[Users] Can I simulate this exotic static topological spacetime with the ET?

[Peter Diener]

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
>      > _______________________________________________
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>
>
>
>      --
>      Erik Schnetter <schnetter at cct.lsu.edu>
>      http://www.perimeterinstitute.ca/personal/eschnetter/
>
>
>