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

[Adam Herbst]

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/
>> >
>> >
>> >
>>
>
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