[Users] QuasiLocalMeasures BH Spin

[Eloisa Bentivegna]

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.

> 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