Le temps peut encore être trompé express

Résumé original

tel que rédigé par les auteurs, en anglais arXiv:2605.26696

We ask whether two classical time-machine geometries, the Ori (2005) compact-vacuum-core metric and the Ahmed (2018) four-dimensional generalisation of Misner space, remain admissible exact solutions when the gravitational sector is enlarged to the recently proposed $$f(R,\mathcal{L}_{m},\Phi,X)$$ class, an extension of $$f(R,\mathcal{L}_{m})$$ that couples curvature, the matter Lagrangian density, a scalar field $$\Phi$$, and its kinetic invariant $$X = g^{\mu u} abla_{\mu}\Phi abla_{ u}\Phi$$. Working with the explicit model $$f = R + \mathcal{L}_{m} + (\lambda/2)\,X$$ and a vanishing scalar potential, we compute the curvature invariants, the modified field equations, and the effective stress-energy components produced by the harmonic scalar profile $$\Phi(x,y) = a(x^{2}-y^{2})/2$$ in both backgrounds. The Ricci scalar vanishes for the Ori metric and obeys $$R = e^{f}(f_{,xx}+f_{,yy})$$ for the Ahmed metric; the kinetic invariant takes the explicit forms $$X = a^{2}(x^{2}+y^{2})$$ and $$X = a^{2}e^{f}(x^{2}+y^{2})$$, respectively. Both metrics solve the field equations of the modified theory with anisotropic matter sources, and the chronology-violating regions $$g_{zz}<0$$ (Ori) and $$g_{\psi\psi}<0$$ (Ahmed) survive the modification. Energy-density profiles measured by a closed-timelike-curve observer match those measured by a static observer outside the chronology horizon, so the additional scalar degree of freedom in $$f(R,\mathcal{L}_{m},\Phi,X)$$ gravity does not enforce a chronology-protection mechanism in either background. The conclusion mirrors the parallel result for the Li time-machine and supplies a consistency test for scalar-extended modified gravity in non-globally-hyperbolic settings.