Demontration 1(nariai): The Nariai 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`

> qload(nariai);

`Default spacetime` = nariai

`For the nariai spacetime:`

Coordinates

x(up)

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

`Line element`

` ds`^2 = L^2*` d`*x^`2 `/(x^2+y^2+z^2)+L^2*` d`*y^...

Constraints = [P(x,y,z,t) = a(t)*cos(ln(sqrt(x^2+y^...

> grcalc(R(dn,up));

Created definition for R(dn,up)

`CPU Time ` = .211

> gralter(_,13);

Component simplification of a GRTensorII object:

Applying routine `Apply constraints repeatedly` to object R(dn,up)

`CPU Time ` = .90e-1

> grdisplay(_);

`For the nariai spacetime:`

`R(dn,up)`

R(dn,up)

`R `[a]^b = matrix([[1/(L^2), 0, 0, 0], [0, 1/(L^2)...

In a basis:

> qload(nariaib);

`Default spacetime` = nariaib

`For the nariaib spacetime:`

Coordinates

x(up)

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

`Basis inner product`

eta(bup,bup)

eta^`(a)`*``^`(b)` = matrix([[-1, 0, 0, 0], [0, -1,...

`Basis (covariant components)`

w1(dn)

omega1[a] = vector([L/(sqrt(x^2+y^2+z^2)), 0, 0, 0]...

w2(dn)

omega2[a] = vector([0, L/(sqrt(x^2+y^2+z^2)), 0, 0]...

w3(dn)

omega3[a] = vector([0, 0, L/(sqrt(x^2+y^2+z^2)), 0]...

w4(dn)

omega4[a] = vector([0, 0, 0, P(x,y,z,t)])

Constraints = [P(x,y,z,t) = a(t)*cos(ln(sqrt(x^2+y^...

> grcalc(R(bdn,bdn));

Created definition for rot(bdn,bup,bdn)

Created a definition for e(bdn,dn,pdn)

`CPU Time ` = .211

> gralter(_,13);

Component simplification of a GRTensorII object:

Applying routine `Apply constraints repeatedly` to object R(bdn,bdn)

`CPU Time ` = .50e-1

> grdisplay(_);

R(bdn,bdn)

R[`(a)`]*``[`(b)`] = matrix([[1/(L^2), 0, 0, 0], [0...

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