From Eqs. (\[10082013h\]) and (\[10082013hh\]), we see that the trace of the extrinsic curvature vanishes at $r_m$. In turn, by applying $\partial_{n_{\pm}}$ to Eq. (\[24062014h\]), one obtains $$\frac{\partial tr(K)}{\partial n_{\pm}}=\frac{2}{R^2} \left[\frac{R}{2}\frac{d}{dR}bR^{-1}+1-b/R \right]. \label{24062014i}$$ Evaluating this expression at $r_m$ for the Wyman metric \[see Eqs. (\[10082013f\]) and (\[10082013h\])\], we find that $$\frac{\partial tr(K)}{\partial n_{\pm}}\Biggl|_{r_m}=-\frac{2}{r_mR_m}(1-r_0/r_m)^{(S-1)/2}<0. \label{24062014j}$$ We have now proved that the “strong flare-out condition” is satisfied and, therefore, we have a wormhole in Wyman spacetime for $S>1$.

Conditions \[9082013dd\], \[9082013e\], \[9082013f\], \[9082013g\], (\[9082013h\]), (\[9082013i\]), (\[9082013j\]), (\[9082013l\]), e (\[9082013m\]) {#13122013a}
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Now we analyze the conditions that are “necessary” to ensure that the wormhole is traversable by humans.

### Conditions \[9082013dd\], \[9082013e\], \[9082013f\], \[9082013g\] {#23122013a}

The condition \[9082013dd\] is just a matter of convenience, hence it is not a problem. With respect to the conditions \[9082013e\], \[9082013f\] and \[9082013g\], we can assume that $r_0/r_2=2M/(Sr_2) \ll 1$ with $w_2=w_2(r_2)$ being the place where the trip ends. From Eqs. (\[10082013g\]) and (\[10082013h\]), one gets $|\Phi|\approx M/r_2$ and $b/R\approx 2M/r_2$, where all these values have been evaluated at $r_2$. Deriving Eq. (\[10082013g\]) with respect to $R$, one finds that $$\Phi'=\frac{r_0S}{2r^2} \frac{W^{(S-1)/2}}{W_m}, \label{15082013c}$$ which yields the approximation $\Phi'\approx M/r^2_2$ (Remember that we have defined $W_m=1-r_m/r$). If 