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

[Adam Herbst]

 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/
>
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