Once again on thin-shell wormholes in scalar-tensor gravity

It has been recently proved in a general form [1] that no wormholes can be formed in any scalartensor theory (STT) of gravity in which the non-minimal coupling function f(Φ) is everywhere positive and the scalar field Φ itself is not a ghost if matter sources of gravity respect the Null Energy Condition (NEC). In order to construct any viable and stable wormhole solution, attempts have been recently made to find such a solution in STT of gravity using thin shells instead of extended matter sources, and it has been claimed that at least in some cases (namely, in the Brans-Dicke STT for a particular range of values of the coupling constant ω ) the shell at the wormhole throat may satisfy the weak and null energy conditions [2]. In this short note we will show explicitly that, in any STT with a massless non-ghost scalar field, in all thin-shell wormholes built from two identical regions of vacuum static, spherically symmetric space-times, the shell has negative surface energy density (note that we here do not consider numerous toy wormhole models which do not represent solutions of any initially fixed equations of some metric theory of gravity). Let us begin with presenting the nonzero components of the Einstein tensor Gμ = R ν μ − 1 2δν μR for a general static, spherically symmetric space-time with the metric ds = edt − edu − edΩ, (1)