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

[Erik Schnetter]

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/