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@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@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@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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