hole_bhns_extr_curv.C

/*
 *  Method of class Hole_bhns to compute the extrinsic curvature tensor
 *
 *    (see file hole_bhns.h for documentation).
 *
 */
 
/*
 *   Copyright (c) 2005-2007 Keisuke Taniguchi
 *
 *   This file is part of LORENE.
 *
 *   LORENE is free software; you can redistribute it and/or modify
 *   it under the terms of the GNU General Public License version 2
 *   as published by the Free Software Foundation.
 *
 *   LORENE is distributed in the hope that it will be useful,
 *   but WITHOUT ANY WARRANTY; without even the implied warranty of
 *   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *   GNU General Public License for more details.
 *
 *   You should have received a copy of the GNU General Public License
 *   along with LORENE; if not, write to the Free Software
 *   Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
 *
 */
 
 
 
/*
 * $Id: hole_bhns_extr_curv.C,v 1.5 2016/12/05 16:17:55 j_novak Exp $
 * $Log: hole_bhns_extr_curv.C,v $
 * Revision 1.5  2016/12/05 16:17:55  j_novak
 * Suppression of some global variables (file names, loch, ...) to prevent redefinitions
 *
 * Revision 1.4  2014/10/13 08:53:00  j_novak
 * Lorene classes and functions now belong to the namespace Lorene.
 *
 * Revision 1.3  2014/10/06 15:13:10  j_novak
 * Modified #include directives to use c++ syntax.
 *
 * Revision 1.2  2008/05/15 19:05:49  k_taniguchi
 * Change of some parameters.
 *
 * Revision 1.1  2007/06/22 01:24:56  k_taniguchi
 * *** empty log message ***
 *
 *
 * $Header: /cvsroot/Lorene/C++/Source/Hole_bhns/hole_bhns_extr_curv.C,v 1.5 2016/12/05 16:17:55 j_novak Exp $
 *
 */
 
// C++ headers
//#include <>
 
// C headers
#include <cmath>
 
// Lorene headers
#include "hole_bhns.h"
#include "utilitaires.h"
#include "unites.h"
//#include "graphique.h"
 
namespace Lorene {
void Hole_bhns::extr_curv_bhns(double omega_orb, double x_rot, double y_rot) {
 
    //----------------------------------
    // Total extrinsic curvature tensor
    //----------------------------------
 
    // Fundamental constants and units
    // -------------------------------
    using namespace Unites ;
 
    // Coordinates
    // -----------
 
    double mass = ggrav * mass_bh ;
 
    Scalar rr(mp) ;
    rr = mp.r ;
    rr.std_spectral_base() ;
    Scalar st(mp) ;
    st = mp.sint ;
    st.std_spectral_base() ;
    Scalar ct(mp) ;
    ct = mp.cost ;
    ct.std_spectral_base() ;
    Scalar sp(mp) ;
    sp = mp.sinp ;
    sp.std_spectral_base() ;
    Scalar cp(mp) ;
    cp = mp.cosp ;
    cp.std_spectral_base() ;
 
    Vector ll(mp, CON, mp.get_bvect_cart()) ;
    ll.set_etat_qcq() ;
    ll.set(1) = st % cp ;
    ll.set(2) = st % sp ;
    ll.set(3) = ct ;
    ll.std_spectral_base() ;
 
    Scalar divshift(mp) ;
    divshift = shift_auto_rs(1).deriv(1) + shift_auto_rs(2).deriv(2)
      + shift_auto_rs(3).deriv(3) + d_shift_comp(1,1)
      + d_shift_comp(2,2) + d_shift_comp(3,3) ;
    divshift.std_spectral_base() ;
 
    if (kerrschild) {
 
    Scalar orb_rot_x(mp) ;
    orb_rot_x = omega_orb * (mp.get_ori_x() - x_rot) ;
    orb_rot_x.std_spectral_base() ;
 
    Scalar orb_rot_y(mp) ;
    orb_rot_y = omega_orb * (mp.get_ori_y() - y_rot) ;
    orb_rot_y.std_spectral_base() ;
 
