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Francisco S. N. Lobo

Publications and source records attributed to Francisco S. N. Lobo.

At least 19 recordsLinked to original sources
arXiv2026

Beyond the Dolgov-Kawasaki instability: Covariant scalaron stability, entropy transport, and inner horizons in f(R) gravity

The Dolgov-Kawasaki instability tests the local scalaron response in metric $f(R)$ gravity, but does not determine strong-field horizon dynamics. We separate these questions using $F=f_R$ and the Legendre potential $\mathcal U(F)=FR-f$. The scalaron restoring coefficient is $m_{\rm sc}^2=(F-Rf_{RR})/(3f_{RR})$, whereas exact spherical Wald-area transport identities cancel the scalaron Hessian. Their Einstein-frame interpretation distinguishes longitudinal kinetic focusing from potential-dependent cross-focusing. We prove that, at fixed constant-curvature background with tracefree matter, transport depends only on $(f,f_R)$, while the scalaron mass depends on the independently adjustable $f_{RR}$. Reissner-Nordström in $f(R)=R+αR^2$ gives an exact discriminator: identical background transport coexists with opposite scalaron mass signs. A charged solution with varying $F$ and a cosmological check test the identities beyond this constant-$F$ sector. For a nonzero massive scalaron tail transmitted to a nondegenerate Cauchy horizon, a conditional fixed-background estimate gives a nonintegrable contribution to generalized entropy focusing. This estimate does not establish nonlinear mass inflation or a particular extendibility class. The results provide a layered framework for coupling regularity, background transport, scalaron stability, and nonlinear inner-horizon evolution.

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arXiv2026

Intrinsic spatial-strain shortcuts in general relativity

We analyze spacetime shortcuts generated by longitudinal contraction and transverse expansion of the intrinsic spatial metric at fixed reference volume. Smooth bounded ultrastatic corridors are globally hyperbolic and geodesically complete and produce an exact travel-time advance between exterior clocks. An axial maximum makes the axial light ray globally minimizing; a strict maximum on an interior interval establishes Olum's endpoint-surface criterion for exactly matched corridors. Exact Einstein-source identities distinguish integrated Eulerian matter energy from asymptotic mass: infinite tubes have negative integrated energy, whereas finite corridors admit competing gradient contributions. A transverse Ricci identity proves local null-energy violation in every nontrivial localized finite corridor. The axial averaged null energy is strictly negative for localized axial-maximum profiles, and quantitative bounds relate its magnitude to the reference time advance and peak strain. We also obtain the complete finite-profile Ricci spectrum and the axial passenger tidal tensor. These results provide explicit causal and source constraints on a prepared geometry without specifying its physical formation or matter dynamics.

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arXiv2026

Planck-scale black hole remnants inside neutron stars

We examine a Planck-scale regular black-hole remnant in neutron-star matter using a T-duality-inspired effective geometry and explicitly separate geometrical results from assumptions about absorption and emission. Within the static family, the surface-gravity temperature grows as $(M-M_{\rm ext})^{1/2}$ after a small mass increase. A constant-greybody Stefan--Boltzmann approximation gives a Hawking power proportional to $(M-M_{\rm ext})^2$ and can produce a stable lower balance point and an unstable growth threshold under stated hypotheses. For $l_0=\ell_{\rm Pl}$, however, the formal lower equilibrium has an excess mass of only $10^{-45}$--$10^{-46}\,\mathrm{kg}$ for order-unity absorption efficiency, so the canonical description is inconsistent with individual neutron captures. For a fiducial area-scale absorption cross section with unit coefficient, the mean capture interval is $5.4\times10^8\,\mathrm{yr}$ and the initial nonradiating growth timescale is $9.1\times10^{27}\,\mathrm{yr}$. Under this assumption, a Planck-mass remnant inside a neutron star is therefore effectively inert and, without additional feeding, cannot destroy the star within the age of the Universe. A stationary capture--relaxation state additionally requires an effective microscopic emission law; it is not established by zero temperature alone. Hydrodynamic accretion requires control of wave effects and a mean-free-path criterion; its onset corresponds to masses of order $10^{11}$--$10^{12}\,\mathrm{kg}$ for representative matter parameters. Direct zero-point-length corrections there are negligible, of order $10^{-40}$ at a femtometre. We discuss conditional growth thresholds, critical slowing near an unstable balance, dark-matter feeding, angular-momentum transfer, and the requirements for neutron-star survival constraints.

