)$ therefore vanishes. Consequently ${\varphi }_{\epsilon }=0$ on both $\rho \le a$ and $\rho \ge b$. The integrand is smooth and vanishes to all orders at the ends of its compact support. The potential extends smoothly over the hyperbolic origin, where it is identically zero. It follows that ${\varphi }_{\epsilon }$, $V_{\epsilon }$, $w_{\epsilon }$ and all departures of (19) from the background are smooth and supported in the stated annulus. In particular $Y_{\epsilon }>0$ by (17).

The momentum constraint and constant mean curvature are exact for every such $w_{\epsilon }$ and every divergence-free $V_{\epsilon }={*}_{\sigma }\mathrm{d}{\varphi }_{\epsilon }$ by Proposition 3.1. It remains to check the Hamiltonian constraint without any asymptotic expansion. For a radial function, the $\sigma$-gradient has norm ${\left|DY\right|}_{\sigma }^2=r_0^{-2}{\left(Y^{\prime }\right)}^2$. From (18),

$$\begin{array}{cc}\hfill Y_{\epsilon }^{\prime }& =-2{\epsilon }^{2}p{p}^{\prime },\hfill \\ \hfill S_{\epsilon }& =|T_{\epsilon }{|}_{{\gamma }_0}^2={\displaystyle \frac{2}{f_0{r}_0^2}}{\left({\varphi }_{\epsilon }^{\prime }\right)}^2=c_0{p}^2\left(1-{\epsilon }^2(p^2+p^{\prime 2})\right).\hfill \end{array}$$

(23)

Substituting these expressions into (13) gives the exact pointwise identity

$$\begin{array}{cc}\hfill Y_{\epsilon }\mathcal{H}& =c_0{\epsilon }^2{p}^2(1-{\epsilon }^2{p}^2)-{\displaystyle \frac{2{\epsilon }^4}{r_0^2}}{p}^2{p}^{\prime 2}-c_0{\epsilon }^2{p}^2(1-{\epsilon }^2{p}^2-{\epsilon }^2{p}^{\prime 2})\hfill \\ \hfill & ={\epsilon }^4{p}^2{p}^{\prime 2}\left(c_0-{\displaystyle \frac{2}{r_0^2}}\right)=0.\hfill \end{array}$$

(24)

This holds even at the zeros of p, including the annular midpoint and the two boundary collars. No division by p is involved. Since $Y_{\epsilon }>0$, the original Hamiltonian constraint vanishes exactly.

Finally ${\varphi }_{\epsilon }={\varphi }_0+O_{C^m}\left({\epsilon }^2\right)$, $Y_{\epsilon }=1+O_{C^m}\left({\epsilon }^2\right)$ and $Y_{\epsilon }^{-1/2}=1+O_{C^m}\left({\epsilon }^2\right)$ for every fixed m. Differentiating (19) at zero therefore gives (21). The tangent is nonzero since $p\ne 0$ and ${\varphi }_0^{\prime }=\sqrt{f_0}p$. □

4.3. Exact Assembly with the Inherited Ori Dust Initial Data

We match the initial data across unchanged vacuum collars. This does not solve the perturbed spacetime junction problem or prescribe an evolution of the dust interface.

Proposition 4.2. 

Let $({\gamma }^{\mathrm{O}},K^{\mathrm{O}},{\mathfrak{m}}^{\mathrm{O}})$ denote the smooth global Einstein–dust initial data of Ori (Sections IV–VI [1]). Assume that its chosen core contains the annulus $a<\rho <b<{\theta }_0$. Replace the original vacuum data on that annulus by (19) and leave ${\mathfrak{m}}^{\mathrm{O}}$, the remainder of the vacuum core, the entire dust envelope and the asymptotically flat exterior unchanged. Then the resulting global initial data satisfy the full Einstein–dust constraint equations exactly. They have the original global topology, asymptotics and matter data.

Proof. 

The new data agree with the originals on open collars at either end of the annulus. They therefore define smooth, globally matched tensors without interpolation, a surface layer, or a junction correction. In the annulus the matter density and momentum vanish and the exact vacuum constraints hold by Proposition 4.1. Elsewhere, the unchanged Ori data already satisfy the Einstein–dust constraints. The constraint equations are local in the initial data and their derivatives, so the globally matched data satisfy them everywhere. The dust fields and the asymptotically flat end remain unchanged. This is a statement only about the initial hypersurface. □

The inherited local Killing initial datum associated with ${\partial }_{\tau }$ is still present in the untouched collar. Its existence is not contradicted by our construction. In particular, the no-local-KID hypothesis of the general local gluing theorem of Chruściel, Isenberg and Pollack (Theorem 1.1 [13]) does not hold in this collar. Our construction is an explicit solution within a special family, not an application of that theorem. Both the axial potential ${\varphi }_{\epsilon }$ and its quadratic source $S_{\epsilon }$ vary with $\epsilon$, as shown in (18) and (25). We have not solved the fixed-source equation for an arbitrarily prescribed $\varphi \in C_c^{\infty }\left({\mathbb{H}}^2\right)$.

4.4. Scope of the Explicit Constraint Family

The reflection condition (16) ensures that the primitive defining ${\varphi }_{\epsilon }$ vanishes on both sides of the annulus. Without a cancellation condition, that primitive need not return to zero outside the annulus. The present construction does not establish a completion for an arbitrary compactly supported scalar seed. Although ${\varphi }_{\epsilon }={\varphi }_0+O\left({\epsilon }^2\right)$, the potential generally depends on the amplitude. The associated quadratic source satisfies

$$S_{\epsilon }=c_0{p}^2-c_0{\epsilon }^2{p}^2(p^2+p^{\prime 2}),$$

(25)

so this finite-amplitude family must not be confused with a formal expansion for a fixed prescribed source. The family’s first variation is nevertheless the nonzero axial datum stated above.

Exact initial matching and persistence of the original vacuum–dust interface under evolution are distinct issues. The interface is unperturbed at the initial slice, but later causal propagation may transmit changes into it. Nor does the existence of KIDs rule out our explicit solution. It only precludes the unqualified use of a gluing theorem that assumes their absence.

The construction uses a restricted ansatz for which both vacuum constraint equations are solved exactly. This establishes existence within that ansatz, not a gluing theorem for arbitrarily prescribed compactly supported axial data. The role of Killing initial data in integrability of linearized constraints is classical. The theorem of Fischer, Marsden and Moncrief (Theorem 3.1 [17]), also discussed in (Section 4, Theorem 4.7 [16]), gives integrability of linearized vacuum deformations on compact CMC slices with one Killing field, subject to the quadratic Taub condition. Noncompactness of Ori’s angular factor alone is not decisive. Fix a closed hyperbolic surface ${\mathbb{H}}^2/\Gamma$ and retain the origin $o\in {\mathbb{H}}^2$ of the radial coordinate used above. For each $R>0$, only finitely many nonidentity elements $\gamma \in \Gamma$ satisfy $d(o,\gamma o)\le 4R$. By residual finiteness of the surface group (p. 323, Theorem [18]), there is a finite-index normal subgroup ${\Gamma }_R\le \Gamma$ containing none of them. Thus $B_R\left(o\right)$ embeds isometrically in the closed hyperbolic surface ${\Sigma }_R={\mathbb{H}}^2/{\Gamma }_R$. The local Ori vacuum metric admits a corresponding compact angular quotient containing this embedded ball. This auxiliary quotient is used only for the integrability comparison and does not replace Ori’s matched vacuum–dust spacetime.

For the compact-slice comparison, choose R large enough that $supp{\varphi }_0\subset B_R\left(o\right)$. The axial tangent $(\dot{\gamma },\dot{K})=(0,T_0)$ of (21) then extends by zero from the embedded ball to $S^1\times {\Sigma }_R$. The tensor remains trace free and divergence free. The quotient spacetime has precisely one Killing field up to scale. Its Kretschmann scalar $R_{abcd}{R}^{abcd}=48{\mu }^2/r^6$ forces Killing fields to preserve r. On a constant-r slice, the circle and hyperbolic factors have distinct Ricci eigenvalues 0 and $-1/r^2$. The closed hyperbolic factor has no nonzero Killing field, and the mixed Killing equations leave only ${\partial }_{\tau }$ up to scale. This field is tangent to the initial slice, so its Killing lapse vanishes. Along $({\gamma }_0,K_0+t{T}_0)$ the momentum constraint residual is zero and the Hamiltonian residual is $-t^2{\left|T_0\right|}_{{\gamma }_0}^2$. Thus the quadratic Taub charge vanishes. The compact-quotient datum is therefore integrable by the theorem of Fischer, Marsden and Moncrief (Theorem 3.1 [17]). This does not itself integrate the prescribed compactly supported datum on Ori’s full vacuum–dust initial slice. Lifting the quotient datum to ${\mathbb{H}}^2$ repeats it on all deck translates and loses global compact support.

Moncrief also studied all-order symmetry-preserving formal perturbations of generalized Taub–NUT backgrounds [30]. These earlier results do not, on their own, establish a vacuum-constraint family agreeing with Ori’s data outside the prescribed annulus, its matching to the unchanged dust envelope, or the protected Weyl–Pontryagin lower bound. We do not use these results in our explicit construction, nor do we claim a solution to general KID obstructions. The construction itself neither predicts an evolved chronology horizon nor rules one out.

