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#include "cctk.h"
#include "cctk_Arguments.h"
#include "cctk_Functions.h"
#include "cctk_Parameters.h"
subroutine qlm_analyse (CCTK_ARGUMENTS, hn)
use adm_metric_simple
use cctk
use constants
use qlm_derivs
use qlm_variables
use tensor
use tensor2
implicit none
DECLARE_CCTK_ARGUMENTS
DECLARE_CCTK_FUNCTIONS
DECLARE_CCTK_PARAMETERS
integer :: hn
CCTK_REAL, parameter :: zero=0, two=2
integer, parameter :: rk = kind(zero)
CCTK_REAL :: theta, phi
CCTK_REAL :: ll(0:3), nn(0:3)
CCTK_COMPLEX :: mm(0:3)
CCTK_REAL :: ss(0:3), tt(0:3)
CCTK_COMPLEX :: npalpha, npbeta
CCTK_REAL :: gg(3,3), dgg(3,3,3), kk(3,3), gg_dot(3,3)
CCTK_REAL :: g4(0:3,0:3), dg4(0:3,0:3,0:3)
CCTK_REAL :: h4(0:3,0:3), dh4(0:3,0:3,0:3)
CCTK_REAL :: dtg, gu(3,3), trk
CCTK_REAL :: xx(3), ee(3,2)
CCTK_REAL :: xi(2), xi1(3)
CCTK_REAL :: qq(2,2), dtq
CCTK_REAL :: adm_energy, adm_mom(3), adm_amom(3,3)
CCTK_COMPLEX :: ev
CCTK_REAL :: spin
CCTK_REAL :: npspin
CCTK_REAL :: wsspin
CCTK_REAL :: tx, ty, tz
CCTK_REAL :: xi1_x(3), xi1_y(3), xi1_z(3)
CCTK_REAL :: coordspinx, coordspiny, coordspinz
CCTK_REAL :: delta_space(2)
CCTK_REAL, allocatable :: weights(:)
CCTK_REAL :: sum1
integer :: ntheta_inner, nphi_inner
integer :: i_eq, j_p0, j_p2
integer :: i, j, l
integer :: a, b, c
character :: msg*1000
if (veryverbose/=0) then
call CCTK_INFO ("Calculating spin")
end if
delta_space(:) = (/ qlm_delta_theta(hn), qlm_delta_phi(hn) /)
! Equatorial circumference
i_eq = (qlm_ntheta(hn) + 1) / 2
! Polar circumference at phi=0
j_p0 = 1+qlm_nghostsphi(hn)
! Polar circumference at phi=pi/2
j_p2 = 1+qlm_nghostsphi(hn) + (qlm_nphi(hn) - 2*qlm_nghostsphi(hn) - 1) / 4
! Initial values
qlm_equatorial_circumference(hn) = 0
qlm_polar_circumference_0(hn) = 0
qlm_polar_circumference_pi_2(hn) = 0
qlm_area(hn) = 0
qlm_spin(hn) = 0
qlm_cvspin(hn) = 0
qlm_npspin(hn) = 0
qlm_wsspin(hn) = 0
qlm_coordspinx(hn) = 0
qlm_coordspiny(hn) = 0
qlm_coordspinz(hn) = 0
qlm_adm_energy(hn) = 0
qlm_adm_momentum_x(hn) = 0
qlm_adm_momentum_y(hn) = 0
qlm_adm_momentum_z(hn) = 0
qlm_adm_angular_momentum_x(hn) = 0
qlm_adm_angular_momentum_y(hn) = 0
qlm_adm_angular_momentum_z(hn) = 0
! Compute weights for spherical integration (see Driscoll and Healy).
! These are the correct weights in a Gauss-Legendre-Senc.
! Also compare qlm_area with AHFinderDirect's area, they are in much
! better agreement than with the previous method using
! delta_theta*delta_phi.