    Vector uv(mp, CON, mp.get_bvect_cart()) ; // unit vector
    uv.set_etat_qcq() ;
    uv.set(1) = - orb_rot_y ;
    uv.set(2) = orb_rot_x ;
    uv.set(3) = 0. ;
    uv.std_spectral_base() ;
 
    // Computation of \tilde{A}^{ij}
    // -----------------------------
 
    Sym_tensor flat_taij(mp, CON, mp.get_bvect_cart()) ;
    flat_taij.set_etat_qcq() ;
 
    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            flat_taij.set(i,j) = shift_auto_rs(j).deriv(i)
          + shift_auto_rs(i).deriv(j) + d_shift_comp(i,j)
          + d_shift_comp(j,i)
          - 2. * divshift * flat.con()(i,j) / 3. ;
        }
    }
 
    flat_taij.std_spectral_base() ;
 
    Sym_tensor curv_taij(mp, CON, mp.get_bvect_cart()) ;
    curv_taij.set_etat_qcq() ;
 
    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            curv_taij.set(i,j) =
          -2. * lapconf_auto_bh * lapconf_auto_bh * mass
          * (ll(i) * (shift_auto_rs(j).dsdr()
                  + ll(1)*d_shift_comp(1,j)
                  + ll(2)*d_shift_comp(2,j)
                  + ll(3)*d_shift_comp(3,j))
             + ll(j) * (shift_auto_rs(i).dsdr()
                + ll(1)*d_shift_comp(1,i)
                + ll(2)*d_shift_comp(2,i)
                + ll(3)*d_shift_comp(3,i))
             - 2. * ll(i) * ll(j) * divshift / 3.) / rr ;
        }
    }
 
    curv_taij.std_spectral_base() ;
 
    Sym_tensor resi_taij(mp, CON, mp.get_bvect_cart()) ;
    resi_taij.set_etat_qcq() ;
 
    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            resi_taij.set(i,j) =
          2. * lapconf_auto_bh * lapconf_auto_bh * mass
          * ( ll(i) * (shift_auto_rs(j) + shift_comp(j))
              + ll(j) * (shift_auto_rs(i) + shift_comp(i))
              + ( flat.con()(i,j)
              - lapconf_auto_bh*lapconf_auto_bh
              *(9.+14.*mass/rr)*ll(i)*ll(j) )
              * ( ll(1) * (shift_auto_rs(1) + shift_comp(1))
              + ll(2) * (shift_auto_rs(2) + shift_comp(2))
              + ll(3) * (shift_auto_rs(3) + shift_comp(3)) ) / 3. )
          / rr / rr ;
        }
    }
 
    resi_taij.std_spectral_base() ;
    resi_taij.inc_dzpuis(2) ;
 
    taij_tot_rs = 0.5 * pow(confo_tot, 7.)
      * (flat_taij + curv_taij + resi_taij) / lapconf_tot ;
 
    taij_tot_rs.std_spectral_base() ;
    taij_tot_rs.annule_domain(0) ;
 
    taij_tot_rot.set_etat_qcq() ;
 
    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            taij_tot_rot.set(i,j) = pow(confo_tot,7.)
          * lapconf_auto_bh * lapconf_auto_bh * mass
          * ( ll(i) * uv(j) + ll(j) * uv(i)
              + ( flat.con()(i,j)
              - lapconf_auto_bh*lapconf_auto_bh
              *(9.+14.*mass/rr)*ll(i)*ll(j) )
              * (ll(2)*orb_rot_x - ll(1)*orb_rot_y) / 3. )
          / lapconf_tot / rr / rr ;
        }
    }
 
    taij_tot_rot.std_spectral_base() ;
    taij_tot_rot.annule_domain(0) ;
    taij_tot_rot.inc_dzpuis(2) ;
 
    taij_tot_bh.set_etat_qcq() ;
 
    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            taij_tot_bh.set(i,j) = 2. * pow(confo_tot,7.)
          * pow(lapconf_auto_bh,6.) * mass * (2.+3.*mass/rr)
          * ( (1.+2.*mass/rr)*flat.con()(i,j)
              - (3.+2.*mass/rr) * ll(i) * ll(j) )
          / 3. / lapconf_tot / rr / rr ;
        }
    }
 
    taij_tot_bh.std_spectral_base() ;
    taij_tot_bh.annule_domain(0) ;
    taij_tot_bh.inc_dzpuis(2) ;
 
    taij_tot = taij_tot_rs + taij_tot_rot + taij_tot_bh ;
 
    taij_tot.std_spectral_base() ;
    taij_tot.annule_domain(0) ;
 
