[Users] QuasiLocalMeasures BH Spin

[Gwyneth Allwright]

Thanks for the help Eloisa and Vassili! I really appreciate it.

Gwyneth


On Thu, Mar 2, 2017 at 6:51 PM, Vassilios Mewes <vassilios.mewes at uv.es>
wrote:

> 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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