<div dir="ltr"><div>Hi Erik,</div><div>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!</div><div><br></div><div>Adam<br></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Mon, Mar 8, 2021 at 6:16 PM Erik Schnetter <<a href="mailto:schnetter@cct.lsu.edu">schnetter@cct.lsu.edu</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">Adam<br>
<br>
To solve a problem numerically, one must first have a well-posed<br>
formulation of the problem, and then choose a well-posed<br>
discretization. Both are difficult to obtain from the equations. If<br>
one just implements an equation, some boundary and/or initial<br>
conditions, and then runs a solver, most likely things won't work, and<br>
one won't have the slightest idea what is going wrong. In addition to<br>
the above, you'll need some intuition for length scales, time scales,<br>
curvature scales, etc. Starting with a 4d spacetime is probably<br>
hopeless.<br>
<br>
Is there a way to simplify the problem to fewer dimensions? To simpler<br>
equations? Maybe to simpler physics even, solving a strawman problem?<br>
<br>
For example, when learning how to solve the Einstein equations (which<br>
are nonlinear tensorial wave equations), we started with solving the<br>
linear scalar wave equation in one dimension. If there is no<br>
one-dimensional case, then maybe assuming axisymmetry or stationarity<br>
will help, or maybe one can study a linearization of the equations<br>
about some background, etc.<br>
<br>
Even when simulating binary black holes (which is, by now, a well<br>
understood problem, since we have been simulating them for 15 years),<br>
it is difficult to get started from scratch. Most people start by<br>
taking an existing simulation and making small variations (masses,<br>
spins, initial velocities, etc.), or they take a known physical<br>
scenario and change numerical parameters (resolution, boundaries,<br>
numerical methods). The "original" black hole simulations were quite<br>
difficult to obtain and were based on years of experience, including<br>
experience from axisymmetric black hole simulations from many years<br>
earlier.<br>
<br>
Since you are interested in studying a completely new set of<br>
equations, I suggest to consider first a much simplified problem. I<br>
wish that tools such as the Einstein Toolkit were black box solvers<br>
(similar to Mathematica's "Integrate" function), but in truth we're<br>
far away from that...<br>
<br>
-erik<br>
<br>
<br>
<br>
<br>
<br>
On Mon, Mar 8, 2021 at 5:19 PM Adam Herbst <<a href="mailto:adamdrewherbst@gmail.com" target="_blank">adamdrewherbst@gmail.com</a>> wrote:<br>
><br>
> Hi Erik / Peter,<br>
> 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.<br>
><br>
> <a href="https://adamdrewherbst.pythonanywhere.com/welcome/spacetime/index?language=english&section=brief" rel="noreferrer" target="_blank">https://adamdrewherbst.pythonanywhere.com/welcome/spacetime/index?language=english&section=brief</a><br>
><br>
> 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!<br>
><br>
> Thank you,<br>
> Adam<br>
><br>
><br>
><br>
> On Fri, Mar 5, 2021 at 5:10 PM Adam Herbst <<a href="mailto:adamdrewherbst@gmail.com" target="_blank">adamdrewherbst@gmail.com</a>> wrote:<br>
>><br>
>> 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.<br>
>><br>
>> Adam<br>
>><br>
>> On Fri, Mar 5, 2021 at 8:39 AM Peter Diener <<a href="mailto:diener@cct.lsu.edu" target="_blank">diener@cct.lsu.edu</a>> wrote:<br>
>>><br>
>>> Hi Adam,<br>
>>><br>
>>> If it indeed turns out that your problem can be cast as a 4th order<br>
>>> elliptical PDE, I don't see any reason why this could not be simulated.<br>
