Hi all, Before tackling the learning curve, I want to see if there's any chance I can do what I'm hoping to, because it seems unlikely, but with something as highly developed as the ET appears to be, you never know!
I want to find a stationary spacetime, in which each time-slice has a topological defect anchored at the origin. Specifically, we take an "extruded sphere" (S^2 x [0,1]), set the metric such that the radii of the end-spheres goes to zero, and attach each end to one "half-space" of the origin (theta in [0, pi/2] and theta in [pi/2, pi]). This can be done "smoothly" by having g_{theta,theta} from outside approach sin^2(2 * theta) instead of sin^2(theta), so that a radial cross-section becomes a pair of spheres, one for each half-space, instead of a single sphere. Thus the defect is actually a "bridge" between these two half-spaces, and geodesics through the origin traverse this loop. But the curvature does become infinite at the origin.
Now the thing is, what I really want to do is start with the ansatz described above (I already have a formula for the metric), and make it converge to a solution of the Einstein-Hilbert action, while keeping it stationary. But in this case it is NOT the same as the vacuum field equation, because the "boundary condition" of the topological singularity will not allow the Ricci curvature to disappear, even when we minimize total curvature. Or so I believe. So that's why it has to be a purely action-based approach, if that even makes sense.
So I hope this was coherent. And if it is possible, can you let me know which modules I should start getting familiar with in order to give it a shot?
Thank you for reading! Cheers,
Adam