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arXiv2026

Stochastic Thermodynamics of Dynamical Thin-Shell Wormholes: Effective Temperatures, Fluctuations and Conditional Quasi-Horizon Unification

We develop an effective stochastic and semiclassical framework for dynamical thin-shell wormholes connecting two asymptotic spacetime regions. The Israel junction geometry is separated from the phenomenological open-system description: stochastic dynamics is formulated in the radial phase space $(a,\dot a)$, whereas $σ$, $P$, and the effective temperature scales are induced observables. A one-dimensional Langevin--Fokker--Planck model is obtained only after an explicit overdamped/adiabatic approximation. Local acceleration scales are interpreted as reservoir temperatures only under additional KMS and detailed-balance assumptions. For a symmetric Schwarzschild--Schwarzschild wormhole, the transparent conservative shell defines the reference potential and restoring frequency, while dissipation and noise are introduced as open-system corrections. The stable linear response is closed by the constitutive slope $β_0^2=(dP/dσ)_0$. For $2M<a_0<3M$, stability requires a lower bound on $β_0^2$ that diverges as $a_0\to2M^+$. Moreover, the derivative along the family of static configurations equals the marginal-stability slope and must not be identified with the constitutive derivative of a stable stochastic shell. The reduced model determines stationary distributions, fluctuations, correlations, Markovian and non-Markovian spectra, and entropy production. Near the horizon, the junction conditions yield $T_{\rm stress}=T_{\rm acc}+O(\hbarγ/a)$, with equality at leading order under a specified dominance condition. A peeling temperature participates only if the ray-tracing map satisfies generalized Unruh/peeling and adiabaticity conditions. Thus, the three-temperature correspondence is conditional, while the stochastic coefficients remain phenomenological pending a microscopic shell--reservoir model.

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arXiv2026

Quantum-gravity-inspired Alcubierre warp-drive geometries

String T-duality, through its correspondence with path-integral duality, endows the low-energy propagator with a zero-point length $l_0=2π\sqrt{α'}$ that softens the short-distance behavior of gravitational fields. Motivated by the regular black-hole construction obtained from the corresponding smeared source, we formulate a T-duality-inspired Alcubierre geometry by identifying the warp profile with the complementary cumulative mass fraction of that source. For the effective one-scale choice $f(r_s)=1-r_s^3/(r_s^2+l^2)^{3/2}$, with $l^2=R^2+l_0^2$, the metric is $C^2$ at the bubble center and smooth elsewhere, while the energy density, its volume integral, and the York expansion are available in closed form. In particular, $E=-(15π/1024)v_s^2 l$ in geometric units and $|ρ_E|$ is bounded by a constant times $v_s^2/l^2$. Within this one-scale family, the model therefore removes the divergence associated with the profile-contraction limit $R\to 0$ at fixed velocity, because $l$ cannot fall below $l_0$. This should not be confused with a regularization of the conventional Alcubierre thin-wall limit at fixed macroscopic bubble radius, in which an independent wall thickness is taken to zero. We emphasize, however, that the construction is an effective ansatz rather than a derivation from string-corrected field equations: $R$ is a macroscopic profile scale, the choice $l^2=R^2+l_0^2$ is an interpolation, and the classical $l_0\to 0$ limit remains a smooth thick-walled profile rather than the distributional Alcubierre top hat. Exotic matter remains necessary, and neither semiclassical stability nor a modified quantum energy inequality is claimed without an explicit renormalized stress-tensor calculation.

