blackhole_extr_curv.C

/*
 *  Method of class Black_hole to compute the extrinsic curvature tensor
 *
 *    (see file blackhole.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
 *
 */

char blackhole_extr_curv_C[] = "$Header: /cvsroot/Lorene/C++/Source/Black_hole/blackhole_extr_curv.C,v 1.2 2008/05/15 19:27:14 k_taniguchi Exp $" ;

/*
 * $Id: blackhole_extr_curv.C,v 1.2 2008/05/15 19:27:14 k_taniguchi Exp $
 * $Log: blackhole_extr_curv.C,v $
 * Revision 1.2  2008/05/15 19:27:14  k_taniguchi
 * Change of some parameters.
 *
 * Revision 1.1  2007/06/22 01:19:32  k_taniguchi
 * *** empty log message ***
 *
 *
 * $Header: /cvsroot/Lorene/C++/Source/Black_hole/blackhole_extr_curv.C,v 1.2 2008/05/15 19:27:14 k_taniguchi Exp $
 *
 */

// C++ headers
//#include <>

// C headers
#include <math.h>

// Lorene headers
#include "blackhole.h"
#include "utilitaires.h"
#include "unites.h"

void Black_hole::extr_curv_bh() {

    // Fundamental constants and units
    // -------------------------------
    using namespace Unites ;

    if (kerrschild) {

        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() ;


    // Computation of \tilde{A}^{ij}
    // -----------------------------

    Scalar divshift(mp) ;
    divshift = shift_rs(1).dsdx() + shift_rs(2).dsdy()
      + shift_rs(3).dsdz() ;
    divshift.std_spectral_base() ;

    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_rs(j).deriv(i)
          + shift_rs(i).deriv(j)
          - 2. * divshift * flat.con()(i,j) / 3. ;
        }
    }

    flat_taij.std_spectral_base() ;

    Scalar lapse_bh(mp) ;
    lapse_bh = 1. / sqrt(1. + 2. * mass / rr) ;
    lapse_bh.std_spectral_base() ;
    lapse_bh.annule_domain(0) ;
    lapse_bh.raccord(1) ;

    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. * lapse_bh * lapse_bh * mass
          * (ll(i)*(shift_rs(j).dsdr()) + ll(j)*(shift_rs(i).dsdr())
             - 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. * lapse_bh * lapse_bh * mass
          * ( ll(i) * shift_rs(j) + ll(j) * shift_rs(i)
              + ( flat.con()(i,j)
              - lapse_bh*lapse_bh*(9.+14.*mass/rr)*ll(i)*ll(j) )
              * ( ll(1)*shift_rs(1)+ll(2)*shift_rs(2)
              +ll(3)*shift_rs(3) )/ 3. )
          / rr / rr ;
        }
    }

    resi_taij.std_spectral_base() ;
    resi_taij.inc_dzpuis(2) ;

    taij_rs = 0.5 * pow(confo, 7.)
      * (flat_taij + curv_taij + resi_taij) / lapconf ;

    taij_rs.std_spectral_base() ;
    taij_rs.annule_domain(0) ;

    Sym_tensor taij_bh(mp, CON, mp.get_bvect_cart()) ;
    taij_bh.set_etat_qcq() ;

    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            taij_bh.set(i,j) = 2.*pow(lapse_bh,6.)*mass*(2.+3.*mass/rr)
          *( (1.+2.*mass/rr) * flat.con()(i,j)
             - (3.+2.*mass/rr) * ll(i) * ll(j) )
          *pow(confo, 7.)/lapconf/3./rr/rr ;
        }
    }

    taij_bh.std_spectral_base() ;
    taij_bh.inc_dzpuis(2) ;
    taij_bh.annule_domain(0) ;

    taij = taij_rs + taij_bh ;
    taij.std_spectral_base() ;
    taij.annule_domain(0) ;

    /*
    Sym_tensor taij_ks_con(mp, CON, mp.get_bvect_cart()) ;
    taij_ks_con.set_etat_qcq() ;

