Package examples

<<"~/Desktop/EinsteinVariation.m"

Spherical Coordinates

For Minkowski space-time, in spherical coordinates:

gll = DiagonalMatrix[{-1,1,r^2,r^2 Sin[T]^2}];
guu = Inverse[gll];
Xu = {t,r,T,P};

GetCCull[gll,Xu]

GetRiemann[gll,Xu]

Sphere

On the surface of a sphere :

gll = {{r^2,0},{0,r^2 Sin[T]^2}};
guu = Inverse[gll];

Xu = {T,P};

Rulll = GetRiemann[gll,Xu]

GetRicci[gll,Xu]

GetRicciS[gll,Xu]

The connection, in its one-up, two-down form:

CCull = GetCCull[gll,Xu]

CCull[[1,2,2]]

CCull[[2,1,2]]

CCull[[2,2,1]]

The Riemann tensor

Rulll

Here is the lower form:

Rllll = Table[Sum[Rulll[[s1,b,u,v]] gll[[s1,a]] ,{s1,1,2}],{a,1,2},{b,1,2},{u,1,2},{v,1,2}]