Dr. Sarah's Christoffel Symbols and Curvature Tensors of Spacetime Metrics
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Wormhole Metric
restart:
with(Physics):
Setup(dimension = 4, quiet):
Coordinates(X = [t,r, theta,phi], quiet): Setup(coordinates = X, quiet):
g := Matrix(4, 4,
[[-1 + r0/r - epsilon/r^2, 0, 0, 0],
[0, 1/(1 - r0/r), 0, 0],
[0, 0, r^2, 0],
[0, 0, 0, r^2*sin(theta)^2]]
):
Setup(metric = g, quiet):
Christoffel_nonzero := ArrayElems( rhs( Christoffel[~alpha, beta, gamma, array] ) );
Riemann_array := Riemann[array];
Riemann_nonzero := Riemann[nonzero];
Ricci_matrix := simplify( Ricci[matrix] );
Ricci_nonzero := Ricci[nonzero];
R_scalar := simplify( Ricci[scalar] );
Einstein_matrix := simplify( Einstein[matrix] );
Einstein_nonzero := Einstein[nonzero];
Weak-Field Metric
restart:
with(Physics):
Setup(dimension = 4, quiet):
Coordinates(X = [t,r, theta,phi], quiet): Setup(coordinates = X, quiet):
Phi := -G*M/r:
g := Matrix(4, 4,
[[-(1 + 2*Phi/c^2)*c^2, 0, 0],
[0, (1 - 2*Phi/c^2), 0, 0],
[0, 0, (1 - 2*Phi/c^2)*r^2, 0],
[0, 0, 0, (1 - 2*Phi/c^2)*r^2*sin(theta)^2]]
):
Setup(metric = g, quiet):
Christoffel_nonzero := ArrayElems( rhs( Christoffel[~alpha, beta, gamma, array] ) );
Riemann_nonzero := Riemann[nonzero];
Ricci_matrix := simplify( Ricci[matrix] );
Ricci_nonzero := Ricci[nonzero];
R_scalar := simplify( Ricci[scalar] );
Einstein_matrix := simplify( Einstein[matrix] );
Einstein_nonzero := Einstein[nonzero];
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Rotating Minkowski Metric
restart:
with(Physics):
Setup(dimension = 4, quiet):
Coordinates(X = [t,r, phi,z], quiet): Setup(coordinates = X, quiet):
g := Matrix(4, 4,
[[(1 - Omega^2*r^2/c^2)*c^2, 0, Omega*r^2, 0],
[0, -1, 0, 0],
[Omega*r^2, 0, -r^2, 0],
[0, 0, 0, -1]]
):
Setup(metric = g, quiet):
Christoffel_nonzero := ArrayElems( rhs( Christoffel[~alpha, beta, gamma, array] ) );
Riemann_array := Riemann[array];
Riemann_nonzero := Riemann[nonzero];
Ricci_matrix := simplify( Ricci[matrix] );
Ricci_nonzero := Ricci[nonzero];
R_scalar := simplify( Ricci[scalar] );
Einstein_matrix := simplify( Einstein[matrix] );
Einstein_nonzero := Einstein[nonzero];
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