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arXiv2026

Accretion dynamics of thin and thick disks in charged Kalb-Ramond black holes

In this work, we investigate the accretion properties of an electrically charged, static, spherically symmetric black hole in Kalb-Ramond (KR) gravity, where a dimensionless parameter $l$ characterizes spontaneous Lorentz symmetry-breaking. We first constrain the free charge and KR parameter space, $(Q,l)$, by comparing the predicted shadow radius of this family of solutions with the bounds inferred from Event Horizon Telescope observations of Sagittarius A*. The inclusion of electric charge introduces a degeneracy between $Q$ and $l$, enlarging the region compatible with the observed shadow size and yielding the broad constraint $-0.3900 \leq l \leq 0.1898$, with part of the compatible parameter space extending into the naked-singularity sector. We then investigate relativistic accretion disks around the charged KR black hole within the above constraints. For geometrically thin disks, we compute the time averaged energy flux, temperature distribution, emission spectrum, time averaged torque, and mass-to-radiation conversion efficiency, finding that the radiative output increases for higher values of $l$, while lower values of $l$ suppress it. Furthermore, when coupled with the electric charge this effect is amplified. For geometrically thick, constant-angular momentum tori-shaped disks, Lorentz symmetry-breaking shifts the characteristic orbital radii, compresses the radial extent of equilibrium configurations, narrows the interval supporting finite equilibrium tori, and shifts the stationary-fluid topology toward open and unconfined configurations outside that interval. These results show how charge and spontaneous Lorentz symmetry-breaking jointly affect observational constraints and the structure and emission properties of black hole accretion flows, providing potential signatures of KR gravity in the strong-gravity regime.

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arXiv2026

Gravitational Baryogenesis in Energy-Momentum Squared~Gravity

We demonstrate that the matter sector itself can drive baryogenesis in energy-momentum squared gravity, $f(R,\mathcal T^2)$ with $\mathcal T^2\equiv T_{μν}T^{μν}$. High-density matter corrections provide new time-dependent sources for the baryon asymmetry through derivative couplings to $\mathcal T^2$ and to the full combination $f(R,\mathcal T^2)$. Notably, the decoupling temperature is not treated as a free parameter; instead, it is fixed by the freeze-out of the Weinberg $B-L$-violating operator, directly linking the asymmetry to the light-neutrino mass scale and to the modified expansion history. Analyzing representative powers $n=1/4$, $n=1/2$, $n=5/8$, and $n=1$, we find that entropy evolution sharply distinguishes the models. The $n=1/4$ branch cannot serve as a self-contained radiation-era model, the $n=1/2$ branch is entropy-safe but too weak to reproduce the observed asymmetry, and the $n=1$ branch survives only after entropy dilution. Remarkably, the $n=5/8$ branch provides the cleanest viable realization, generating the observed baryon-to-entropy ratio with controlled effective-field-theory hierarchies.

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arXiv2026

A regularized photon-pair geometry motivated by the ER=EPR conjecture

We regularize the Aichelburg-Sexl shock-wave geometry for massless particles by smearing the point-like source over a string-inspired length scale $l_0$. A radial extension of the transverse geometry admits a zero-throat Einstein-Rosen interpretation. For a photon wave packet of longitudinal extent $L$, the regularized gravitational self-energy of a transversely separated pair is $E_{\rm GSE}\sim \frac{4G(\hbarω)^2}{c^4L} \ln\!\left(\frac{d^2}{l_0^2}\right)$, where $d$ is the transverse separation. The factor $1/L$ strongly suppresses the interaction, giving gravitational stability times greater than $10^{30}$ years for optical photons. We also examine an effective two-dimensional entanglement-entropy description of the shock-wave geometry. The entropy construction reproduces the same dependence on the scales $d$, $L$, and $l_0$, and can be calibrated to reproduce the normalization of the direct self-energy calculation. However, we find that the gravitationally induced excess entropy, and hence the self-energy inferred from this prescription, is the same for Bell-entangled and unentangled photon pairs having identical energy-momentum distributions. This agrees with the direct geometrical calculation, since the classical two-shock metric depends on the stress-energy tensor but not on the polarization entanglement. The present construction should therefore be understood as a semiclassical ER-like geometry motivated by ER=EPR, rather than as a complete realization of the conjecture. A geometry whose connectivity tracks the amount of quantum entanglement would require a quantum-gravitational description sensitive to state-dependent correlations beyond the one-point stress-energy tensor.

