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! $Header$
#include "cctk.h"
module adm_metric
use tensor
implicit none
private
public calc_4metric
public calc_4metricderivs
public calc_4metricderivs2
public calc_3metric
public calc_3metricderivs
public calc_3metricdot
public calc_extcurv
contains
subroutine calc_4metric (gg, alfa, beta, g4)
CCTK_REAL, intent(in) :: gg(3,3), alfa, beta(3)
CCTK_REAL, intent(out) :: g4(0:3,0:3)
CCTK_REAL :: betal(3)
! ds^2 = -alpha^2 dt^2 + g_ij (dx^i + beta^i dt) (dx^j + beta^j dt)
betal = matmul(gg, beta)
g4(0 ,0 ) = -alfa**2 + sum(betal*beta)
g4(1:3,0 ) = betal
g4(0 ,1:3) = betal
g4(1:3,1:3) = gg
end subroutine calc_4metric
subroutine calc_4metricderivs (gg,alfa,beta, dgg,dalfa,dbeta, &
gg_dot,alfa_dot,beta_dot, g4,dg4)
CCTK_REAL, intent(in) :: gg(3,3),alfa,beta(3)
CCTK_REAL, intent(in) :: dgg(3,3,3),dalfa(3),dbeta(3,3)
CCTK_REAL, intent(in) :: gg_dot(3,3),alfa_dot,beta_dot(3)
CCTK_REAL, intent(out) :: g4(0:3,0:3),dg4(0:3,0:3,0:3)
CCTK_REAL :: d4gg(3,3,0:3),d4alfa(0:3),d4beta(3,0:3)
CCTK_REAL :: betal(3),d4betal(3,0:3)
integer :: i,j
! 4-metric
forall (i=1:3)
betal(i) = sum(gg(i,:) * beta(:))
end forall
g4(0 ,0 ) = -alfa**2 + sum(betal(:) * beta(:))
g4(1:3,0 ) = betal
g4(0 ,1:3) = betal
g4(1:3,1:3) = gg
! derivatives
d4gg (:,:,0 ) = gg_dot(:,:)
d4gg (:,:,1:3) = dgg(:,:,:)
d4alfa( 0 ) = alfa_dot
d4alfa( 1:3) = dalfa(:)
d4beta(:, 0 ) = beta_dot(:)
d4beta(:, 1:3) = dbeta(:,:)
forall (i=1:3, j=0:3)
d4betal(i,j) = sum(d4gg(i,:,j) * beta(:) + gg(i,:) * d4beta(:,j))
end forall
forall (i=0:3)
dg4(0 ,0 ,i) = - 2 * alfa * d4alfa(i) &
& + sum(d4betal(:,i) * beta(:) + betal(:) * d4beta(:,i))
dg4(1:3,0 ,i) = d4betal(:,i)
dg4(0 ,1:3,i) = d4betal(:,i)
dg4(1:3,1:3,i) = d4gg(:,:,i)
end forall
end subroutine calc_4metricderivs
subroutine calc_4metricderivs2 (gg,alfa,beta, dgg,dalfa,dbeta, &
ddgg,ddalfa,ddbeta, gg_dot,alfa_dot,beta_dot, &
gg_dot2,alfa_dot2,beta_dot2, dgg_dot,dalfa_dot,dbeta_dot, g4,dg4,ddg4)
CCTK_REAL, intent(in) :: gg(3,3),alfa,beta(3)
CCTK_REAL, intent(in) :: dgg(3,3,3),dalfa(3),dbeta(3,3)
CCTK_REAL, intent(in) :: ddgg(3,3,3,3),ddalfa(3,3),ddbeta(3,3,3)
CCTK_REAL, intent(in) :: gg_dot(3,3),alfa_dot,beta_dot(3)
CCTK_REAL, intent(in) :: gg_dot2(3,3),alfa_dot2,beta_dot2(3)
CCTK_REAL, intent(in) :: dgg_dot(3,3,3),dalfa_dot(3),dbeta_dot(3,3)
CCTK_REAL, intent(out) :: g4(0:3,0:3),dg4(0:3,0:3,0:3)
CCTK_REAL, intent(out) :: ddg4(0:3,0:3,0:3,0:3)
CCTK_REAL :: d4gg(3,3,0:3),d4alfa(0:3),d4beta(3,0:3)
CCTK_REAL :: dd4gg(3,3,0:3,0:3),dd4alfa(0:3,0:3),dd4beta(3,0:3,0:3)
