function of $S$ from $1$ to $\infty$, where we have set $M=1$ and used one hundred terms of the series in Eq. (\[22122013a\]). One can verify that the qualitative behavior of the curve will not change if we increase the number of terms. []{data-label="22012014a"}](f311.jpg)

### Choosing a value for $w_2$ {#20022014b}

From the metric (\[10082013c\]), it is clear that $r_2$ has to be big enough to decrease significantly the values of terms like $M/r$. However, it cannot be too big because of the conditions (\[9082013h\]) and (\[9082013i\]). To accommodate these requirements, we use $r_2=10^AM$ with $A{ \put(2,0){$>$} \put(2,-5){$\sim$}\quad \ }6$ (for more details about this constraint, see Sec. \[23122013a\]). By substituting $r_2$ in Eq. (\[18122013b\]), one finds that $w_2\approx r_2$, which together with the maximum value of $r_1$ (see Sec. \[23122013e\]) leads to $w_1+w_2\approx w_2$. This result will be used later.

The behavior of the curvature far from the throat {#04022014b}
-------------------------------------------------

From Eq. (8) in Ref. [@citeulike:5196630], we see that the only nonvanishing components of the Riemann tensor in the basis (\[10082013b\]) have the form $$\begin{aligned}
Form_1= \left(1-b/R \right)\left[ -\Phi^{\prime\prime}+\frac{1}{2}\frac{(b'-b/R)}{R-b}\Phi'-(\Phi')^2 \right], \label{23112013a} \\
\nonumber \\
Form_2=-\frac{(1-b/R)}{R}\Phi', \label{23112013b}\\
\nonumber \\
Form_3=\frac{b'R-b}{2R^3} \label{23112013c}\\
\nonumber \\
Form_4=\frac{b}{R^3}. \label{23112013d}\end{aligned}$$ From Eqs. (\[10082013f\])-(\[10082013h\]), one 