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

Mon Mar 8 20:16:37 CST 2021

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/