Hi Erik, I am elated to receive such a detailed answer, and it appears you have understood my problem perfectly, maybe better than I understand it myself. I'll see if I can clear up the write-up I had and send it over. But I think you are right that I have not developed this enough to be tested numerically yet. After reading more, I think the Hilbert action approach doesn't make sense anyway. Also, as far as I can tell, the curvature singularity is unavoidable due to the topological transition to the loop.
I had previously based the idea on a "curvature wave equation", which might be an elliptic PDE but it would be fourth-order in the metric. Could a 4th-order PDE be simulated? Thank you kindly, Adam On Tue, Mar 2, 2021 at 1:00 PM Erik Schnetter <[email protected]> wrote: > 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/ >
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