limittwospecial.mws

limittwospecial.mws

v(theta)=1,dv(theta)/dtheta not 0

(dv(theta)/dtheta)^2=exp(P[infinity](theta))^2*diff(Q[infinity](theta),`$`(theta,2))^2

>    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)

with

>    qq(theta):=diff(Q[infinity](theta),`$`(theta,2))/2;

qq(theta) := 1/2*diff(Q[infinity](theta),`$`(theta,2))

>    V(theta):=exp(P[infinity](theta))^2*diff(Q[infinity](theta),`$`(theta,2))^2/4;

V(theta) := 1/4*exp(P[infinity](theta))^2*diff(Q[infinity](theta),`$`(theta,2))^2

>    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 but keep all derivatives

>    grmap(_,subs,diff(Q[infinity](theta),theta$4)=Q[4],`x`); 

Applying routine subs to WeylSq

>    grmap(_,subs,diff(Q[infinity](theta),theta$3)=Q[3],`x`); 

Applying routine subs to WeylSq

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

Applying routine subs to WeylSq

>    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$4)=v[4],`x`); 

Applying routine subs to WeylSq

>    grmap(_,subs,diff(v(theta),theta$3)=v[3],`x`); 

Applying routine subs to WeylSq

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

Applying routine subs to WeylSq

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

Applying routine subs to WeylSq

>    grmap(_,subs,v(theta)=1,`x`); 

Applying routine subs to WeylSq

To do the special case we also insert the following (it turns out that the sign does not matter)

>    grmap(_,subs,v[1]=Q[2]*exp(P[infinity](theta)),`x`); 

Applying routine subs to WeylSq

grmap(_,subs,v[1]=-Q[2]*exp(P[infinity](theta)),`x`);

>    gralter(_,6); 

Component simplification of a GRTensorII object:

Applying routine expand to object WeylSq

`CPU Time ` = .438

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

>    limit(%,t=0);

16*V(theta)^2+16*exp(P[infinity](theta))^4*qq(theta)^4-32*V(theta)*qq(theta)^2*exp(P[infinity](theta))^2+3*Q[2]^4*exp(P[infinity](theta))^4

Put in qq(theta)

>    subs(qq(theta)=diff(Q[infinity](theta),`$`(theta,2))/2,%);

16*V(theta)^2+exp(P[infinity](theta))^4*diff(Q[infinity](theta),`$`(theta,2))^4-8*V(theta)*diff(Q[infinity](theta),`$`(theta,2))^2*exp(P[infinity](theta))^2+3*Q[2]^4*exp(P[infinity](theta))^4

Put in Q[2]

>    subs(Q[2]=diff(Q[infinity](theta),theta$2),%);

16*V(theta)^2+4*exp(P[infinity](theta))^4*diff(Q[infinity](theta),`$`(theta,2))^4-8*V(theta)*diff(Q[infinity](theta),`$`(theta,2))^2*exp(P[infinity](theta))^2

>    Ww:=t*log(t)^4*exp(gamma(t,theta))*%;

Ww := t*ln(t)^4*exp(gamma(t,theta))*(16*V(theta)^2+4*exp(P[infinity](theta))^4*diff(Q[infinity](theta),`$`(theta,2))^4-8*V(theta)*diff(Q[infinity](theta),`$`(theta,2))^2*exp(P[infinity](theta))^2)

Note that the form of V(theta) must be inserted.

Put in V(theta)

>    V(theta):=exp(P[infinity](theta))^2*diff(Q[infinity](theta),`$`(theta,2))^2/4;

V(theta) := 1/4*exp(P[infinity](theta))^2*diff(Q[infinity](theta),`$`(theta,2))^2

>    factor(Ww);

3*t*ln(t)^4*exp(gamma(t,theta))*exp(P[infinity](theta))^4*diff(Q[infinity](theta),`$`(theta,2))^4

>    latex(%); 

3\,t \left( \ln  \left( t \right)  \right) ^{4}{e^{\gamma \left( t,

\theta \right) }} \left( {e^{P_{{\infty }} \left( \theta \right) }}

 \right) ^{4} \left( {\frac {d^{2}}{d{\theta}^{2}}}Q_{{\infty }}

 \left( \theta \right)  \right) ^{4}

>    kernelopts(cputime);

127.048

>