"Kerr Newman, kartesische Kerr Schild Koordinaten"

(* |||||| Mathematica Syntax | GEODESIC SOLVER | geodesics.yukterez.net | Version 10.01.2020 ||||||*)

F=-(2M r^3-℧^2 r^2)/(r^4+a^2 Z^2);
l=dt+Z/r dZ+r/(a^2+r^2)(X dX+Y dY)-a/(a^2+r^2)(X dY-Y dX);
d=dt^2-dX^2-dY^2-dZ^2+F l^2
s=Expand[FullSimplify[d]];M=1;

g11->gtt->FullSimplify[s/.(dX^2)->0/.(dY^2)->0/.(dZ^2)->0/.dX->0/.dY->0/.dZ->0/.dt->1]
g22->gXX->FullSimplify[s/.(dt^2)->0/.(dY^2)->0/.(dZ^2)->0/.dt->0/.dY->0/.dZ->0/.dX->1]
g33->gYY->FullSimplify[s/.(dt^2)->0/.(dX^2)->0/.(dZ^2)->0/.dt->0/.dX->0/.dZ->0/.dY->1]
g44->gZZ->FullSimplify[s/.(dt^2)->0/.(dX^2)->0/.(dY^2)->0/.dt->0/.dX->0/.dY->0/.dZ->1]
g12->gtX->FullSimplify[s/.(dt^2)->0/.(dX^2)->0/.(dY^2)->0/.(dZ^2)->0/.dY->0/.dZ->0/.dt->1/.dX->1]/2
g13->gtY->FullSimplify[s/.(dt^2)->0/.(dX^2)->0/.(dY^2)->0/.(dZ^2)->0/.dX->0/.dZ->0/.dt->1/.dY->1]/2
g14->gtZ->FullSimplify[s/.(dt^2)->0/.(dX^2)->0/.(dY^2)->0/.(dZ^2)->0/.dX->0/.dY->0/.dt->1/.dZ->1]/2
g23->gXY->FullSimplify[s/.(dt^2)->0/.(dX^2)->0/.(dY^2)->0/.(dZ^2)->0/.dt->0/.dZ->0/.dX->1/.dY->1]/2
g24->gXZ->FullSimplify[s/.(dt^2)->0/.(dX^2)->0/.(dY^2)->0/.(dZ^2)->0/.dt->0/.dY->0/.dX->1/.dZ->1]/2
g34->gYZ->FullSimplify[s/.(dt^2)->0/.(dX^2)->0/.(dY^2)->0/.(dZ^2)->0/.dt->0/.dX->0/.dY->1/.dZ->1]/2

r=Solve[(X^2+Y^2)/(r^2+a^2)+Z^2/r^2==1,r][[4,1,2]]; "r"->r

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"Schwarzschild, kartesische Droste Koordinaten"

(* |||||| Mathematica Syntax | GEODESIC SOLVER | geodesics.yukterez.net | Version 10.01.2020 ||||||*)

F=1/r;
A=1-2F;d=A dt^2-(X^2 dX^2+Y^2 dY^2+Z^2 dZ^2+2X Y dX dY+2X Z dX dZ+2Y Z dY dZ)/(A r^2)-((X dX+Y dY)^2Z^2-2(X dX+Y dY)(X^2+Y^2)Z dZ+(X^2+Y^2)^2 dZ^2)/(X^2+Y^2)/r^2-(Y^2 dX^2-2X Y dX dY+X^2 dY^2)/(X^2+Y^2)
s=Expand[FullSimplify[d]]; M=1;

g11->gtt->FullSimplify[s/.(dX^2)->0/.(dY^2)->0/.(dZ^2)->0/.dX->0/.dY->0/.dZ->0/.dt->1]
g22->gXX->FullSimplify[s/.(dt^2)->0/.(dY^2)->0/.(dZ^2)->0/.dt->0/.dY->0/.dZ->0/.dX->1]
g33->gYY->FullSimplify[s/.(dt^2)->0/.(dX^2)->0/.(dZ^2)->0/.dt->0/.dX->0/.dZ->0/.dY->1]
g44->gZZ->FullSimplify[s/.(dt^2)->0/.(dX^2)->0/.(dY^2)->0/.dt->0/.dX->0/.dY->0/.dZ->1]
g12->gtX->FullSimplify[s/.(dt^2)->0/.(dX^2)->0/.(dY^2)->0/.(dZ^2)->0/.dY->0/.dZ->0/.dt->1/.dX->1]/2
g13->gtY->FullSimplify[s/.(dt^2)->0/.(dX^2)->0/.(dY^2)->0/.(dZ^2)->0/.dX->0/.dZ->0/.dt->1/.dY->1]/2
g14->gtZ->FullSimplify[s/.(dt^2)->0/.(dX^2)->0/.(dY^2)->0/.(dZ^2)->0/.dX->0/.dY->0/.dt->1/.dZ->1]/2
g23->gXY->FullSimplify[s/.(dt^2)->0/.(dX^2)->0/.(dY^2)->0/.(dZ^2)->0/.dt->0/.dZ->0/.dX->1/.dY->1]/2
g24->gXZ->FullSimplify[s/.(dt^2)->0/.(dX^2)->0/.(dY^2)->0/.(dZ^2)->0/.dt->0/.dY->0/.dX->1/.dZ->1]/2
g34->gYZ->FullSimplify[s/.(dt^2)->0/.(dX^2)->0/.(dY^2)->0/.(dZ^2)->0/.dt->0/.dX->0/.dY->1/.dZ->1]/2

r=Sqrt[X^2+Y^2+Z^2]; "r"->r

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