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V0Equations.c
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V0Equations.c
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/* Numerically solve the transport equation for V_0
* along a geodesic. This file contains the RHS's for all the transport equations
* required to calculate V_0 from Delta^1/2. Most are derivatives of sigma and in
* that case they are named sigma_n_m_RHS to indicate n unprimed and m primed derivatives
* of sigma and u means upstairs, d downstairs.
*
* Copyright (C) 2009 Barry Wardell
*
* This program is free software; you can redistribute it and/or
* modify it under the terms of the GNU General Public License
* as published by the Free Software Foundation; either version 2
* of the License, or (at your option) any later version.
*
* This software is distributed on an "AS IS" basis, WITHOUT WARRANTY OF ANY
* KIND, either express or implied.
*/
#include <stdio.h>
#include <string.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_odeiv.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_blas.h>
#include <gsl/gsl_linalg.h>
#include "SpacetimeTensors.h"
/* Bivector of parallel displacement, g_{a'}^{~ a}. We use the defining equation
\sigma^{b'} g_{a'}{}^{a}{}_{;b'} = 0 */
int IRHS (double tau, const gsl_vector * y, const gsl_vector * yp, const gsl_matrix * I, gsl_matrix * f, void * params)
{
/* Gamma*u */
gsl_matrix * gu = gsl_matrix_calloc(4,4);
Gu(y, yp, gu, params);
/* RHS */
gsl_blas_dgemm(CblasTrans, CblasNoTrans, 1.0, gu, I, 0., f);
gsl_matrix_free(gu);
return GSL_SUCCESS;
}
int etaRHS (double tau, const gsl_vector * y, const gsl_vector * yp, const gsl_matrix * q, const gsl_matrix * eta, gsl_matrix * f, void * params)
{
int i;
/* eta*Gu */
gsl_matrix * gu = gsl_matrix_calloc(4,4);
gsl_matrix * eta_gu = gsl_matrix_calloc(4,4);
Gu(y, yp, gu, params);
gsl_blas_dgemm(CblasNoTrans, CblasNoTrans, 1.0, eta, gu, 0., eta_gu);
/* Xi */
gsl_matrix * xi = gsl_matrix_calloc(4,4);
gsl_matrix_memcpy(xi, q);
for( i=0; i<4; i++)
{
gsl_matrix_set(xi,i,i, gsl_matrix_get(xi,i,i) + 1.);
}
/* RHS */
gsl_matrix * eta_rhs = gsl_matrix_calloc(4,4);
if(tau!=0.0)
{
gsl_matrix_memcpy(eta_rhs, eta);
gsl_blas_dgemm(CblasNoTrans, CblasNoTrans, -1.0, eta, xi, 1.0, eta_rhs);
gsl_matrix_scale(eta_rhs, 1.0/tau);
}
gsl_matrix_add(eta_rhs, eta_gu);
gsl_matrix_memcpy(f, eta_rhs);
gsl_matrix_free(gu);
gsl_matrix_free(eta_gu);
gsl_matrix_free(xi);
gsl_matrix_free(eta_rhs);
return GSL_SUCCESS;
}
int dIinvRHS (double tau, const gsl_vector * y, const gsl_vector * yp, const gsl_matrix * I, const gsl_matrix *q, const gsl_vector * dIinv, gsl_vector * f, void * params)
{
int i, j, k, l;
/* First, we need I^-1 */
int signum=0;
gsl_permutation * p = gsl_permutation_calloc (4);
gsl_matrix * I_inv = gsl_matrix_calloc(4,4);
gsl_matrix * lu = gsl_matrix_calloc(4,4);
gsl_matrix_memcpy(lu, I);
gsl_linalg_LU_decomp (lu, p, &signum);
gsl_linalg_LU_invert (lu, p, I_inv);
/* And calculate sigma_R */
gsl_vector * sigma_R = gsl_vector_calloc(4*4*4);
R_sigma(y, yp, sigma_R, params);
gsl_vector_set_zero(f);
/* Now, calculate sigma_R * I^(-1) */
