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 are Killing vector vields. 

Schwarzschild metric 

>	restart: with(atlas):

 

Constants:
Constants(r[g],Lambda); 

(2.1)

 

Vector fields:
Vectors(E[i],X,Y,Z); 

(2.2)

 

Differential p-forms:
Forms(e[j]=1); 

(2.3)

 

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

(2.4)

 

Frame vector fields:
Frame(E[i]); 

(2.5)

 

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)) ); 

(2.6)

 

Connection 1-forms:
Connection(omega); 

(2.7)

 

>	eval(omega);

 

(2.8)

 

Curvature 2-forms:
Curvature(Omega); 

(2.9)

 

>	eval(Omega);

 

(2.10)

 

Curvature tensor field:
Riemann(R); 

(2.11)

 

Ricci tensor field calculation: 

>	Ricci(ric);

 

(2.12)

 

Verify that metric is Einstein one: 

>	'ric'-Lambda*g=simplify(ric-Lambda*ToBasis(g));

 

(2.13)

 

Verify that and are Killing vector fields: 

>	'L[E[1]](g)'=L[E[1]](g);

 

(2.14)

 

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

 

(2.15)

 

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