# [Users] Isotropic Initial Data Evolving Towards Anisotropy

Erik Schnetter schnetter at gmail.com
Fri May 27 13:33:13 CDT 2022

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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