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
for(l=0; l<4; l++)
gsl_vector_set(f, 16*i+4*j+k, gsl_vector_get(f, 16*i + 4*j + k) + gsl_vector_get(sigma_R, 16*j + 4*l +k)*gsl_matrix_get(I_inv, i, l));
if(tau!=0.0)
{
/* And calculate dIinv * xi*/
gsl_matrix * xi = gsl_matrix_calloc(4,4);
gsl_matrix_memcpy(xi, q);
for( i=0; i<4; i++)
{
gsl_matrix_set(xi,i,i, gsl_matrix_get(xi,i,i) + 1.);
}
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
for(l=0; l<4; l++)
gsl_vector_set(f, 16*i+4*j+k, gsl_vector_get(f, 16*i + 4*j + k) - gsl_vector_get(dIinv, 16*i + 4*j + l)*gsl_matrix_get(xi, l, k)/tau);
gsl_matrix_free(xi);
} else {
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
gsl_vector_set(f, 16*i+4*j+k, gsl_vector_get(f, 16*i + 4*j + k) - 0.5 * gsl_vector_get(sigma_R, 16*j + 4*i + k));
}
/* Christoffel terms */
gsl_matrix * gu = gsl_matrix_calloc(4,4);
Gu(y, yp, gu, params);
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
for(l=0; l<4; l++)
gsl_vector_set(f, 16*i+4*j+k, gsl_vector_get(f, 16*i + 4*j + k) - gsl_vector_get(dIinv, 16*i + 4*l + k)*gsl_matrix_get(gu, j, l)
+ gsl_vector_get(dIinv, 16*i + 4*j + l)*gsl_matrix_get(gu, l, k));
gsl_permutation_free(p);
gsl_matrix_free(I_inv);
gsl_matrix_free(lu);
gsl_vector_free(sigma_R);
gsl_matrix_free(gu);
return GSL_SUCCESS;
}
/* RHS of transport equation for dxi. */
int dxiRHS (double tau, const gsl_vector * y, const gsl_vector * yp, const gsl_matrix *q, const gsl_vector * dxi, gsl_vector * f, void * params)
{
int i, j, k, l;
/* Calculate sigma_R */
gsl_vector * sigma_R = gsl_vector_calloc(4*4*4);
R_sigma(y, yp, sigma_R, params);
/* Calculate sigma2_dR */
gsl_vector * sigma2_dR = gsl_vector_calloc(4*4*4);
dR_sigma2(y, yp, sigma2_dR, params);
/* First, we need xi from q */
gsl_matrix * xi = gsl_matrix_calloc(4,4);
gsl_matrix_memcpy(xi, q);
for( i=0; i<4; i++)
{
gsl_matrix_set(xi,i,i, gsl_matrix_get(xi,i,i) + 1.);
}
gsl_vector_set_zero(f);
if(tau!=0.0){
/* The first term on the RHS is just dxi/tau */
gsl_vector_memcpy(f, dxi);
/* Now we work out the three xi*dxi terms */
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
for(l=0; l<4; l++)
gsl_vector_set(f, 16*i + 4*j + k, gsl_vector_get(f, 16*i + 4*j + k)
- gsl_matrix_get(xi, i, l)*gsl_vector_get(dxi, 16*l + 4*j + k)
- gsl_matrix_get(xi, l, j)*gsl_vector_get(dxi, 16*i + 4*l + k)
- gsl_matrix_get(xi, l, k)*gsl_vector_get(dxi, 16*i + 4*l + j)
);
gsl_vector_scale(f, 1./tau);
} else {
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
gsl_vector_set(f, 16*i + 4*j + k, gsl_vector_get(f, 16*i + 4*j + k)
-(gsl_vector_get(sigma_R, 16*i + 4*j + k)
+gsl_vector_get(sigma_R, 16*i + 4*k + j))/3.0
);
}
/* Now, calculate the three sigma_R * xi terms */
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
for(l=0; l<4; l++)
gsl_vector_set(f, 16*i + 4*j + k, gsl_vector_get(f, 16*i + 4*j + k)
- gsl_matrix_get(xi, i, l)*gsl_vector_get(sigma_R, 16*l + 4*j + k)
+ gsl_matrix_get(xi, l, j)*gsl_vector_get(sigma_R, 16*i + 4*l + k)
+ gsl_matrix_get(xi, l, k)*gsl_vector_get(sigma_R, 16*i + 4*l + j)
);
/* Now, calculate the sigma2_dR term */