ntheta_inner = qlm_ntheta(hn) - 2*qlm_nghoststheta(hn)
nphi_inner = qlm_nphi(hn) - 2*qlm_nghostsphi(hn)
allocate (weights(1+qlm_nghoststheta(hn) : qlm_ntheta(hn)-qlm_nghoststheta(hn)))
do i = 1+qlm_nghoststheta(hn), qlm_ntheta(hn)-qlm_nghoststheta(hn)
theta = qlm_origin_theta(hn) + (i-1)*qlm_delta_theta(hn)
sum1 = 0
do l = 0, (ntheta_inner-1)/2
sum1 = sum1 + sin((2*l+1)*theta)/(2*l+1)
end do
weights(i) = 8*pi * sum1 / (nphi_inner * ntheta_inner)
end do
do j = 1+qlm_nghostsphi(hn), qlm_nphi(hn)-qlm_nghostsphi(hn)
do i = 1+qlm_nghoststheta(hn), qlm_ntheta(hn)-qlm_nghoststheta(hn)
theta = qlm_origin_theta(hn) + (i-1)*qlm_delta_theta(hn)
phi = qlm_origin_phi(hn) + (j-1)*qlm_delta_phi(hn)
! 2-metric on the horizon
qq(1,1) = qlm_qtt(i,j,hn)
qq(1,2) = qlm_qtp(i,j,hn)
qq(2,2) = qlm_qpp(i,j,hn)
qq(2,1) = qq(1,2)
call calc_2det (qq, dtq)
ll(0) = qlm_l0(i,j,hn)
ll(1) = qlm_l1(i,j,hn)
ll(2) = qlm_l2(i,j,hn)
ll(3) = qlm_l3(i,j,hn)
nn(0) = qlm_n0(i,j,hn)
nn(1) = qlm_n1(i,j,hn)
nn(2) = qlm_n2(i,j,hn)
nn(3) = qlm_n3(i,j,hn)
mm(0) = qlm_m0(i,j,hn)
mm(1) = qlm_m1(i,j,hn)
mm(2) = qlm_m2(i,j,hn)
mm(3) = qlm_m3(i,j,hn)
tt = (ll + nn) / sqrt(two)
ss = (ll - nn) / sqrt(two)
npalpha = qlm_npalpha(i,j,hn)
npbeta = qlm_npbeta(i,j,hn)
gg(1,1) = qlm_gxx(i,j)
gg(1,2) = qlm_gxy(i,j)
gg(1,3) = qlm_gxz(i,j)
gg(2,2) = qlm_gyy(i,j)
gg(2,3) = qlm_gyz(i,j)
gg(3,3) = qlm_gzz(i,j)
gg(2,1) = gg(1,2)
gg(3,1) = gg(1,3)
gg(3,2) = gg(2,3)
dgg(1,1,1) = qlm_dgxxx(i,j)
dgg(1,2,1) = qlm_dgxyx(i,j)
dgg(1,3,1) = qlm_dgxzx(i,j)
dgg(2,2,1) = qlm_dgyyx(i,j)
dgg(2,3,1) = qlm_dgyzx(i,j)
dgg(3,3,1) = qlm_dgzzx(i,j)
dgg(2,1,1) = dgg(1,2,1)
dgg(3,1,1) = dgg(1,3,1)
dgg(3,2,1) = dgg(2,3,1)
dgg(1,1,2) = qlm_dgxxy(i,j)
dgg(1,2,2) = qlm_dgxyy(i,j)
dgg(1,3,2) = qlm_dgxzy(i,j)
dgg(2,2,2) = qlm_dgyyy(i,j)
dgg(2,3,2) = qlm_dgyzy(i,j)
dgg(3,3,2) = qlm_dgzzy(i,j)
dgg(2,1,2) = dgg(1,2,2)
dgg(3,1,2) = dgg(1,3,2)
dgg(3,2,2) = dgg(2,3,2)
dgg(1,1,3) = qlm_dgxxz(i,j)
dgg(1,2,3) = qlm_dgxyz(i,j)
dgg(1,3,3) = qlm_dgxzz(i,j)
dgg(2,2,3) = qlm_dgyyz(i,j)
dgg(2,3,3) = qlm_dgyzz(i,j)
dgg(3,3,3) = qlm_dgzzz(i,j)
dgg(2,1,3) = dgg(1,2,3)
dgg(3,1,3) = dgg(1,3,3)
dgg(3,2,3) = dgg(2,3,3)
kk(1,1) = qlm_kxx(i,j)
kk(1,2) = qlm_kxy(i,j)
kk(1,3) = qlm_kxz(i,j)
kk(2,2) = qlm_kyy(i,j)
kk(2,3) = qlm_kyz(i,j)
kk(3,3) = qlm_kzz(i,j)
kk(2,1) = kk(1,2)
kk(3,1) = kk(1,3)
kk(3,2) = kk(2,3)
call calc_det (gg, dtg)
call calc_inv (gg, dtg, gu)
call calc_trace (gu, kk, trk)
! Calculate time derivatives
call calc_3metricdot_simple (kk, gg_dot)
! Calculate 4-metric
call calc_4metricderivs_simple (gg, dgg, gg_dot, g4,dg4)
! "Perturbative" 4-metric
h4 = g4 - eta4
dh4 = dg4
xx(1) = qlm_x(i,j,hn)
xx(2) = qlm_y(i,j,hn)
xx(3) = qlm_z(i,j,hn)
ee(1,1:2) = deriv (qlm_x(:,:,hn), i, j, delta_space)
