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ultimate nil Christoffel formulas in model

This commit is contained in:
Zeno Rogue 2023-08-07 20:48:55 +02:00
parent d2c601834a
commit 301236d154
1 changed files with 25 additions and 48 deletions
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73
nonisotropic.cpp
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@ -786,54 +786,31 @@ EX namespace nilv {
}
hyperpoint christoffel(const hyperpoint Position, const hyperpoint Velocity, const hyperpoint Transported, ld model = model_used) {
/* to do: write these formulas in model */
if(model == nmSym) {
hyperpoint c; c[3] = 0;
ld x = Position[0]; ld y = Position[1]; ld diag = (y*y-x*x)/8;
c[ 0 ] = 0
+ Velocity[ 0 ] * Transported[ 1 ] * ( -y/4 )
+ Velocity[ 1 ] * Transported[ 0 ] * ( -y/4 )
+ Velocity[ 1 ] * Transported[ 1 ] * ( x/2 )
+ Velocity[ 1 ] * Transported[ 2 ] * ( -1/2. )
+ Velocity[ 2 ] * Transported[ 1 ] * ( -1/2. );
c[ 1 ] = 0
+ Velocity[ 0 ] * Transported[ 0 ] * ( y/2 )
+ Velocity[ 0 ] * Transported[ 1 ] * ( -x/4 )
+ Velocity[ 0 ] * Transported[ 2 ] * ( 1/2. )
+ Velocity[ 1 ] * Transported[ 0 ] * ( -x/4 )
+ Velocity[ 2 ] * Transported[ 0 ] * ( 1/2. );
c[ 2 ] = 0
+ Velocity[ 0 ] * Transported[ 0 ] * ( x*y/4 )
+ Velocity[ 0 ] * Transported[ 1 ] * diag
+ Velocity[ 0 ] * Transported[ 2 ] * ( x/4 )
+ Velocity[ 1 ] * Transported[ 0 ] * diag
+ Velocity[ 1 ] * Transported[ 1 ] * ( -x*y/4 )
+ Velocity[ 1 ] * Transported[ 2 ] * ( y/4 )
+ Velocity[ 2 ] * Transported[ 0 ] * ( x/4 )
+ Velocity[ 2 ] * Transported[ 1 ] * ( y/4 );
return c;
}
if(model == nmHeis) {
ld x = Position[0];
return point3(
x * Velocity[1] * Transported[1] - 0.5 * (Velocity[1] * Transported[2] + Velocity[2] * Transported[1]),
-.5 * x * (Velocity[1] * Transported[0] + Velocity[0] * Transported[1]) + .5 * (Velocity[2] * Transported[0] + Velocity[0] * Transported[2]),
-.5 * (x*x-1) * (Velocity[1] * Transported[0] + Velocity[0] * Transported[1]) + .5 * x * (Velocity[2] * Transported[0] + Velocity[0] * Transported[2])
);
}
else {
hyperpoint P = Position;
hyperpoint V = Velocity;
hyperpoint T = Transported;
convert_ref(P, model_used, nmHeis);
convert_tangent_ref(P, V, model_used, nmHeis);
convert_tangent_ref(P, T, model_used, nmHeis);
auto res = christoffel(P, V, T, nmHeis);
auto T1 = T + res;
convert_tangent_ref(P + V, T1, nmHeis, model_used);
return T1 - Transported;
}
hyperpoint c; c[3] = 0;
ld x = Position[0]; ld y = Position[1];
auto mu = model;
c[ 0 ] = 0
+ Velocity[ 0 ] * Transported[ 1 ] * ( y*(mu - 1)/4 )
+ Velocity[ 1 ] * Transported[ 0 ] * ( y*(mu - 1)/4 )
+ Velocity[ 1 ] * Transported[ 1 ] * ( x*(mu + 1)/2 )
+ Velocity[ 1 ] * Transported[ 2 ] * ( -1/2. )
+ Velocity[ 2 ] * Transported[ 1 ] * ( -1/2. );
c[ 1 ] = 0
+ Velocity[ 0 ] * Transported[ 0 ] * ( y*(1 - mu)/2 )
+ Velocity[ 0 ] * Transported[ 1 ] * ( -x*(mu + 1)/4 )
+ Velocity[ 0 ] * Transported[ 2 ] * ( 1/2. )
+ Velocity[ 1 ] * Transported[ 0 ] * ( -x*(mu + 1)/4 )
+ Velocity[ 2 ] * Transported[ 0 ] * ( 1/2. );
c[ 2 ] = 0
+ Velocity[ 0 ] * Transported[ 0 ] * ( x*y*(1 - mu*mu)/4 )
+ Velocity[ 0 ] * Transported[ 1 ] * ( -mu*mu*x*x/8 + mu*mu*y*y/8 - mu*x*x/4 - mu*y*y/4 + mu/2 - x*x/8 + y*y/8 )
+ Velocity[ 0 ] * Transported[ 2 ] * ( x*(mu + 1)/4 )
+ Velocity[ 1 ] * Transported[ 0 ] * ( -mu*mu*x*x/8 + mu*mu*y*y/8 - mu*x*x/4 - mu*y*y/4 + mu/2 - x*x/8 + y*y/8 )
+ Velocity[ 1 ] * Transported[ 1 ] * ( x*y*(mu*mu - 1)/4 )
+ Velocity[ 1 ] * Transported[ 2 ] * ( y*(1 - mu)/4 )
+ Velocity[ 2 ] * Transported[ 0 ] * ( x*(mu + 1)/4 )
+ Velocity[ 2 ] * Transported[ 1 ] * ( y*(1 - mu)/4 );
return c;
}
EX hyperpoint formula_exp(hyperpoint v) {