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
gsl_vector_set(f, 16*i + 4*j + k, gsl_vector_get(f, 16*i + 4*j + k)
- tau*gsl_vector_get(sigma2_dR, 16*i + 4*j + k));
/* Christoffel terms */
gsl_matrix * gu = gsl_matrix_calloc(4,4);
Gu(y, yp, gu, params);
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
for(l=0; l<4; l++)
gsl_vector_set(f, 16*i + 4*j + k, gsl_vector_get(f, 16*i + 4*j + k)
- gsl_matrix_get(gu, i, l)*gsl_vector_get(dxi, 16*l + 4*j + k)
+ gsl_matrix_get(gu, l, j)*gsl_vector_get(dxi, 16*i + 4*l + k)
+ gsl_matrix_get(gu, l, k)*gsl_vector_get(dxi, 16*i + 4*j + l));
gsl_vector_free(sigma_R);
gsl_vector_free(sigma2_dR);
gsl_matrix_free(xi);
gsl_matrix_free(gu);
return GSL_SUCCESS;
}
/* RHS of transport equation for deta. The matrix elements are the first two indices. */
int detaRHS (double tau, const gsl_vector * y, const gsl_vector * yp, const gsl_matrix *q, const gsl_vector * dxi, const gsl_matrix * eta, const gsl_vector * deta, gsl_vector * f, void * params)
{
int i, j, k, l;
/* Calculate sigma_R */
gsl_vector * sigma_R = gsl_vector_calloc(4*4*4);
R_sigma(y, yp, sigma_R, params);
gsl_vector_set_zero(f);
if(tau!=0.0)
{
/* First, we need xi from q */
gsl_matrix * xi = gsl_matrix_calloc(4,4);
gsl_matrix_memcpy(xi, q);
for( i=0; i<4; i++)
{
gsl_matrix_set(xi,i,i, gsl_matrix_get(xi,i,i) + 1.);
}
/* The first term on the RHS is just deta/tau */
gsl_vector_memcpy(f, deta);
/* Now we work out the two xi*deta terms and one eta dxi term*/
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
for(l=0; l<4; l++)
gsl_vector_set(f, 16*i + 4*j + k, gsl_vector_get(f, 16*i + 4*j + k)
- gsl_matrix_get(xi, l, j)*gsl_vector_get(deta, 16*i + 4*l + k)
- gsl_matrix_get(xi, l, k)*gsl_vector_get(deta, 16*i + 4*l + j)
- gsl_matrix_get(eta, i, l)*gsl_vector_get(dxi, 16*l + 4*j + k)
);
gsl_vector_scale(f, 1./tau);
gsl_matrix_free(xi);
} else {
/* Calculate sigma_R_alt */
gsl_vector * sigma_R_alt = gsl_vector_calloc(4*4*4);
R_sigma_alt(y, yp, sigma_R_alt, params);
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
gsl_vector_set(f, 16*i + 4*j + k, gsl_vector_get(f, 16*i + 4*j + k)
- gsl_vector_get(sigma_R_alt, 16*i + 4*j + k)/2.0
- (gsl_vector_get(sigma_R, 16*i + 4*j + k) +
gsl_vector_get(sigma_R, 16*i + 4*k + j))/6.0
);
gsl_vector_free(sigma_R_alt);
}
/* Now, calculate the sigma_R * eta term */
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
for(l=0; l<4; l++)
gsl_vector_set(f, 16*i + 4*j + k, gsl_vector_get(f, 16*i + 4*j + k)
- gsl_matrix_get(eta, i, l)*gsl_vector_get(sigma_R, 16*l + 4*j + k));
/* Christoffel terms */
gsl_matrix * gu = gsl_matrix_calloc(4,4);
Gu(y, yp, gu, params);
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
for(l=0; l<4; l++)
gsl_vector_set(f, 16*i + 4*j + k, gsl_vector_get(f, 16*i + 4*j + k)
+ gsl_matrix_get(gu, l, j)*gsl_vector_get(deta, 16*i + 4*l + k)
+ gsl_matrix_get(gu, l, k)*gsl_vector_get(deta, 16*i + 4*j + l));
gsl_vector_free(sigma_R);
gsl_matrix_free(gu);
return GSL_SUCCESS;
}
int d2IinvRHS (double tau, const gsl_vector * y, const gsl_vector * yp, const gsl_matrix *q, const gsl_vector * dxi, const gsl_matrix * I, const gsl_vector * dIinv, const gsl_vector * d2Iinv, gsl_vector * f, void * params)
