Kerr black hole

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Kerr black hole

Problem:

Kerr black hole is 4-dimentional Lorentz manifold M with zero Ricci curvature and group U(1) = S^`1` 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 Diff(``,t), Diff(``,phi) 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);

Declare constants and :
Constants(rg,a);

${Pi, Catalan, _Z, I, a, rg, -I}$

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

Declare forms:
Forms(e[j]=1);

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

Declare frame vectors:
Frame(E[i]);

For Kerr metric we use well known aliases Delta = r^2-rg*r+a^2, rho = r^2+a^2-a^2*sin(theta)^2 :
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));

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

Connection calculation:
Connection(omega);

Let us see a 1-form:
omega[4,2];

1/2*a*rg*(r-a*cos(theta))*(r+a*cos(theta))/(r^2+a^2*cos(theta)^2)^2/Delta*e[1]+1/2*(2*r^5-2*r^4*rg+4*r^3*a^2*cos(theta)^2-r^2*a^2*rg*cos(theta)^2-r^2*a^2*rg+2*r*a^4*cos(theta)^4-a^4*cos(theta)^4*rg+a^4...

Curvature calculation:
Curvature(Omega);

Let as see a 2-form:
Omega[2,3];

-1/2*a*rg*Delta*cos(theta)*sin(theta)*(3*r^2-a^2*cos(theta)^2)/(r^2+a^2*cos(theta)^2)^3*`&^`(e[1],e[4])-1/2*rg*r*(-3*a^2*cos(theta)^2+r^2)/(r^2+a^2*cos(theta)^2)^2*`&^`(e[2],e[3])

Riemannian tensor calculation:
Riemann(R);

R = 1/2*(3*r^2-a^2*cos(theta)^2)*sin(theta)*cos(theta)*rg*a/(r^2+a^2*cos(theta)^2)^2*`&.`(`&^`(e[1],e[4]),`&^`(e[2],e[3]))-1/2*(-1+cos(theta))*(cos(theta)+1)*(-3*a^2*cos(theta)^2+r^2)*(3*r^2-2*rg*r+3*a...