Kerr black hole 

Problem: 

Kerr black hole  is a 4-dimentional Lorentz manifold M with zero Ricci curvature and group as a subgroup of the manifold isometry group. 

For Kerr metric calculate the following: 

• connetion 1-forms

 

• curvature 2-forms

 

• Riemannin tensor field

 

• Ricci tensor field

 

Verify that are Killing vector vields. 

Kerr black hole 

First of all we load atlas package: 

>	restart: with(atlas):

 

Redefine `atlas/simp` procedure to simplify the results:
`atlas/simp`:=proc(a) factor(simplify(a)) end: 

Total space 

Declare domain M - black hole space:
Domain(M); 

(2.1.1)

 

Declare constants and :
Constants(rg,a); 

(2.1.2)

 

Declare vectors:
Vectors(E[i],X,Y,Z); 

(2.1.3)

 

Declare forms:
Forms(e[j]=1); 

(2.1.4)

 

Declare coframe:
Coframe(e[1]=d(t),e[2]=d(r),e[3]=d(theta),e[4]=d(phi)); 

(2.1.5)

 

Declare frame vectors:
Frame(E[i]); 

(2.1.6)

 

For Kerr metric we use well known aliases : 

>	alias(Delta=r^2-rg*r+a^2,rho=r^2+a^2-a^2*sin(theta)^2,                            -rho=-r^2-a^2+a^2*sin(theta)^2):

 

Now we declare Kerr metric: 

>	Metric(g=(Delta-a^2*sin(theta)^2)/rho*d(t)&.d(t)        +a*sin(theta)^2*(r^2+a^2-Delta)/rho*(d(t)&.d(phi)+d(phi)&.d(t))    -rho/Delta*d(r)&.d(r)-rho*d(theta)&.d(theta) +((a^2*sin(theta)^2*Delta)-(r^2+a^2)^2)*sin(theta)^2/rho*d(phi)&.d(phi));

 

(2.1.7)

 

Connection calculation:
Connection(omega); 

(2.1.8)

 

Let us see a 1-form:
omega[4,2]; 

(2.1.9)

 

Curvature calculation:
Curvature(Omega); 

(2.1.10)

 

Let as see a 2-form:
Omega[2,3]; 

(2.1.11)

 

Riemannian tensor calculation:
Riemann(R); 

(2.1.12)

 

>

 

Ricci tensor calculation:
Ricci(ric); 

(2.1.13)

 

E[1] is Killing vector field:
'L[E[1]](g)' = L[E[1]](g); 

(2.1.14)

 

E[4] is Killing vector field:
'L[E[4]](g)' = L[E[4]](g); 

(2.1.15)

 

Killing vector fields:
&@(t)=ToBasis(&@(t));
&@(phi)=ToBasis(&@(phi)); 

 

	(2.1.16)

 

>