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@gmail.com> wrote:
On Thu, May 12, 2022 at 3:28 AM Nick Olsen <n.olsen.3.711@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.PNGgxxz.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
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