5. Axial Linearized Vacuum Equations on the Ori Core

For the circle-independent linearization of the exact family, we use the established vector-type reduction of the vacuum Einstein equations due to Kodama and Ishibashi (Section 5.2 [5]). Version 3 of their arXiv preprint presents the invariant reconstruction in Equation (5.12), the master variable in Equation (5.13), and the effective potential for vector-type perturbations in Equation (5.15) (Section 5.2, Eqs. (5.10)–(5.15) [5]). The specialization to the negative-curvature orbit factor uses $n=2$, $K=-1$, $M=-\mu$ and $\mathsf{\Lambda }=0$ in that reference. In the Kodama–Ishibashi convention, the static metric coefficient is $f_{\mathrm{KI}}=K-2M/r=-f$, where $f=1-2\mu /r$ in (2). Thus the orbit metric in Ori’s chronological region is the same Lorentzian metric, with r as the time coordinate. This sign correspondence must be made before specializing the master equation. Set $\mathcal{A}=-{\Delta }_{{\mathbb{H}}^2}$. A scalar generalized harmonic $Y_{\lambda }$ has $\mathcal{A}{Y}_{\lambda }=\lambda Y_{\lambda }$, where $\lambda \ge 1/4$, and the coexact one-form $\mathbb{V}={*}_{{\mathbb{H}}^2}\mathrm{d}{Y}_{\lambda }$ satisfies $k_V^2=\lambda +1$. The hyperbolic Weitzenböck commutator and $Ric\left(h_{{\mathbb{H}}^2}\right)=-h_{{\mathbb{H}}^2}$ give

$${\widehat{D}}^B{\widehat{D}}_B{\left({*}_{h_{{\mathbb{H}}^2}}\mathrm{d}{Y}_{\lambda }\right)}_A=-(\lambda +1){\left({*}_{h_{{\mathbb{H}}^2}}\mathrm{d}{Y}_{\lambda }\right)}_A.$$

(26)

All spectral equalities below are in the $L^2$ functional calculus. Individual generalized harmonics are not assumed square integrable.

The gauge-invariant axial master variable $\mathsf{\Phi }=\mathsf{\Omega }/r$ satisfies the specialized Kodama–Ishibashi equation

$${\partial }_r\left(f{\partial }_r\mathsf{\Phi }\right)-f^{-1}{\partial }_{\tau }^2\mathsf{\Phi }-r^{-2}{\Delta }_{{\mathbb{H}}^2}\mathsf{\Phi }+{\displaystyle \frac{6\mu }{r^3}}\mathsf{\Phi }=0.$$

(27)

In the horizon-regular background chart $v=-(\tau +r_{*})$, where $r_{*}^{\prime }=f^{-1}$, the equation becomes

$${\partial }_r\left(f{\partial }_r\mathsf{\Phi }\right)-2{\partial }_v{\partial }_r\mathsf{\Phi }-r^{-2}{\Delta }_{{\mathbb{H}}^2}\mathsf{\Phi }+{\displaystyle \frac{6\mu }{r^3}}\mathsf{\Phi }=0.$$

(28)

For the vector metric perturbation $h_{ai}=r{F}_a{\mathbb{V}}_i$, the invariant orbit one-form is obtained from

$$r{F}^a={ϵ}^{ab}{\partial }_b\mathsf{\Omega },\mathsf{\Omega }=r\mathsf{\Phi }.$$

(29)

With the sign convention ${ϵ}^{vr}=1$ in the EF orbit chart $g_{\left(2\right)}=f\mathrm{d}{v}^2+2\mathrm{d}v\mathrm{d}r$, this gives

$$r{F}_v=f{\partial }_r\mathsf{\Omega }-{\partial }_v\mathsf{\Omega },r{F}_r={\partial }_r\mathsf{\Omega }.$$

(30)

The reconstruction gives a linearized vacuum Einstein field, not merely an unconstrained metric perturbation. The orbital amplitudes and their coexact angular reconstruction are invariant under the admissible smooth vector-type infinitesimal gauges. The nonzero $L^2$ harmonic one-form sector of ${\mathbb{H}}^2$ is classical (p. 396, Theorem (iii) [19]). For example, the one-form $Re\left(\mathrm{d}\zeta \right)$ in a Poincaré disk coordinate $\zeta$ is $L^2$ harmonic. Its squared $L^2$ norm is $\pi$ by conformal invariance in two dimensions. It lies outside the scalar-derived positive-spectral sector used here and is not considered.

6. Cauchy Transfer for Compactly Supported Axial Data

For the axial tangent of the exact family, the physical initial data are compactly supported while the auxiliary master potential need not be. We compute the physical-to-master Cauchy map and the regularity obstruction at the background horizon.

6.1. The Spectral Cauchy Map for Compactly Supported Axial Data

We use the vector reconstruction stated in Section 5. In particular, $r_h=2\mu$, $r_0=3\mu$, $f=1-r_h/r$, and ${\gamma }_0=f_0\mathrm{d}{\tau }^2+\sigma$ with $f_0=1/3$. Fix $0\ne \varphi \in C_c^{\infty }\left({\mathbb{H}}^2\right)$. In the combined theorem, we take $\varphi ={\varphi }_0$ from (22). Consider the transverse-traceless axial seed

$$\dot{\gamma }=0,{\dot{K}}_{\tau A}={\left({*}_{\sigma }\mathrm{d}\varphi \right)}_A,{\dot{K}}_{\tau \tau }={\dot{K}}_{AB}=0.$$

(31)

The tensor is trace free. Its divergence vanishes since $D^{\sigma A}{\left({*}_{\sigma }\mathrm{d}\varphi \right)}_A=0$ and it is independent of $\tau$. The initial slice is umbilical, so $(\dot{\gamma },\dot{K})$ satisfies the linearized vacuum constraints. The map from this physical datum to the Kodama–Ishibashi master scalar need not be local. We explicitly compute it rather than presuming a compact master field.

In the Regge–Wheeler vector gauge and the original diagonal $(r,\tau )$ chart, a $\tau$-independent mode has

$$h_{\tau A}=-f{\partial }_r{\mathsf{\Omega }}_{\lambda }{\mathbb{V}}_A,h_{rA}=0,\mathbb{V}={*}_{\sigma }\mathrm{d}{Y}_{\lambda },{\mathsf{\Omega }}_{\lambda }=r{u}_{\lambda }.$$

(32)

These follow from the vector reconstruction (29) and the coordinate change from $\tau$ to v. All the formulas in this subsection are restricted to the scalar-derived, nonexceptional vector sector with $\lambda \ge 1/4$. The static master equation for each $-{\Delta }_{{\mathbb{H}}^2}$ spectral fiber is

$${\left(f{u}_{\lambda }^{\prime }\right)}^{\prime }+\left({\displaystyle \frac{\lambda }{r^2}}+{\displaystyle \frac{6\mu }{r^3}}\right)u_{\lambda }=0,\lambda \in [1/4,\infty ).$$

(33)

Upon substituting $u_{\lambda }={\mathsf{\Omega }}_{\lambda }/r$, this becomes the exact identity

$$f{\mathsf{\Omega }}_{\lambda }^{\prime \prime }+\left(f^{\prime }-{\displaystyle \frac{2f}{r}}\right){\mathsf{\Omega }}_{\lambda }^{\prime }+{\displaystyle \frac{\lambda +2}{r^2}}{\mathsf{\Omega }}_{\lambda }=0.$$

(34)

On the initial slice the Regge–Wheeler gauge has vanishing shift perturbation and lapse perturbation. The chosen future normal is $n=-\sqrt{f}{\partial }_r$, so $K=-\frac{1}{2}{\mathcal{L}}_n\gamma =(\sqrt{f}/2){\partial }_r\gamma$ when the shift vanishes. Consequently, in the extrinsic-curvature convention of (3),

$${\dot{K}}_{\tau A}\left(r_0\right)={\displaystyle \frac{\sqrt{f_0}}{2}}{\left.{\partial }_r{h}_{\tau A}\right|}_{r_0}={\displaystyle \frac{\sqrt{f_0}(\lambda +2)}{2r_0^2}}{\mathsf{\Omega }}_{\lambda }\left(r_0\right){\mathbb{V}}_A,$$

(35)

where ${\dot{\gamma }}_{\tau A}=0$ has imposed ${\mathsf{\Omega }}_{\lambda }^{\prime }\left(r_0\right)=0$. Here and in the next display, angular spectral coefficients are understood in the $L^2$ functional calculus, not as normalizable individual plane waves. The exact Cauchy map is therefore

$$\mathsf{\Omega }(r_0,·)={\displaystyle \frac{2r_0^2}{\sqrt{f_0}}}{(-{\Delta }_{{\mathbb{H}}^2}+2)}^{-1}\varphi ,{\partial }_r\mathsf{\Omega }(r_0,·)=0.$$

(36)

The inverse in (36) is a bounded elliptic inverse on $L^2\left({\mathbb{H}}^2\right)$, but it is nonlocal and does not generally preserve compact support. For example, if $\varphi \ge 0$ and $\varphi \neg \equiv 0$, positivity of the resolvent kernel implies that $\mathsf{\Omega }(r_0,y)>0$ at every $y\in {\mathbb{H}}^2$. Thus the KI master datum associated with a compactly supported axial constraint perturbation need not be compactly supported. This nonlocality does not, by itself, imply that the physical perturbation reconstructed in a regular gauge has noncompact angular support.

