<div dir="ltr">Nick<div><br></div><div>Here is an example:</div><div><br></div><div>Take the 3-metric ds^2 = a dr^2 + dθ^2 + (sin θ)^2 dϕ . It is spherically symmetric.</div><div><br></div><div>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.</div><div><br></div><div>-erik</div><div><br></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Fri, May 27, 2022 at 10:56 AM Nick Olsen <<a href="mailto:n.olsen.3.711@gmail.com">n.olsen.3.711@gmail.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;border-left-color:rgb(204,204,204);padding-left:1ex"><div dir="auto">Hello Erik <div dir="auto"><br></div><div dir="auto">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. </div><div dir="auto"><br></div><div dir="auto">Nicholas Olsen </div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Fri., May 13, 2022, 2:28 p.m. Erik Schnetter, <<a href="mailto:schnetter@gmail.com" target="_blank">schnetter@gmail.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;border-left-color:rgb(204,204,204);padding-left:1ex"><div dir="ltr"><div dir="ltr">On Thu, May 12, 2022 at 3:28 AM Nick Olsen <<a href="mailto:n.olsen.3.711@gmail.com" rel="noreferrer" target="_blank">n.olsen.3.711@gmail.com</a>> wrote:<br></div><div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;border-left-color:rgb(204,204,204);padding-left:1ex"><div dir="ltr">Hello Everyone<div><br></div><div>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.</div></div></blockquote><div><br></div><div>Nils</div><div><br></div><div>What you describe sounds isotropic.</div><div><br></div><div>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.</div><div><br></div><div>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.</div><div><br></div><div>The tensor itself does not need to remain spherically symmetric. (From your description above it sounds as if you assumed this was the case.)</div><div><br></div><div>-erik</div><div><br></div><div><br></div><div> </div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;border-left-color:rgb(204,204,204);padding-left:1ex"><div dir="ltr"><img src="cid:ii_l32opx3n0" alt="gxxx.PNG" width="454" height="336"><img src="cid:ii_l32oqeht1" alt="gxxz.PNG" width="454" height="341"><br><div>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.</div><div><br></div><div>Thanks,</div><div>Nicholas Olsen</div></div>
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</blockquote></div><br clear="all"><div><br></div>-- <br><div dir="ltr"><div dir="ltr"><div><div dir="ltr"><div>Erik Schnetter <<a href="mailto:schnetter@gmail.com" rel="noreferrer" target="_blank">schnetter@gmail.com</a>></div><div><a href="http://www.perimeterinstitute.ca/personal/eschnetter/" rel="noreferrer" target="_blank">http://www.perimeterinstitute.ca/personal/eschnetter/</a><br></div><div><br></div></div></div></div></div></div>
</blockquote></div>
</blockquote></div><br clear="all"><div><br></div>-- <br><div dir="ltr" class="gmail_signature"><div dir="ltr"><div><div dir="ltr"><div>Erik Schnetter <<a href="mailto:schnetter@gmail.com" target="_blank">schnetter@gmail.com</a>></div><div><a href="http://www.perimeterinstitute.ca/personal/eschnetter/" target="_blank">http://www.perimeterinstitute.ca/personal/eschnetter/</a><br></div><div><br></div></div></div></div></div>