    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            taij_ks_con.set(i,j) = 2.*pow(lap_bh2,2.5)*mass
          * (2.+3.*mass/rr)
          * ( (1.+2.*mass/rr)*flat.con()(i,j)
              - (3.+2.*mass/rr)*ll(i)*ll(j) ) / 3. / rr / rr ;
        }
    }
    taij_ks_con.std_spectral_base() ;
    taij_ks_con.annule_domain(0) ;
    taij_ks_con.inc_dzpuis(2) ;

    cout << "taij(1,1) - taij_ks_con(1,1) : " << endl ;
    cout << taij(1,1) - taij_ks_con(1,1) << endl ;
    arrete() ;

    cout << "taij_ks_con(1,1) : " << endl ;
    cout << taij_ks_con(1,1) << endl ;
    arrete() ;

    cout << "taij(1,1) : " << endl ;
    cout << taij(1,1) << endl ;
    arrete() ;
    */

    // 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_rs(1).deriv(i))
          + flat.cov()(j,2)*(shift_rs(2).deriv(i))
          + flat.cov()(j,3)*(shift_rs(3).deriv(i))
          + flat.cov()(i,1)*(shift_rs(1).deriv(j))
          + flat.cov()(i,2)*(shift_rs(2).deriv(j))
          + flat.cov()(i,3)*(shift_rs(3).deriv(j))
          - 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_rs(1).deriv(i))
                  + ll(2)*(shift_rs(2).deriv(i))
                  + ll(3)*(shift_rs(3).deriv(i)) )
             + ll(i) *( ll(1)*(shift_rs(1).deriv(j))
                + ll(2)*(shift_rs(2).deriv(j))
                + ll(3)*(shift_rs(3).deriv(j)) )
             - 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_rs(1)
                + (flat.cov()(i,2)+2.*mass*ll(i)*ll(2)/rr)
                *shift_rs(2)
                + (flat.cov()(i,3)+2.*mass*ll(i)*ll(3)/rr)
                *shift_rs(3)
                )
            + ll(i)*( (flat.cov()(j,1)+2.*mass*ll(j)*ll(1)/rr)
                  *shift_rs(1)
                  + (flat.cov()(j,2)+2.*mass*ll(j)*ll(2)/rr)
                  *shift_rs(2)
                  + (flat.cov()(j,3)+2.*mass*ll(j)*ll(3)/rr)
                  *shift_rs(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 * lapse_bh * lapse_bh
          * (ll(1)*shift_rs(1)+ll(2)*shift_rs(2)+ll(3)*shift_rs(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_rs_down(mp, COV, mp.get_bvect_cart()) ;
    taij_rs_down.set_etat_qcq() ;

    taij_rs_down = 0.5 * pow(confo, 7.)
      * (flat_dshift + curv_dshift + tmp1 + tmp2) / lapconf ;

    taij_rs_down.std_spectral_base() ;
    taij_rs_down.annule_domain(0) ;

    Sym_tensor taij_bh_down(mp, COV, mp.get_bvect_cart()) ;
    taij_bh_down.set_etat_qcq() ;

    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
          taij_bh_down.set(i,j) = 2.*pow(lapse_bh,4.)*mass*(2.+3.*mass/rr)
        *pow(confo,7.)*(flat.cov()(i,j)-(3.+4.*mass/rr)*ll(i)*ll(j))
        /lapconf/3./rr/rr ;
        }
    }

    taij_bh_down.std_spectral_base() ;
    taij_bh_down.inc_dzpuis(2) ;
    taij_bh_down.annule_domain(0) ;

    /*
    Sym_tensor taij_ks_cov(mp, COV, mp.get_bvect_cart()) ;
    taij_ks_cov.set_etat_qcq() ;

    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
          taij_ks_cov.set(i,j) = 2.*pow(lap_bh2,1.5)*mass * (2.+3.*mass/rr)
        * ( flat.cov()(i,j)
            - (3.+4.*mass/rr)*ll(i)*ll(j) ) / 3. / rr / rr ;
        }
    }
    taij_ks_cov.std_spectral_base() ;
    taij_ks_cov.annule_domain(0) ;
    taij_ks_cov.inc_dzpuis(2) ;

    cout << "taij_down(1,1) - taij_ks_cov(1,1) : " << endl ;
    cout << taij_down(1,1) - taij_ks_cov(1,1) << endl ;
    arrete() ;

    cout << "taij_ks_cov(1,1) : " << endl ;
    cout << taij_ks_cov(1,1) << endl ;
    arrete() ;

    cout << "taij_down(1,1) : " << endl ;
    cout << taij_down(1,1) << endl ;
    arrete() ;
    */