6.2. A Logarithmic Horizon Coefficient for Circle-Independent Axial Data

The static fiber equation (33) has a repeated horizon indicial root. Denote by $F_{\lambda }\left(r\right)$ the solution analytic at $r=r_h$, normalized by

$$F_{\lambda }\left(r_h\right)=1,F_{\lambda }^{\prime }\left(r_h\right)=-{\displaystyle \frac{\lambda +3}{r_h}}.$$

(37)

The radial weighted Wronskian of $F_{\lambda }$ and any other solution $u_{\lambda }$ is constant. The regular Frobenius coefficients depend analytically on the spectral parameter, including at the auxiliary value $\lambda =-6$ used below. The regular series is obtained by a recursion whose jth denominator for $j\ge 1$ is a nonzero multiple of $j^2$, whereas its numerator is polynomial in $\lambda$. The regular series converges uniformly for $\lambda$ in compact sets and $|r-r_h|\le z_I{r}_h$ for any fixed $0<z_I<1$. For the particular data (36),

$$\begin{array}{cc}\hfill W_{\lambda }& =f(F_{\lambda }{u}_{\lambda }^{\prime }-F_{\lambda }^{\prime }{u}_{\lambda })\hfill \\ \hfill & =-f_0\left(F_{\lambda }^{\prime }\left(r_0\right)+{\displaystyle \frac{F_{\lambda }\left(r_0\right)}{r_0}}\right)u_{\lambda }\left(r_0\right),\hfill \\ \hfill u_{\lambda }\left(r_0\right)& ={\displaystyle \frac{2r_0}{\sqrt{f_0}(\lambda +2)}}\widehat{\varphi }(\lambda ,\omega ).\hfill \end{array}$$

(38)

Here $\omega$ is the multiplicity parameter in the hyperbolic spectral transform. It does not denote a circle frequency. Writing $s=r-r_h$, the coefficient of $\log (s/r_h)$ is

$$C_{\lambda }\left(\omega \right)=r_h{W}_{\lambda }=-r_h{f}_0\left(F_{\lambda }^{\prime }\left(r_0\right)+{\displaystyle \frac{F_{\lambda }\left(r_0\right)}{r_0}}\right){\displaystyle \frac{2r_0\widehat{\varphi }(\lambda ,\omega )}{\sqrt{f_0}(\lambda +2)}}.$$

(39)

The passage from the fiberwise logarithmic coefficient to an $L^2$ spectral lower bound requires estimates uniform in the spectral parameter. We establish these estimates explicitly rather than taking a pointwise Frobenius limit through a Plancherel integral.

Lemma 1. 

Fix a compact interval $I⋐(1/4,\infty )$ and let $u_{\lambda }$ be specified by the Cauchy pair in (36). Set $z=(r-r_h)/r_h$. There are bandwise $L^2$ amplitudes $A_{\lambda }\left(\omega \right)$ and $C_{\lambda }\left(\omega \right)$, linear in $\widehat{\varphi }(\lambda ,\omega )$, with $C_{\lambda }$ given by (39), such that for some $z_I>0$ and $K_I<\infty$,

$$\begin{array}{cc}\hfill u_{\lambda }(r,\omega )& =C_{\lambda }\left(\omega \right)\log z+A_{\lambda }\left(\omega \right)+R_{\lambda }(z,\omega ),\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill |R_{\lambda }|+|z{\partial }_z{R}_{\lambda }|& \le K_{I}z(1+|\log z\left|\right)|\widehat{\varphi }(\lambda ,\omega )|(0<z<z_I),\hfill \end{array}$$

(41)

for almost every $(\lambda ,\omega )$ in the spectral band over I. In particular, as $z\downarrow 0$, $u_{\lambda }/\log z$ and $z{\partial }_z{u}_{\lambda }$ both converge strongly to $C_{\lambda }$ in the band-limited spectral $L^2$ space.

Proof. 

With $x=r/r_h=1+z$, the static equation (33) is exactly

$$x^2(x-1)u_{xx}+x{u}_x+(\lambda x+3)u=0.$$

(42)

For the horizon-regular solution write $F_{\lambda }={\sum }_{j\ge 0}{a}_j\left(\lambda \right)z^j$ with $a_0=1$ and $a_{-1}=0$. Comparing coefficients gives the explicit recurrence

$${(j+1)}^2{a}_{j+1}+(2j^2-j+\lambda +3)a_j+\left((j-1)(j-2)+\lambda \right)a_{j-1}=0,j\ge 0.$$

(43)

In particular $a_1=-(\lambda +3)$, agreeing with (37) after converting back from z to r. The nonzero factors ${(j+1)}^2$ show that every $a_j$ is a polynomial in $\lambda$. Standard regular-singular series theory, or a majorant applied to this recurrence, gives convergence for $\left|z\right|<1$ and uniform analytic bounds for $\lambda$ in compact sets and $\left|z\right|\le z_I<1$. Thus $F_{\lambda }\left(z\right)=1+O_I\left(z\right)$ and we may shrink $z_I$ so that $F_{\lambda }$ has no zero in this interval for every $\lambda \in I$.

Let $W_{\lambda }=f(F_{\lambda }{u}_{\lambda }^{\prime }-F_{\lambda }^{\prime }{u}_{\lambda })$. Reduction of order and $f=z/(1+z)$ imply

$${\partial }_z\left({\displaystyle \frac{u_{\lambda }}{F_{\lambda }}}\right)={\displaystyle \frac{r_h{W}_{\lambda }(1+z)}{z{F}_{\lambda }^2}}={\displaystyle \frac{C_{\lambda }}{z}}+C_{\lambda }{q}_{\lambda }\left(z\right),$$

(44)

where $q_{\lambda }$ is analytic near $z=0$ and bounded there uniformly on I. Integrating gives $u_{\lambda }=F_{\lambda }(C_{\lambda }\log z+B_{\lambda }\left(z\right))$ with $B_{\lambda }$ regular at $z=0$. The Cauchy pair at $r_0$ and smooth ODE propagation down to any fixed positive $z_I$ show that $B_{\lambda }$ and its first derivative are bounded by $K_I\left|\widehat{\varphi }(\lambda ,\omega )\right|$ uniformly on the band. Taking $A_{\lambda }=B_{\lambda }\left(0\right)$ and using $F_{\lambda }=1+O_I\left(z\right)$ proves both bounds in (). Plancherel on the compact band then proves the asserted strong limits. □

The horizon coefficient need not be nonzero at every spectral parameter. It is nonzero on a set of positive hyperbolic spectral measure for every $0\ne \varphi \in C_c^{\infty }\left({\mathbb{H}}^2\right)$.

Proposition 6.1. 

For every $0\ne \varphi \in C_c^{\infty }\left({\mathbb{H}}^2\right)$, take the physical axial constraint datum (31) on the $r=r_0$ slice of the auxiliary untruncated Ori vacuum background with periodic τ and angular factor ${\mathbb{H}}^2$. The corresponding circle-invariant master Cauchy pair is given by (36). For the resulting master potential Ω, define the reconstructed coexact angular one-form

$${\mathfrak{F}}_r(r,·):={*}_{\sigma }{\mathrm{d}}_{{\mathbb{H}}^2}\left({\partial }_r\mathsf{\Omega }(r,·)\right).$$

There exist a compact interval $I\subset (1/4,\infty )$ and $c_I>0$ such that the gauge-invariant vector reconstruction in the smooth EF chart satisfies

$${∥{\mathsf{\Pi }}_I{\mathfrak{F}}_r(r,·)∥}_{L^2\left(T^{*}{\mathbb{H}}^2\right)}\ge {\displaystyle \frac{c_I}{r-r_h}}(0<r-r_h\ll 1).$$

(45)

Here ${\mathsf{\Pi }}_I$ is the spectral-band projector for $-{\Delta }_{{\mathbb{H}}^2}$ on scalars and the induced projector on scalar-derived coexact one-forms through ${*}_{\sigma }{\mathrm{d}}_{{\mathbb{H}}^2}$. The assertion is a fixed-backgroundlinearized vacuumresult in the scalar-derived axial sector. It is not a claim about the globally matched spacetime or a theorem on the linearized Weyl curvature.

Proof. 

The regular Frobenius solution $F_{\lambda }$, evaluated at a fixed $r>r_h$, is real-analytic in $\lambda$. Therefore

$$b\left(\lambda \right)=F_{\lambda }^{\prime }\left(r_0\right)+{\displaystyle \frac{F_{\lambda }\left(r_0\right)}{r_0}}$$

(46)

is real-analytic. It is not identically zero. At the nonphysical value $\lambda =-6$ one has $F_{-6}\left(r\right)={(r/r_h)}^3$, whence $b(-6)=4r_0^2/r_h^3\ne 0$. This value is used only to rule out the zero analytic function and is not a normalizable hyperbolic mode. The zero set of b in the physical spectral interval is discrete. Helgason’s Plancherel theorem for the hyperbolic Fourier transform (Theorem 2.2 and Equation (3.5) [20]) gives an absolutely continuous spectral resolution with positive density. The disk metric in that source has curvature $-4$. After rescaling to curvature $-1$, the spectral variable in our notation is $\lambda =({\zeta }^2+1)/4$, where $\zeta >0$ is the Fourier parameter. Since $\varphi \ne 0$, its transform cannot be supported on the discrete zero set of b. Consequently, the coefficient in (39) is nonzero on a set of positive spectral measure.

