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 <[email protected]> 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 <[email protected]> > 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§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/ > >>> > > >>> > > >>> > > > > > -- > Erik Schnetter <[email protected]> > http://www.perimeterinstitute.ca/personal/eschnetter/ >
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