Simple traversable wormholes violating energy conditions only near the Planck scale

We present a static and axisymmetric traversable wormhole spacetime with vanishing Arnowitt-Deser-Misner (ADM) mass which is characterized by a length parameter $l$ and a deformation parameter $a$ and reduces to the massless Kerr vacuum wormhole as $l\to 0$. The spacetime is analytic everywhere and regularizes a ring-like conical singularity of the massless Kerr wormhole by virtue of a localized exotic matter which violates the standard energy conditions only near the wormhole throat. In the spherically symmetric case ($a=0$), the areal radius of the wormhole throat is exactly $l$ and all the standard energy conditions are respected outside the proper radial distance approximately $1.60l$ from the throat. While the curvature at the throat is beyond the Planck scale if $l$ is identical to the Planck length $l_{\rm p}$, our wormhole may be a semi-classical model for $l\simeq 10l_{\rm p}$. With $l=10l_{\rm p}$, the total amount of the negative energy supporting this wormhole is only $E\simeq -26.5m_{\rm p}c^2$, which is the rest mass energy of about $-5.77\times 10^{-4}{\rm g}$. It is shown that the geodesic behavior on the equatorial plane does not qualitatively change by the localization of an exotic matter.

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