Choose a compact interval I of positive spectral mass avoiding the discrete zero set. Lemma 1 establishes the uniform remainder bounds on this band, so

$$(r-r_h)u_{\lambda }^{\prime }(r,\omega )⟶C_{\lambda }\left(\omega \right),(r-r_h){\partial }_r{\mathsf{\Omega }}_{\lambda }(r,\omega )⟶r_h{C}_{\lambda }\left(\omega \right)$$

(47)

strongly in the corresponding finite-band spectral $L^2$ space. The EF reconstruction (30) gives $r{F}_r={\partial }_r\mathsf{\Omega }$ for the scalar orbit coefficient when the field is independent of v. The coexact angular reconstruction and spectral Plancherel identity give

$$\parallel {\mathsf{\Pi }}_I{\mathfrak{F}}_r{(r,·)\parallel }_{L^2\left(T^{*}{\mathbb{H}}^2\right)}^2={\int }_I\lambda {\left|{\partial }_r{\mathsf{\Omega }}_{\lambda }(r,\omega )\right|}^2\mathrm{d}\nu (\lambda ,\omega ).$$

The bandwise strong limit then gives

$$\underset{r\downarrow r_h}{\lim }(r-r_h){∥{\mathsf{\Pi }}_I{\mathfrak{F}}_r(r,·)∥}_{L^2\left(T^{*}{\mathbb{H}}^2\right)}=r_h{\left({\int }_I\lambda {\left|C_{\lambda }\left(\omega \right)\right|}^2\mathrm{d}\nu (\lambda ,\omega )\right)}^{1/2}>0.$$

(48)

This proves (45). □

Remark 1. 

The excluded spectral zeros are not merely a formal possibility. With $z=(r-r_h)/r_h$, the recurrence (43) and $r_0=3r_h/2$ give

$$r_{h}b\left(\lambda \right)=\sum _{j\ge 0}\left(2j+{\displaystyle \frac{2}{3}}\right)2^{-j}{a}_j\left(\lambda \right).$$

For $6.74\le \lambda \le 6.75$, set $c_j={(-1)}^j{a}_j$ and $d_j=c_j-c_{j-1}$. The recurrence implies

$$d_{j+1}={\displaystyle \frac{(j-1)(j-2)+\lambda }{{(j+1)}^2}}{d}_j(j\ge 1).$$

Here $d_1=\lambda +2>0$, while $0<d_{j+1}\le d_j$ for $j\ge 2$. The first two coefficients then give $0<c_j\le 16j$ for $j\ge 2$. Truncating the displayed series after $j=150$ has absolute error at most

$$482^{-151}(2·{151}^2+4·151+6)<8·{10}^{-40}.$$

Set

$$\begin{array}{cc}\hfill {\lambda }_{-}& =6.74673670987912949775,\hfill \\ \hfill {\lambda }_{+}& =6.74673670987912949776.\hfill \end{array}$$

Exact rational evaluation of the partial sum at these two endpoints gives values in $(-1.780,-1.779)\times {10}^{-21}$ and $(2.355,2.356)\times {10}^{-21}$, respectively. Both signs are certified by the error bound. Thus b has an actual zero ${\lambda }_{*}\in ({\lambda }_{-},{\lambda }_{+})$. This observation is not needed for the positive-measure spectral-band argument.

The conclusion in the zero circle-frequency sector follows from the physical Cauchy data, not from independently prescribed master-field data. The $F_a$ in this argument are invariant under smooth vector-type infinitesimal coordinate changes. The $(v,r)$ chart is regular on the background. The growth of $F_r$ alone is not a curvature statement. The next section computes a gauge-invariant Weyl response and shows how the nonlocal auxiliary master potential is compatible with finite propagation inside Ori’s vacuum core.

7. A Localized Linearized Weyl Obstruction in the Protected Ori Core

The fact that the auxiliary master potential need not have compact angular support does not imply that the physical perturbation has noncompact angular support. Support-sensitive solvability questions for the sourced linearized Einstein equations have Killing-charge conditions (Theorem 1.1 [21]). That theorem is not used here. We work with source-free axial data and establish physical localization through a local hyperbolic Cauchy problem. For the circle-independent datum in (31), we then prove logarithmic growth of a local curvature observable within a protected portion of Ori’s finite vacuum core. The resulting statement remains a linearized vacuum result and does not control an exact perturbed Einstein–dust development.

We use the established relation between axial gravitational master fields and the first variation of the Weyl Pontryagin density. The analogous Schwarzschild identity appears in the Chern–Simons perturbation literature [7] and explicitly in (Appendix F, Eq. (F.2) [10]). The odd-parity curvature interpretation has also been developed in (Sections IV and VI [9]). Dotti and Fernández Tío developed related invariant methods for Schwarzschild and Reissner–Nordström perturbations [11,12]. In the charged case they also analyzed the divergence of derivatives of the curvature invariants near the Cauchy horizon (Section IV [12]). We do not claim originality for the curvature diagnostic or the general phenomenon of linear horizon growth. We verify the negative-curvature Ori-core specialization and connect it to the explicit constraint datum within the causally protected vacuum region.

7.1. The Direct Curvature Identity and Its Initial-Data Check

Write $\mathcal{A}=-{\Delta }_{{\mathbb{H}}^2}$ and $\mathcal{L}=\mathcal{A}+2$. Let Y be a smooth angular scalar. In the diagonal chart of (32), set

$$\mathbb{V}={*}_{{\mathbb{H}}^2}\mathrm{d}Y,h_{\tau A}=H\left(r\right){\mathbb{V}}_A,h_{rA}=0,H=-f{\partial }_r\mathsf{\Omega }.$$

(49)

All other first-order metric components vanish in this vector gauge. We use the Weyl Pontryagin pseudoscalar ${\mathcal{P}=}^{★}{C}_{abcd}{C}^{abcd}$, with a fixed orientation. It vanishes identically on the unperturbed core. For an explicit local contraction, take upper-half-plane coordinates $(x,y)$ with $y>0$ and $h_{{\mathbb{H}}^2}=y^{-2}(\mathrm{d}{x}^2+\mathrm{d}{y}^2)$. With a suitable choice of orientation, the vector form is ${\mathbb{V}}_x={\partial }_{y}Y$ and ${\mathbb{V}}_y=-{\partial }_{x}Y$. Direct first variation of the four-dimensional Riemann tensor gives

$$\dot{\mathcal{P}}={\epsilon }_{\mathrm{or}}{\displaystyle \frac{24\mu }{r^6}}(r{H}^{\prime }-2H){\Delta }_{{\mathbb{H}}^2}Y,{\epsilon }_{\mathrm{or}}\in \{+1,-1\}.$$

(50)

The direct tensor calculation can be performed without fixing an angular eigenmode. With h denoting the off-diagonal perturbation and ${\nabla }^0$ the background connection, the first variations are obtained from

$$\begin{array}{cc}\hfill {{\dot{\Gamma }}^a}_{bc}& ={\displaystyle \frac{1}{2}}{\left(g_0\right)}^{ad}({\nabla }_b^0{h}_{dc}+{\nabla }_c^0{h}_{db}-{\nabla }_d^0{h}_{bc}),\hfill \\ \hfill {{\dot{R}}^a}_{bcd}& ={\nabla }_c^0{{\dot{\Gamma }}^a}_{db}-{\nabla }_d^0{{\dot{\Gamma }}^a}_{cb}.\hfill \end{array}$$

(51)

We raise the Riemann indices with the background inverse and its first variation. Appendix A lists the coordinate formulas for a finite-jet check. The surviving angular factor is $y^2({\partial }_x^2+{\partial }_y^2)Y={\Delta }_{{\mathbb{H}}^2}Y$. The sign ${\epsilon }_{\mathrm{or}}$ depends only on orientation and is immaterial to the norm estimates. As ${tr}_{g_0}h=0$, there is no first-order determinant correction to the volume tensor. The four-dimensional Pontryagin contraction equals its Weyl counterpart, and the background is vacuum. Thus (50) follows without imposing an equation of motion on H.

The static axial master equation (34) can also be written as

$$f{\mathsf{\Omega }}^{\prime \prime }+\left(f^{\prime }-{\displaystyle \frac{2f}{r}}\right){\mathsf{\Omega }}^{\prime }+{\displaystyle \frac{\mathcal{L}}{r^2}}\mathsf{\Omega }=0.$$

(52)

Consequently

$$r{H}^{\prime }-2H=-rf{\mathsf{\Omega }}^{\prime \prime }-(r{f}^{\prime }-2f){\mathsf{\Omega }}^{\prime }={\displaystyle \frac{1}{r}}\mathcal{L}\mathsf{\Omega }.$$

(53)

Superposition in the angular spectral representation therefore gives the local curvature-transfer identity

$$\dot{\mathcal{P}}(r,·)=-{\epsilon }_{\mathrm{or}}{\displaystyle \frac{24\mu }{r^7}}\mathcal{A}\mathcal{L}\mathsf{\Omega }(r,·)=-{\epsilon }_{\mathrm{or}}{\displaystyle \frac{24\mu }{r^7}}\mathcal{A}\mathsf{\Psi }(r,·),\mathsf{\Psi }:=\mathcal{L}\mathsf{\Omega }.$$

(54)

Equivalently, in a $\lambda$-mode its magnitude is $24\mu \lambda (\lambda +2)|{\mathsf{\Phi }}_{\lambda }|/r^6$. The factor $\lambda (\lambda +2)$ is the hyperbolic-curvature counterpart of the classical spherical $\ell (\ell +1)(\ell -1)(\ell +2)$ factor. The angular operator in (54) is differential. Unlike $\mathsf{\Omega }$ itself, its physical curvature combination requires no elliptic inversion to be evaluated from $\mathsf{\Psi }$.

An independent initial-slice calculation checks the normalization. Since (36) gives $\mathsf{\Psi }\left(r_0\right)=2r_0^2\varphi /\sqrt{f_0}$, one obtains

$$\dot{\mathcal{P}}(r_0,·)=-{\epsilon }_{\mathrm{or}}{\displaystyle \frac{48\mu }{r_0^5\sqrt{f_0}}}\mathcal{A}\varphi .$$

(55)

Using $r_0=3\mu$ and ${\Delta }_{\sigma }=r_0^{-2}{\Delta }_{{\mathbb{H}}^2}$, this agrees in magnitude with the initial-slice Gauss–Codazzi calculation of Subsection 8.3. In particular it detects every nonzero compactly supported $\varphi$ because the only compactly supported harmonic scalar is zero. This agreement checks the coefficient’s magnitude without identifying a master amplitude with curvature by definition.

