Demonstration 1 (rnds): Calculation of the invariants $R$,$R_{ab}^{~~ab}$,$R_{abcd}^{~~~~abcd}$,$R_{,a}^{~~,a}$, and $R_{ab;c}^{~~~~ab;c}$

in the Reissner-Nordstrom-de Sitter 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...

> grlib(dinvar):

`Differential Invariants`

`Last modified Jan. 20, 1995`

> qload(rnds);

`Default spacetime` = rnds

`For the rnds spacetime:`

Coordinates

x(up)

`x `^a = vector([r, theta, phi, t])

`Line element`

` ds`^2 = ` d`*r^`2 `/(1-2*m/r-1/3*Lambda*r^2+e^2/(...

`The Reissner-Nordstrom-de Sitter metric`

> grcalc(Ricciscalar,RicciSq,RiemSq,Dsq[Ricciscalar],diRicci);

Created definition for R(up,dn)

Created definition for R(dn,dn,up,up)

Created definition for R(dn,up)

Created a definition for R(dn,up,cdn)

`CPU Time ` = .450

> gralter(_,2,7);

Component simplification of a GRTensorII object:

Applying routine `simplify[trig]` to object Ricciscalar

Applying routine `simplify[trig]` to object RicciSq

Applying routine `simplify[trig]` to object RiemSq

Applying routine `simplify[trig]` to object Dsq[Ricciscalar]

Applying routine `simplify[trig]` to object diRicci

Applying routine factor to object Ricciscalar

Applying routine factor to object RicciSq

Applying routine factor to object RiemSq

Applying routine factor to object Dsq[Ricciscalar]

Applying routine factor to object diRicci

`CPU Time ` = .70e-1

> grdisplay(_);

`For the rnds spacetime:`

`Ricci scalar`

`R ` = 4*Lambda

`Full Contraction of Ricci`

`RicciSq ` = 4*(Lambda^2*r^8+e^4)/(r^8)

`Full Contraction of Riemann`

`K ` = 8/3*(18*m^2*r^2-36*m*r*e^2+Lambda^2*r^8+21*e...

` g{^a ^b} X{,a} X{,b}`

`Dsq ` = 0

` R_{a b ; e} R^{a b ; e}`

`diRicci ` = -80/3*(-3*r^2+6*m*r+Lambda*r^4-3*e^2)*...

>