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.

Figure 1

For $0<\zeta <1/4$, ${\lambda }_1>0$, and ${\lambda }_2>0$, the potential has a global minimum at $\varphi =0$ if $\zeta \ge {\zeta }_D$. For $0<\zeta <{\zeta }_D$, it has a local maximum at $\varphi =0$ and a global minimum at $v=[4(D-1){\lambda }_2{\lambda }_1^{-1}(\zeta -{\zeta }_D)/(2\zeta -1){]}^{2\zeta /(1-4\zeta )}$. The critical value for symmetry breaking occurs at the conformal coupling $\zeta ={\zeta }_D$. For $\zeta =1/4$ this potential describes again a symmetry breaking self-interaction with the same qualitative features, where the minimum is now at the vacuum expectation value $v=\exp [1/2(-{\lambda }_1/{\lambda }_2-D+1)]$.