    // Computation of \tilde{A}^{ij} \tilde{A}_{ij}
    // --------------------------------------------
 
    Sym_tensor flat_dshift(mp, COV, mp.get_bvect_cart()) ;
    flat_dshift.set_etat_qcq() ;
 
    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            flat_dshift.set(i,j) =
          flat.cov()(j,1)*(shift_auto_rs(1).deriv(i)
                   + d_shift_comp(i,1))
          + flat.cov()(j,2)*(shift_auto_rs(2).deriv(i)
                     + d_shift_comp(i,2))
          + flat.cov()(j,3)*(shift_auto_rs(3).deriv(i)
                     + d_shift_comp(i,3))
          + flat.cov()(i,1)*(shift_auto_rs(1).deriv(j)
                     + d_shift_comp(j,1))
          + flat.cov()(i,2)*(shift_auto_rs(2).deriv(j)
                     + d_shift_comp(j,2))
          + flat.cov()(i,3)*(shift_auto_rs(3).deriv(j)
                     + d_shift_comp(j,3))
          - 2. * divshift * flat.cov()(i,j) / 3. ;
        }
    }
 
    flat_dshift.std_spectral_base() ;
 
    Sym_tensor curv_dshift(mp, COV, mp.get_bvect_cart()) ;
    curv_dshift.set_etat_qcq() ;
 
    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            curv_dshift.set(i,j) = 2. * mass
          *( ll(j) * ( ll(1)*(shift_auto_rs(1).deriv(i)
                      + d_shift_comp(i,1))
                   + ll(2)*(shift_auto_rs(2).deriv(i)
                    + d_shift_comp(i,2))
                   + ll(3)*(shift_auto_rs(3).deriv(i)
                    + d_shift_comp(i,3)))
             + ll(i) * ( ll(1)*(shift_auto_rs(1).deriv(j)
                    + d_shift_comp(j,1))
                 + ll(2)*(shift_auto_rs(2).deriv(j)
                      + d_shift_comp(j,2))
                 + ll(3)*(shift_auto_rs(3).deriv(j)
                      + d_shift_comp(j,3)))
             - 2. * divshift * ll(i) * ll(j) / 3. ) / rr ;
        }
    }
 
    curv_dshift.std_spectral_base() ;
 
    Sym_tensor tmp1(mp, COV, mp.get_bvect_cart()) ;
    tmp1.set_etat_qcq() ;
 
    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            tmp1.set(i,j) = 2. * mass
          *(ll(j)*( (flat.cov()(i,1)+2.*mass*ll(i)*ll(1)/rr)
                * (shift_auto_rs(1) + shift_comp(1))
                + (flat.cov()(i,2)+2.*mass*ll(i)*ll(2)/rr)
                * (shift_auto_rs(2) + shift_comp(2))
                + (flat.cov()(i,3)+2.*mass*ll(i)*ll(3)/rr)
                * (shift_auto_rs(3) + shift_comp(3))
                )
            + ll(i)*( (flat.cov()(j,1)+2.*mass*ll(j)*ll(1)/rr)
                  * (shift_auto_rs(1) + shift_comp(1))
                  + (flat.cov()(j,2)+2.*mass*ll(j)*ll(2)/rr)
                  * (shift_auto_rs(2) + shift_comp(2))
                  + (flat.cov()(j,3)+2.*mass*ll(j)*ll(3)/rr)
                  * (shift_auto_rs(3) + shift_comp(3)) )
            ) / rr / rr ;
        }
    }
    tmp1.std_spectral_base() ;
    tmp1.inc_dzpuis(2) ;
 
    Sym_tensor tmp2(mp, COV, mp.get_bvect_cart()) ;
    tmp2.set_etat_qcq() ;
 