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arXiv2026

Thin-shell wormholes in cosmic voids

Cosmic voids are underdense regions that can provide an effective large-scale environment with a de Sitter-like gravitational behavior. Motivated by recent black-hole solutions embedded in void density profiles, we construct a symmetric thin-shell wormhole by gluing two copies of the positive-lapse region of a black hole inside a cosmic void. The surface stresses are obtained from the Darmois--Israel junction conditions, and the corresponding null, weak, dominant, and strong energy-condition combinations are written directly in terms of the void mass function and density profile. We further develop the thermodynamics of the static shell, deriving a first law that relates the shell entropy to the black-hole and cosmological-like horizon entropies. We then formulate the radial dynamics of the throat through an effective potential and study the local stability of static configurations when the exotic matter on the shell obeys either a generalized cosmic Chaplygin gas or a modified cosmic Chaplygin gas equation of state. In both models the Chaplygin parameter $B$ is fixed by the static junction condition, so that the remaining stability test is governed by the void geometry and by the equation-of-state parameters. Numerical results reveal that GCCG-supported configurations are generically unstable, whereas MCCG-supported wormholes can be stable for a sufficiently large linear term in the equation of state. The resulting framework connects the de Sitter-like structure of cosmic voids with the standard thin-shell wormhole formalism and provides a starting point for identifying stable or unstable wormhole configurations located between the black-hole and cosmological-like horizons of the void spacetime.

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arXiv2026

A general static and spherically symmetric black hole in traceless metric-affine bumblebee gravity

We construct and analyze a general exact static and spherically symmetric black hole solution in traceless metric-affine bumblebee gravity by considering a vacuum expectation value with temporal and radial components. The temporal component remains constant and the auxiliary geometry is Ricci flat, while Lorentz symmetry breaking is transferred to the physical metric through the metric-affine deformation. After introducing the areal radius and normalizing the Killing time at spatial infinity, the geometry is controlled by a single invariant deformation parameter. The resulting spacetime is asymptotically conical, possesses a regular Killing horizon, and retains the causal structure of the Schwarzschild black hole, with a curvature singularity located at the origin. Initially, we examine its thermodynamic properties, gravitational Doppler effect, and tidal forces. Lorentz symmetry breaking modifies the Hawking temperature and the magnitude of the tidal field, although the characteristic ratio between radial stretching and transverse compression is preserved. The thermodynamic-topology analysis reveals no critical point within the admissible black hole sector. Furthermore, massless scalar perturbations are also investigated through WKB and characteristic time domain solution, showing decaying profiles throughout the sampled parameter range and deformation-dependent quasinormal frequencies. Finally, light deflection and perihelion advance are used to establish stringent Solar System bounds on the invariant deformation, which, at leading order, depends on a combination of the temporal and norm sectors of the bumblebee field.

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arXiv2026

A Buchdahl solution generator for Einstein-sigma gravity and multiscalar-tensor gravity

Buchdahl transformations provide a geometric route from vacuum spacetimes to scalar-field solutions. We formulate this mechanism as a target-covariant solution generator for Einstein gravity coupled to nonlinear sigma models in $D>3$ dimensions. On any connected region in which a Ricci-flat seed admits a non-null hypersurface-orthogonal cyclic coordinate with fixed-sign nonzero norm, the logarithm of that norm defines a Buchdahl potential $Σ$. A one-parameter redistribution of the cyclic norm then generates the exact Ricci source $R_{AB}=C_D(β)\partial_AΣ\partial_BΣ$, with $C_D(β)=\frac{D-2}{D-3}(1-β^2)$, while $Σ$ remains harmonic in the deformed metric. Composing $Σ$ with an affinely parametrized geodesic of a non-degenerate target metric whose squared speed is $C_D(β)$ therefore yields an exact Einstein--sigma-model solution. For positive-definite targets, the corresponding pullback condition also implies that the nontrivial scalar map has one-dimensional image; for indefinite targets, the geodesic ansatz remains a sufficient solution-generating sector but need not exhaust all possible realizations of the same rank-one spacetime source. Through the Einstein/Jordan-frame correspondence, the construction gives vacuum tensor--multiscalar solutions as well. We illustrate the unified generator with flat, spherical, hyperbolic and axion--dilaton targets and with Tangherlini, Weyl, Kasner, Rosen-wave, Gowdy, Kaluza--Klein bubble and C-metric seeds. The result separates cleanly the geometric Buchdahl source from its target-space realization and extends this rank-one cyclic sector across static, cosmological, wave and accelerating geometries.

