Gravitational Cheshire effect: Nonminimally coupled scalar fields may not curve spacetime

It is shown that flat spacetime can be dressed with a real scalar field that satisfies the nonlinear Klein-Gordon equation without curving spacetime. Surprisingly, this possibility arises from the nonminimal coupling of the scalar field with the curvature, since a footprint of the coupling remains in the energy-momentum tensor even when gravity is switched off. Requiring the existence of solutions with vanishing energy-momentum tensor fixes the self-interaction potential as a local function of the scalar field depending on two coupling constants. The solutions describe shock waves and, in the Euclidean continuation, instanton configurations in any dimension. As a consequence of this effect, the tachyonic solutions of the free massive Klein-Gordon equation become part of the vacuum.