    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            tmp2.set(i,j) = 2. * mass * lapconf_auto_bh * lapconf_auto_bh
          * ( ll(1) * (shift_auto_rs(1) + shift_comp(1))
              + ll(2) * (shift_auto_rs(2) + shift_comp(2))
              + ll(3) * (shift_auto_rs(3) + shift_comp(3)) )
          * (flat.cov()(i,j)
             - (9.+28.*mass/rr+24.*mass*mass/rr/rr)*ll(i)*ll(j))
          / 3. / rr / rr ;
        }
    }
    tmp2.std_spectral_base() ;
    tmp2.inc_dzpuis(2) ;
 
    Sym_tensor taij_down_rs(mp, COV, mp.get_bvect_cart()) ;
    taij_down_rs.set_etat_qcq() ;
 
    taij_down_rs = 0.5 * pow(confo_tot,7.)
      * (flat_dshift + curv_dshift + tmp1 + tmp2) / lapconf_tot ;
 
    taij_down_rs.std_spectral_base() ;
    taij_down_rs.annule_domain(0) ;
 
    Sym_tensor taij_down_rot(mp, COV, mp.get_bvect_cart()) ;
    taij_down_rot.set_etat_qcq() ;
 
    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
          taij_down_rot.set(i,j) = mass * pow(confo_tot,7.)
        * ( ll(j)*uv(i) + ll(i)*uv(j)
            + lapconf_auto_bh * lapconf_auto_bh
            * (ll(2)*orb_rot_x - ll(1)*orb_rot_y)
            * ( flat.cov()(i,j) - (9.+16.*mass/rr)*ll(i)*ll(j) ) / 3.
            ) / lapconf_tot / rr / rr ;
        }
    }
    taij_down_rot.std_spectral_base() ;
    taij_down_rot.annule_domain(0) ;
    taij_down_rot.inc_dzpuis(2) ;
 
    Sym_tensor taij_down_bh(mp, COV, mp.get_bvect_cart()) ;
    taij_down_bh.set_etat_qcq() ;
 
    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
          taij_down_bh.set(i,j) = 2. * mass * pow(confo_tot,7.)
        * pow(lapconf_auto_bh,4.) * (2.+3.*mass/rr)
        * (flat.cov()(i,j) - (3.+4.*mass/rr) * ll(i) * ll(j))
        / 3. / lapconf_auto / rr / rr ;
        }
    }
    taij_down_bh.std_spectral_base() ;
    taij_down_bh.annule_domain(0) ;
    taij_down_bh.inc_dzpuis(2) ;
 
    Scalar taij_quad_rstot(mp) ;
    taij_quad_rstot = 0. ;
 
    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            taij_quad_rstot += taij_down_rs(i,j) * taij_tot(i,j) ;
        }
    }
    taij_quad_rstot.std_spectral_base() ;
 
    Scalar taij_quad_rsrotbh(mp) ;
    taij_quad_rsrotbh = 0. ;
 
    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            taij_quad_rsrotbh += taij_tot_rs(i,j)
          * (taij_down_rot(i,j) + taij_down_bh(i,j)) ;
        }
    }
    taij_quad_rsrotbh.std_spectral_base() ;
 
    taij_quad_tot_rs = taij_quad_rstot + taij_quad_rsrotbh ;
    taij_quad_tot_rs.std_spectral_base() ;
 
    taij_quad_tot_rot = 8.*mass*mass*pow(confo_tot,14.)
      * pow(lapconf_auto_bh,6.) * (2.+3.*mass/rr)
      * (ll(2)*orb_rot_x - ll(1)*orb_rot_y)
      / 3. / lapconf_tot / lapconf_tot / pow(rr,4.)
      + 2.*mass*mass*pow(confo_tot,14.)*pow(lapconf_auto_bh,4.)
      * (3.*(1.+2.*mass/rr)*(orb_rot_x*orb_rot_x+orb_rot_y*orb_rot_y)
         -2.*(1.+3.*mass/rr)*(ll(2)*orb_rot_x-ll(1)*orb_rot_y)
         *(ll(2)*orb_rot_x-ll(1)*orb_rot_y))
      / 3. / lapconf_tot / lapconf_tot / pow(rr,4.) ;
 
    taij_quad_tot_rot.std_spectral_base() ;
    taij_quad_tot_rot.inc_dzpuis(4) ;
 
    taij_quad_tot_bh = 8.*mass*mass*pow(confo_tot,14.)
      * pow(lapconf_auto_bh,8.) * (2.+3.*mass/rr) * (2.+3.*mass/rr)
      / 3. / lapconf_tot / lapconf_tot / pow(rr,4.) ;
 
    taij_quad_tot_bh.std_spectral_base() ;
    taij_quad_tot_bh.inc_dzpuis(4) ;
 
    taij_quad_tot = taij_quad_tot_rs + taij_quad_tot_rot
      + taij_quad_tot_bh ;
    taij_quad_tot.std_spectral_base() ;
 