ee(2,1:2) = deriv (qlm_y(:,:,hn), i, j, delta_space)
ee(3,1:2) = deriv (qlm_z(:,:,hn), i, j, delta_space)
xi(1) = qlm_xi_t(i,j,hn)
xi(2) = qlm_xi_p(i,j,hn)
if (i == i_eq) then
qlm_equatorial_circumference(hn) = qlm_equatorial_circumference(hn) &
+ sqrt(qq(2,2)) * qlm_delta_phi(hn)
end if
if (j == j_p0) then
qlm_polar_circumference_0(hn) = qlm_polar_circumference_0(hn) &
+ sqrt(qq(1,1)) * qlm_delta_theta(hn)
end if
if (j == j_p2) then
qlm_polar_circumference_pi_2(hn) = qlm_polar_circumference_pi_2(hn) &
+ sqrt(qq(1,1)) * qlm_delta_theta(hn)
end if
qlm_area(hn) = qlm_area(hn) &
+ sqrt(dtq) * weights(i)
! s^i: outward spacelike normal
! K_ij: extrinsic curvature
! phi^i: rotational Killing vector
do a=1,3
xi1(a) = 0
do b=1,2
xi1(a) = xi1(a) + ee(a,b) * xi(b)
end do
end do
! phi^i omega_i = - phi^i s^j K_ij
spin = 0
do a=1,3
do b=1,3
spin = spin + xi1(a) * ss(b) * kk(a,b)
end do
end do
! phi^i omega_i = (alpha + ~beta) phi^i m_i + complex conjugate
npspin = 0
do a=1,3
do b=1,3
npspin = npspin + 2 * real((npalpha + conjg(npbeta)) * xi1(a) * gg(a,b) * mm(b))
end do
end do
! phi^i omega_i = 1/2 f Im Psi_2
! (or is it phi^i omega_i = - f Im Psi_2 ?)
wsspin = - qlm_inv_z(i,j,hn) * aimag(qlm_psi2(i,j,hn))
! x = sin theta cos phi
! y = sin theta sin phi
! z = cos theta
! xi_x = (0,-z,y)
! xi_y = (z,0,-x)
! xi_z = (-y,x,0)
tx = qlm_x(i,j,hn) - qlm_origin_x(hn)
ty = qlm_y(i,j,hn) - qlm_origin_y(hn)
tz = qlm_z(i,j,hn) - qlm_origin_z(hn)
xi1_x(:) = (/ zero, -tz, ty /)
xi1_y(:) = (/ tz, zero, -tx /)
xi1_z(:) = (/ -ty, tx, zero /)
coordspinx = 0
coordspiny = 0
coordspinz = 0
do a=1,3
do b=1,3
coordspinx = coordspinx + xi1_x(a) * ss(b) * kk(a,b)
coordspiny = coordspiny + xi1_y(a) * ss(b) * kk(a,b)
coordspinz = coordspinz + xi1_z(a) * ss(b) * kk(a,b)
end do
end do
xi1_x(:) = (/ 1, 0, 0 /)
xi1_y(:) = (/ 0, 1, 0 /)
xi1_z(:) = (/ 0, 0, 1 /)
qlm_spin(hn) = qlm_spin(hn) &
+ spin * sqrt(dtq) * weights(i)
qlm_npspin(hn) = qlm_npspin(hn) &
+ npspin * sqrt(dtq) * weights(i)
qlm_wsspin(hn) = qlm_wsspin(hn) &
+ wsspin * sqrt(dtq) * weights(i)
qlm_coordspinx(hn) = qlm_coordspinx(hn) &
+ coordspinx * sqrt(dtq) * weights(i)
qlm_coordspiny(hn) = qlm_coordspiny(hn) &
+ coordspiny * sqrt(dtq) * weights(i)
qlm_coordspinz(hn) = qlm_coordspinz(hn) &
+ coordspinz * sqrt(dtq) * weights(i)
! ADM quantities
! Weinberg, chapter 7.6, pp. 165 ff, eqns. (7.6.22) - (7.6.24):
! ADM energy
! E_adm = (-1/16 pi) int_S(r) [h_jj,i - hij,j] n_i r^2 dOmega
adm_energy = 0
do a=1,3
do b=1,3
adm_energy = adm_energy + (dh4(b,b,a) - dh4(a,b,b)) * ss(a)
end do
end do
qlm_adm_energy(hn) = qlm_adm_energy(hn) &
& + adm_energy / (-16*pi) &
& * sqrt(dtq) * weights(i)
! ADM momentum
! P_adm^j = (-1/16 pi) int_S(r)
! [- h_kk,0 delta_ij + h_k0,k delta_ij
! - h_j0,i + h_ij,0 ] n_i r^2 dOmega
!
