Can the universe change signature when gravity is dynamical?

Abstract

Can a universe undergo a regular Euclidean–Lorentzian signature transition when the gravitational coupling itself is dynamical? We address this question in scalar–tensor gravity with nonminimal coupling F(ϕ)R. Although the spatially flat Friedmann–Lemaître–Robertson–Walker sector can be mapped to the Einstein frame for F > 0 and a nondegenerate scalar redefinition, not all transition properties are frame independent. For a finite, positive, and sufficiently regular conformal factor, the existence and transverse character of the type change are preserved, whereas the extrinsic geometry is not. We therefore use the Einstein frame as a solution-generating representation and impose total geodesy in the physical Jordan frame. We construct two exactly integrable classes of solutions. In the oscillator–ghost-oscillator branch, the scalar field is stationary at the transition and Jordan-frame total geodesy follows under suitable regularity assumptions on the conformal factor. In the critical exponential-potential branch, the scalar is generically nonstationary and total geodesy instead requires HEΣ=12d⁡ln⁡FdψΣψ̇Σ. We also distinguish this condition from the stronger smoothness requirements of a Kossowski–Kriele-type transverse metric. Finally, while the canonical Einstein-frame scalar does not allow effective phantom evolution, the dynamical nonminimal coupling can generate a locally superaccelerating Jordan-frame regime near the transition when d⁡ln⁡FdψΣ>0 for the chosen oscillator branch. Thus signature change persists beyond Einstein gravity, with regularity and effective cosmological behavior remaining intrinsically frame sensitive.