Wormhole

Remarks

Example: the Ellis wormhole, with $\Phi = 0$ and $r(\ell) = \sqrt{\ell^2 + n^2}$ for a constant $n > 0$. Check: $r$ has its minimum $n$ at $\ell = 0$, $\sqrt{\ell^2 + n^2}/\abs{\ell} \to 1$, and $\Phi = 0$ is bounded.

Non-example: $\Phi = 0$ and $r(\ell) = \ell^2 + 1$ satisfy clauses 1 and 3 but fail clause 2, since $r(\ell)/\abs{\ell} \to \infty$.

Consequence for the matter content: for the null vector $k = e^{-\Phi}\partial_t + \partial_\ell$, a computation gives $R_{ab}k^a k^b = \frac{2}{r}\left(\Phi' r' - r''\right)$. At the throat $r' = 0$ and $r'' \ge 0$, since $\ell = 0$ is a minimum. If $r''(0) > 0$, then $R_{ab}k^ak^b < 0$, and the Einstein equations force $T_{ab}k^ak^b < 0$: the null energy condition fails there. The ends $\ell \to +\infty$ and $\ell \to -\infty$ may be two universes, or, after identifications far away, two regions of one universe.