7.2. Localization of the Physical Perturbation Despite a Nonlocal Master Potential

The potential $\mathsf{\Omega }\left(r_0\right)=\mathrm{const}·{\mathcal{L}}^{-1}\varphi$ need not have compact angular support. Nevertheless, the combination $\mathsf{\Psi }=\mathcal{L}\mathsf{\Omega }$ satisfies the same local hyperbolic equation

$$f{\mathsf{\Psi }}^{\prime \prime }+\left(f^{\prime }-{\displaystyle \frac{2f}{r}}\right){\mathsf{\Psi }}^{\prime }-{\displaystyle \frac{1}{r^2}}{\Delta }_{{\mathbb{H}}^2}\mathsf{\Psi }+{\displaystyle \frac{2}{r^2}}\mathsf{\Psi }=0,$$

(56)

with compactly supported initial data

$$\mathsf{\Psi }(r_0,y)={\displaystyle \frac{2r_0^2}{\sqrt{f_0}}}\varphi \left(y\right),{\mathsf{\Psi }}^{\prime }(r_0,y)=0.$$

(57)

This initial-value problem is local. Rauch and Rendall used compactly supported test-field data and finite propagation in certain Cauchy-horizon geometries with noncompact transverse sections ( Section 3, concluding paragraphs [31]). For the present Ori core, equation (56) gives the principal propagation speed $1/\left(r\sqrt{f\left(r\right)}\right)$ in the unit hyperbolic metric. If $supp\varphi \subset B_R\left(o\right)$, finite propagation gives

$$supp\mathsf{\Psi }(r,·)\subseteq \overline{B_{R+d\left(r\right)}\left(o\right)},d\left(r\right)={\int }_r^{r_0}{\displaystyle \frac{\mathrm{d}s}{s\sqrt{f\left(s\right)}}},0<d\left(r\right)\le {\theta }_c,$$

(58)

where ${\theta }_c=\log (2+\sqrt{3})$ and $r_h<r<r_0=3\mu$. The lower-order potential term in (56) does not change the characteristic cones.

For this circle-independent Cauchy pair, the physical metric perturbation can be reconstructed from the solution $\mathsf{\Psi }$ of (56) without angular elliptic inversion. In divergence form, (52) reads

$${\left({\displaystyle \frac{f}{r^2}}{\mathsf{\Omega }}^{\prime }\right)}^{\prime }=-{\displaystyle \frac{\mathsf{\Psi }}{r^4}}.$$

(59)

The condition ${\mathsf{\Omega }}^{\prime }\left(r_0\right)=0$ therefore gives

$$H(r,y)=-f\left(r\right){\mathsf{\Omega }}^{\prime }(r,y)=-r^2{\int }_r^{r_0}{\displaystyle \frac{\mathsf{\Psi }(s,y)}{s^4}}\mathrm{d}s.$$

(60)

The right-hand side is supported in $\overline{B_{R+d\left(r\right)}\left(o\right)}$, since $d\left(s\right)\le d\left(r\right)$ for $r\le s\le r_0$. It follows that both components of the vector-type metric perturbation, including the EF component $r{F}_r={\mathsf{\Omega }}^{\prime }=-H/f$ for each fixed $r_h<r\le r_0$, vanish outside the same angular propagation cone. Even when ${\mathcal{L}}^{-1}\varphi$ has a noncompact angular tail, that tail belongs to an auxiliary variable and does not force noncompact support of the reconstructed physical linear perturbation for $r_h<r\le r_0$. This concerns support alone and gives no uniform bound on reconstructed metric components as $r\downarrow r_h$. Equation (54) also shows directly that

$$supp\dot{\mathcal{P}}(r,·)\subseteq \overline{B_{R+d\left(r\right)}\left(o\right)}.$$

(61)

This conclusion is specific to the circle-independent axial Cauchy pair and its local field equation. It is not a localization theorem for arbitrary polar modes or for the nonlinear CMC correction.

7.3. A Protected-Patch Linearized Curvature Growth Estimate

Proposition 7.1. 

Let $\varphi \in C_c^{\infty }\left({\mathbb{H}}^2\right)$ be nonzero, supported in $B_R\left(o\right)$. Consider the smooth circle-independent axial constraint datum (31) on $r_0=3\mu$ and its linearized vacuum Einstein development for $r_h<r\le r_0$, approaching the background horizon $r_h=2\mu$. Assume Ori’s truncated core has angular radius ${\theta }_0$ with

$$R+2{\theta }_c<{\theta }_0,{\theta }_c=\log (2+\sqrt{3}).$$

(62)

Then the reconstructed metric perturbation in the stated vector gauge and the invariant $\dot{\mathcal{P}}$ are supported within the protected vacuum domain for every $r_h<r\le r_0$. There exist $c>0$ and $\delta >0$, depending on the seed and background parameters, such that

$$\parallel \dot{\mathcal{P}}(r,·){\parallel }_{L^2(B_{{\theta }_0-{\theta }_c}\left(o\right),\mathrm{d}{V}_{{\mathbb{H}}^2})}\ge c\left|\log {\displaystyle \frac{r-r_h}{r_h}}\right|(0<r-r_h<\delta ).$$

(63)

Thus the first-order, spacetime-gauge-invariant Weyl Pontryagin response does not extend as a bounded continuous scalar to the protected horizon patch. No assertion is made about a curvature singularity in any exact, finite-amplitude Einstein–dust solution.

Proof. 

The physical initial constraints and zero-frequency transfer have been established in (31) and (36). By (57)–(61), finite propagation confines the physical perturbation and $\dot{\mathcal{P}}$ to $B_{R+{\theta }_c}\left(o\right)$. More precisely, Ori’s causal-distance bound leaves the known vacuum region $B_{{\theta }_0-d\left(r\right)}\left(o\right)$ on each pre-horizon slice, while our support lies in $B_{R+d\left(r\right)}\left(o\right)$. Since $d\left(r\right)\le {\theta }_c$, the strict inequality $R+2{\theta }_c<{\theta }_0$ gives $R+d\left(r\right)<{\theta }_0-d\left(r\right)$ at every such slice. At the horizon, Ori’s protected sector contains $B_{{\theta }_0-{\theta }_c}\left(o\right)$ by (Section III.C–D, Eq. (7) [1]). The bound $R+2{\theta }_c<{\theta }_0$ is a sufficient support condition for this specified datum, not a necessary condition for all perturbations of Ori’s model. The size inequality keeps the perturbation strictly within this protected vacuum region. Local Cauchy uniqueness then identifies the linearized field in the stated vector gauge with the solution from the same initial data on the untruncated Ori vacuum background with periodic $\tau$ and angular factor ${\mathbb{H}}^2$. There is no need to prescribe a disturbance of the dust shell to obtain this interior linear conclusion.

In the hyperbolic spectral resolution the initial-data Wronskian calculation (38)–(39) gives

$$u_{\lambda }(r,\omega )=C_{\lambda }\left(\omega \right)\log {\displaystyle \frac{r-r_h}{r_h}}+O_I\left(|\widehat{\varphi }(\lambda ,\omega )|\right)$$

(64)

on any fixed compact angular spectral band I away from the zeros of $b\left(\lambda \right)$. This uniform error bound follows from the bound on $A_{\lambda }$ in the proof of Lemma 1 and the remainder estimate (). Since the seed is nonzero and b is analytic, there is a compact interval $I⋐(1/4,\infty )$ on which

$$0<{\int }_I{\lambda }^2{(\lambda +2)}^2{\left|C_{\lambda }\left(\omega \right)\right|}^2\mathrm{d}\nu (\lambda ,\omega )<\infty .$$

(65)

Substituting $\mathsf{\Omega }=ru$ into (54), Plancherel and the uniform Frobenius expansion give

$$\underset{r\downarrow r_h}{\lim }{\displaystyle \frac{{∥{\mathsf{\Pi }}_I\dot{\mathcal{P}}\left(r\right)∥}_{L^2\left({\mathbb{H}}^2\right)}}{\left|\log ((r-r_h)/r_h)\right|}}={\displaystyle \frac{24\mu }{r_h^6}}{\left({\int }_I{\lambda }^2{(\lambda +2)}^2{\left|C_{\lambda }\left(\omega \right)\right|}^2\mathrm{d}\nu \right)}^{1/2}>0.$$

(66)

Although the spectral projection ${\mathsf{\Pi }}_I$ is nonlocal, it is used only to bound a global norm from below. The unprojected curvature scalar is compactly supported in the fixed protected angular ball by (61). Hence

$$\parallel \dot{\mathcal{P}}{\left(r\right)\parallel }_{L^2\left(B_{{\theta }_0-{\theta }_c}\left(o\right)\right)}=\parallel \dot{\mathcal{P}}{\left(r\right)\parallel }_{L^2\left({\mathbb{H}}^2\right)}\ge {\parallel {\mathsf{\Pi }}_I\dot{\mathcal{P}}\left(r\right)\parallel }_{L^2\left({\mathbb{H}}^2\right)},$$

(67)

which proves (63). Since the protected angular ball has finite hyperbolic area, the $L^{\infty }$ norm of $\dot{\mathcal{P}}$ on that ball is also unbounded as $r\downarrow r_h$. Finally the background $\mathcal{P}\left[g_O\right]$ vanishes identically, so its first variation is invariant under smooth spacetime infinitesimal diffeomorphisms. It cannot be removed as a coordinate effect. □

7.4. Scope and Limits of the Linearized Curvature Estimate

This proposition concerns a fixed background and the first-order Einstein equations for compactly supported, circle-invariant axial data with transverse-traceless extrinsic-curvature variation. The estimate concerns a curvature scalar that vanishes on the background, not merely a divergent Regge–Wheeler reconstruction coefficient. It does not prove a generic theorem for all polarizations or arbitrary circle dependence. The spectral argument explicitly allows exceptional individual fibers where $b\left(\lambda \right)=0$. Such isolated fibers have zero Plancherel measure on the full hyperbolic plane. The spatial support condition (62) is an additional restriction and is not automatic for every Ori envelope.

