Schwarzschild black hole with cosmological constant

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Schwarzschild black hole with cosmological constant

Problem:

Schwarzschild black hole with cosmological constant is 4-dimentional Lorentz manifold with constant Ricci curvature , timelike Killing vector field and group as a subgroup of the manifold isometry group (with spacelike orbits).
For Schwarzschild metric calculate the following:

connetion 1-forms
curvature 2-forms
Riemannin tensor field
Ricci tensor field

Verify that Diff(``,t), Diff(``,phi) are Killing vector vields.

Schwarzschild metric

restart:
with(atlas):

Constants:
Constants(r[g],Lambda);

${Pi, I, _Z, Catalan, Lambda, r[g], -I}$

Vector fields:
Vectors(E[i],X,Y,Z);

Differential p-forms:
Forms(e[j]=1);

Coframe 1-forms:
Coframe(e[1]=d(t),e[2]=d(rho),e[3]=d(theta),e[4]=d(phi));

Frame vector fields:
Frame(E[i]);

Metric tensor field :
Metric( g=(1-r[g]/rho+Lambda/3*rho^2)*d(t)&.d(t)-1/(1-r[g]/rho+Lambda/3*rho^2)*d(rho)&.d(rho)-rho^2*(d(theta)&.d(theta)+sin(theta)^2*d(phi)&.d(phi)) );

g = (1-r[g]/rho+1/3*Lambda*rho^2)*`&.`(e[1],e[1])-1/(1-r[g]/rho+1/3*Lambda*rho^2)*`&.`(e[2],e[2])-rho^2*(`&.`(e[3],e[3])+sin(theta)^2*`&.`(e[4],e[4]))

Connection 1-forms:
Connection(omega);

eval(omega);

TABLE([(4, 3) = 1/sin(theta)*cos(theta)*e[4], (3, 4) = -sin(theta)*cos(theta)*e[4], (4, 4) = 1/rho*e[2]+1/sin(theta)*cos(theta)*e[3], (1, 2) = 1/2*1/rho/(3*rho-3*r[g]+Lambda*rho^3)*(3*r[g]+2*Lambda*rho...

Curvature 2-forms:
Curvature(Omega);

eval(Omega);

TABLE([(4, 3) = 1/3*(Lambda*rho^3-3*r[g])/rho*`&^`(e[3],e[4]), (3, 4) = -1/3*sin(theta)^2*(Lambda*rho^3-3*r[g])/rho*`&^`(e[3],e[4]), (4, 4) = 0, (1, 2) = -(Lambda*rho^3-3*r[g])/rho^2/(3*rho-3*r[g]+Lamb...

Curvature tensor field:
Riemann(R);

R = -1/3*1/rho^3*(Lambda*rho^3-3*r[g])*`&.`(`&^`(e[1],e[2]),`&^`(e[1],e[2]))+1/2*1/(3*rho-3*r[g]+Lambda*rho^3)*(3*r[g]+2*Lambda*rho^3)*`&.`(`&^`(e[2],e[3]),`&^`(e[2],e[3]))-1/18*1/rho^2*(3*r[g]+2*Lambd...

Ricci tensor field calculation:
Ricci(ric);

ric = 1/3*Lambda/rho*(3*rho-3*r[g]+Lambda*rho^3)*`&.`(e[1],e[1])-3*rho/(3*rho-3*r[g]+Lambda*rho^3)*Lambda*`&.`(e[2],e[2])-Lambda*rho^2*`&.`(e[3],e[3])-sin(theta)^2*Lambda*rho^2*`&.`(e[4],e[4])

Verify that metric is Einstein one:
'ric'-Lambda*g=simplify(ric-Lambda*ToBasis(g));

Verify that E[1] = Diff(``,t) and E[4] = Diff(``,phi) are Killing vector fields:
'L[E[1]](g)'=L[E[1]](g);

'L[E[4]](g)'=L[E[4]](g);