[Users] QuasiLocalMeasures BH Spin

[Vassilios Mewes]

Hello Gwyneth and Eloisa,

On Thu, Mar 2, 2017 at 8:31 AM, Eloisa Bentivegna <
eloisa.bentivegna at ct.infn.it> wrote:

> On 28/02/17 09:43, Gwyneth Allwright wrote:
> > Hi All,
> >
> > I'm trying to reproduce the testbed BBH results in Etienne et al. 2009:
> > https://arxiv.org/abs/0812.2245
> >
> > I'd like to calculate the final Kerr black hole spin using the ratio of
> > the polar and equatorial circumferences. QuasiLocalMeasures qlm_scalars
> > gives several spin-related quantities:
> >
> > qlm_spin_guess
> > qlm_spin
> > qlm_npspin
> > qlm_wsspin
> > qlm_cvspin
> > qlm_coordspinx, qlm_coordspiny and qlm_coordspinz.
> >
> > How are these related? Are any of them calculated using the Kerr formula?
>
> Dear Gwyneth,
>
> as you've noticed, QLM implements various measures of a surface spin
> (some better tested than others). Unfortunately the references to the
> corresponding formalisms are scattered around, but here's a primer:
>
> 1) qlm_spin_guess is a spin estimate which assumes the spacetime is
> Kerr, and uses the area and equatorial circumference of the surface to
> build the spin according to
>
> ! equatorial circumference L, area A
>
> ! L = 2 pi (r^2 + a^2) / r
> ! A = 4 pi (r^2 + a^2)
> ! r = M + sqrt (M^2 - a^2)
>
> ! r = A / (2 L)
> ! a^2 = A / (4 pi) - r^2   ("spin" a = J/M = specific angular momentum)
> ! M = (r^2 + a^2) / (2 r)
>
> ! J = a M   (angular momentum)
>
> (this is from the thorn's qlm_analyse.F90)
>
> If the assumption is fine with you, you can just use this estimate.
>
> 2) qlm_spin is equation (25) in http://arxiv.org/pdf/gr-qc/0206008.pdf
> (in a nutshell, it involves identifying a rotational symmetry on the
> surface and constructing the corresponding conserved charge);
>
> 3) qlm_npspin and qlm_wsspin are measures of angular momentum based on
> the Newman-Penrose coefficients and Weyl scalars, respectively (for an
> example of what the integrands look like on e.g. Kerr, you can take a
> look at Chapter 6 of Chandrasekhar's book);
>
> 4) qlm_cvspin is, as far as I can tell, currently not set;
>
> 5) qlm_coordspin* is the same as 2), but assuming that the generators of
> the rotational symmetry are the x, y, and z axis, respectively.
>

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 (https://arxiv.org/pdf/1505.07225.pdf).

>
> > Also: what's the difference between qlm_polar_circumference_0 and
> > qlm_polar_circumference_pi_2?
>
> These are the length of the meridians at phi=0 and phi=pi/2, respectively.
>
> Best,
> Eloisa
>

Best wishes,

Vassili

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