{
int i, j, k, l, m;
/* Christoffel terms */
gsl_matrix * gu = gsl_matrix_calloc(4,4);
Gu(y, yp, gu, params);
/* xi */
gsl_matrix * xi = gsl_matrix_calloc(4,4);
gsl_matrix_memcpy(xi, q);
for( i=0; i<4; i++)
{
gsl_matrix_set(xi,i,i, gsl_matrix_get(xi,i,i) + 1.);
}
/* Calculate sigma_R */
gsl_vector * sigma_R = gsl_vector_calloc(4*4*4);
R_sigma(y, yp, sigma_R, params);
/* Calculate sigma_dR */
gsl_vector * sigma_dR = gsl_vector_calloc(4*4*4*4);
dR_sigma(y, yp, sigma_dR, params);
/* I^-1 */
int signum;
gsl_permutation * p = gsl_permutation_alloc (4);
gsl_matrix * lu = gsl_matrix_calloc(4,4);
gsl_matrix * Iinv = gsl_matrix_calloc(4,4);
gsl_matrix_memcpy(lu, I);
gsl_linalg_LU_decomp (lu, p, &signum);
gsl_linalg_LU_invert (lu, p, Iinv);
gsl_vector_set_zero(f);
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
for(l=0; l<4; l++)
for(m=0; m<4; m++)
gsl_vector_set(f, 64*i+16*j+4*k+l, gsl_vector_get(f, 64*i+16*j+4*k+l)
/* gu * d2I */
+ gsl_vector_get(d2Iinv, 64*i+16*j+4*m+l) * gsl_matrix_get(gu, m, k)
+ gsl_vector_get(d2Iinv, 64*i+16*j+4*k+m) * gsl_matrix_get(gu, m, l)
- gsl_vector_get(d2Iinv, 64*i+16*m+4*k+l) * gsl_matrix_get(gu, j, m)
/* R_sigma * dIinv */
+ gsl_vector_get(dIinv, 16*i+4*m+k)*gsl_vector_get(sigma_R, 16*j + 4*m + l)
+ gsl_vector_get(dIinv, 16*i+4*m+l)*gsl_vector_get(sigma_R, 16*j + 4*m + k)
- gsl_vector_get(dIinv, 16*i+4*j+m)*gsl_vector_get(sigma_R, 16*m + 4*k + l)
/* dR_sigma * Iinv */
+ gsl_matrix_get(Iinv, i, m)*gsl_vector_get(sigma_dR, 64*j + 16*m + 4*k + l)
);
if(tau!=0.0)
{
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
for(l=0; l<4; l++)
for(m=0; m<4; m++)
gsl_vector_set(f, 64*i+16*j+4*k+l, gsl_vector_get(f, 64*i+16*j+4*k+l)
/* xi * d2I */
- (
+ gsl_vector_get(d2Iinv, 64*i+16*j+4*m+l) * gsl_matrix_get(xi, m, k)
+ gsl_vector_get(d2Iinv, 64*i+16*j+4*m+k) * gsl_matrix_get(xi, m, l)
/* dxi * dIinv */
+ gsl_vector_get(dIinv, 16*i+4*j+m) * gsl_vector_get(dxi, 16*m+4*k+l)
)/tau);
} else {
/* init^{~ b'}_{a ~ c' d'} (0) = -2/3* R^{b'}_{ a' \alpha (c' ; d')}*/
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
for(l=0; l<4; l++)
gsl_vector_set(f, 64*i+16*j+4*k+l, gsl_vector_get(f, 64*i+16*j+4*k+l)
-(gsl_vector_get(sigma_dR, 64*j + 16*i + 4*k + l)
+gsl_vector_get(sigma_dR, 64*j + 16*i + 4*l + k)
)/3.0);
}
gsl_vector_free(sigma_R);
gsl_vector_free(sigma_dR);
gsl_permutation_free(p);
gsl_matrix_free(lu);
gsl_matrix_free(Iinv);
gsl_matrix_free(xi);
gsl_matrix_free(gu);
return GSL_SUCCESS;
}
int d2xiRHS (double tau, const gsl_vector * y, const gsl_vector * yp, const gsl_matrix *q, const gsl_vector * dxi, const gsl_matrix * I, const gsl_vector * dIinv, const gsl_vector * d2xi, gsl_vector * f, void * params)
{
int i, j, k, l, m;
/* Christoffel terms */
gsl_matrix * gu = gsl_matrix_calloc(4,4);
Gu(y, yp, gu, params);
/* xi */
gsl_matrix * xi = gsl_matrix_calloc(4,4);
gsl_matrix_memcpy(xi, q);
for( i=0; i<4; i++)
{
gsl_matrix_set(xi,i,i, gsl_matrix_get(xi,i,i) + 1);
}
/* Calculate sigma_R */
gsl_vector * sigma_R = gsl_vector_calloc(4*4*4);
R_sigma(y, yp, sigma_R, params);