    Scalar taij_quad_rsrs(mp) ;
    taij_quad_rsrs = 0. ;

    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            taij_quad_rsrs += taij_rs_down(i,j) * taij_rs(i,j) ;
        }
    }
    taij_quad_rsrs.std_spectral_base() ;

    Scalar taij_quad_rsbh1(mp) ;
    taij_quad_rsbh1 = 0. ;

    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            taij_quad_rsbh1 += taij_rs_down(i,j) * taij_bh(i,j) ;
        }
    }
    taij_quad_rsbh1.std_spectral_base() ;

    Scalar taij_quad_rsbh2(mp) ;
    taij_quad_rsbh2 = 0. ;

    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            taij_quad_rsbh2 += taij_bh_down(i,j) * taij_rs(i,j) ;
        }
    }
    taij_quad_rsbh2.std_spectral_base() ;

    taij_quad_rs = taij_quad_rsrs + taij_quad_rsbh1 + taij_quad_rsbh2 ;
    taij_quad_rs.std_spectral_base() ;

    Scalar taij_quad_bh(mp) ;
    taij_quad_bh = 8.*pow(lapse_bh,10.)*mass*mass*(2.+3.*mass/rr)
      *(2.+3.*mass/rr)*pow(confo,12.)/3./pow(rr,4.)/lapconf/lapconf ;
    taij_quad_bh.std_spectral_base() ;
    taij_quad_bh.inc_dzpuis(4) ;

    taij_quad = taij_quad_rs + taij_quad_bh ;

    taij_quad.std_spectral_base() ;

    /*
    Scalar taij_quad_ks(mp) ;
    taij_quad_ks = 8. * pow(lap_bh2,3.) * mass * mass * (2.+3.*mass/rr)
      * (2.+3.*mass/rr) / 3. / pow(rr, 4.) ;
    taij_quad_ks.std_spectral_base() ;
    taij_quad_ks.annule_domain(0) ;
    taij_quad_ks.inc_dzpuis(4) ;

    cout << "taij_quad - taij_quad_ks : " << endl ;
    cout << taij_quad - taij_quad_ks << endl ;
    arrete() ;
    */
    }
    else {  // Isotropic coordinates with the maximal slicing

        // Computation of \tilde{A}^{ij}
    // -----------------------------

    Scalar divshift(mp) ;
    divshift = shift(1).dsdx() + shift(2).dsdy() + shift(3).dsdz() ;
    divshift.std_spectral_base() ;

    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(j).deriv(i) + shift(i).deriv(j)
          - 2. * divshift * flat.con()(i,j) / 3. ;
        }
    }

    flat_taij.std_spectral_base() ;

    taij = 0.5 * pow(confo, 7.) * flat_taij / lapconf ;

    taij.std_spectral_base() ;
    taij.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(1).deriv(i))
          + flat.cov()(j,2)*(shift(2).deriv(i))
          + flat.cov()(j,3)*(shift(3).deriv(i))
          + flat.cov()(i,1)*(shift(1).deriv(j))
          + flat.cov()(i,2)*(shift(2).deriv(j))
          + flat.cov()(i,3)*(shift(3).deriv(j))
          - 2. * divshift * flat.cov()(i,j) / 3. ;
        }
    }

    Sym_tensor taij_down(mp, COV, mp.get_bvect_cart()) ;
    taij_down.set_etat_qcq() ;

    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
          taij_down.set(i,j) = 0.5 * pow(confo, 7.) * flat_dshift(i,j)
          / lapconf ;
        }
    }

    taij_down.std_spectral_base() ;
    taij_down.annule_domain(0) ;

    taij_quad = 0. ;

    for (int i=1; i<=3; i++) {
        for (int j=1; j<=3; j++) {
            taij_quad += taij_down(i,j) * taij(i,j) ;
        }
    }
    taij_quad.std_spectral_base() ;

    }

}

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