>>> In fact in the thorn NoExcision, we actually use up to a 6th order<br>
>>> ellitpical PDE to fill in the interior of a black hole with constraint<br>
>>> violating data that smoothly matches the exterior data. In this thorn<br>
>>> we implemented a conjugate gradient method to solve the equations and<br>
>>> didn't see any issues with the fact the the equations involved 6th<br>
>>> derivatives.<br>
>>><br>
>>> Cheers,<br>
>>><br>
>>> Peter<br>
>>><br>
>>> On Wednesday 2021-03-03 14:52, Adam Herbst wrote:<br>
>>><br>
>>> >Date: Wed, 3 Mar 2021 14:52:53<br>
>>> >From: Adam Herbst <<a href="mailto:adamdrewherbst@gmail.com" target="_blank">adamdrewherbst@gmail.com</a>><br>
>>> >To: Erik Schnetter <<a href="mailto:schnetter@cct.lsu.edu" target="_blank">schnetter@cct.lsu.edu</a>><br>
>>> >Cc: Einstein Toolkit Users <<a href="mailto:users@einsteintoolkit.org" target="_blank">users@einsteintoolkit.org</a>><br>
>>> >Subject: Re: [Users] Can I simulate this exotic static topological spacetime<br>
>>> > with the ET?<br>
>>> ><br>
>>> >Hi Erik,<br>
>>> >I am elated to receive such a detailed answer, and it appears you have<br>
>>> >understood my problem perfectly, maybe better than I understand it myself.<br>
>>> >I'll see if I can clear up the write-up I had and send it over. But I think<br>
>>> >you are right that I have not developed this enough to be tested numerically<br>
>>> >yet. After reading more, I think the Hilbert action approach doesn't make<br>
>>> >sense anyway. Also, as far as I can tell, the curvature singularity is<br>
>>> >unavoidable due to the topological transition to the loop.<br>
>>> ><br>
>>> >I had previously based the idea on a "curvature wave equation", which might<br>
>>> >be an elliptic PDE but it would be fourth-order in the metric. Could a<br>
>>> >4th-order PDE be simulated?<br>
>>> ><br>
>>> >Thank you kindly,<br>
>>> >Adam<br>
>>> ><br>
>>> >On Tue, Mar 2, 2021 at 1:00 PM Erik Schnetter <<a href="mailto:schnetter@cct.lsu.edu" target="_blank">schnetter@cct.lsu.edu</a>> wrote:<br>
>>> > Adam<br>
>>> ><br>
>>> > The setup you described seems to have singularities on the<br>
>>> > boundary.<br>
>>> > This is usually a very elegant ansatz for an analytic study, but<br>
>>> > is<br>
>>> > disastrous in a numerical study. As a first step, it will be<br>
>>> > necessary<br>
>>> > to convert this ansatz to a setup that has no singularities,<br>
>>> > i.e.<br>
>>> > metric is non-zero and non-infinite everywhere, and the<br>
>>> > curvature also<br>
>>> > needs to be finite everywhere. There are several generic methods<br>
>>> > for<br>
>>> > that (e.g. "subtracting" or "dividing by" singular terms), but<br>
>>> > it<br>
>>> > remains a non-trivial task.<br>
>>> ><br>
>>> > Most people use the Einstein Toolkit to evolve a dynamical<br>
>>> > spacetime.<br>
>>> > Looking for a stationary solution would be called "setting up<br>
>>> > initial<br>
>>> > conditions" in our lingo. While the Einstein Toolkit has many<br>
>>> > kinds of<br>
>>> > initial conditions built in, it's usually a bit involved to set<br>
>>> > up a<br>
>>> > new kind of initial condition.<br>
>>> ><br>
>>> > Even so, the Einstein Toolkit is geared towards solving R_ab = 0<br>
>>> > (in<br>
>>> > vacuum). What you describe sounds like a very different method.<br>
>>> > I<br>
>>> > don't know how one would formulate allowing for non-zero Ricci<br>
>>> > curvature without prescribing a matter content in terms of an<br>
>>> > elliptic<br>
>>> > PDE.<br>
>>> ><br>
>>> > If you can formulate your problem in terms of elliptic PDEs then<br>
>>> > I (or<br>
