Stability for a formally determined Lorentzian inverse problem

We consider the inverse problem of recovering a potential $q$, depending on space and time, in the wave equation $({\mathrm{\square }}_g+q)u=0$ on a Lorentzian manifold. We prove Lipschitz stability for a formally determined version of this problem under a $\mathit{\text{null-cone}}\mathit{\text{foliation}}$ assumption. As a consequence, we obtain uniqueness in the Lorentzian Calderón problem for $q$ under this assumption. Our approach is based on a modification of the Bukhgeim–Klibanov method using distorted plane waves.