[Users] Isotropic Initial Data Evolving Towards Anisotropy

[Nick Olsen]

Hello everyone

Working from the general spherically symmetric form of a metric I verified
the data is in fact isotropic. I appreciate both of your inputs on this, it
helped a lot.

Nick

On Fri, May 27, 2022 at 3:53 PM Peter Diener <diener at cct.lsu.edu> wrote:

> Hi Nick,
>
> If I understand your notation g_yy(0,1,0) correctly to mean g_yy along the
> y axis and g_yy(1,0,0) to g_yy along the x-axis, the statement you make is
> not correct. What you would expect in a spherically symmetric spacetime is
> that g_xx(1,0,0)=g_yy(0,1,0)=g_zz(0,0,1), i.e. gxx along x should be the
> same as g_yy along y and g_zz along z.
>
> Cheers,
>
>    Peter
>
>
> On Fri, 27 May 2022, Nick Olsen wrote:
>
> > Hello Erik
> > I see what you're getting at here, but as I understand it the metric
> should
> > also be invariant under rotations and so for example gyy should not
> change
> > depending on the axis you look at. More explicitly, isotropy should have
> > that g_yy(0,1,0)=g_yy(1,0,0) or any similar combination, but the data I
> have
> > is showing otherwise.
> >
> > Nick
> >
> > On Fri., May 27, 2022, 2:33 p.m. Erik Schnetter, <schnetter at gmail.com>
> > wrote:
> >       Nick
> > Here is an example:
> >
> > Take the 3-metric ds^2 = a dr^2 + dθ^2 + (sin θ)^2 dϕ . It is
> > spherically symmetric.
> >
> > Along the z axis, you have gxx = gyy = 1, but there is gzz = a. The
> > metric tensor itself (as object in tangent space) is not spherically
> > symmetric. It is only spherically symmetric as object on the manifold.
> >
> > -erik
> >
> >
> > On Fri, May 27, 2022 at 10:56 AM Nick Olsen <n.olsen.3.711 at gmail.com>
> > wrote:
> >       Hello Erik
> > Forgive the late reply, it's been a busy few days. As I
> > understand things isotropic and spherically symmetric should be
> > the same thing in this case, with an isotropic metric taking the
> > form -a(r)^2 dt^2+b(r)^2 ds^2, so the fact that g_xx=/=g_yy and
> > its value depends on direction is what has me worried.
> >
> > Nicholas Olsen
> >
> > On Fri., May 13, 2022, 2:28 p.m. Erik Schnetter,
> > <schnetter at gmail.com> wrote:
> >       On Thu, May 12, 2022 at 3:28 AM Nick Olsen
> >       <n.olsen.3.711 at gmail.com> wrote:
> >       Hello Everyone
> > I am running into a problem where I evolve a
> > Gaussian shell scalar field alongside the BSSN
> > equations using the Scalar/ScalarInit/ScalarBase
> > thorns, where the initial data is isotropic but
> > evolves to an anisotropic solution. More
> > specifically, along the x axis I have g_yy=g_zz and
> > along the z axis I have g_xx=g_yy, with g_xx along
> > the x axis equal to g_zz along the z axis, despite
> > having isotropic initial conditions. The point is
> > illustrated by the first image being the plot of
> > g_xx and g_zz along their respective axes at a later
> > time, and the rest of the diagonal metric values
> > being shown in the second image. Additionally, T_ij
> > shows a similar problem, where If so, T_xx along the
> > x axis and T_zz along the z axis are equal to
> > eachother, but not the rest of the diagonal entries
> > of T_ij, which are all equal.
> >
> >
> > Nils
> >
> > What you describe sounds isotropic.
> >
> > I assume that by saying "isotropic" you mean "spherically
> > symmetric", i.e. the solution only depends on the radius r
> > and not on the angles \theta or \phi.
> >
> > If so, then scalars should be the same in every direction,
> > vectors should point in the radial direction, and tensors
> > will look a bit more complicated. but "g_xx in the x
> > direction is the same as g_zz in the z direction" sounds
> > correct: If you rotate this tensor from the x to the z
> > axis, then you're essentially exchanging x and z
> > directions.
> >
> > The tensor itself does not need to remain spherically
> > symmetric. (From your description above it sounds as if
> > you assumed this was the case.)
> >
> > -erik
> >
> >
> >
> >       gxxx.PNG gxxz.PNG
> >       I have attached the parameter file used to get
> >       these results, which is a modified version of
> >       the test parameter file found in the Scalar
> >       thorn bundle.
> >
> > Thanks,
> > Nicholas Olsen
> > _______________________________________________
> > Users mailing list
> > Users at einsteintoolkit.org
> > http://lists.einsteintoolkit.org/mailman/listinfo/users
> >
> >
> >
> > --
> > Erik Schnetter <schnetter at gmail.com>
> > http://www.perimeterinstitute.ca/personal/eschnetter/
> >
> >
> >
> > --
> > Erik Schnetter <schnetter at gmail.com>
> > http://www.perimeterinstitute.ca/personal/eschnetter/
> >
> >
> >
-------------- next part --------------
An HTML attachment was scrubbed...
URL: http://lists.einsteintoolkit.org/pipermail/users/attachments/20220530/e79bd281/attachment.html