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§ion=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 <[email protected]> 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 <[email protected]> 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 <[email protected]> >> >To: Erik Schnetter <[email protected]> >> >Cc: Einstein Toolkit Users <[email protected]> >> >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 <[email protected]> >> 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 >> > <[email protected]> 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 >> > > [email protected] >> > > http://lists.einsteintoolkit.org/mailman/listinfo/users >> > >> > >> > >> > -- >> > Erik Schnetter <[email protected]> >> > http://www.perimeterinstitute.ca/personal/eschnetter/ >> > >> > >> > >> >
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