Even within the protected patch, the amplitude measured here is $\dot{\mathcal{P}}$, not $\mathcal{P}\left[g_{\epsilon }\right]$ for a family of exact evolved metrics. A first-order term of size $\left|\epsilon \right|\left|\log \right((r-r_h)/r_h\left)\right|$ leaves the perturbative regime arbitrarily close to the horizon at each fixed nonzero $\epsilon$. The calculation neither determines the nonlinear endpoint nor proves that the original chronology horizon cannot be replaced by a different geometry. Our exact annular constraint family is matched to the unmodified dust initial data, but no corresponding claim about its full coupled nonlinear evolution follows. Nothing here proves finite-amplitude curvature blow-up, inextendibility of the coupled solution, or a universal chronology-protection principle.

The background geometry is due to Ori [1], and the axial master equation to Kodama and Ishibashi [5]. Pontryagin curvature diagnostics also appear in earlier axial perturbation studies [7,10]. The statement here is restricted to the specified compactly supported transverse-traceless data, the resulting gauge-invariant first variation, and the causally protected region of the Ori vacuum core. We do not claim a general Cauchy-horizon instability theorem.

8. Finite-Amplitude Twist Data and Local Vacuum Developments

Proposition 4.1 constructs exact finite-amplitude vacuum constraint data, not merely a formal perturbation series. We work with the same annular family and do not substitute new free data. Wherever the circle Killing field is spacelike, the local vacuum equations admit the $2+1$ Einstein–wave-map reduction developed by Geroch and Moncrief [22,23]. Moncrief gives the local formulation explicitly in his later exposition (p. 168 [24]).

8.1. Persistence of the Inherited Symmetry in the Vacuum Development

Let $\xi ={\partial }_{\tau }$ on the original core initial surface. Since $Y_{\epsilon }$ and ${\varphi }_{\epsilon }$ depend only on $\rho$, the tensors ${\gamma }_{\epsilon }$ and $K_{\epsilon }$ in (19) are invariant under the $\tau$-circle action and the rotations generated by ${\partial }_{\vartheta }$. They are smooth across $\rho =0$, where the latter vector field vanishes. By local geometric uniqueness for vacuum Einstein evolution, these initial symmetries induce Killing fields on the corresponding vacuum domain of dependence Section 2.2 [25]). The inherited periodic coordinate $\tau$ gives a free circle action on the core initial slice. Its Killing generator is spacelike there because

$${\lambda }_{\epsilon }:=g_{\epsilon }{(\xi ,\xi )|}_{{\Sigma }_0}=f_0{Y}_{\epsilon }=f_0(1-{\epsilon }^2{p}^2)>0.$$

(68)

It remains spacelike in a sufficiently small vacuum neighborhood of that slice. We make no claim that it stays spacelike up to a perturbed chronology horizon. In particular the symmetry is asserted only in the vacuum region inherited from the core, not through the entire non-symmetric dust envelope.

Where $\lambda =g_{\epsilon }(\xi ,\xi )>0$, use the Einstein-frame metric h on the three-dimensional orbit space,

$$h_{ab}=\lambda \left(g_{ab}-{\lambda }^{-1}{\xi }_a{\xi }_b\right).$$

(69)

The induced two-metric q on the quotient of the vacuum-core portion of ${\Sigma }_0$ by the $\tau$-circle action satisfies the exact identity

$$q_{\epsilon }={\lambda }_{\epsilon }\left(Y_{\epsilon }^{-1}\sigma \right)=f_0\sigma =q_0.$$

(70)

Thus the finite-amplitude deformation has no effect on the Einstein-frame quotient spatial metric at the initial instant. This is an identity for this particular ansatz. It says nothing about the subsequent quotient metric or its second fundamental form.

8.2. A Compactly Supported Initial Twist Potential

In the vacuum development define the Killing twist one-form by

$${\varpi }_a={\epsilon }_{abcd}{\xi }^b{\nabla }^c{\xi }^d.$$

(71)

It is closed wherever $Ric\left(g_{\epsilon }\right)=0$. Consequently it is locally $\mathrm{d}{\chi }_{\epsilon }$ for a twist potential ${\chi }_{\epsilon }$. We fix the orientation by $\mathrm{d}t\wedge \mathrm{d}\tau \wedge \mathrm{d}\rho \wedge \mathrm{d}\vartheta >0$ and take the future unit normal to be $n={\partial }_t$ in Gaussian normal coordinates along ${\Sigma }_0$. Our convention for extrinsic curvature is $K_{ij}=-{\displaystyle \frac{1}{2}}{\left({\mathcal{L}}_n\gamma \right)}_{ij}$. A reversal of spacetime orientation reverses all the signs in the next twist identity but not its consequences.

Proposition 8.1. 

For the exact family (19) the restriction of the Killing twist to ${\Sigma }_0$ is

$$\varpi {|}_{{\Sigma }_0}=-2\epsilon \sqrt{f_0}\mathrm{d}{\varphi }_{\epsilon },{\chi }_{\epsilon }{|}_{{\Sigma }_0}=-2\epsilon \sqrt{f_0}{\varphi }_{\epsilon }.$$

(72)

The potential is normalized to vanish in the two vacuum collars. The initial twist potential is smooth, compactly supported, and nonconstant for every sufficiently small nonzero ε. The exact nonlinear vacuum development is therefore not polarized in any open neighborhood intersecting the nonzero-twist part of the annulus.

Proof. 

At $t=0$ the metric is diagonal in $(\tau ,\rho ,\vartheta )$ with spatial coefficients

$$A=f_0{Y}_{\epsilon },B={\displaystyle \frac{r_0^2}{Y_{\epsilon }}},C={\displaystyle \frac{r_0^2{\sinh }^2\rho }{Y_{\epsilon }}}.$$

(73)

Writing $a_{\epsilon }={\varphi }_{\epsilon }^{\prime }$, the only mixed initial component of the second fundamental form is

$$K_{\tau \vartheta }=\epsilon Y_{\epsilon }^{-1/2}{a}_{\epsilon }\sinh \rho .$$

(74)

Gaussian evolution gives ${\partial }_t{g}_{\tau \vartheta }{|}_{t=0}=-2K_{\tau \vartheta }$. The terms in ${\xi }^{♭}\wedge \mathrm{d}{\xi }^{♭}$ involving $\mathrm{d}A$ cancel because ${\xi }^{♭}=A\mathrm{d}\tau$ at $t=0$. The surviving term is

$${\left.{\xi }^{♭}\wedge \mathrm{d}{\xi }^{♭}\right|}_{t=0}=2\epsilon f_0\sqrt{Y_{\epsilon }}{a}_{\epsilon }\sinh \rho \mathrm{d}t\wedge \mathrm{d}\tau \wedge \mathrm{d}\vartheta .$$

(75)

Taking the four-dimensional Hodge dual using $\sqrt{-\det g}=\sqrt{f_0}{r}_0^2\sinh \rho /\sqrt{Y_{\epsilon }}$ gives precisely ${\varpi |}_{{\Sigma }_0}=-2\epsilon \sqrt{f_0}{a}_{\epsilon }\mathrm{d}\rho$ in our orientation. Equation (18) then supplies the compactly supported potential. Where $a_{\epsilon }\ne 0$ the three-form (75) is nonzero. Frobenius’ criterion therefore excludes hypersurface orthogonality of $\xi$ in that region. □

The exact norm and twist data also allow a direct computation of the spatial contribution to the reduced wave-map energy. The target of the standard vacuum $U\left(1\right)$ reduction is hyperbolic (pp. 168–171 [24]). For our norm-and-twist variables we choose the metric

$$G_{\mathbb{H}}={\displaystyle \frac{\mathrm{d}{\lambda }^2+\mathrm{d}{\chi }^2}{2{\lambda }^2}}.$$

(76)

We call the spatial gradient contribution

$${\mathcal{E}}_{\mathrm{sp}}={\displaystyle \frac{|\mathrm{d}{\lambda }_{\epsilon }{|}_{q_0}^2+{\left|\mathrm{d}{\chi }_{\epsilon }\right|}_{q_0}^2}{2{\lambda }_{\epsilon }^2}}$$

(77)

the gradient energy density associated with the quotient initial two-metric. It is not the total wave-map energy and need not be a conserved dynamical charge.

Proposition 8.2. 

For the annular data defined in (19),

$${\mathcal{E}}_{\mathrm{sp}}\left(\rho \right)={\displaystyle \frac{2{\epsilon }^{2}p{\left(\rho \right)}^2}{f_0{r}_0^2[1-{\epsilon }^{2}p{\left(\rho \right)}^2]}},$$

(78)

and its spatial integral is

$${\int }_{{\mathbb{H}}^2}{\mathcal{E}}_{\mathrm{sp}}\mathrm{d}{V}_{q_0}=4\pi {\epsilon }^2{\int }_a^b{\displaystyle \frac{p{\left(\rho \right)}^2\sinh \rho }{1-{\epsilon }^{2}p{\left(\rho \right)}^2}}\mathrm{d}\rho >0$$

(79)

for $\epsilon \ne 0$. In particular the integral is $O\left({\epsilon }^2\right)$ and its leading quadratic coefficient is strictly positive.