/* Calculate sigma_R_alt */
gsl_vector * sigma_R_alt = gsl_vector_calloc(4*4*4);
R_sigma_alt(y, yp, sigma_R_alt, params);
/* Calculate sigma2_d2R */
gsl_vector * sigma2_d2R = gsl_vector_calloc(4*4*4*4);
d2R_sigma2(y, yp, sigma2_d2R, params);
/* Calculate sigma_dR */
gsl_vector * sigma_dR = gsl_vector_calloc(4*4*4*4);
dR_sigma(y, yp, sigma_dR, params);
gsl_vector_set_zero(f);
if(tau!=0.0)
{
gsl_vector_memcpy(f, d2xi);
gsl_vector_scale(f, 1./tau);
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
for(l=0; l<4; l++)
for(m=0; m<4; m++)
gsl_vector_set(f, 64*i+16*j+4*k+l,
gsl_vector_get(f, 64*i+16*j+4*k+l)
+ (
/* dxi * dxi */
- gsl_vector_get(dxi, 16*i+4*m+k) * gsl_vector_get(dxi, 16*m+4*j+l)
- gsl_vector_get(dxi, 16*i+4*m+j) * gsl_vector_get(dxi, 16*m+4*k+l)
- gsl_vector_get(dxi, 16*i+4*m+l) * gsl_vector_get(dxi, 16*m+4*j+k)
/* d2xi * xi */
- gsl_vector_get(d2xi, 64*m+16*j+4*k+l) * gsl_matrix_get(xi, i, m)
- gsl_vector_get(d2xi, 64*i+16*m+4*k+l) * gsl_matrix_get(xi, m, j)
- gsl_vector_get(d2xi, 64*i+16*m+4*j+l) * gsl_matrix_get(xi, m, k)
- gsl_vector_get(d2xi, 64*i+16*m+4*j+k) * gsl_matrix_get(xi, m, l)
)/tau
);
} else {
/* init^{a}_{~ b c' d'} (0) = -3/2* R^{a'}_{~ b |\alpha| c' ; d')}*/
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
for(l=0; l<4; l++)
gsl_vector_set(f, 64*i+16*j+4*k+l, gsl_vector_get(f, 64*i+16*j+4*k+l)
-( gsl_vector_get(sigma_dR, 64*i + 16*j + 4*k + l)
+ gsl_vector_get(sigma_dR, 64*i + 16*j + 4*l + k)
+ gsl_vector_get(sigma_dR, 64*i + 16*k + 4*j + l)
+ gsl_vector_get(sigma_dR, 64*i + 16*k + 4*l + j)
+ gsl_vector_get(sigma_dR, 64*i + 16*l + 4*j + k)
+ gsl_vector_get(sigma_dR, 64*i + 16*l + 4*k + j)
)/4.0);
}
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
for(l=0; l<4; l++)
for(m=0; m<4; m++)
gsl_vector_set(f, 64*i+16*j+4*k+l,
gsl_vector_get(f, 64*i+16*j+4*k+l)
/* gu * d2xi */
+ gsl_vector_get(d2xi, 64*i+16*m+4*k+l) * gsl_matrix_get(gu, m, j)
+ gsl_vector_get(d2xi, 64*i+16*j+4*m+l) * gsl_matrix_get(gu, m, k)
+ gsl_vector_get(d2xi, 64*i+16*j+4*k+m) * gsl_matrix_get(gu, m, l)
- gsl_vector_get(d2xi, 64*m+16*j+4*k+l) * gsl_matrix_get(gu, i, m)
/* r_sigma * r_sigma */
+ tau*(
+ gsl_vector_get(sigma_R_alt, 16*i+4*m+l)*gsl_vector_get(sigma_R, 16*m + 4*j + k)
+ gsl_vector_get(sigma_R_alt, 16*i+4*m+k)*gsl_vector_get(sigma_R, 16*m + 4*j + l)
)
/* r_sigma * dxi */
- gsl_vector_get(dxi, 16*i+4*j+m)*gsl_vector_get(sigma_R, 16*m + 4*k + l)
- gsl_vector_get(dxi, 16*i+4*k+m)*gsl_vector_get(sigma_R, 16*m + 4*j + l)
- gsl_vector_get(dxi, 16*i+4*l+m)*gsl_vector_get(sigma_R, 16*m + 4*j + k)
+ gsl_vector_get(dxi, 16*m+4*j+l)*gsl_vector_get(sigma_R, 16*i + 4*m + k)
+ gsl_vector_get(dxi, 16*m+4*k+l)*gsl_vector_get(sigma_R, 16*i + 4*m + j)
+ gsl_vector_get(dxi, 16*m+4*j+k)*gsl_vector_get(sigma_R, 16*i + 4*m + l)
/* dR_sigma * xi */
- gsl_matrix_get(xi, i, m)*gsl_vector_get(sigma_dR, 64*m + 16*j + 4*k + l)
+ gsl_matrix_get(xi, m, j)*gsl_vector_get(sigma_dR, 64*i + 16*m + 4*k + l)
+ gsl_matrix_get(xi, m, k)*gsl_vector_get(sigma_dR, 64*i + 16*m + 4*j + l)
+ gsl_matrix_get(xi, m, l)*gsl_vector_get(sigma_dR, 64*i + 16*m + 4*j + k)
);
