Great to know, thanks Peter! I will continue to work on it and see if it comes to that point. I really appreciate all the information.
Adam On Fri, Mar 5, 2021 at 8:39 AM Peter Diener <[email protected]> wrote: > Hi Adam, > > If it indeed turns out that your problem can be cast as a 4th order > elliptical PDE, I don't see any reason why this could not be simulated. > In fact in the thorn NoExcision, we actually use up to a 6th order > ellitpical PDE to fill in the interior of a black hole with constraint > violating data that smoothly matches the exterior data. In this thorn > we implemented a conjugate gradient method to solve the equations and > didn't see any issues with the fact the the equations involved 6th > derivatives. > > Cheers, > > Peter > > On Wednesday 2021-03-03 14:52, Adam Herbst wrote: > > >Date: Wed, 3 Mar 2021 14:52:53 > >From: Adam Herbst <[email protected]> > >To: Erik Schnetter <[email protected]> > >Cc: Einstein Toolkit Users <[email protected]> > >Subject: Re: [Users] Can I simulate this exotic static topological > spacetime > > with the ET? > > > >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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