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As of recent, I've been doing a bit of self-education in GR, equipped with a working knowledge of the key elements of the differential geometry in GR, and in looking at the Einstein-Rosen bridge, Einstein-Rosen Bridge

I see that geometrically it is a hyperboloid of one sheet. Now when using this manifold to calculate things like curvature and geodesics, we need a metric, which we can somewhat easily derive from the equation of the aforementioned surface.

Now per $$ R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R = \frac{8\pi G}{c^{4}} T_{\mu\nu} $$ (setting $ \Lambda = 0 $ for the simple case), one should be able to solve for the stress-energy tensor, albeit the grotesque mathematics. By this logic, based on the predetermined shape of the local spacetime, we should be able to calculate the mass and geometry of the object required to create such a metric.

My questions then are:

To what extent is this a valid statement if at all?

Given the fact that there are going to be initial value problems embedded in the PDE's with conditions that require the input of some properties of the geometry of the aforementioned object: How is it that one goes about finding physically viable objects that satisfy the EFE under the constraints that we've applied if any exists. How does one test to see if there are even any solutions that work?

For an attempt at the second questions, I was thinking that if we have to apply constraints to get rid of the unknown constants that resulted from the PDE, we just put constraints on what the solution can be, to ensure that it fits with what we want. Ie I want to make sure my sphere of matter is of this size, but can't exceed this mass.

Am I at least on the right track with this?

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    $\begingroup$ This is a perfectly good way of approaching the problem. The Alcubierre drive was formulated by starting with the desired metric and working out what stress-energy tensor was required to create it. The problem with working in this direction is that you have no way of knowing whether the stress-energy tensor you end up with is physically reasonable. For the Alcubierre drive and every wormhole I've seen the stress-energy tensor requires exotic matter and therefore (probably) isn't physically meaningful. $\endgroup$ Mar 14, 2014 at 11:11
  • $\begingroup$ Perfect! So knowing that energy density, rho c^2, is a term in the tensor, I'm assuming physicists were running into negative values for that, corresponding to either negative mass or negative volume, since c is always positive, which is where exotic matter is matter that fits the the aforementioned properties? $\endgroup$ Mar 14, 2014 at 11:17
  • $\begingroup$ Exotic matter is matter that violates some or all of the reasonable 'energy conditions' that people impose in order to model physically reasonable systems. Wikipedia lists the most famous/important ones (en.wikipedia.org/wiki/Energy_condition#Mathematical_statement) $\endgroup$
    – Danu
    Mar 14, 2014 at 11:22
  • $\begingroup$ In short then, yes! $\endgroup$ Mar 14, 2014 at 12:05

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After a little bit more research, I verified this is the exact method employed to determine whether a given metric is feasible to create with a non-exotic energy-momentum tensor, $ T_{\mu\nu} $.

Resources: General Relativity Lecture Notes by Sean M. Carroll and the arXiv research paper Passing The Einstein-Rosen Bridge, by M. O. Katanaev, published at Mod. Phys. Lett. A 29, 17, 1450090 (2014). The research paper was an excellent read, and I definitely recommend giving it a read!

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