! P_adm^j = (1/8 pi) int_S(r) [K_ij - K delta_ij] n_i r^2 dOmega
do a=1,3
adm_mom(a) = 0
do b=1,3
adm_mom(a) = adm_mom(a) &
+ (- dh4(b,b,0) + dh4(b,0,b)) * ss(a) &
+ (- dh4(a,0,b) + dh4(b,a,0)) * ss(b)
end do
end do
qlm_adm_momentum_x(hn) = qlm_adm_momentum_x(hn) &
& + adm_mom(1) / (-16*pi) &
& * sqrt(dtq) * weights(i)
qlm_adm_momentum_y(hn) = qlm_adm_momentum_y(hn) &
& + adm_mom(2) / (-16*pi) &
& * sqrt(dtq) * weights(i)
qlm_adm_momentum_z(hn) = qlm_adm_momentum_z(hn) &
& + adm_mom(3) / (-16*pi) &
& * sqrt(dtq) * weights(i)
! ADM angular momentum
! J_adm^jk = (-1/16 pi) int_S(r)
! [- x_j h_0k,i + x_k h_0j,i
! + x_j h_ki,0 - x_k h_ji,0
! + h_0k delta_ij - h_0j delta_ik] n_i r^2 dOmega
do a=1,3
do b=1,3
adm_amom(a,b) = 0
do c=1,3
adm_amom(a,b) = adm_amom(a,b) + ss(c) * ( &
+ (- xx(a) * dh4(0,b,c) + xx(b) * dh4(0,a,c)) &
+ (+ xx(a) * dh4(b,c,0) - xx(b) * dh4(a,c,0)) &
+ (h4(0,b) * delta4(c,a) - h4(0,a) * delta4(c,b)))
end do
end do
end do
qlm_adm_angular_momentum_x(hn) = qlm_adm_angular_momentum_x(hn) &
& + adm_amom(2,3) / (-16*pi) &
& * sqrt(dtq) * weights(i)
qlm_adm_angular_momentum_y(hn) = qlm_adm_angular_momentum_y(hn) &
& + adm_amom(3,1) / (-16*pi) &
& * sqrt(dtq) * weights(i)
qlm_adm_angular_momentum_z(hn) = qlm_adm_angular_momentum_z(hn) &
& + adm_amom(1,2) / (-16*pi) &
& * sqrt(dtq) * weights(i)
end do
end do
deallocate(weights)
qlm_polar_circumference_0(hn) = qlm_polar_circumference_0(hn) * 2
qlm_polar_circumference_pi_2(hn) = qlm_polar_circumference_pi_2(hn) * 2
! A = 4 pi R^2
! R = 2 M
qlm_radius(hn) = sqrt(qlm_area(hn) / (4*pi))
qlm_irreducible_mass(hn) = qlm_radius(hn) / 2
qlm_spin(hn) = qlm_spin(hn) / (8*pi)
qlm_mass(hn) = 1/(2*qlm_radius(hn)) * sqrt(qlm_radius(hn)**4 + 4*qlm_spin(hn)**2)
qlm_cvspin(hn) = qlm_cvspin(hn) / (8*pi)
qlm_npspin(hn) = qlm_npspin(hn) / (-8*pi)
qlm_wsspin(hn) = qlm_wsspin(hn) / (-4*pi)
qlm_coordspinx(hn) = qlm_coordspinx(hn) / (8*pi)
qlm_coordspiny(hn) = qlm_coordspiny(hn) / (8*pi)
qlm_coordspinz(hn) = qlm_coordspinz(hn) / (8*pi)
! The event horizon is at r = M + sqrt (M^2 - a^2)
! with x^2 + y^2 + z^2 = rho^2 = r^2 + a^2 (1 - z^2 / r^2)
call guess_mass_spin &
(qlm_area(hn), qlm_equatorial_circumference(hn), qlm_mass_guess(hn), qlm_spin_guess(hn))
if (verbose/=0) then
write (msg, '("Geometric quantities for surface ",i4,":")') hn-1
call CCTK_INFO (msg)
write (msg, '(" Area A: ",g16.6)') qlm_area(hn)
call CCTK_INFO (msg)
write (msg, '(" Irreducible mass M = R/2: ",g16.6)') qlm_irreducible_mass(hn)
call CCTK_INFO (msg)
write (msg, '(" Areal radius R = sqrt(A/4pi): ",g16.6)') qlm_radius(hn)
call CCTK_INFO (msg)
write (msg, '("Coordinate-dependent quantities for surface ",i4,":")') hn-1
call CCTK_INFO (msg)
write (msg, '(" Equatorial circumference: ",g16.6)') &
qlm_equatorial_circumference(hn)
call CCTK_INFO (msg)
write (msg, '(" Polar circumference at phi=0: ",g16.6)') &
qlm_polar_circumference_0(hn)
call CCTK_INFO (msg)
write (msg, '(" Polar circumference at phi=pi/2: ",g16.6)') &
qlm_polar_circumference_pi_2(hn)
call CCTK_INFO (msg)
if (qlm_spin_guess(hn) >= 0) then
write (msg, '(" Spin guess J from distortion: ",g16.6)') &
qlm_spin_guess(hn)
call CCTK_INFO (msg)
else
call CCTK_INFO (" No valid spin guess from distortion.")