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
for(l=0; l<4; l++)
gsl_vector_set(f, 64*i+16*j+4*k+l,
gsl_vector_get(f, 64*i+16*j+4*k+l)
/* d2R_sigma */
- tau * gsl_vector_get(sigma2_d2R, 64*i + 16*j + 4*k + l)
);
gsl_vector_free(sigma_R);
gsl_vector_free(sigma_R_alt);
gsl_vector_free(sigma2_d2R);
gsl_vector_free(sigma_dR);
gsl_matrix_free(xi);
gsl_matrix_free(gu);
return GSL_SUCCESS;
}
/* d2eta */
int d2etaRHS (double tau, const gsl_vector * y, const gsl_vector * yp, const gsl_matrix *eta, const gsl_matrix *q, const gsl_vector * dxi, const gsl_vector * d2xi, const gsl_vector * deta, const gsl_vector * d2eta, gsl_vector * f, void * params)
{
int i, j, k, l, m;
/* Christoffel terms */
gsl_matrix * gu = gsl_matrix_calloc(4,4);
Gu(y, yp, gu, params);
/* xi */
gsl_matrix * xi = gsl_matrix_calloc(4,4);
gsl_matrix_memcpy(xi, q);
for( i=0; i<4; i++)
{
gsl_matrix_set(xi,i,i, gsl_matrix_get(xi,i,i) + 1.);
}
/* Calculate sigma_R */
gsl_vector * sigma_R = gsl_vector_calloc(4*4*4);
R_sigma(y, yp, sigma_R, params);
/* Calculate sigma_dR */
gsl_vector * sigma_dR = gsl_vector_calloc(4*4*4*4);
dR_sigma(y, yp, sigma_dR, params);
gsl_vector_set_zero( f );
if(tau!=0.0)
{
gsl_vector_memcpy(f, d2eta);
gsl_vector_scale(f, 1./tau);
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
for(l=0; l<4; l++)
for(m=0; m<4; m++)
gsl_vector_set(f, 64*i+16*j+4*k+l,
gsl_vector_get(f, 64*i+16*j+4*k+l) + (
/* deta * dxi */
- gsl_vector_get(deta, 16*i+4*m+j) * gsl_vector_get(dxi, 16*m+4*k+l)
- gsl_vector_get(deta, 16*i+4*m+k) * gsl_vector_get(dxi, 16*m+4*j+l)
- gsl_vector_get(deta, 16*i+4*m+l) * gsl_vector_get(dxi, 16*m+4*j+k)
/* d2xi * eta and d2eta * xi */
- gsl_vector_get(d2xi, 64*m+16*j+4*k+l) * gsl_matrix_get(eta, i, m)
- gsl_vector_get(d2eta, 64*i+16*m+4*k+l) * gsl_matrix_get(xi, m, j)
- gsl_vector_get(d2eta, 64*i+16*m+4*j+l) * gsl_matrix_get(xi, m, k)
- gsl_vector_get(d2eta, 64*i+16*m+4*j+k) * gsl_matrix_get(xi, m, l)
)/tau);
} else {
/* init^{a}_{~ b c' d'} (0) = ...*/
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
for(l=0; l<4; l++)
gsl_vector_set(f, 64*i+16*j+4*k+l, gsl_vector_get(f, 64*i+16*j+4*k+l)
+( gsl_vector_get(sigma_dR, 64*i + 16*k + 4*l + j)
+ gsl_vector_get(sigma_dR, 64*i + 16*l + 4*k + j))/4.0
-5.0*( gsl_vector_get(sigma_dR, 64*i + 16*j + 4*k + l)
+ gsl_vector_get(sigma_dR, 64*i + 16*j + 4*l + k))/12.0
+7.0*(gsl_vector_get(sigma_dR, 64*i + 16*l + 4*j + k)
+ gsl_vector_get(sigma_dR, 64*i + 16*k + 4*j + l))/12.0
);
}
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
for(l=0; l<4; l++)
for(m=0; m<4; m++)
gsl_vector_set(f, 64*i+16*j+4*k+l,
gsl_vector_get(f, 64*i+16*j+4*k+l) +
/* gu * d2eta */
+ gsl_vector_get(d2eta, 64*i+16*m+4*k+l) * gsl_matrix_get(gu, m, j)
+ gsl_vector_get(d2eta, 64*i+16*j+4*m+l) * gsl_matrix_get(gu, m, k)
+ gsl_vector_get(d2eta, 64*i+16*j+4*k+m) * gsl_matrix_get(gu, m, l)
/* r_sigma * deta */
- gsl_vector_get(deta, 16*i+4*j+m)*gsl_vector_get(sigma_R, 16*m + 4*k + l)
- gsl_vector_get(deta, 16*i+4*k+m)*gsl_vector_get(sigma_R, 16*m + 4*j + l)
- gsl_vector_get(deta, 16*i+4*l+m)*gsl_vector_get(sigma_R, 16*m + 4*j + k)
/* dr_sigma * eta */
