The metrics to describe a wormhole

[PeterDonis]

It's necessary (but not sufficient) for a metric to have a negative determinant for it to be Lorentzian, I believe.

This is correct, and it means the line element as it's written down in the OP can only be valid for $r > 2M$. But that, in itself, does not show that the spacetime geometry itself does not extend to a region that corresponds to $r \le 2M$; it just shows that, if there is such a region, it can't be described by a coordinate chart of the form given in the OP.

I'm pretty sure the stress-energy tensor is really a tensor density.

Not if you're referring to the thing that appears on the RHS of the Einstein Field Equation. That's a tensor.

 

[PeterDonis]

We might also be interested in whether or not the strong and weak energy conditions are satisfied

The weak energy condition will always be violated close enough to the throat in any wormhole metric (the Morris-Thorne paper discusses this on p. 405). For this particular metric, I think @Ibix is correct in post #14 that the WEC is violated everywhere. I would expect the null energy condition to be violated everywhere as well. Since this particular metric has a traceless stress-energy tensor, the dominant energy condition is the same as the weak; and since it has $T^t{}_t = 0$, the strong energy condition is obviously violated since the vector $T^a{}_b X^b$ will vanish for the timelike $X^b = \partial_t$. So it looks like all of the energy conditions are violated by this particular wormhole.

 

[pervect]

Oh, there's something else I wanted to say. I'm guessing that it's possible that the motivation of the OP might be to understand how the Schwarzschild geometry can be viewed as an Einstein-Rosen bridge, a sort of non-traversable wormhole.

MTW has a discussion of this view of the Schwarzschild geometry in their textbook "Gravitation". And I suppose there's also the Einstein-Rosen paper.

If this was the intent, then the discussion of the underlying issue is getting confused by introducing alternative metrics that aren't the Schwarzschild metric and turn out to have some fundamental issues such as not being Lorentzian manifolds in the interior region. But perhaps understanding the wormhole nature of the Schwarzschild geometery wasn't the original intent. More could be said on this, but it may be a digression so I won't belabor it anymore.