<div dir="ltr">
<div id="gmail-:265" class="gmail-a3s gmail-aiL"><div dir="ltr"><div>Hi Erik,</div><div>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.</div><div><br></div><div>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?</div><div><br></div><div>Thank you kindly,<br></div><div>Adam</div></div></div>
</div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Tue, Mar 2, 2021 at 1:00 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>
The setup you described seems to have singularities on the boundary.<br>
This is usually a very elegant ansatz for an analytic study, but is<br>
disastrous in a numerical study. As a first step, it will be necessary<br>
to convert this ansatz to a setup that has no singularities, i.e.<br>
metric is non-zero and non-infinite everywhere, and the curvature also<br>
needs to be finite everywhere. There are several generic methods for<br>
that (e.g. "subtracting" or "dividing by" singular terms), but it<br>
remains a non-trivial task.<br>
<br>
Most people use the Einstein Toolkit to evolve a dynamical spacetime.<br>
Looking for a stationary solution would be called "setting up initial<br>
conditions" in our lingo. While the Einstein Toolkit has many kinds of<br>
initial conditions built in, it's usually a bit involved to set up a<br>
new kind of initial condition.<br>
<br>
Even so, the Einstein Toolkit is geared towards solving R_ab = 0 (in<br>
vacuum). What you describe sounds like a very different method. 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 elliptic<br>
PDE.<br>
<br>
If you can formulate your problem in terms of elliptic PDEs then I (or<br>
others!) can point you towards thorns or modules to study. Otherwise<br>
you're probably still a step away from using a numerical method. I<br>
might have misunderstood your problem description, though. Do 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 <<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 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!<br>
><br>
> 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.<br>
><br>
> 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.<br>
><br>
> 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?<br>
><br>
> Thank you for reading! Cheers,<br>
><br>
> Adam<br>
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<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>