- gsl_matrix_get(eta, i, m)*gsl_vector_get(sigma_dR, 64*m + 16*j + 4*k + l)
);
gsl_vector_free(sigma_R);
gsl_vector_free(sigma_dR);
gsl_matrix_free(xi);
gsl_matrix_free(gu);
return GSL_SUCCESS;
}
/* gamma is the matrix inverse of eta */
int gammaBitensor ( const gsl_matrix * eta, gsl_matrix * gamma )
{
/* Gamma is the matrix inverse of eta */
int signum;
gsl_permutation * p = gsl_permutation_alloc (4);
gsl_matrix * lu = gsl_matrix_calloc(4,4);
gsl_matrix_memcpy(lu, eta);
gsl_linalg_LU_decomp (lu, p, &signum);
gsl_linalg_LU_invert (lu, p, gamma);
gsl_matrix_free(lu);
gsl_permutation_free(p);
return GSL_SUCCESS;
}
/* Box SqrtDelta */
int boxSqrtDelta (double tau, const double * y, double * f, void * params)
{
int i, j, k, l, m, n;
const double * I = y+5+16+1;
const double * dI_Inv = y+5+16+1+16+16;
const double * dEta = y+5+16+1+16+16+64+64;
const double * d2I_Inv = y+5+16+1+16+16+64+64+64;
const double * d2Eta = y+5+16+1+16+16+64+64+64+256+256;
const double * SqrtDelta = y+5+16;
gsl_matrix_const_view eta = gsl_matrix_const_view_array(y+5+16+1+16,4,4);
gsl_matrix * gamma = gsl_matrix_calloc(4,4);
gsl_matrix * metric = gsl_matrix_calloc(4,4);
/* Compute gamma matrix */
gammaBitensor( &eta.matrix, gamma );
/* Compute metric (in primed coordinates) */
metric_up_up(y, metric, params);
/* Initialize the RHS to 0 */
*f = 0.;
/* The vector tr(I dI^-1) */
double trIdI_inv[4] = {0, 0, 0, 0};
for(i=0; i<4; i++)
{
for(j=0; j<4; j++)
{
for(k=0; k<4; k++)
{
trIdI_inv[i] += I[j*4+k]*dI_Inv[k*16+j*4+i];
}
}
}
/* The vector tr(gamma dEta^-1) */
double trGammadEta[4] = {0, 0, 0, 0};
for(i=0; i<4; i++)
{
for(j=0; j<4; j++)
{
for(k=0; k<4; k++)
{
trGammadEta[i] += gsl_matrix_get(gamma, j, k) * dEta[16*k+4*j+i];
}
}
}
/* Now contracting over the free index) */
double tr2 = 0.;
for(i=0; i<4; i++)
{
for(j=0; j<4; j++)
{
tr2 += (trIdI_inv[i]+trGammadEta[i])*(trIdI_inv[j]+trGammadEta[j])*gsl_matrix_get(metric,i,j);
}
}
/* We really need half of this */
tr2 /= 2.;
/* Next, we need tr(I dI^-1 I dI^-1) */
double trIdI_invIdI_inv = 0.;
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
for(l=0; l<4; l++)
for(m=0; m<4; m++)
for(n=0; n<4; n++)
trIdI_invIdI_inv += I[i*4+j]*dI_Inv[16*j+4*k+m]*I[4*k+l]*dI_Inv[16*l+4*i+n]*gsl_matrix_get(metric,m,n);
/* Next, we need tr(gamma deta gamma deta) */
double trGammadEtaGammadEta = 0.;
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
for(l=0; l<4; l++)
for(m=0; m<4; m++)
for(n=0; n<4; n++)
trGammadEtaGammadEta += gsl_matrix_get(gamma,i,j)*dEta[16*j+4*k+m]*gsl_matrix_get(gamma,k,l)*dEta[16*l+4*i+n]*gsl_matrix_get(metric,m,n);
/* Now, tr(I * d2I_Inv) */
double trId2I_Inv = 0.;
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
for(l=0; l<4; l++)
trId2I_Inv += I[4*i+j]*d2I_Inv[64*j+16*i+4*k+l]*gsl_matrix_get(metric, k, l);
/* Now, tr(gamma * d2I_Inv) */
double trGammad2Eta = 0.;
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
for(l=0; l<4; l++)
trGammad2Eta += gsl_matrix_get(gamma, i, j)*d2Eta[64*j+16*i+4*k+l]*gsl_matrix_get(metric, k, l);
/* We have everything we need, now just calculate Box SqrtDelta */
*(f) = (*SqrtDelta)*(tr2-trIdI_invIdI_inv-trGammadEtaGammadEta+trId2I_Inv+trGammad2Eta)/2.;
