limitthree.mws

limitthree.mws

The special case v(theta)=1 ,  diff(Q[infinify](theta),theta)=0 for consistency

Notation diff(v(thtea),theta$n)=v[n]

Here dv(theta)/dtheta=0

>    restart:

>   

The following substitutions for P(t,theta) and Q(t,theta) are made:

>    P(t,theta)=P[infinity](theta)-v(theta)*log(t)+V(theta)*t^2*(log(t))^2+V[1](theta)*t^2+V[2](theta)*t^2*(log(t)),Q(t,theta)=Q[infinity](theta)+q(theta)*t^(2*(v(theta)))+qq(theta)*t^(2*(v(theta)))*log(t);

P(t,theta) = P[infinity](theta)-v(theta)*ln(t)+V(theta)*t^2*ln(t)^2+V[1](theta)*t^2+V[2](theta)*t^2*ln(t), Q(t,theta) = Q[infinity](theta)+q(theta)*t^(2*v(theta))+qq(theta)*t^(2*v(theta))*ln(t)

>    restart:

>    grtw(); 

Scalar invariant library.

Last modified 25 March 1997.

`Differential Invariants`

`Last modified Jan. 20, 1995`

`Basis/tetrad related object definitions`

`Last modified 23 January 2001`

`Last built 27 May, 1999`

`Last built 27 May, 1999`

`GRTensorII Version 1.79 (R4)`

`6 February 2001`

`Developed by Peter Musgrave, Denis Pollney and Kayll Lake`

`Copyright 1994-2001 by the authors.`

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

`c:/Grtii(6)/Metrics`

>    qload(gowdy);

`Default spacetime` = gowdy

`For the gowdy spacetime:`

Coordinates

x(up)

`x `^a = vector([t, theta, x1, x2])

`Line element`

` ds`^2 = -exp(-1/2*gamma(t,theta))/t^(1/2)*` d`*t^`2 `+exp(-1/2*gamma(t,theta))/t^(1/2)*` d`*theta^`2 `+t*exp(P(t,theta))*` d`*x1^`2 `+2*t*exp(P(t,theta))*Q(t,theta)*` d`*x1^` `*`d `*x2^` `+(t*exp(P(t...

Constraints = [diff(gamma(t,theta),t) = -t*exp(P(t,theta))^2*diff(Q(t,theta),t)^2-t*diff(P(t,theta),t)^2-t*exp(P(t,theta))^2*diff(Q(t,theta),theta)^2-t*diff(P(t,theta),theta)^2, diff(gamma(t,theta),the...
Constraints = [diff(gamma(t,theta),t) = -t*exp(P(t,theta))^2*diff(Q(t,theta),t)^2-t*diff(P(t,theta),t)^2-t*exp(P(t,theta))^2*diff(Q(t,theta),theta)^2-t*diff(P(t,theta),theta)^2, diff(gamma(t,theta),the...

>    grcalc(WeylSq);

`CPU Time ` = .94e-1

>    gralter(_,13,6,7); 

Component simplification of a GRTensorII object:

Applying routine `Apply constraints repeatedly` to object WeylSq

Applying routine expand to object WeylSq

Applying routine factor to object WeylSq

`CPU Time ` = .31e-1

>    grmap(_,subs,P(t,theta)=P[infinity](theta)-v(theta)*log(t)+V(theta)*t^2*(log(t))^2+V[1](theta)*t^2+V[2](theta)*t^2*(log(t)),Q(t,theta)=Q[infinity](theta)+q(theta)*t^(2*(v(theta)))+qq(theta)*t^(2*(v(theta)))*log(t),`x`); 

Applying routine subs to WeylSq

Apply consistency relation

>    grmap(_,subs,diff(Q[infinity](theta),theta)=0,`x`); 

Applying routine subs to WeylSq

Make sure you keep all derivatives of v(theta)

>    grmap(_,subs,diff(v(theta),theta$2)=v[2],`x`); 

Applying routine subs to WeylSq

>    grmap(_,subs,diff(v(theta),theta$1)=0,`x`); 

Applying routine subs to WeylSq

>    grmap(_,subs,v(theta)=1,`x`); 

Applying routine subs to WeylSq

>    gralter(_,6,7); 

Component simplification of a GRTensorII object:

Applying routine expand to object WeylSq

Applying routine factor to object WeylSq

`CPU Time ` = 3.219

>    core1:=expand(grcomponent(WeylSq,[])/(t*log(t)^4*exp(gamma(t,theta)))):

>    core2:=factor(limit(core1,t=0));

core2 := 16*(-V(theta)+exp(P[infinity](theta))^2*qq(theta)^2)^2

>    t*log(t)^4*exp(gamma(t,theta))*core2;

16*t*ln(t)^4*exp(gamma(t,theta))*(-V(theta)+exp(P[infinity](theta))^2*qq(theta)^2)^2

>    kernelopts(cputime);

35.326

>