Curvature invariants with F(t).ipynb

Kernel: SageMath 8.0

version()

'SageMath version 8.0, Release Date: 2017-07-21'

%display latex

General case: F(t)

Spacetime

M = Manifold(4, 'M')
print(M)

4-dimensional differentiable manifold M

XBL.<t,x,th,ph> = M.chart(r't x th:(0,pi):\theta ph:(0,2*pi):\phi')
XBL

g = M.lorentzian_metric('g')
F=function('F')(t)
g[0,0]=-1/F
g[1,1]=F
g[2,2]=t^2
g[3,3]=t^2*sin(th)^2
g.display()

Ricci's scalar

Ric = g.ricci_scalar() ; print(Ric)

Scalar field r(g) on the 4-dimensional differentiable manifold M

Ric.expr().factor()

Kretschmann's scalar

R = g.riemann() ; print(R)

Tensor field Riem(g) of type (1,3) on the 4-dimensional differentiable manifold M

dR = R.down(g); print(dR)

Tensor field of type (0,4) on the 4-dimensional differentiable manifold M

uR = R.up(g); print(uR)

Tensor field of type (4,0) on the 4-dimensional differentiable manifold M

Kr_scalar = uR['^{abcd}']*dR['_{abcd}']
K=Kr_scalar.expr().factor()
K

Case 1 (Hayward): F_1(t)=\frac{2t^2}{t^3+2b^2}-1 --> nonsingular

var('b');

Spacetime

g1 = M.lorentzian_metric('g1')
F1=2*t^2/(t^3+2*b)-1
g1[0,0]=-1/F1
g1[1,1]=F1
g1[2,2]=t^2
g1[3,3]=t^2*sin(th)^2
g1.display()

Ricci's scalar

Ric1 = g1.ricci_scalar() ; print(Ric1)

Scalar field r(g1) on the 4-dimensional differentiable manifold M

Ric1.expr().factor()

Kretschmann's scalar

R1 = g1.riemann() ; print(R1)

Tensor field Riem(g1) of type (1,3) on the 4-dimensional differentiable manifold M

dR1 = R1.down(g1); print(dR1)

Tensor field of type (0,4) on the 4-dimensional differentiable manifold M

uR1 = R1.up(g1); print(uR1)

Tensor field of type (4,0) on the 4-dimensional differentiable manifold M

Kr_scalar1 = uR1['^{abcd}']*dR1['_{abcd}']
K1=Kr_scalar1.expr().factor()
K1

Case 2: F_2(t)=\frac{2t}{t^2+2b}-1 --> singular

Spacetime

g2 = M.lorentzian_metric('g2')
F2=2*t/(t^2+2*b)-1
g2[0,0]=-1/F2
g2[1,1]=F2
g2[2,2]=t^2
g2[3,3]=t^2*sin(th)^2
g2.display()

Ricci's scalar

Ric2 = g2.ricci_scalar() ; print(Ric2)

Scalar field r(g2) on the 4-dimensional differentiable manifold M

Ric2.expr().factor()

Kretschmann's scalar

R2 = g2.riemann() ; print(R2)

Tensor field Riem(g2) of type (1,3) on the 4-dimensional differentiable manifold M

dR2 = R2.down(g2); print(dR2)

Tensor field of type (0,4) on the 4-dimensional differentiable manifold M

uR2 = R2.up(g2); print(uR2)

Tensor field of type (4,0) on the 4-dimensional differentiable manifold M

Kr_scalar2 = uR2['^{abcd}']*dR2['_{abcd}']
K2=Kr_scalar2.expr().factor()
K2