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arXiv2026

Mass inflation or scalaron rigidity: Cauchy horizons in metric f(R)

We develop a double-null framework for Cauchy-horizon dynamics in metric $f(R)$ gravity that separates ordinary mass inflation from scalaron cancellation and genuine escape from the blueshift instability. Starting from the exact spherical Hawking-mass transport equation, we show that a regular nondegenerate Cauchy horizon undergoes mass inflation whenever the effective longitudinal source is eventually nonnegative, uniformly dominates the mixed channel, and has a divergent blueshift-weighted integral. Hence bounded mass requires sufficiently strong blueshift-integrable suppression of this source or an independent competing channel. We then analyze the weaker branch in which the scalaron cancels only the leading Price-tail contribution. Under non-cancelling transverse and trace asymptotics, this forces $|R|\to\infty$ and an asymptotically linear high-curvature theory, $f(R)/R\to F_->0$, where $F_-\equiv\lim_{v\to\infty}f_R$. The curvature coefficient is fixed explicitly by the transverse asymptotics. For regular model classes with a finite high-curvature $f_{RR}$ limit one obtains $f_{RR}\to0$. More sharply, every eventually viable branch with $f_{RR}>0$ is forced to $R\to-\infty$ and $Λ_\infty>-1$; the standard scalaron mass parameter then diverges, while an additional differentiable rate condition yields $m_{\rm sc}^2\sim F/(3f_{RR})\to+\infty$. In the Einstein frame the scalaron null kinetic contribution is nonnegative, so the Jordan-frame cancellation cannot be interpreted as negative scalaron null energy. The result is a local rigidity classification rather than a global theorem of strong cosmic censorship.

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arXiv2026

Embedding Wormholes and Dyonic Black Strings in Warped Braneworlds via Local Sum Rules

Building on our previous work [1], where the Local Sum Rules (LSR) were established, we investigate the construction of compact objects in Randall-Sundrum braneworlds supported by matter fields that are dynamically consistent and localizable. We begin by revisiting the Chamblin et al. black string, highlighting its role as a foundational higher-dimensional solution. We then show that the Ellis-Bronnikov wormhole can be consistently embedded in this framework via a localized free scalar field, providing a simple yet nontrivial example of a braneworld compact object. Finally, we derive two novel black string solutions sourced by a localized nonlinear electrodynamics (NED) theory with Lagrangian $\mathcal{L}(\mathcal{F}) = -β\sqrt{\mathcal{F}}$, corresponding to purely magnetic and dyonic configurations. The purely magnetic solution reproduces the classical Letelier string cloud on the brane, while the dyonic solution generalizes it to include electric charge, closely paralleling the Letelier-Alencar construction. Both NED solutions reduce smoothly to the Chamblin et al. black string in the limit $β\to 0$, illustrating how localized higher-dimensional matter fields can consistently support braneworld compact objects and connect higher-dimensional physics with well-known four-dimensional solutions.

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arXiv2026

Differential Obstructions to Curvature-Dependent Conformal Transformations

Curvature-dependent conformal rules are local forward assignments on known metrics, but they are not generically local changes of metric variables. We study the nondegenerate class $\widetilde g_{μν}=F(R[g])g_{μν}$, with $F>0$ and $F_R\neq0$. Introducing an independent auxiliary scalar localizes the forward map, while recovering the original metric requires a differential constraint. The complete inverse metric tangent map contains the nonpolynomial projector $ξ^μξ^ν/ξ^2$; hence no differentiable finite-jet inverse, i.e., a formula involving only finitely many derivatives at the same point, exists on an open set of unrestricted metrics. Branchwise functional inverses may nevertheless exist after boundary or Cauchy data are specified. Metric $f(R)$ gravity gives an explicit realization: its local Einstein-frame scalar--tensor representation is a parent theory, whereas a metric-only Einstein-side description requires a differential section governed by a normal operator. We derive the pulled-back classical Hessian and its off-shell embedding term, and in the quadratic model show how constrained Gaussian elimination produces the scalaron nonlocal kernel, the corresponding normal determinant for the displayed measure, and the zero-mode compatibility condition. Exact parent and metric solutions remain equivalent; the obstruction concerns locality and the off-shell variational and fluctuation domains. These results provide a precise framework for assessing curvature-dependent frame transformations in modified gravity and clarify their implications for effective actions, semiclassical analyses, and quantum frame equivalence.