    }
    else { // Isotropic coordinates with the maximal slicing
 
        // Sets C/M^2 for each case of the lapse boundary condition
        // --------------------------------------------------------
        double cc ;
 
    if (bc_lapconf_nd) {  // Neumann boundary condition
        if (bc_lapconf_fs) {  // First condition
            // d(\alpha \psi)/dr = 0
            // ---------------------
            cc = 2. * (sqrt(13.) - 1.) / 3. ;
        }
        else {  // Second condition
            // d(\alpha \psi)/dr = (\alpha \psi)/(2 rah)
            // -----------------------------------------
            cc = 4. / 3. ;
        }
    }
    else {  // Dirichlet boundary condition
        if (bc_lapconf_fs) {  // First condition
            // (\alpha \psi) = 1/2
            // -------------------
            cout << "!!!!! WARNING: Not yet prepared !!!!!" << endl ;
        abort() ;
        }
        else {  // Second condition
            // (\alpha \psi) = 1/sqrt(2.) \psi_KS
            // ----------------------------------
            cout << "!!!!! WARNING: Not yet prepared !!!!!" << endl ;
        abort() ;
        //          cc = 2. * sqrt(2.) ;
        }
    }
 
    Scalar r_are(mp) ;
    r_are = r_coord(bc_lapconf_nd, bc_lapconf_fs) ;
    r_are.std_spectral_base() ;
 
        // Computation of \tilde{A}^{ij}
        // -----------------------------
 
    Sym_tensor flat_taij(mp, CON, mp.get_bvect_cart()) ;
    flat_taij.set_etat_qcq() ;
 
    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            flat_taij.set(i,j) = shift_auto_rs(j).deriv(i)
          + shift_auto_rs(i).deriv(j) + d_shift_comp(i,j)
          + d_shift_comp(j,i)
          - 2. * divshift % flat.con()(i,j) / 3. ;
        }
    }
 
    flat_taij.std_spectral_base() ;
 
    taij_tot_rs = 0.5 * pow(confo_tot, 7.) * flat_taij / lapconf_tot ;
 
    taij_tot_rs.std_spectral_base() ;
    taij_tot_rs.annule_domain(0) ;
 
    if (taij_tot_rs(1,2).get_etat() == ETATQCQ) {
      for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
          taij_tot_rs.set(i,j).raccord(1) ;
        }
      }
    }
 
    taij_tot_bh.set_etat_qcq() ;
 
    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            taij_tot_bh.set(i,j) = pow(confo_tot,7.)*mass*mass*cc
          * sqrt(1. - 2.*mass/r_are/rr + cc*cc*pow(mass/r_are/rr,4.))
          * (flat.con()(i,j) - 3.*ll(i)*ll(j)) / lapconf_tot
          / pow(r_are*rr,3.) ;
        }
    }
 
    taij_tot_bh.std_spectral_base() ;
    taij_tot_bh.annule_domain(0) ;
 
    for (int i=1; i<=3; i++) {
      for (int j=1; j<=3; j++) {
        taij_tot_bh.set(i,j).raccord(1) ;
      }
    }
 
    taij_tot_bh.inc_dzpuis(2) ;
 
    taij_tot = taij_tot_rs + taij_tot_bh ;
 
    taij_tot.std_spectral_base() ;
    taij_tot.annule_domain(0) ;
 
    for (int i=1; i<=3; i++) {
      for (int j=1; j<=3; j++) {
        taij_tot.set(i,j).raccord(1) ;
      }
    }
 
    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
        taij_tot_rot.set(i,j) = 0. ;
        }
    }
    taij_tot_rot.std_spectral_base() ;
 