gsl_matrix_free(metric);
gsl_matrix_free(gamma);
return GSL_SUCCESS;
}
/* tr2 term of Box SqrtDelta */
int tr2term (double tau, const double * y, double * f, void * params)
{
int i, j, k;
const double * I = y+5+16+1;
const double * dI_Inv = y+5+16+1+16+16;
const double * dEta = y+5+16+1+16+16+64+64;
gsl_matrix_const_view eta = gsl_matrix_const_view_array(y+5+16+1+16,4,4);
gsl_matrix * gamma = gsl_matrix_calloc(4,4);
gsl_matrix * metric = gsl_matrix_calloc(4,4);
/* Compute gamma matrix */
gammaBitensor( &eta.matrix, gamma );
/* Compute metric (in primed coordinates) */
metric_up_up(y, metric, params);
/* Initialize the RHS to 0 */
*f = 0.;
/* The vector tr(I dI^-1) */
double trIdI_inv[4] = {0, 0, 0, 0};
for(i=0; i<4; i++)
{
for(j=0; j<4; j++)
{
for(k=0; k<4; k++)
{
trIdI_inv[i] += I[j*4+k]*dI_Inv[k*16+j*4+i];
}
}
}
/* The vector tr(gamma dEta^-1) */
double trGammadEta[4] = {0, 0, 0, 0};
for(i=0; i<4; i++)
{
for(j=0; j<4; j++)
{
for(k=0; k<4; k++)
{
trGammadEta[i] += gsl_matrix_get(gamma, j, k) * dEta[16*k+4*j+i];
}
}
}
/* Now contracting over the free index) */
double tr2 = 0.;
for(i=0; i<4; i++)
{
for(j=0; j<4; j++)
{
tr2 += (trIdI_inv[i]+trGammadEta[i])*(trIdI_inv[j]+trGammadEta[j])*gsl_matrix_get(metric,i,j);
}
}
/* We have everything we need, now just calculate Box SqrtDelta */
*(f) = tr2;
gsl_matrix_free(metric);
gsl_matrix_free(gamma);
return GSL_SUCCESS;
}
/* V0: D'V_0 = -V0 - 1/2 V_0 ( xi - 4 ) - 1/2 Box (Delta^1/2) */
int V0RHS (double tau, const gsl_vector * y, const gsl_vector * yp, const gsl_matrix * q, const double * sqrt_delta, const double * dal_sqrt_delta, const double * v0, double * f, void * params)
{
int i;
double rhs = 0.;
struct geodesic_params p = *(struct geodesic_params *)params;
if(tau!=0.)
{
for(i=0; i<4; i++)
{
rhs -= gsl_matrix_get(q, i, i);
}
rhs = (rhs*(*v0)/2. - (*v0) - (*dal_sqrt_delta - (p.xi*(*sqrt_delta)*RicciScalar()) )/2.)/(tau);
*f = rhs;
} else {
rhs = d_RicciScalar(y, yp, params);
/* FIXME: xi missing */
*f = rhs*( p.xi - 1.0/6.0 )/4.0;
}
return GSL_SUCCESS;
}
/* Initial conditions */
int d2IinvInit(double * d2Iinv, double r0, void * params)
{
int i, j, k, l;
gsl_vector * R = gsl_vector_calloc(4*4*4*4);
gsl_vector_set_zero(R);
Riemann(R->data, r0, params);
/* d2Iinv^{~ b'}_{a' ~ c' d'} (0) = -1/2* R^{b'}_{ a' c' d'}*/
for(i=0; i<4; i++)
for(j=0; j<4; j++)
for(k=0; k<4; k++)
for(l=0; l<4; l++)
d2Iinv[64*i+16*j+4*k+l] = -gsl_vector_get(R, 64*j + 16*i + 4*k + l)/2.;
gsl_vector_free(R);
return GSL_SUCCESS;
}
int d2xiInit(double * d2xi, double r0, void * params)
{
int i;
/* d2xi^{a'}_{ b' c' d'} (0) = -2/3* R^{a'}_{ (c' | b' | d')}*/
RiemannSym(d2xi, r0, params);
for(i=0; i<4*4*4*4; i++)
d2xi[i] *= -2./3.;
return GSL_SUCCESS;
}
int d2etaInit(double * d2eta, double r0, void * params)
{
int i;
/* d2xi^{a'}_{ b' c' d'} (0) = -2/3* R^{a'}_{ (c' | b' | d')}*/
RiemannSym(d2eta, r0, params);
gsl_vector * d2eta2 = gsl_vector_calloc(4*4*4*4);
Riemann(d2eta2->data, r0, params);
for(i=0; i<4*4*4*4; i++)
d2eta[i] = -d2eta[i]/3. - d2eta2->data[i]/2.;
gsl_vector_free(d2eta2);
return GSL_SUCCESS;
}