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