# [Users] Isotropic Initial Data Evolving Towards Anisotropy

Nick Olsen n.olsen.3.711 at gmail.com
Fri May 27 14:16:33 CDT 2022

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
>>>
>>>
>>>
>>>
>>>> [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 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/
>
>
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