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, <schnet...@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....@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
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
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