[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/
> >
> >
> >
>
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