Gravitational collapse of the dust spheres

Maple and Mathematica packages - math tools for professionals

Shopping Cart

Products

Contacts

Support

Updates

atlas™ - modern differential geometry for Maple™

> Features List & Examples

> Template Worksheets

> Screenshots

> Documentation & Downloads

> License & Pricing

> Buy Online

Features List & Examples

atlasWizard - Maplet™

Gravitational collapse of the dust sphere

Problem:

Energy-momentum tensor for spherically symmetric matter is T[i]^j = matrix([[epsilon, 0, 0, 0], [0, -p, 0, 0], [0, 0, -p, 0], [0, 0, 0, -p]]) , here is energy density function and pressure (just dust). Einstein equations are r-s*g/2 = 8*Pi*k*T , 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);

${_Z, Pi, k, Catalan, I, -I}$

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

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

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

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

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

Metric tensor fie ld:
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)) );

Connection 1-forms:
Connection(omega);

Curvature 2-forms:
Curvature(Omega);

Curvature tensor field:
Riemann(Riem);

Riem = (r*Diff(r,tau,R)-1/2*Diff(lambda,tau)*r*Diff(r,R))*`&.`(`&^`(e[1],e[3]),`&^`(e[2],e[3]))+r*Diff(r,`$`(tau,2))*`&.`(`&^`(e[1],e[3]),`&^`(e[1],e[3]))+r*Diff(r,`$`(tau,2))*sin(theta)^2*`&.`(`&^`(e[...

Ricci tensor field calculation:
Ricci(ric);

ric = (-2*Diff(r,tau,R)+Diff(lambda,tau)*Diff(r,R))/r*`&.`(e[1],e[2])+1/2*(Diff(lambda,tau,R)*r-Diff(lambda,R,tau)*r+2*Diff(lambda,tau)*Diff(r,R)-4*Diff(r,R,tau))/r*`&.`(e[2],e[1])-1/4*(2*Diff(lambda,`...

RicciScalar(scal);

scal = -1/2*(2*exp(lambda)*r^2*Diff(lambda,`$`(tau,2))+exp(lambda)*r^2*Diff(lambda,tau)^2+8*r*Diff(r,`$`(tau,2))*exp(lambda)+4*r*Diff(r,R)*Diff(lambda,R)-8*r*Diff(r,`$`(R,2))+4*r*Diff(r,tau)*exp(lambda...

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

Einstein := (-2*Diff(r,tau,R)+Diff(lambda,tau)*Diff(r,R))/r*`&.`(e[1],e[2])+1/2*(Diff(lambda,tau,R)*r-Diff(lambda,R,tau)*r+2*Diff(lambda,tau)*Diff(r,R)-4*Diff(r,R,tau))/r*`&.`(e[2],e[1])+(r*Diff(r,R)*D...

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

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

"eqs" tensor components:
eq[1,1]:=iota[E[1],E[1]](eqs);

eq[1,1] := (r*Diff(r,R)*Diff(lambda,R)-2*r*Diff(r,`$`(R,2))+r*Diff(r,tau)*exp(lambda)*Diff(lambda,tau)+exp(lambda)+Diff(r,tau)^2*exp(lambda)-Diff(r,R)^2)/exp(lambda)/r^2 = 8*Pi*k*epsilon
eq[2,2]:=iota[E[2],E[2]](eqs);

eq[2,2] := (-2*r*Diff(r,`$`(tau,2))*exp(lambda)-exp(lambda)-Diff(r,tau)^2*exp(lambda)+Diff(r,R)^2)/r^2 = 0

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

eq[3,3] := -1/4*r*(4*Diff(r,`$`(tau,2))*exp(lambda)+2*Diff(r,R)*Diff(lambda,R)-4*Diff(r,`$`(R,2))+2*Diff(lambda,tau)*Diff(r,tau)*exp(lambda)+2*exp(lambda)*Diff(lambda,`$`(tau,2))*r+exp(lambda)*Diff(lam...

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

eq[1,2] := (-2*Diff(r,tau,R)+Diff(lambda,tau)*Diff(r,R))/r = 0

Verify that E[4] = Diff(``,phi) are Killing vector field but E[1] = Diff(``,tau) is not:

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

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

Using covariant derivative:
'cov(E[i],e[1]&^e[2])'=cov(E[i],e[1]&^e[2]);

cov(E[i],`&^`(e[1],e[2])) = r*Diff(r,tau)*delta[3,i]*`&^`(e[2],e[3])+r*Diff(r,tau)*sin(theta)^2*delta[4,i]*`&^`(e[2],e[4])+(-1/2*Diff(lambda,tau)*delta[1,i]-1/2*Diff(lambda,R)*delta[2,i])*`&^`(e[1],e[2...