call CCTK_INFO (" (Spin guess is imaginary.)")
write (msg, '(" Magnitude of invalid spin guess: ",g16.6)') &
abs(qlm_spin_guess(hn))
call CCTK_INFO (msg)
end if
write (msg, '(" Mass guess M from distortion: ",g16.6)') &
qlm_mass_guess(hn)
call CCTK_INFO (msg)
write (msg, '("Isolated Horizon quantities for surface ",i4,":")') hn-1
call CCTK_INFO (msg)
ev = cmplx(qlm_killing_eigenvalue_re(hn), qlm_killing_eigenvalue_im(hn),rk)
write (msg, '(" Killing vector field eigenvalue norm: ",g14.6)') abs(ev)
call CCTK_INFO (msg)
write (msg, '(" Spin J: ",g14.6)') qlm_spin(hn)
call CCTK_INFO (msg)
write (msg, '(" Kerr spin parameter a = J/M: ",g14.6)') qlm_spin(hn) / qlm_mass(hn)
call CCTK_INFO (msg)
write (msg, '(" Dimensionless spin parameter a = J/M^2: ",g14.6)') qlm_spin(hn) / qlm_mass(hn)**2
call CCTK_INFO (msg)
write (msg, '(" Spin J from NP: ",g14.6)') qlm_npspin(hn)
call CCTK_INFO (msg)
write (msg, '(" Spin J from phi-coordinate-vector: ",g14.6)') qlm_cvspin(hn)
call CCTK_INFO (msg)
write (msg, '(" Mass M: ",g14.6)') qlm_mass(hn)
call CCTK_INFO (msg)
write (msg, '("Global quantities for surface ",i4,":")') hn-1
call CCTK_INFO (msg)
write (msg, '(" ADM energy: ",g16.6)') qlm_adm_energy(hn)
call CCTK_INFO (msg)
write (msg, '(" ADM momentum x: ",g16.6)') qlm_adm_momentum_x(hn)
call CCTK_INFO (msg)
write (msg, '(" ADM momentum y: ",g16.6)') qlm_adm_momentum_y(hn)
call CCTK_INFO (msg)
write (msg, '(" ADM momentum z: ",g16.6)') qlm_adm_momentum_z(hn)
call CCTK_INFO (msg)
write (msg, '(" ADM angular momentum x: ",g16.6)') qlm_adm_angular_momentum_x(hn)
call CCTK_INFO (msg)
write (msg, '(" ADM angular momentum y: ",g16.6)') qlm_adm_angular_momentum_y(hn)
call CCTK_INFO (msg)
write (msg, '(" ADM angular momentum z: ",g16.6)') qlm_adm_angular_momentum_z(hn)
call CCTK_INFO (msg)
end if
contains
subroutine guess_mass_spin (area, circumference, mass, spin)
CCTK_REAL, intent(in) :: area, circumference
CCTK_REAL, intent(out) :: mass, spin
CCTK_REAL :: radius, amom2, amom
! 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)
radius = area / (2*circumference)
amom2 = area / (4*pi) - radius**2
amom = sign(sqrt(abs(amom2)), amom2)
mass = (radius**2 + amom2) / (2*radius)
spin = amom * mass
! equatorial circumference L, area A
! mass M, areal radius R, spin a = J/M^2
! A = 4 pi R^2
! L = 4 pi M
! a^2 = (R/M)^2 - 1/4 (R/M)^4
! J = a M^2
end subroutine guess_mass_spin
end subroutine qlm_analyse
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