CCTK_REAL :: betal(3),d4betal(3,0:3),dd4betal(3,0:3,0:3)
integer :: i,j,k
! 4-metric
forall (i=1:3)
betal(i) = sum(gg(i,:) * beta(:))
end forall
g4(0 ,0 ) = -alfa**2 + sum(betal(:) * beta(:))
g4(1:3,0 ) = betal
g4(0 ,1:3) = betal
g4(1:3,1:3) = gg
! first derivatives
d4gg (:,:,0 ) = gg_dot(:,:)
d4gg (:,:,1:3) = dgg(:,:,:)
d4alfa( 0 ) = alfa_dot
d4alfa( 1:3) = dalfa(:)
d4beta(:, 0 ) = beta_dot(:)
d4beta(:, 1:3) = dbeta(:,:)
forall (i=1:3, j=0:3)
d4betal(i,j) = sum(d4gg(i,:,j) * beta(:) + gg(i,:) * d4beta(:,j))
end forall
forall (i=0:3)
dg4(0 ,0 ,i) = - 2 * alfa * d4alfa(i) &
& + sum(d4betal(:,i) * beta(:) + betal(:) * d4beta(:,i))
dg4(1:3,0 ,i) = d4betal(:,i)
dg4(0 ,1:3,i) = d4betal(:,i)
dg4(1:3,1:3,i) = d4gg(:,:,i)
end forall
! second derivatives
dd4gg (:,:,0 ,0 ) = gg_dot2(:,:)
dd4gg (:,:,1:3,0 ) = dgg_dot(:,:,:)
dd4gg (:,:,0 ,1:3) = dgg_dot(:,:,:)
dd4gg (:,:,1:3,1:3) = ddgg(:,:,:,:)
dd4alfa( 0 ,0 ) = alfa_dot2
dd4alfa( 1:3,0 ) = dalfa_dot(:)
dd4alfa( 0 ,1:3) = dalfa_dot(:)
dd4alfa( 1:3,1:3) = ddalfa(:,:)
dd4beta(:, 0 ,0 ) = beta_dot2(:)
dd4beta(:, 1:3,0 ) = dbeta_dot(:,:)
dd4beta(:, 0 ,1:3) = dbeta_dot(:,:)
dd4beta(:, 1:3,1:3) = ddbeta(:,:,:)
! betal(i) = gg(i,m) * beta(m)
! d4betal(i,j) = d4gg(i,m,j) * beta(m) + gg(i,m) * d4beta(m,j)
! dd4betal(i,j,k) = dd4gg(i,m,j,k) * beta(m) + d4gg(i,m,j) * d4beta(m,k)
! + d4gg(i,m,k) * d4beta(m,j) + gg(i,m) * dd4beta(m,j,k)
forall (i=1:3, j=0:3, k=0:3)
dd4betal(i,j,k) = sum(+ dd4gg(i,:,j,k) * beta(:) &
& + d4gg(i,:,j) * d4beta(:,k) &
& + d4gg(i,:,k) * d4beta(:,j) &
& + gg(i,:) * dd4beta(:,j,k))
end forall
! g4(0 ,0 ) = -alfa**2 + sum(betal*beta)
! g4(1:3,0 ) = betal
! g4(0 ,1:3) = betal
! g4(1:3,1:3) = gg
! dg4(0 ,0 ,i) = -2*alfa*d4alfa(i) &
! & + d4betal(m,i)*beta(m) + betal(m)*d4beta(m,i)
! dg4(1:3,0 ,i) = d4betal(:,i)
! dg4(0 ,1:3,i) = d4betal(:,i)
! dg4(1:3,1:3,i) = d4gg(:,:,i)
forall (i=0:3, j=0:3)
ddg4(0 ,0 ,i,j) = - 2 * d4alfa(j) * d4alfa(i) &
& - 2 * alfa * dd4alfa(i,j) &
& + sum(+ dd4betal(:,i,j) * beta(:) &
& + d4betal(:,i) * d4beta(:,j) &
& + d4betal(:,j) * d4beta(:,i) &
& + betal(:) * dd4beta(:,i,j))
ddg4(1:3,0 ,i,j) = dd4betal(:,i,j)
ddg4(0 ,1:3,i,j) = dd4betal(:,i,j)
ddg4(1:3,1:3,i,j) = dd4gg(:,:,i,j)
end forall
end subroutine calc_4metricderivs2
subroutine calc_3metric (g4, gg, alfa, beta)
CCTK_REAL, intent(in) :: g4(0:3,0:3)
CCTK_REAL, intent(out) :: gg(3,3), alfa, beta(3)
CCTK_REAL :: betal(3)
CCTK_REAL :: dtg, gu(3,3)
! ds^2 = -alpha^2 dt^2 + g_ij (dx^i + beta^i dt) (dx^j + beta^j dt)
betal = g4(1:3,0)
gg = g4(1:3,1:3)
call calc_det (gg, dtg)
call calc_inv (gg, dtg, gu)
beta = matmul(gu, betal)
