<div dir="ltr">Thanks for the help Eloisa and Vassili! I really appreciate it.<div><br></div><div>Gwyneth</div><div><br></div><div class="gmail_extra"><br><div class="gmail_quote">On Thu, Mar 2, 2017 at 6:51 PM, Vassilios Mewes <span dir="ltr"><<a href="mailto:vassilios.mewes@uv.es" target="_blank">vassilios.mewes@uv.es</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div>
<div dir="ltr">Hello Gwyneth and Eloisa,
<div class="gmail_extra"><br>
<div class="gmail_quote"><div><div class="h5">On Thu, Mar 2, 2017 at 8:31 AM, Eloisa Bentivegna <span dir="ltr">
<<a href="mailto:eloisa.bentivegna@ct.infn.it" target="_blank">eloisa.bentivegna@ct.infn.it</a>></span> wrote:<br>
<blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
<span class="m_-1652715801038431917gmail-">On 28/02/17 09:43, Gwyneth Allwright wrote:<br>
> Hi All,<br>
><br>
> I'm trying to reproduce the testbed BBH results in Etienne et al. 2009:<br>
> <a href="https://arxiv.org/abs/0812.2245" rel="noreferrer" target="_blank">https://arxiv.org/abs/0812.224<wbr>5</a><br>
><br>
> I'd like to calculate the final Kerr black hole spin using the ratio of<br>
> the polar and equatorial circumferences. QuasiLocalMeasures qlm_scalars<br>
> gives several spin-related quantities:<br>
><br>
> qlm_spin_guess<br>
> qlm_spin<br>
> qlm_npspin<br>
> qlm_wsspin<br>
> qlm_cvspin<br>
> qlm_coordspinx, qlm_coordspiny and qlm_coordspinz.<br>
><br>
> How are these related? Are any of them calculated using the Kerr formula?<br>
<br>
</span>Dear Gwyneth,<br>
<br>
as you've noticed, QLM implements various measures of a surface spin<br>
(some better tested than others). Unfortunately the references to the<br>
corresponding formalisms are scattered around, but here's a primer:<br>
<br>
1) qlm_spin_guess is a spin estimate which assumes the spacetime is<br>
Kerr, and uses the area and equatorial circumference of the surface to<br>
build the spin according to<br>
<br>
! equatorial circumference L, area A<br>
<br>
! L = 2 pi (r^2 + a^2) / r<br>
! A = 4 pi (r^2 + a^2)<br>
! r = M + sqrt (M^2 - a^2)<br>
<br>
! r = A / (2 L)<br>
! a^2 = A / (4 pi) - r^2 ("spin" a = J/M = specific angular momentum)<br>
! M = (r^2 + a^2) / (2 r)<br>
<br>
! J = a M (angular momentum)<br>
<br>
(this is from the thorn's qlm_analyse.F90)<br>
<br>
If the assumption is fine with you, you can just use this estimate.<br>
<br>
2) qlm_spin is equation (25) in <a href="http://arxiv.org/pdf/gr-qc/0206008.pdf" rel="noreferrer" target="_blank">
http://arxiv.org/pdf/gr-qc/020<wbr>6008.pdf</a><br>
(in a nutshell, it involves identifying a rotational symmetry on the<br>
surface and constructing the corresponding conserved charge);<br>
<br>
3) qlm_npspin and qlm_wsspin are measures of angular momentum based on<br>
the Newman-Penrose coefficients and Weyl scalars, respectively (for an<br>
example of what the integrands look like on e.g. Kerr, you can take a<br>
look at Chapter 6 of Chandrasekhar's book);<br>
<br>
4) qlm_cvspin is, as far as I can tell, currently not set;<br>
<br>
5) qlm_coordspin* is the same as 2), but assuming that the generators of<br>
the rotational symmetry are the x, y, and z axis, respectively.<br>
</blockquote>
<div><br>
</div>
</div></div><div>Just to add, this measure is identical to the angular momentum calculated using the Weinberg pseudotensor in qlm_analyse.f90 (as the calculations are performed with the lapse =1 and shift =0 in the thorn). In case of an axisymmetric horizon, this is equal
to to the Komar angular momentum of the BH (<a href="https://arxiv.org/pdf/1505.07225.pdf" target="_blank">https://arxiv.org/pdf/1505.<wbr>07225.pdf</a>).</div><span class="">
<blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
<span class="m_-1652715801038431917gmail-"><br>
> Also: what's the difference between qlm_polar_circumference_0 and<br>
> qlm_polar_circumference_pi_2?<br>
<br>
</span>These are the length of the meridians at phi=0 and phi=pi/2, respectively.<br>
<br>
Best,<br>
Eloisa<br>
</blockquote>
<div><br>
</div>
</span><div>Best wishes,</div>
<div><br>
</div>
<div>Vassili </div><span class="">
<blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
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