<div dir="auto">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.</div><div dir="auto"><br></div><div dir="auto">Adam</div><div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Fri, Mar 5, 2021 at 8:39 AM Peter Diener <<a href="mailto:diener@cct.lsu.edu">diener@cct.lsu.edu</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">Hi Adam,<br>
<br>
If it indeed turns out that your problem can be cast as a 4th order <br>
elliptical PDE, I don't see any reason why this could not be simulated.<br>
In fact in the thorn NoExcision, we actually use up to a 6th order <br>
ellitpical PDE to fill in the interior of a black hole with constraint<br>
violating data that smoothly matches the exterior data. In this thorn<br>
we implemented a conjugate gradient method to solve the equations and <br>
didn't see any issues with the fact the the equations involved 6th<br>
derivatives.<br>
<br>
Cheers,<br>
<br>
Peter<br>
<br>
On Wednesday 2021-03-03 14:52, Adam Herbst wrote:<br>
<br>
>Date: Wed, 3 Mar 2021 14:52:53<br>
>From: Adam Herbst <<a href="mailto:adamdrewherbst@gmail.com" target="_blank">adamdrewherbst@gmail.com</a>><br>
>To: Erik Schnetter <<a href="mailto:schnetter@cct.lsu.edu" target="_blank">schnetter@cct.lsu.edu</a>><br>
>Cc: Einstein Toolkit Users <<a href="mailto:users@einsteintoolkit.org" target="_blank">users@einsteintoolkit.org</a>><br>
>Subject: Re: [Users] Can I simulate this exotic static topological spacetime<br>
> with the ET?<br>
><br>
>Hi Erik,<br>
>I am elated to receive such a detailed answer, and it appears you have<br>
>understood my problem perfectly, maybe better than I understand it myself. <br>
>I'll see if I can clear up the write-up I had and send it over. But I think<br>
>you are right that I have not developed this enough to be tested numerically<br>
>yet. After reading more, I think the Hilbert action approach doesn't make<br>
>sense anyway. Also, as far as I can tell, the curvature singularity is<br>
>unavoidable due to the topological transition to the loop.<br>
><br>
>I had previously based the idea on a "curvature wave equation", which might<br>
>be an elliptic PDE but it would be fourth-order in the metric. Could a<br>
>4th-order PDE be simulated?<br>
><br>
>Thank you kindly,<br>
>Adam<br>
><br>
>On Tue, Mar 2, 2021 at 1:00 PM Erik Schnetter <<a href="mailto:schnetter@cct.lsu.edu" target="_blank">schnetter@cct.lsu.edu</a>> wrote:<br>
> Adam<br>
><br>
> The setup you described seems to have singularities on the<br>
> boundary.<br>
> This is usually a very elegant ansatz for an analytic study, but<br>
> is<br>
> disastrous in a numerical study. As a first step, it will be<br>
> necessary<br>
> to convert this ansatz to a setup that has no singularities,<br>
> i.e.<br>
> metric is non-zero and non-infinite everywhere, and the<br>
> curvature also<br>
> needs to be finite everywhere. There are several generic methods<br>
> for<br>
> that (e.g. "subtracting" or "dividing by" singular terms), but<br>
> it<br>
> remains a non-trivial task.<br>
><br>
> Most people use the Einstein Toolkit to evolve a dynamical<br>
> spacetime.<br>
> Looking for a stationary solution would be called "setting up<br>
> initial<br>
> conditions" in our lingo. While the Einstein Toolkit has many<br>
> kinds of<br>
> initial conditions built in, it's usually a bit involved to set<br>
> up a<br>
> new kind of initial condition.<br>
><br>
> Even so, the Einstein Toolkit is geared towards solving R_ab = 0<br>
> (in<br>
> vacuum). What you describe sounds like a very different method.<br>
> I<br>
> don't know how one would formulate allowing for non-zero Ricci<br>
> curvature without prescribing a matter content in terms of an<br>
> elliptic<br>
> PDE.<br>
><br>
> If you can formulate your problem in terms of elliptic PDEs then<br>
> I (or<br>
> others!) can point you towards thorns or modules to study.<br>
> Otherwise<br>
> you're probably still a step away from using a numerical method.<br>
> I<br>
> might have misunderstood your problem description, though. Do<br>
> you have<br>
> a pointer to a write-up that gives more details?<br>
><br>
> -erik<br>
><br>
><br>
><br>
> On Tue, Mar 2, 2021 at 11:40 AM Adam Herbst<br>
> <<a href="mailto:adamdrewherbst@gmail.com" target="_blank">adamdrewherbst@gmail.com</a>> wrote:<br>
> ><br>
> > Hi all,<br>
> > Before tackling the learning curve, I want to see if there's<br>
> any chance I can do what I'm hoping to, because it seems<br>
> unlikely, but with something as highly developed as the ET<br>
> appears to be, you never know!<br>
> ><br>
> > I want to find a stationary spacetime, in which each<br>
> time-slice has a topological defect anchored at the origin. <br>
> Specifically, we take an "extruded sphere" (S^2 x [0,1]), set<br>
> the metric such that the radii of the end-spheres goes to zero,<br>
> and attach each end to one "half-space" of the origin (theta in<br>
> [0, pi/2] and theta in [pi/2, pi]). This can be done "smoothly"<br>
> by having g_{theta,theta} from outside approach sin^2(2 * theta)<br>
> instead of sin^2(theta), so that a radial cross-section becomes<br>
> a pair of spheres, one for each half-space, instead of a single<br>
> sphere. Thus the defect is actually a "bridge" between these<br>
> two half-spaces, and geodesics through the origin traverse this<br>
> loop. But the curvature does become infinite at the origin.<br>
> ><br>
> > Now the thing is, what I really want to do is start with the<br>
> ansatz described above (I already have a formula for the<br>
> metric), and make it converge to a solution of the<br>
> Einstein-Hilbert action, while keeping it stationary. But in<br>
> this case it is NOT the same as the vacuum field equation,<br>
> because the "boundary condition" of the topological singularity<br>
> will not allow the Ricci curvature to disappear, even when we<br>
> minimize total curvature. Or so I believe. So that's why it<br>
> has to be a purely action-based approach, if that even makes<br>
> sense.<br>
> ><br>
> > So I hope this was coherent. And if it is possible, can you<br>
> let me know which modules I should start getting familiar with<br>
> in order to give it a shot?<br>
> ><br>
> > Thank you for reading! Cheers,<br>
> ><br>
> > Adam<br>
> > _______________________________________________<br>
> > Users mailing list<br>
> > <a href="mailto:Users@einsteintoolkit.org" target="_blank">Users@einsteintoolkit.org</a><br>
> > <a href="http://lists.einsteintoolkit.org/mailman/listinfo/users" rel="noreferrer" target="_blank">http://lists.einsteintoolkit.org/mailman/listinfo/users</a><br>
><br>
><br>
><br>
> --<br>
> Erik Schnetter <<a href="mailto:schnetter@cct.lsu.edu" target="_blank">schnetter@cct.lsu.edu</a>><br>
> <a href="http://www.perimeterinstitute.ca/personal/eschnetter/" rel="noreferrer" target="_blank">http://www.perimeterinstitute.ca/personal/eschnetter/</a><br>
><br>
><br>
><br>
</blockquote></div></div>