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