Proof. 

From (68) and (72),

$${\lambda }_{\epsilon }^{\prime }=-2f_0{\epsilon }^{2}p{p}^{\prime },{\chi }_{\epsilon }^{\prime }=-2\epsilon \sqrt{f_0}{\varphi }_{\epsilon }^{\prime },q_0=f_0{r}_0^2(\mathrm{d}{\rho }^2+{\sinh }^2\rho \mathrm{d}{\vartheta }^2).$$

(80)

Use ${\left({\varphi }_{\epsilon }^{\prime }\right)}^2=f_0{p}^2[1-{\epsilon }^2(p^2+{\left(p^{\prime }\right)}^2)]$. The numerator in (77) is

$${\displaystyle \frac{4f_0{\epsilon }^2{p}^2}{r_0^2}}\left[{\epsilon }^2{\left(p^{\prime }\right)}^2+1-{\epsilon }^2(p^2+{\left(p^{\prime }\right)}^2)\right]={\displaystyle \frac{4f_0{\epsilon }^2{p}^2}{r_0^2}}(1-{\epsilon }^2{p}^2).$$

(81)

Division by $2{\lambda }_{\epsilon }^2$ gives (78). Integrating with $\mathrm{d}{V}_{q_0}=f_0{r}_0^2\sinh \rho \mathrm{d}\rho \mathrm{d}\vartheta$ proves (79). □

8.3. A Finite-Amplitude Curvature Diagnostic at the Initial Slice

The initial twist rules out polarization wherever it is nonzero. We also derive an initial-slice expansion of the four-dimensional Weyl Pontryagin scalar for the exact finite-amplitude data. The electric and magnetic Weyl formulas and their Pontryagin contraction are standard (Appendix B, Eqs. (B6)–(B8) [8]). On a vacuum initial slice, our curvature conventions give

$$E_{ij}=Ric{\left(\gamma \right)}_{ij}+(trK)K_{ij}-{K_i}^k{K}_{kj},B_{ij}={{ϵ}_i}^{kl}{D}_k{K}_{lj}.$$

(82)

We use the standard identity $\mathcal{P}=\pm 16{E_i}^j{B_j}^i$, with orientation-dependent sign. The background Pontryagin scalar vanishes, so its first variation is gauge invariant.

Proposition 8.3. 

Let $\left(g_{\epsilon }\right)$ be any local smooth vacuum development of the exact family in (19), with a consistent orientation and identification of initial surfaces. For every fixed nonnegative integer m, the exact initial Weyl Pontryagin scalar satisfies

$$\mathcal{P}\left[g_{\epsilon }\right]{|}_{{\Sigma }_0}=\pm {\displaystyle \frac{16\epsilon }{\sqrt{f_0}{r}_0^2}}{\Delta }_{\sigma }{\varphi }_0+O_{C^m}\left({\epsilon }^3\right).$$

(83)

For each sufficiently small nonzero ε, the scalar is not identically zero. It is an invariant of a finite-amplitude vacuum metric, not a formal linear curvature perturbation.

Proof. 

The diagonal background electric Weyl eigenvalues are

$${{\left(E_0\right)}_{\tau }}^{\tau }={\displaystyle \frac{2}{3r_0^2}},{{\left(E_0\right)}_{\rho }}^{\rho }={{\left(E_0\right)}_{\vartheta }}^{\vartheta }=-{\displaystyle \frac{1}{3r_0^2}},$$

(84)

and $B_0=0$. With $y=Y_{\epsilon }$ and $a_{\epsilon }={\varphi }_{\epsilon }^{\prime }$, computing ${B_i}^j$ from the diagonal metric (73) and the mixed tensor (74) gives

$$\begin{array}{cc}\hfill {B_{\tau }}^{\tau }& =\pm {\displaystyle \frac{\epsilon }{\sqrt{f_0}{r}_0^2}}\left(a_{\epsilon }^{\prime }+\coth \rho a_{\epsilon }-{\displaystyle \frac{y^{\prime }}{y}}{a}_{\epsilon }\right),\hfill \\ \hfill {B_{\rho }}^{\rho }& =\pm {\displaystyle \frac{\epsilon }{\sqrt{f_0}{r}_0^2}}\left({\displaystyle \frac{y^{\prime }}{y}}-\coth \rho \right)a_{\epsilon },\hfill \\ \hfill {B_{\vartheta }}^{\vartheta }& =\mp {\displaystyle \frac{\epsilon }{\sqrt{f_0}{r}_0^2}}{a}_{\epsilon }^{\prime }.\hfill \end{array}$$

(85)

There are no off-diagonal magnetic components for this radial annular ansatz. The displayed trace is exactly zero. Both y and $a_{\epsilon }$ are even smooth functions of $\epsilon$, with $y=1+O_{C^m}\left({\epsilon }^2\right)$ and $a_{\epsilon }={\varphi }_0^{\prime }+O_{C^m}\left({\epsilon }^2\right)$. The diagonal electric components are even in $\epsilon$, whereas the only off-diagonal electric terms are odd and have no magnetic counterpart. Therefore ${E_i}^j{B_j}^i$ is odd in $\epsilon$ and its next term after the linear one is cubic. By the trace-free identity for B, the eigenvalue gap in (84) gives

$${E_i}^j{B_j}^i=\pm {\displaystyle \frac{\epsilon }{\sqrt{f_0}{r}_0^4}}({\varphi }_0^{\prime \prime }+\coth \rho {\varphi }_0^{\prime })+O_{C^m}\left({\epsilon }^3\right).$$

(86)

This is (83) because ${\Delta }_{\sigma }{\varphi }_0=r_0^{-2}({\varphi }_0^{\prime \prime }+\coth \rho {\varphi }_0^{\prime })$. If ${\Delta }_{\sigma }{\varphi }_0$ vanished everywhere, integration by parts on ${\mathbb{H}}^2$ and compact support would force ${\varphi }_0=0$, contrary to (22). Consequently, for every sufficiently small nonzero $\epsilon$, the finite-amplitude Pontryagin scalar is nonzero at some point of the annulus. □

There is a second, geometric explanation for the absence of even powers in (83). The reflection $J(\tau ,\rho ,\vartheta )=(-\tau ,\rho ,\vartheta )$ leaves ${\gamma }_{\epsilon }$ unchanged and pulls $K_{\epsilon }$ back to $K_{-\epsilon }$. It reverses spatial orientation. Under geometric uniqueness the corresponding local vacuum developments are related by an orientation-reversing isometry, so their Pontryagin scalars have opposite signs after the canonical initial-data identification. The oddness conclusion concerns the Pontryagin pseudoscalar $\mathcal{P}$. It does not assert oddness of orientation-even curvature invariants such as the Kretschmann scalar.

8.4. Local Perturbative Control on Fixed Pre-Horizon Slabs

The general local existence and Cauchy stability results for the vacuum Einstein equations are classical (Section 2.2 [25]). Damour and Schmidt studied the relation between smooth one-parameter families of exact Einstein solutions and their perturbative expansions [27]. Our use of local Cauchy stability is confined to the domain of dependence of the vacuum core. No well-posedness assertion for the moving vacuum–dust interface follows from these vacuum-only results.

Proposition 8.4. 

Fix a relatively compact smooth pre-horizon vacuum subslab $\mathfrak{K}$ of the background Ori development. Assume that its causal antecedent lies in the interior of the original vacuum core and is separated from the core–dust interface, the initial-slice boundary and the background chronology horizon by positive margins. After decreasing ${\epsilon }_{*}$ if necessary, the annular data have vacuum developments on a common neighborhood of $\mathfrak{K}$ for all $\left|\epsilon \right|<{\epsilon }_{*}$. In a consistent smooth equivariant identification of these developments,

$${∥\mathcal{P}\left[g_{\epsilon }\right]-\epsilon \dot{\mathcal{P}}∥}_{C^m\left(\mathfrak{K}\right)}\le C_{\mathfrak{K},m}{\left|\epsilon \right|}^3,$$

(87)

for any specified finite m and sufficiently small $\left|\epsilon \right|$. The derivative $\dot{\mathcal{P}}$ is the linearized vacuum curvature response of the same annular tangent data. In particular, near any initial point where $\dot{\mathcal{P}}\ne 0$, the exact scalar $\mathcal{P}\left[g_{\epsilon }\right]$ is nonzero for every sufficiently small nonzero amplitude.

Proof. 

The initial data depend smoothly on $\epsilon$ in every local Sobolev norm. Fix m and choose an integer Sobolev order $s>m+5$. Use a generalized wave gauge relative to the background metric. Set the initial lapse to one and the initial shift to zero, and determine their normal derivatives from the wave-gauge conditions. These choices depend smoothly on the data and commute with the reflection J. On compact vacuum Cauchy neighborhoods covering the causal past of $\mathfrak{K}$, the first-order variables $U_{\epsilon }=(g_{\epsilon },\partial g_{\epsilon })$ obey a smooth quasilinear symmetric-hyperbolic reduced system. The classical symmetric-hyperbolic formulation of Einstein evolution is established in [26]. Local Cauchy stability, constraint propagation and finite speed give a common regular neighborhood after restricting the amplitude range (Section 2.2–2.3 [25]).

