Demonstration 1 (godel): An elementary study of the G\"{o}del metric.

> restart:

> grtw();

`GRTensorII Version 1.79 (R6)`

`2 February 2001`

`Developed by Peter Musgrave, Denis Pollney and Kay...

`Copyright 1994-2001 by the authors.`

`Latest version available from: http://grtensor.phy...

`e:/Grtii(6)/Metrics`

> grOptionTrace:=false:

> qload(godel1);

`Default spacetime` = godel1

`For the godel1 spacetime:`

Coordinates

x(up)

`x `^a = vector([t, x, y, z])

`Line element`

` ds`^2 = -` d`*t^`2 `-2*exp(sqrt(2)*omega*x)*` d`*...

`The Godel universe (c.f. Hawking and Ellis Section...

First we define a velocity field and then examine the kinematics

> grdef(`u{^a}:=[1,0,0,0]`);

Components assigned for metric: godel1

Created definition for u(up)

> grcalc(acc[u](up),expsc[u],shear[u](up,up),vor[u]);

Created a definition for u(up,cdn)

Created definition for shear(up,up)

Created definition for u(dn)

Created a definition for u(dn,cdn)

Created definition for acc(dn)

Created definition for vor(up,dn)

`CPU Time ` = .419

> gralter(_,radical,expand,factor);

Component simplification of a GRTensorII object:

Applying routine `simplify[radical]` to object acc(up)[u]

Applying routine `simplify[radical]` to object expsc[u]

Applying routine `simplify[radical]` to object shear(up,up)[u]

Applying routine `simplify[radical]` to object vor[u]

Applying routine expand to object acc(up)[u]

Applying routine expand to object expsc[u]

Applying routine expand to object shear(up,up)[u]

Applying routine expand to object vor[u]

Applying routine factor to object acc(up)[u]

Applying routine factor to object expsc[u]

Applying routine factor to object shear(up,up)[u]

Applying routine factor to object vor[u]

`CPU Time ` = .80e-1

> grdisplay(_);

`For the godel1 spacetime:`

`Acceleration vector`

acc(up)

a^a = `All components are zero`

`Expansion scalar`

Theta = 0

`shear(up,up)`

shear(up,up)

sigma^a*``^b = `All components are zero`

`Vorticity scalar`

`omega[u]` = sqrt(2)*sqrt(omega^2)

Einstein tensor is augmented with $\lambda$ (the cosmological constant).

> grdef(`En{a b}:=G{a b}+lambda*g{a b}`);

Created definition for En(dn,dn)

> grcalc(En(up,up));

Created definition for En(up,up)

`CPU Time ` = .40e-1

> gralter(_,expand,factor);

Component simplification of a GRTensorII object:

Applying routine expand to object En(up,up)

Applying routine factor to object En(up,up)

`CPU Time ` = .10e-1

> grdisplay(_);

`For the godel1 spacetime:`

`En(up,up)`

En(up,up)

En^a*``^b = matrix([[3*omega^2+lambda, 0, -2*(omega...

It is clear that for dust

> lambda:=-omega^2;

lambda := -omega^2

> grdisplay(_);

`For the godel1 spacetime:`

`En(up,up)`

En(up,up)

En^a*``^b = matrix([[2*omega^2, 0, 0, 0], [0, 0, 0,...

We now calculate the electric and magnetic components of the Weyl tensor associated with the velocity field u.

> grcalc(E[u](dn,dn),H[u](dn,dn));

`CPU Time ` = .80e-1

> gralter(_,6,7);

Component simplification of a GRTensorII object:

Applying routine expand to object E(dn,dn)[u]

Applying routine expand to object H(dn,dn)[u]

Applying routine factor to object E(dn,dn)[u]

Applying routine factor to object H(dn,dn)[u]

`CPU Time ` = .20e-1

> grdisplay(_);

`For the godel1 spacetime:`

`Electric part of Weyl`

E(dn,dn)

E[a]*``[b] = matrix([[0, 0, 0, 0], [0, 1/3*omega^2,...

`Magnetic part of Weyl`

H(dn,dn)

H[a]*``[b] = `All components are zero`

It is clear that the space has a high degree of symmetry. Here is a quick look for Killing vectors:

> KillingCoords();

Testing Killing coordinates for godel1

Created definition for coord1(dn)

Created a definition for coord1(dn,cdn)

Created a definition for coord1(up,cdn)

Created definition for coord2(dn)

Created a definition for coord2(dn,cdn)

Created a definition for coord2(up,cdn)

Created definition for coord3(dn)

Created a definition for coord3(dn,cdn)

Created a definition for coord3(up,cdn)

Created definition for coord4(dn)

Created a definition for coord4(dn,cdn)

Created a definition for coord4(up,cdn)

`CPU Time ` = .842

`Killing Coordinate Test Results`

`Coordinate vector ` = [t, x, y, z]

coord1(up) = [1, 0, 0, 0], ` a Killing vector.`

coord2(up) = [0, 1, 0, 0], ` not a Killing vector.`...

coord3(up) = [0, 0, 1, 0], ` a Killing vector.`

coord4(up) = [0, 0, 0, 1], ` a Killing vector.`

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