alfa = sqrt(sum(betal*beta) - g4(0,0))
end subroutine calc_3metric
subroutine calc_3metricderivs (g4,dg4, gg,alfa,beta, dgg,dalfa,dbeta, &
gg_dot,alfa_dot,beta_dot)
CCTK_REAL, intent(in) :: g4(0:3,0:3),dg4(0:3,0:3,0:3)
CCTK_REAL, intent(out) :: gg(3,3),alfa,beta(3)
CCTK_REAL, intent(out) :: dgg(3,3,3),dalfa(3),dbeta(3,3)
CCTK_REAL, intent(out) :: gg_dot(3,3),alfa_dot,beta_dot(3)
CCTK_REAL :: betal(3),d4betal(3,0:3)
CCTK_REAL :: dtg,gu(3,3),dgu(3,3,3),gu_dot(3,3)
CCTK_REAL :: d4gg(3,3,0:3),d4gu(3,3,0:3)
CCTK_REAL :: d4alfa(0:3),d4beta(3,0:3)
integer :: i,j
! ds^2 = -alpha^2 dt^2 + g_ij (dx^i + beta^i dt) (dx^j + beta^j dt)
gg = g4(1:3,1:3)
call calc_det (gg, dtg)
call calc_inv (gg, dtg, gu)
betal = g4(1:3,0)
beta = matmul(gu, betal)
alfa = sqrt(sum(betal*beta) - g4(0,0))
forall (i=0:3)
d4gg(:,:,i) = dg4(1:3,1:3,i)
end forall
gg_dot = d4gg(:,:,0)
dgg = d4gg(:,:,1:3)
call calc_invderiv (gu, dgg, dgu)
call calc_invdot (gu, gg_dot, gu_dot)
d4gu(:,:,0) = gu_dot
d4gu(:,:,1:3) = dgu
forall (i=0:3)
d4betal(:,i) = dg4(1:3,0,i)
end forall
forall (i=0:3, j=1:3)
d4beta(j,i) = sum(d4gu(j,:,i)*betal) + sum(gu(j,:)*d4betal(:,i))
end forall
forall (i=0:3)
d4alfa(i) = 1/(2*alfa) &
* (sum(d4betal(:,i)*beta) + sum(betal*d4beta(:,i)) - dg4(0,0,i))
end forall
alfa_dot = d4alfa(0)
dalfa = d4alfa(1:3)
beta_dot = d4beta(:,0)
dbeta = d4beta(:,1:3)
end subroutine calc_3metricderivs
subroutine calc_3metricdot (gg, dgg, kk, alfa, beta, dbeta, gg_dot)
CCTK_REAL, intent(in) :: gg(3,3), dgg(3,3,3)
CCTK_REAL, intent(in) :: kk(3,3)
CCTK_REAL, intent(in) :: alfa
CCTK_REAL, intent(in) :: beta(3), dbeta(3,3)
CCTK_REAL, intent(out) :: gg_dot(3,3)
integer :: i,j,k
! d/dt g_ij = -2 alpha K_ij + g_kj beta^k,i + g_ik beta^k,j + beta^k g_ij,k
do i=1,3
do j=1,3
gg_dot(i,j) = -2*alfa * kk(i,j)
do k=1,3
gg_dot(i,j) = gg_dot(i,j) &
+ gg(k,j) * dbeta(k,i) + gg(i,k) * dbeta(k,j) &
+ beta(k) * dgg(i,j,k)
end do
end do
end do
end subroutine calc_3metricdot
subroutine calc_extcurv (gg, dgg, gg_dot, alfa, beta, dbeta, kk)
CCTK_REAL, intent(in) :: gg(3,3), dgg(3,3,3), gg_dot(3,3)
CCTK_REAL, intent(in) :: alfa
CCTK_REAL, intent(in) :: beta(3), dbeta(3,3)
CCTK_REAL, intent(out) :: kk(3,3)
integer :: i,j,k
! d/dt g_ij = -2 alpha K_ij + g_kj beta^k,i + g_ik beta^k,j + beta^k g_ij,k
do i=1,3
do j=1,3
kk(i,j) = - gg_dot(i,j)
do k=1,3
kk(i,j) = kk(i,j) &
+ gg(k,j) * dbeta(k,i) + gg(i,k) * dbeta(k,j) &
+ beta(k) * dgg(i,j,k)
end do
kk(i,j) = kk(i,j) / (2*alfa)
end do
end do
end subroutine calc_extcurv
end module adm_metric
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