<p>In our previous work [H. Zejli, Int. J. Mod. Phys. D 34, 2550052 (2025), arXiv:2508.00035], we introduced a <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathcal {P}\mathcal {T}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">P</mi> <mi mathvariant="script">T</mi> </mrow> </math></EquationSource> </InlineEquation>-symmetric wormhole model based on a bimetric geometry, capable of generating closed timelike curves (CTCs). In this paper, we extend the analysis to the null hypersurface at the throat of this modified Einstein-Rosen bridge, where two regular Eddington-Finkelstein metrics render the geometry traversable. Using the Barrabés-Israël formalism in Poisson’s reformulation, we evaluate the null shell’s surface stress-energy tensor <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(S^{\alpha \beta }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>S</mi> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> from the jump of the transverse curvature, revealing a violation of the null energy condition: a lightlike membrane of exotic matter with negative surface energy density and positive tangential pressure. This exotic fluid acts as a repulsive source stabilizing the throat, ensuring consistency with the Einstein field equations, including conservation laws on the shell. Beyond the local characterization, we outline potential observational signatures: (i) gravitational-wave echoes from the photon-sphere cavity; (ii) horizon-scale imaging with duplicated and through-throat photon rings, and non-Kerr asymmetries; (iii) quantum effects such as <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mathcal{P}\mathcal{T}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">P</mi> <mi mathvariant="script">T</mi> </mrow> </math></EquationSource> </InlineEquation>-induced frequency pairing with possible QNM doublets and partial suppression of vacuum flux at the throat; and (iv) a relic cosmological population yielding an effective <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\Lambda _\textrm{eff}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Λ</mi> <mtext>eff</mtext> </msub> </math></EquationSource> </InlineEquation> and seeding voids. Compared with timelike thin-shell constructions, our approach is based on a null junction interpreted as a lightlike membrane, combined with <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\mathcal{P}\mathcal{T}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">P</mi> <mi mathvariant="script">T</mi> </mrow> </math></EquationSource> </InlineEquation> symmetry, providing a distinct route to traversability and clarifying the conditions under which CTCs can arise in a self-consistent framework.</p>

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Characterization of exotic matter in \(\mathcal{P}\mathcal{T}\)-symmetric wormholes

  • Hicham Zejli

摘要

In our previous work [H. Zejli, Int. J. Mod. Phys. D 34, 2550052 (2025), arXiv:2508.00035], we introduced a \(\mathcal {P}\mathcal {T}\) P T -symmetric wormhole model based on a bimetric geometry, capable of generating closed timelike curves (CTCs). In this paper, we extend the analysis to the null hypersurface at the throat of this modified Einstein-Rosen bridge, where two regular Eddington-Finkelstein metrics render the geometry traversable. Using the Barrabés-Israël formalism in Poisson’s reformulation, we evaluate the null shell’s surface stress-energy tensor \(S^{\alpha \beta }\) S α β from the jump of the transverse curvature, revealing a violation of the null energy condition: a lightlike membrane of exotic matter with negative surface energy density and positive tangential pressure. This exotic fluid acts as a repulsive source stabilizing the throat, ensuring consistency with the Einstein field equations, including conservation laws on the shell. Beyond the local characterization, we outline potential observational signatures: (i) gravitational-wave echoes from the photon-sphere cavity; (ii) horizon-scale imaging with duplicated and through-throat photon rings, and non-Kerr asymmetries; (iii) quantum effects such as \(\mathcal{P}\mathcal{T}\) P T -induced frequency pairing with possible QNM doublets and partial suppression of vacuum flux at the throat; and (iv) a relic cosmological population yielding an effective \(\Lambda _\textrm{eff}\) Λ eff and seeding voids. Compared with timelike thin-shell constructions, our approach is based on a null junction interpreted as a lightlike membrane, combined with \(\mathcal{P}\mathcal{T}\) P T symmetry, providing a distinct route to traversability and clarifying the conditions under which CTCs can arise in a self-consistent framework.