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

[Adam Herbst]

Hi Erik,
This is quite sobering.  I am very grateful for the in-depth response, and
frankly in awe of all you folks are doing.  Thank you so much!

Adam

On Mon, Mar 8, 2021 at 6:16 PM Erik Schnetter <schnetter at cct.lsu.edu> wrote:

> Adam
>
> To solve a problem numerically, one must first have a well-posed
> formulation of the problem, and then choose a well-posed
> discretization. Both are difficult to obtain from the equations. If
> one just implements an equation, some boundary and/or initial
> conditions, and then runs a solver, most likely things won't work, and
> one won't have the slightest idea what is going wrong. In addition to
> the above, you'll need some intuition for length scales, time scales,
> curvature scales, etc. Starting with a 4d spacetime is probably
> hopeless.
>
> Is there a way to simplify the problem to fewer dimensions? To simpler
> equations? Maybe to simpler physics even, solving a strawman problem?
>
> For example, when learning how to solve the Einstein equations (which
> are nonlinear tensorial wave equations), we started with solving the
> linear scalar wave equation in one dimension. If there is no
> one-dimensional case, then maybe assuming axisymmetry or stationarity
> will help, or maybe one can study a linearization of the equations
> about some background, etc.
>
> Even when simulating binary black holes (which is, by now, a well
> understood problem, since we have been simulating them for 15 years),
> it is difficult to get started from scratch. Most people start by
> taking an existing simulation and making small variations (masses,
> spins, initial velocities, etc.), or they take a known physical
> scenario and change numerical parameters (resolution, boundaries,
> numerical methods). The "original" black hole simulations were quite
> difficult to obtain and were based on years of experience, including
> experience from axisymmetric black hole simulations from many years
> earlier.
>
> Since you are interested in studying a completely new set of
> equations, I suggest to consider first a much simplified problem. I
> wish that tools such as the Einstein Toolkit were black box solvers
> (similar to Mathematica's "Integrate" function), but in truth we're
> far away from that...
>
> -erik
>
>
>
>
>
> On Mon, Mar 8, 2021 at 5:19 PM Adam Herbst <adamdrewherbst at gmail.com>
> wrote:
> >
> > Hi Erik / Peter,
> > Here is the write-up of the idea I'd like to simulate.  I know it is
> pretty outlandish and not very likely to be true at the end of the day, but
> I can't shake the fact that it seems to explain the baryons so naturally.
> So I'd be ecstatic if you'd take a look and see if you think it would be
> possible to simulate this model of the electron.  Even if I could just use
> the Toolkit for something like calculating the d'Alembertian of the Riemann
> tensor, so I could play with the metric and try to get it to converge to
> zero.
> >
> >
> https://adamdrewherbst.pythonanywhere.com/welcome/spacetime/index?language=english&section=brief
> >
> > But honestly, I would really appreciate it if any of you spacetime
> experts could tell me your reaction to the model as a whole, because it's
> hard to get that kind of feedback!  If you see a multitude of reasons it
> should be dumped without further ado, well, that would be valuable too.
> But I understand you may not have the time for that.  In any case, looking
> forward to a response!
> >
> > Thank you,
> > Adam
> >
> >
> >
> > On Fri, Mar 5, 2021 at 5:10 PM Adam Herbst <adamdrewherbst at gmail.com>
> wrote:
> >>
> >> 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/
> >>> >
> >>> >
> >>> >
>
>
>
> --
> Erik Schnetter <schnetter at cct.lsu.edu>
> http://www.perimeterinstitute.ca/personal/eschnetter/
>
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