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

TL;DR

This work analyzes the exponential Papapetrou metric as a static antiscalar solution to the Einstein–Klein–Gordon system and interprets it as a traversable wormhole with throat at $r=M$, while exploring curvature and thermodynamic properties of the scalar background. It shows that the topological Gauss–Bonnet invariant changes sign at the throat, the Keplerian frequency for circular orbits diverges as $r\to M$, and curvature invariants peak around $r=M/2$, linking these effects to a geometrized thermodynamics framework in which the scalar background has a stiff equation of state and a temperature related to curvature via $\Theta \propto \sqrt{R}$. A ξ-scheme of thermodynamics is developed to connect local temperature to thermodynamic quantities, and the scalar background is interpreted as a neutral superposition of quasistatic electric fields, yielding a stable electrovacuum-like medium. The results offer a route to identifying a massless scalar background and suggest observational signatures in strong-field regimes alongside a thermodynamic picture of scalar fields in curved spacetime.

Abstract

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 \textit{r=M}. 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 \textit{r=M/2} (rather than \textit{r=M}) 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.

Figures (10)

• Figure 1: Two-fold representation of the effective Newtonian potential $\varphi(R)$ in curvature coordinates (orange dashed branches) in scalar field background (SF, left) and in vacuum (right) which never covers a region near the origin (marked with red dotted lines) making it inaccessible for comparison with observations. Only the lower blue branches $\varphi_0 (W_0)$ and $\varphi_+$ might be considered as meaningful.
• Figure 2: The profiles of the Kretschmann invariants $K$\ref{['kretch']}, Ricci $R$\ref{['ricci']}, and the area $A^{-1}$ of Morris-Thorn TWH resulting from \ref{['area']}. Profiles are transformed to dimensions $[r^ 2]$ (top, left), $[r^{-2}]$ (top, right), $[r]$ (bottom, left), $[r^{-1}]$ (bottom, right) and scaled in accord with \ref{['euler_char']}. Vertical dotted lines fix the critical scales at $r_K=0.442 M$ (green), $r_R=M/2$ (blue) and $r_A= M$ (red) which correspond to minima or maxima of corresponding profiles.
• Figure 3: The Gauss-Bonnet invariant $\mathcal{G}$ inside the interval $(r=M/3, \,\, r=M)$ becomes negative, which may indicate the concavity of the corresponding generating curve, in contrast to convexity outside this interval, caused by the intersection points of the generating curves $K(r)$ and $\Xi(r)$.
• Figure 4: Left: The Kretschmann invariants in isotropic coordinates on a scalar background ($K_P$) and in vacuum ($K_S$) are limited, and both at the origin and for $r>2M$ they asymptotically tend to zero, and are maximum at $r/M= 0.442$ (for $K_P(r)$, dotted line, blue) and for $r/M=0.5$ (for $K_S(r)$, dotted line, green). Right: In curvature coordinates in vacuum, the Kretschmann invariant is singular at the origin (naked singularity). For scalar background $K_P(e)=0.21978$, but information about the invariant $K_P(R)$ in the region $R/M<e=2.71$ is absent in principle, because the mapping \ref{['trasform']} does not extend to this region . The Newtonian potential in the indicated coordinates behaves similarly.
• Figure 5: Keplerian frequencies in the isotropic Schwarzschild metric $\Omega_\phi^S$\ref{['SchwKepler']} and in the Papapetrou metric $\Omega_\phi^P$\ref{['O']}, for which the vertical asymptote (dotted line) coincides with the topological limit for the Morris-Thorn type TWH, $r=M$.