[Users] Can I simulate this exotic static topological spacetime with the ET?
Adam Herbst
adamdrewherbst at gmail.com
Fri Mar 5 19:10:16 CST 2021
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 <diener at cct.lsu.edu> 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 <adamdrewherbst at gmail.com>
> >To: Erik Schnetter <schnetter at cct.lsu.edu>
> >Cc: Einstein Toolkit Users <users at einsteintoolkit.org>
> >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 <schnetter at cct.lsu.edu>
> 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
> > <adamdrewherbst at gmail.com> 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
> > > Users at einsteintoolkit.org
> > > http://lists.einsteintoolkit.org/mailman/listinfo/users
> >
> >
> >
> > --
> > Erik Schnetter <schnetter at cct.lsu.edu>
> > http://www.perimeterinstitute.ca/personal/eschnetter/
> >
> >
> >
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: http://lists.einsteintoolkit.org/pipermail/users/attachments/20210305/e2e80c86/attachment-0001.html
More information about the Users
mailing list