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arXiv2026

Horizon-regular cross-focusing and inner-horizon obstructions in spherical f(R) gravity

We formulate a horizon-regular double-null criterion for regular inner marginal horizons in spherically symmetric metric $f(R)$ gravity. Using normalized outgoing and ingoing radial null vectors $\ell^μ$ and $n^μ$, with $n^μ$ affinely parametrized, we derive an exact evolution law for the area-weighted outgoing expansion $r^2θ_{(\ell)}$. Its source is controlled by the scalaron $F\equiv f_R>0$ and by a mixed quantity $\mathcal{P}_{\ell n}$ containing matter, scalaron derivatives, and the curvature potential. If $\mathcal{P}_{\ell n}\leq F/r^2$ along a regular ingoing null segment issuing from a nondegenerate future outer marginal sphere, then the outgoing expansion cannot return to zero, and no second regular marginal sphere of the same family can occur on that generator. Conversely, a nondegenerate future inner marginal sphere requires the reverse inequality, so an outer--inner pair necessarily entails a source reversal and an exact integral balance. No trapped-region assumption is required. In the static limit, the criterion reduces to a horizon-regular relation involving the radial derivative of the metric function and remains valid in the degenerate case under the stated regularity conditions. It reproduces the Reissner--Nordström classification and is verified in an exact charged, nonconstant-curvature $f(R)$ black hole with a nonconstant scalaron. The resulting Cauchy-horizon statement is conditional and applies only when the candidate boundary is also a regular nondegenerate future inner marginal horizon.

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arXiv2026

Exact general relativistic solutions for a cylindrically symmetric stiff fluid matter source

In this work, we derive the general solutions for a cylindrically symmetric space-time filled with a cosmological perfect fluid obeying $p=γρ$ ($0\leq γ\leq 1$), where $γ=1$ represents a stiff or Zeldovich fluid. Using Marder's metric with coefficients depending on $t$ and $r$, we obtain explicit solutions of the gravitational field equations for the three cases $δ= 1, 0, -1$, corresponding to exponential, power-law, and trigonometric behaviors of the metric functions. The resulting space-times exhibit anisotropic evolution, nontrivial expansion and shear, and curvature singularities, with energy density and pressure profiles determined by the integration constants. These solutions provide a comprehensive framework for modeling cylindrically symmetric cosmologies, offering insights into early-universe dynamics and anisotropic gravitational phenomena. The versatility of the solutions also opens avenues for extensions to higher-dimensional or modified gravity scenarios, making them a valuable tool for both theoretical and phenomenological studies in general relativity.

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arXiv2026

Scalaron-modified null focusing and radial monotonicity in static f(R) gravity

We derive an exact radial monotonicity law for static, spherically symmetric spacetimes in metric \(f(R)\) gravity. For \(ds^2=A(r)dt^2-dr^2/B(r)-r^2dΩ^2\), the matter contribution and the scalaron Hessian combine into an effective radial-convergence numerator that fixes the derivative of \(\Q=B/A\). On every connected static interval with \(A>0\), \(B>0\), and \(f_R\equiv df/dR>0\), its sign therefore determines the monotonicity of \(\Q\). The integrated identity retains the finite, generally nonzero value of \(B/A\) at a regular nondegenerate Killing horizon; consequently, a fixed-sign convergence condition orders the horizon endpoint ratios rather than excluding two horizons. A zero-integral obstruction arises for equal endpoint values, including boundaries where \(B\to0\) while \(A\) remains finite and nonzero. Saturation is equivalent to \(B/A=\mathrm{const}\), and in vacuum requires a scalaron profile linear in the areal radius. We illustrate the strict non-saturated branch, within the General-Relativity sector, using the exact constant-density stellar interior, and apply the equality and consistency diagnostics to the constant-\(X\) and power-law solutions of Multamäki and Vilja. In particular, in the Schwarzschild--de Sitter two-horizon parameter range, one constant-\(X\) solution crosses \(f_R=0\) inside the complete static patch, while a direct substitution into the original radial equation exposes an unresolved exponent mismatch in the displayed power-law family. The results provide a model-independent static-sector diagnostic and a precise starting point for a future horizon-regular treatment of black-hole interiors.

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arXiv2026

Black bounce sourced by non-minimally coupled linear electrodynamics and a canonical scalar field through a thin shell at the throat

We construct a novel class of black bounce solutions within General Relativity, sourced by a canonical scalar field non-minimally coupled to linear electrodynamics, establishing a self-consistent framework in which regular black holes and traversable wormholes are supported by ordinary bulk matter, with the necessary exoticity confined to an infinitesimally thin defect at the bounce.

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