To obtain the stated cubic remainder, expand the smooth gauged initial data through order two in $\epsilon$. Let $U_1$ and $U_2$ solve the first and second variational reduced equations along the background solution $U_0$, with those Taylor data. These are linear symmetric-hyperbolic equations with smooth coefficients and, at second order, a smooth quadratic source. The second-order approximate field $U_{\epsilon }^{\left[2\right]}=U_0+\epsilon U_1+{\epsilon }^2{U}_2$ therefore has an initial discrepancy and a reduced-equation residual bounded by $C_s{\left|\epsilon \right|}^3$ in $H^s$ on the fixed regular neighborhood. Applying the symmetric-hyperbolic energy estimate to the difference between the exact and approximate reduced systems gives

$$\underset{t\in [0,T]}{\sup }\parallel U_{\epsilon }\left(t\right)-U_{\epsilon }^{\left[2\right]}{\left(t\right)\parallel }_{H^s}\le C_s{\left|\epsilon \right|}^3.$$

(88)

Here the smooth initial family supplies as many higher Sobolev norms as required to bound the coefficients and the residual. Sobolev embedding and the local curvature formula consequently give a $C^m$ expansion of $\mathcal{P}\left[g_{\epsilon }\right]$ through order two with an $O_{C^m}{\left(\right|\epsilon |}^3)$ remainder.

The reflection J from the preceding subsection maps the initial data at amplitude $\epsilon$ isometrically to those at $-\epsilon$. Equivariance of the chosen gauge and uniqueness give $g_{-\epsilon }=J^{*}{g}_{\epsilon }$ in the common identification. Since J reverses spacetime orientation while each development is $\tau$-invariant, the corresponding Pontryagin pseudoscalars satisfy

$$\mathcal{P}\left[g_{-\epsilon }\right](t,\tau ,\rho ,\vartheta )=-\mathcal{P}\left[g_{\epsilon }\right](t,-\tau ,\rho ,\vartheta )=-\mathcal{P}\left[g_{\epsilon }\right](t,\tau ,\rho ,\vartheta ).$$

(89)

Comparing the second-order expansions for the two amplitudes using (89) shows that both the zeroth and quadratic coefficients vanish. The cubic residual estimate (88) then gives (87). The first variation of $g_{\epsilon }$ is the linearized Einstein development of (21) in the chosen gauge. Its induced Weyl Pontryagin variation is the scalar coefficient $\dot{\mathcal{P}}$, which is gauge invariant because the background Pontryagin scalar vanishes. □

The estimate above is established only on fixed compact slabs. The argument gives no bound for $C_{\mathfrak{K},m}$ that is uniform as the slab approaches $r=2\mu$, nor does it ensure that a fixed nonzero $\epsilon$-development reaches an analogue of the background Cauchy horizon. The odd function $\tanh \left(\epsilon \dot{\mathcal{P}}\right)$ illustrates why a cubic Taylor remainder on every fixed compact slab need not control an endpoint where the linear coefficient grows without bound. Thus neither the oddness nor (87) proves finite-amplitude singularity of a chronology horizon. The $2+1$ Einstein–wave-map formulation could be useful for studying the nonlinear endpoint, but our argument does not give estimates uniform as the Killing norm $\lambda$ approaches zero.

8.5. Related Cauchy-Horizon Results and Limitations

Nonlinear vacuum chronology-horizon problems with Killing symmetry have a long history. Moncrief constructed infinite-dimensional families containing smoothly extendible vacuum Cauchy horizons [28,29]. His analysis also established boundary curvature singularities for generic diagonal Gowdy solutions [28]. For certain extendible backgrounds, he showed that generic linear perturbations blow up near the Cauchy horizons of those backgrounds [29]. The same author studied symmetry-preserving perturbations of generalized Taub–NUT models to all formal orders [30]. In a separate paper, he analyzed a cubic nonlinear wave equation on fixed vacuum Cauchy-horizon backgrounds [33].

Rauch and Rendall proved test-field nonextendibility results on Taub and Moncrief backgrounds (Section 2–3 [31]). They also explained how their flux argument extends to certain noncompact transverse sections for compactly supported initial data, assuming propagation limited by the light cones and a nonzero flux (Section 3, concluding paragraphs [31]). In their Taub example, the spatially homogeneous linear solution in Eq. (7) contains an integral whose integrand has a simple pole at the Cauchy horizon. Consequently, that example admits logarithmically divergent solutions (Section 2, Eq. (7) [31]). Their Theorem 2 requires a nonzero flux involving a derivative of the test scalar in the circle direction. For circle-invariant scalar modes this flux vanishes, so that theorem does not by itself imply our tensorial Pontryagin estimate.

Grubišić and Moncrief used all-order perturbative expansions to analyze curvature singularities in $T^3\times \mathbb{R}$ Gowdy vacuum spacetimes [32]. Their geometry and boundary differ from those of the matched Ori core.

Moncrief and Isenberg proved that analytic vacuum and electrovacuum spacetimes with compact null hypersurfaces ruled by closed generators satisfying a local product-bundle condition possess nontrivial Killing symmetries [34]. Isenberg and Moncrief subsequently removed the bundle condition while retaining analyticity and the closed-generator assumption [35]. Moncrief also classified analytic vacuum solutions with compact Hopf-fibered $S^3$ Cauchy horizons, obtaining an infinite-dimensional family [36]. In a related construction, he used Geroch transformations to obtain an infinite-dimensional family of curvature-singular circle-symmetric vacuum spacetimes from generalized Taub–NUT solutions [37]. His later exposition describes the accompanying change in circle-bundle topology (p. 173 [24]). For smooth Ricci-flat spacetimes with compact, smooth, totally geodesic Cauchy horizons whose surface gravity can be normalized to a nonzero constant, Petersen and Rácz established the existence of a nontrivial Killing field (Theorem 1.2 [38]).

Under different global assumptions, Maeda and Ishibashi proved global hyperbolicity for vacuum spacetimes with compact $S^1$-fibered spatial sections, provided their null generic and chronological-horizon conditions hold and all occurring curvature singularities are strong (Section IV [39]). Maeda, Ishibashi and Narita excluded a restricted class of chronology-violating regions in maximal finitely vicious spacetimes, assuming their specified boundary, strong-causality, energy and genericity conditions and that every incomplete causal geodesic terminates at a non-naked singularity (Theorem 15 in the cited preprint [40]).

Yurtsever proved instability results for Killing–Cauchy horizons in plane-symmetric spacetimes [41]. For colliding parallel-polarized gravitational plane waves, Yurtsever proved nonlinear instability of Killing–Cauchy horizons under small generic initial-data perturbations [42]. He later extended the result to arbitrary polarizations under generic plane-symmetric initial-data perturbations [43].

For black-hole Cauchy horizons, Luk and Oh proved linear scalar-wave instability at the Reissner–Nordström Cauchy horizon [44], while Sbierski proved a linearized gravitational instability theorem at the Kerr Cauchy horizon [45].

At the nonlinear level, Gurriaran’s 2026 preprint (Theorem 4.5.1 [46]) establishes conditional Lipschitz inextendibility near timelike infinity for vacuum developments from smooth data on a hypersurface slightly inside a dynamical black hole settling down to subextremal Kerr. The hypotheses include a precise nonlinear Price-law-type estimate and a nonvanishing leading-mode condition. Luk and Sbierski’s 2026 preprint [47] establishes weak null singularity formation for dynamical rotating black-hole interiors under characteristic initial-data assumptions involving an event horizon settling down to that of strictly rotating subextremal Kerr and specified upper and lower bounds. These results concern different backgrounds, perturbation classes, initial surfaces, or matter models. None is invoked as a theorem about the matched Ori 2007 vacuum–dust data.

The exact initial observables and fixed-slab control do not establish nonlinear horizon instability. A stronger dynamical result would need to control the $U\left(1\right)$ quotient evolution as $\lambda =g_{\epsilon }(\xi ,\xi )\downarrow 0$, and then to demonstrate a perturbed-horizon regularity obstruction or a geometric obstruction to extendibility. It would also need to address the coupled evolution at the dust interface. None of these conclusions follows from the present argument.

9. Proof of the Combined Theorem and Conclusions

Proof 

(Proof of Theorem 1). Proposition 3.1 gives the exact constraint reduction. Proposition 4.1 constructs the smooth finite-amplitude family and computes its nonzero axial tangent. Proposition 4.2 assembles the global constraint data with the unchanged Ori envelope. For the tangent $\varphi ={\varphi }_0$, the map (36) is the physical axial Cauchy map. The Wronskian proof of Proposition 6.1 selects a finite spectral band on which the logarithmic horizon coefficient has strictly positive spectral $L^2$ norm. Proposition 7.1 converts that coefficient to a lower bound for the gauge-invariant Weyl Pontryagin variation. Its strict support condition is satisfied because $supp{\varphi }_0\subset \{\rho \le b\}$ and $b+2{\theta }_c<{\theta }_0$. This is precisely (1). □

The nonlinear and linearized conclusions must be kept separate. For every sufficiently small amplitude, the matched initial data satisfy the full nonlinear Einstein–dust constraints exactly. The family’s tangent at $\epsilon =0$ has a linearized vacuum development on the fixed background whose Weyl Pontryagin response grows without bound toward the background horizon. The local Taylor estimate on each fixed pre-horizon compact slab has a slab-dependent constant and cannot justify interchanging $\epsilon \to 0$ with $r\downarrow r_h$. In particular, this article does not establish that an exact perturbed Einstein–dust solution forms a Cauchy horizon, that a new Cauchy horizon is necessarily singular, or that chronology violation is prevented.

The linearized curvature estimate applies to every admissible nonzero compactly supported axial potential. The combined theorem specializes it to the tangent of the explicit finite-amplitude constraint family. The constraint, perturbation, and curvature arguments draw substantially on established results. A general nonlinear propagation theorem for the vacuum–dust interface, uniform endpoint estimates, and a global nonlinear horizon theorem remain outside this work.