Gravitational collapse of the dust sphere 

Problem: 

Energy-momentum tensor for spherically symmetric matter is , here is energy density function and pressure (just dust). Einstein equations are , where -Ricci tensor -Ricci scalar, k-gravitational constant (here light velocity ).
Calculate Einstein tensor and corresponding equations for dust sphere metric. Verify that corresponding space-time has axial symmetry and it is not stationary.  

Dust sphere metric 

>	restart: with(atlas):

 

Constants:
Constants(k); 

(2.1)

 

Functions:
Functions(lambda=lambda(tau,R),r=r(tau,R),epsilon=epsilon(tau,R)); 

(2.2)

 

Vector fields:
Vectors(E[i],X,Y,Z); 

(2.3)

 

Differential p-forms:
Forms(e[j]=1); 

(2.4)

 

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

(2.5)

 

Frame vector fields:
Frame(E[i]); 

(2.6)

 

Metric tensor field:
Metric( g=d(tau)&.d(tau)-exp(lambda)*d(R)&.d(R)-r^2*(d(theta)&.d(theta)+sin(theta)^2*d(phi)&.d(phi)) ); 

(2.7)

 

Connection 1-forms:
Connection(omega); 

(2.8)

 

Curvature 2-forms:
Curvature(Omega); 

(2.9)

 

Curvature tensor field:
Riemann(Riem); 

(2.10)

 

Ricci tensor field calculation: 

>	Ricci(ric);

 

(2.11)

 

>	RicciScalar(scal);

 

(2.12)

 

>	Einstein:=collect(ric-1/2*scal*ToBasis(g),`&.`,factor);

 

>

 

(2.13)

 

Energy-momentum tensor for dust sphere:
T:=epsilon*e[1]&.e[1]; 

(2.14)

 

>	eqs:=Einstein=8*Pi*k*T: 'Einstein'=8*Pi*k*T;

 

(2.15)

 

"eqs" tensor components: 

>	eq[1,1]:=iota[E[1],E[1]](eqs);

 

(2.16)

 

>	eq[2,2]:=iota[E[2],E[2]](eqs);

 

(2.17)

 

>	eq[3,3]:=iota[E[3],E[3]](eqs);

 

(2.18)

 

>	eq[1,2]:=iota[E[2],E[1]](eqs);

 

(2.19)

 

Verify that are Killing vector field but is not: 

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

 

(2.20)

 

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

 

(2.21)

 

Using covariant derivative: 

>	'cov(E[i],e[1]&^e[2])'=cov(E[i],e[1]&^e[2]);

 

(2.22)

 

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