    // Computation of \tilde{A}_{BH}^{ij}
    // ----------------------------------
 
    Scalar divshift_auto(mp) ;
    divshift_auto = shift_auto_rs(1).deriv(1)
      + shift_auto_rs(2).deriv(2) + shift_auto_rs(3).deriv(3) ;
    divshift_auto.std_spectral_base() ;
 
    Sym_tensor flat_taij_auto_rs(mp, CON, mp.get_bvect_cart()) ;
    flat_taij_auto_rs.set_etat_qcq() ;
 
    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            flat_taij_auto_rs.set(i,j) = shift_auto_rs(j).deriv(i)
          + shift_auto_rs(i).deriv(j)
          - 2. * divshift_auto % flat.con()(i,j) / 3. ;
        }
    }
 
    flat_taij_auto_rs.std_spectral_base() ;
 
    taij_auto_rs = 0.5 * pow(confo_tot, 7.) * flat_taij_auto_rs
      / lapconf_tot ;
 
    taij_auto_rs.std_spectral_base() ;
    taij_auto_rs.annule_domain(0) ;
 
    if (taij_auto_rs(1,2).get_etat() == ETATQCQ) {
      for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
          taij_auto_rs.set(i,j).raccord(1) ;
        }
      }
    }
 
    taij_auto = taij_auto_rs + taij_tot_bh ;
 
    taij_auto.std_spectral_base() ;
    taij_auto.annule_domain(0) ;
 
    for (int i=1; i<=3; i++) {
      for (int j=1; j<=3; j++) {
        taij_auto.set(i,j).raccord(1) ;
      }
    }
 
    // Computation of \tilde{A}_{NS}^{ij}
    // ----------------------------------
 
    Scalar divshift_comp(mp) ;
    divshift_comp = d_shift_comp(1,1) + d_shift_comp(2,2)
      + d_shift_comp(3,3) ;
    divshift_comp.std_spectral_base() ;
 
    Sym_tensor flat_taij_comp(mp, CON, mp.get_bvect_cart()) ;
    flat_taij_comp.set_etat_qcq() ;
 
    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            flat_taij_comp.set(i,j) = d_shift_comp(i,j)
          + d_shift_comp(j,i)
          - 2. * divshift_comp % flat.con()(i,j) / 3. ;
        }
    }
 
    flat_taij_comp.std_spectral_base() ;
 
    taij_comp = 0.5 * pow(confo_comp+0.5, 7.) * flat_taij_comp
      / (lapconf_comp+0.5) ;
 
    taij_comp.std_spectral_base() ;
    taij_comp.annule_domain(0) ;
 
    if (taij_comp(1,2).get_etat() == ETATQCQ) {
      for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
          taij_comp.set(i,j).raccord(1) ;
        }
      }
    }
 
    // Computation of \tilde{A}^{ij} \tilde{A}_{ij}
    // --------------------------------------------
 
    Sym_tensor flat_dshift(mp, COV, mp.get_bvect_cart()) ;
    flat_dshift.set_etat_qcq() ;
 
    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            flat_dshift.set(i,j) =
          flat.cov()(j,1)*(shift_auto_rs(1).deriv(i)
                   + d_shift_comp(i,1))
          + flat.cov()(j,2)*(shift_auto_rs(2).deriv(i)
                     + d_shift_comp(i,2))
          + flat.cov()(j,3)*(shift_auto_rs(3).deriv(i)
                     + d_shift_comp(i,3))
          + flat.cov()(i,1)*(shift_auto_rs(1).deriv(j)
                     + d_shift_comp(j,1))
          + flat.cov()(i,2)*(shift_auto_rs(2).deriv(j)
                     + d_shift_comp(j,2))
          + flat.cov()(i,3)*(shift_auto_rs(3).deriv(j)
                     + d_shift_comp(j,3))
          - 2. * divshift * flat.cov()(i,j) / 3. ;
        }
    }
 
    Sym_tensor taij_down_rs(mp, COV, mp.get_bvect_cart()) ;
    taij_down_rs.set_etat_qcq() ;
 
    taij_down_rs = 0.5 * pow(confo_tot, 7.) * flat_dshift / lapconf_tot ;
 
    taij_down_rs.std_spectral_base() ;
    taij_down_rs.annule_domain(0) ;
 
