Adam The setup you described seems to have singularities on the boundary. This is usually a very elegant ansatz for an analytic study, but is disastrous in a numerical study. As a first step, it will be necessary to convert this ansatz to a setup that has no singularities, i.e. metric is non-zero and non-infinite everywhere, and the curvature also needs to be finite everywhere. There are several generic methods for that (e.g. "subtracting" or "dividing by" singular terms), but it remains a non-trivial task.
Most people use the Einstein Toolkit to evolve a dynamical spacetime. Looking for a stationary solution would be called "setting up initial conditions" in our lingo. While the Einstein Toolkit has many kinds of initial conditions built in, it's usually a bit involved to set up a new kind of initial condition. Even so, the Einstein Toolkit is geared towards solving R_ab = 0 (in vacuum). What you describe sounds like a very different method. I don't know how one would formulate allowing for non-zero Ricci curvature without prescribing a matter content in terms of an elliptic PDE. If you can formulate your problem in terms of elliptic PDEs then I (or others!) can point you towards thorns or modules to study. Otherwise you're probably still a step away from using a numerical method. I might have misunderstood your problem description, though. Do you have a pointer to a write-up that gives more details? -erik On Tue, Mar 2, 2021 at 11:40 AM Adam Herbst <[email protected]> wrote: > > 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 > _______________________________________________ > Users mailing list > [email protected] > http://lists.einsteintoolkit.org/mailman/listinfo/users -- Erik Schnetter <[email protected]> http://www.perimeterinstitute.ca/personal/eschnetter/ _______________________________________________ Users mailing list [email protected] http://lists.einsteintoolkit.org/mailman/listinfo/users
