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....@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, <schnet...@gmail.com> > wrote: > >> On Thu, May 12, 2022 at 3:28 AM Nick Olsen <n.olsen.3....@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 >> >> >> >> >>> [image: gxxx.PNG][image: 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@einsteintoolkit.org >>> http://lists.einsteintoolkit.org/mailman/listinfo/users >>> >> >> >> -- >> Erik Schnetter <schnet...@gmail.com> >> http://www.perimeterinstitute.ca/personal/eschnetter/ >> >> -- Erik Schnetter <schnet...@gmail.com> http://www.perimeterinstitute.ca/personal/eschnetter/
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