Are Petrov type-N and D spacetimes admitting CTCs valid in $f(R,\mathcal{L}_m,\Phi,X)$ gravity?

Abstract

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 $X = g^{\mu\nu}\nabla_{\mu}\Phi\nabla_{\nu}\Phi$. 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.