>>> > others!) can point you towards thorns or modules to study.<br>
>>> > Otherwise<br>
>>> > you're probably still a step away from using a numerical method.<br>
>>> > I<br>
>>> > might have misunderstood your problem description, though. Do<br>
>>> > you have<br>
>>> > a pointer to a write-up that gives more details?<br>
>>> ><br>
>>> > -erik<br>
>>> ><br>
>>> ><br>
>>> ><br>
>>> > On Tue, Mar 2, 2021 at 11:40 AM Adam Herbst<br>
>>> > <<a href="mailto:adamdrewherbst@gmail.com" target="_blank">adamdrewherbst@gmail.com</a>> wrote:<br>
>>> > ><br>
>>> > > Hi all,<br>
>>> > > Before tackling the learning curve, I want to see if there's<br>
>>> > any chance I can do what I'm hoping to, because it seems<br>
>>> > unlikely, but with something as highly developed as the ET<br>
>>> > appears to be, you never know!<br>
>>> > ><br>
>>> > > I want to find a stationary spacetime, in which each<br>
>>> > time-slice has a topological defect anchored at the origin.<br>
>>> > Specifically, we take an "extruded sphere" (S^2 x [0,1]), set<br>
>>> > the metric such that the radii of the end-spheres goes to zero,<br>
>>> > and attach each end to one "half-space" of the origin (theta in<br>
>>> > [0, pi/2] and theta in [pi/2, pi]). This can be done "smoothly"<br>
>>> > by having g_{theta,theta} from outside approach sin^2(2 * theta)<br>
>>> > instead of sin^2(theta), so that a radial cross-section becomes<br>
>>> > a pair of spheres, one for each half-space, instead of a single<br>
>>> > sphere. Thus the defect is actually a "bridge" between these<br>
>>> > two half-spaces, and geodesics through the origin traverse this<br>
>>> > loop. But the curvature does become infinite at the origin.<br>
>>> > ><br>
>>> > > Now the thing is, what I really want to do is start with the<br>
>>> > ansatz described above (I already have a formula for the<br>
>>> > metric), and make it converge to a solution of the<br>
>>> > Einstein-Hilbert action, while keeping it stationary. But in<br>
>>> > this case it is NOT the same as the vacuum field equation,<br>
>>> > because the "boundary condition" of the topological singularity<br>
>>> > will not allow the Ricci curvature to disappear, even when we<br>
>>> > minimize total curvature. Or so I believe. So that's why it<br>
>>> > has to be a purely action-based approach, if that even makes<br>
>>> > sense.<br>
>>> > ><br>
>>> > > So I hope this was coherent. And if it is possible, can you<br>
>>> > let me know which modules I should start getting familiar with<br>
>>> > in order to give it a shot?<br>
>>> > ><br>
>>> > > Thank you for reading! Cheers,<br>
>>> > ><br>
>>> > > Adam<br>
>>> > > _______________________________________________<br>
>>> > > Users mailing list<br>
>>> > > <a href="mailto:Users@einsteintoolkit.org" target="_blank">Users@einsteintoolkit.org</a><br>
>>> > > <a href="http://lists.einsteintoolkit.org/mailman/listinfo/users" rel="noreferrer" target="_blank">http://lists.einsteintoolkit.org/mailman/listinfo/users</a><br>
>>> ><br>
>>> ><br>
>>> ><br>
>>> > --<br>
>>> > Erik Schnetter <<a href="mailto:schnetter@cct.lsu.edu" target="_blank">schnetter@cct.lsu.edu</a>><br>
>>> > <a href="http://www.perimeterinstitute.ca/personal/eschnetter/" rel="noreferrer" target="_blank">http://www.perimeterinstitute.ca/personal/eschnetter/</a><br>
>>> ><br>
>>> ><br>
>>> ><br>
<br>
<br>
<br>
-- <br>
Erik Schnetter <<a href="mailto:schnetter@cct.lsu.edu" target="_blank">schnetter@cct.lsu.edu</a>><br>
<a href="http://www.perimeterinstitute.ca/personal/eschnetter/" rel="noreferrer" target="_blank">http://www.perimeterinstitute.ca/personal/eschnetter/</a><br>
</blockquote></div>