    Sym_tensor taij_down_bh(mp, COV, mp.get_bvect_cart()) ;
    taij_down_bh.set_etat_qcq() ;
 
    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            taij_down_bh.set(i,j) = pow(confo_tot,7.)*mass*mass*cc
          * sqrt(1. - 2.*mass/r_are/rr + cc*cc*pow(mass/r_are/rr,4.))
          * (flat.cov()(i,j) - 3.*ll(i)%ll(j)) / lapconf_tot
          / pow(r_are*rr,3.) ;
        }
    }
    taij_down_bh.std_spectral_base() ;
    taij_down_bh.annule_domain(0) ;
 
    for (int i=1; i<=3; i++) {
      for (int j=1; j<=3; j++) {
        taij_down_bh.set(i,j).raccord(1) ;
      }
    }
 
    taij_down_bh.inc_dzpuis(2) ;
 
    Scalar taij_quad_rstot(mp) ;
    taij_quad_rstot = 0. ;
 
    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            taij_quad_rstot += taij_down_rs(i,j) % taij_tot(i,j) ;
        }
    }
    taij_quad_rstot.std_spectral_base() ;
 
    Scalar taij_quad_rsbh(mp) ;
    taij_quad_rsbh = 0. ;
 
    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            taij_quad_rsbh += taij_tot_rs(i,j) % taij_down_bh(i,j) ;
        }
    }
    taij_quad_rsbh.std_spectral_base() ;
 
    taij_quad_tot_rs = taij_quad_rstot + taij_quad_rsbh ;
    taij_quad_tot_rs.std_spectral_base() ;
 
    taij_quad_tot_rot = 0. ;
    taij_quad_tot_rot.std_spectral_base() ;
 
    taij_quad_tot_bh = 6.*pow(confo_tot,14.)*pow(mass*mass*cc,2.)
      * (1. - 2.*mass/r_are/rr + cc*cc*pow(mass/r_are/rr,4.))
      / lapconf_tot / lapconf_tot / pow(r_are*rr, 6.) ;
    taij_quad_tot_bh.std_spectral_base() ;
    taij_quad_tot_bh.annule_domain(0) ;
    taij_quad_tot_bh.raccord(1) ;
 
    taij_quad_tot_bh.inc_dzpuis(4) ;
 
    taij_quad_tot = taij_quad_tot_rs + taij_quad_tot_bh ;
    taij_quad_tot.std_spectral_base() ;
    taij_quad_tot.annule_domain(0) ;
    taij_quad_tot.raccord(1) ;
 
    // -------------------------
    Scalar taij_quad_auto1(mp) ;
    taij_quad_auto1 = 0. ;
    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            taij_quad_auto1 += taij_auto_rs(i,j)
          * (taij_down_rs(i,j)
             + pow(confo_tot/(confo_comp+0.5),7.)*(lapconf_comp+0.5)
             * taij_comp(i,j) / lapconf_tot) ;
        }
    }
    taij_quad_auto1.std_spectral_base() ;
 
    Scalar taij_quad_auto2(mp) ;
    taij_quad_auto2 = 0. ;
    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            taij_quad_auto2 += taij_tot_bh(i,j) % taij_down_rs(i,j) ;
        }
    }
    taij_quad_auto2.std_spectral_base() ;
 
    taij_quad_auto = taij_quad_auto1 + 2.*taij_quad_auto2 ;
    taij_quad_auto.std_spectral_base() ;
    taij_quad_auto.annule_domain(0) ;
    if (taij_quad_auto.get_etat() == ETATQCQ) {
        taij_quad_auto.raccord(1) ;
    }
 
    // Computation of \tilde{A}_{NS}^{ij} \tilde{A}^{NS}_{ij}
    // ------------------------------------------------------
 
    taij_quad_comp = 0. ;
    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
          taij_quad_comp += taij_comp(i,j) % taij_comp(i,j) ;
        }
    }
    taij_quad_comp.std_spectral_base() ;
 
    }
 
    // The derived quantities are obsolete
    // -----------------------------------
 
    del_deriv() ;
 
}
}