Building a wormhole

It's a bit hard to exactly construct a stress-energy tensor similar to a wormhole in normal space since part of the assumption is that the topology isn't simply connected, but consider the following scenario :

Take a thin-shell stress-energy tensor such that

$$T_{\mu\nu} = \delta(r - a) S_{\mu\nu}$$

with $S_{\mu\nu}$ the Lanczos surface energy tensor, where the Lanczos tensor is similar to a thin-shell wormhole. For a static spherical wormhole, that would be

\begin{eqnarray} S_{tt} &=& 0\\ S_{rr} &=& - \frac{2}{a}\\ S_{\theta\theta} = S_{\varphi\varphi} &=& - \frac{1}{a}\\ \end{eqnarray}

If we did this by the usual cut and paste method (cutting a ball out of spacetime before putting it back in, making no change to the space), the Lanczos tensor would be zero due to the normal vectors being the same (there's no discontinuity in the derivatives). But we're imposing the stress-energy tensor by hand here. This is a static spherically symmetric spacetime, for which we can use the usual metric

$$ds^2 = -f(r) dt^2 + h(r) dr^2 + r^2 d\Omega^2$$

with the usual Ricci tensor results :

\begin{eqnarray} R_{tt} &=& \frac{1}{2 \sqrt{hf}} \frac{d}{dr} \frac{f'}{\sqrt{hf}} + \frac{f'}{rhf}\\ R_{rr} &=& - \frac{1}{2 \sqrt{hf}} \frac{d}{dr} \frac{f'}{\sqrt{hf}} + \frac{h'}{rh^2}\\ R_{\theta\theta} = R_{\varphi\varphi} &=& -\frac{f'}{2rhf} + \frac{h'}{2rh^2} + \frac{1}{r^2} (1 - \frac{1}{h}) \end{eqnarray}

Using $R_{\mu\nu} = T_{\mu\nu} - \frac 12 T$ (this will be less verbose), we get that $T = -\delta(r - a) [2(ah)^{-1} + 2 (ar^2)^{-1}]$, and then

\begin{eqnarray} \frac{1}{2 \sqrt{hf}} \frac{d}{dr} \frac{f'}{\sqrt{hf}} + \frac{f'}{rhf} &=& \delta(r - a) \frac{1}{a} (\frac{1}{h} + \frac{1}{r^2}) \\ - \frac{1}{2 \sqrt{hf}} \frac{d}{dr} \frac{f'}{\sqrt{hf}} + \frac{h'}{rh^2} &=& \delta(r - a) \frac{1}{a}[\frac{1}{h} + \frac{1}{r^2} - 1]\\ -\frac{f'}{2rhf} + \frac{h'}{2rh^2} + \frac{1}{r^2} (1 - \frac{1}{h}) &=& \delta(r - a) \frac{1}{a}[\frac{1}{h} + \frac{1}{r^2} - 1] \end{eqnarray}

This is fairly involved and I'm not gonna solve such a system, so let's make one simplifying assumption : just as for the Ellis wormhole, we'll assume $f = 1$, which simplifies things to

\begin{eqnarray} 0 &=& \delta(r - a) \frac{1}{a} (\frac{1}{h} + \frac{1}{r^2}) \\ \frac{h'}{rh^2} &=& \delta(r - a) \frac{1}{a}[\frac{1}{h} + \frac{1}{r^2} - 1]\\ \frac{h'}{2rh^2} + \frac{1}{r^2} (1 - \frac{1}{h}) &=& \delta(r - a) \frac{1}{a}[\frac{1}{h} + \frac{1}{r^2} - 1] \end{eqnarray}

The only solution for the first line would be $h = - r^2$, but then this would not be a metric of the proper signature. I don't think there is a solution here (or if there is, it will have to involve a fine choice of the redshift function), which I believe stems from the following problem :

From the Raychaudhuri equation, we know that in a spacetime where the null energy condition is violated, there is a divergence of geodesic congruences. This is an important property of wormholes : in the optical approximation, a wormhole is just a divergent lense, taking convergent geodesic congruence and turning them into divergent ones. This is fine if the other side of the wormhole is actually anoter copy of the spacetime, but if this is leading to inside flat space, this might be a problem (once crossing the wormhole mouth, the area should "grow", not shrink as it would do here).

A better example, and keeping in line with the bag of gold spacetime, is to consider a thin-shell wormholes that still has trivial topology. Take the two manifolds $\mathbb R^3$ and $\mathbb S^3$. By the Gauss Bonnet theorem, a sphere must have a part in which it has positive curvature (hence focusing geodesics). Then perform the cut and paste operation so that we have the spacetime

$$\mathcal M = \mathbb R \times (\mathbb R^3 \# S^3)$$

Through some topological magic, this is actually just $\mathbb R^4$. The thin-shell approximation is easily done here, and it will give you the proper behaviour : geodesics converge onto the mouth, diverge upon crossing the mouth, then go around the inside of the sphere for a bit before possibly getting out.

From there, it's possible to take various other variants, such as smoothing out the mouth to make it more realistic (which will indeed give you a bag of gold spacetime), as well as a time dependancy to obtain this spacetime from flat Minkowski space.

This article deals with passable wormhole

Wormhole Trafficable properties As we have seen, there are several objections to the possibility of come true interstellar travel through holes black or wormholes from Schwarzschild. To make passable a wormhole should have the following properties:

  1. spherically symmetric and static geometry. It is a condition imposed to simplify the calculations.
  2. Being solution of Einstein's equations.
  3. contain a throat (a narrow fragment space-time, highly curved) connecting two asymptotically flat regions of spacetime.
  4. Absence of horizons to allow the trip in two directions.
  5. small tidal forces, not to destroy possible travelers.
  6. Allow a traveler can cross the wormhole in a suitable time and at a time coordinated reasonable. The latter is measured by an observer far away from the sources of the gravity field.
  7. Matter and fields that generate the curvature of spacetime is described by a tensor of energy- -momento with physical meaning.
  8. The solution must be stable for small perturbations during the passage of the traveler.
  9. Finally, the wormhole must be built with a finite amount of material, certainly less than the the material content of the universe, and an interval of finite time, clearly less than the age of the universe.