The exponential metric: Traversable wormhole and possible identification of scalar background

The static antiscalar solution of the Einstein–Klein–Gordon equations in the form of the Papapetrou exponential metric had been interpreted as a traversable wormhole with a throat at [Formula: see text]. We aim to search for the effects which could be associated with this scale and only find that the topological Gauss–Bonnet invariant swaps sign, and the value of the Keplerian frequency for circular geodesics becomes singular. At the same time, the geometric invariants of the curvature tensor have extremal values at the scale twice less than that of the throat, revealing new physical effects. In particular, the Ricci scalar at [Formula: see text] (rather than [Formula: see text]) is associated with the extremal values of thermodynamic characteristics of the scalar background. This approach in combination with the antiscalar static limit of the Einstein–Maxwell equations suggests the interpretation of the scalar background as a stable medium with a stiff equation of state, formed by the neutral superposition of ambient quasistatic electric fields.