Modified gravity is often formulated as an effective field theory (EFT), where higher-order corrections parametrize departures from general relativity (GR). We argue that such corrections should be constrained by the global causal structure of curved spacetime, in addition to the usual flat-space requirements such as positivity and unitarity. We propose that within the domain of validity of the EFT, the onset of closed timelike curves should not happen in a parametrically more accessible region than in the corresponding GR background. We test this diagnostic in the quadratic k-essence sector of scalar-tensor gravity. For stationary and axisymmetric spacetimes, the invariant test for closed axial orbits is the sign of the azimuthal component of the metric $g_{\phi \phi }$. We supplement this test by requiring a local time function in the space of Killing vectors. We apply these conditions to quadratic k-essence on Kerr-(A)dS backgrounds, with and without scalar charge. The zero-charge branch is exact Kerr-(A)dS, and we treat the charged branch perturbatively in scalar charge and in Hartle-Thorne slow rotation. Expanding for small spin $\chi =a/(GM)\ll 1$, frame dragging begins at $\mathcal{O}(\chi )$, while the quadrupolar backreaction relevant for circular closed timelike curves enters at second order in both rotation and charge. We find that, in the truncation used here, any occurrence of $g_{\phi \phi }<0$ also lies outside EFT control. A higher-order calculation or a fully nonlinear treatment is therefore needed. Finally, we discuss how quasinormal modes and black-hole echoes could probe such causal structure.

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• Alternative gravity theories
• Classical black holes
• Gravitation