limitsix.mws

The special case v(theta)=1 ,  diff(Q[infinify](theta),theta)=0 for consistency and diff(v(theta),theta) = 0 diff(Q[infinity](theta),theta$2)=0

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

>

restart:

>

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

>

P(t,theta) = P[infinity](theta)-v(theta)*ln(t)+1/4*exp(P[infinity](theta))^2*diff(Q[infinity](theta),`$`(theta,2))^2*t^2*ln(t)^2+V[1](theta)*t^2+(exp(2*P[infinity](theta))*psi[Q](theta)*diff(Q[infinity](theta),`$`(theta,2))-1/4*diff(Q[infinity](theta),`$`(theta,2))^2-1/4*diff(v(theta),`$`(theta,2)))*t^2*ln(t),Q(t,theta) = Q[infinity](theta)+psi[Q](theta)*t^(2*v(theta))+1/2*diff(Q[infinity](theta),`$`(theta,2))*t^(2*v(theta))*ln(t);

>

V[1](theta):=(exp(P[infinity](theta))^2*diff(Q[infinity](theta),`$`(theta,2))^2/4)*(3/2)-exp(2*P[infinity](theta))*psi[Q](theta)*diff(Q[infinity](theta),`$`(theta,2))-1/4*diff(Q[infinity](theta),`$`(theta,2))^2-1/4*diff(v(theta),`$`(theta,2))+exp(P[infinity](theta))^2*(psi[Q](theta)+diff(Q[infinity](theta),`$`(theta,2))/4)^2+(diff(P[infinity](theta),theta$2)+diff(v(theta),theta$2))/4;

>

latex(%);

3/8\, \left( {e^{P_{{\infty }} \left( \theta \right) }} \right) ^{2}

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

 \right)  \right) ^{2}-{e^{2\,P_{{\infty }} \left( \theta \right) }}

\psi_{{Q}} \left( \theta \right) {\frac {d^{2}}{d{\theta}^{2}}}Q_{{

\infty }} \left( \theta \right) -1/4\, \left( {\frac {d^{2}}{d{\theta}

^{2}}}Q_{{\infty }} \left( \theta \right)  \right) ^{2}+ \left( {e^{P_

{{\infty }} \left( \theta \right) }} \right) ^{2} \left( \psi_{{Q}}

 \left( \theta \right) +1/4\,{\frac {d^{2}}{d{\theta}^{2}}}Q_{{\infty 

}} \left( \theta \right)  \right) ^{2}+1/4\,{\frac {d^{2}}{d{\theta}^{

2}}}P_{{\infty }} \left( \theta \right) 

>

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`

>	qload(gowdy);

>	grcalc(WeylSq);

>	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

>

grmap(_,subs,P(t,theta) = P[infinity](theta)-v(theta)*ln(t)+1/4*exp(P[infinity](theta))^2*diff(Q[infinity](theta),`$`(theta,2))^2*t^2*ln(t)^2+V[1](theta)*t^2+(exp(2*P[infinity](theta))*psi[Q](theta)*diff(Q[infinity](theta),`$`(theta,2))-1/4*diff(Q[infinity](theta),`$`(theta,2))^2-1/4*diff(v(theta),`$`(theta,2)))*t^2*ln(t),Q(t,theta) = Q[infinity](theta)+psi[Q](theta)*t^(2*v(theta))+1/2*diff(Q[infinity](theta),`$`(theta,2))*t^(2*v(theta))*ln(t),`x`);

Applying routine subs to WeylSq

>

Apply consistency relation but keep all other 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)=0,`x`);

Applying routine subs to WeylSq

>	grmap(_,subs,diff(Q[infinity](theta),theta$2)=0,`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)=0,`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

>	core:=factor(simplify(limit(factor(grcomponent(WeylSq,[])/(t*exp(gamma(t,theta)))),t=0)));

>

factor(expand(subs(V[1](theta)=(exp(P[infinity](theta))^2*diff(Q[infinity](theta),`$`(theta,2))^2/4)*(3/2)-exp(2*P[infinity](theta))*psi[Q](theta)*diff(Q[infinity](theta),`$`(theta,2))-1/4*diff(Q[infinity](theta),`$`(theta,2))^2-1/4*diff(v(theta),`$`(theta,2))+exp(P[infinity](theta))^2*(psi[Q](theta)+diff(Q[infinity](theta),`$`(theta,2))/4)^2+(diff(P[infinity](theta),theta$2)+diff(v(theta),theta$2))/4,diff(Q[infinity](theta),theta$2)=0,diff(Q[infinity](theta),theta$3)=0,%)));

>

latex(%);

-3\, \left( {\frac {d^{2}}{d{\theta}^{2}}}P_{{\infty }} \left( \theta

 \right) + \left( {\frac {d}{d\theta}}P_{{\infty }} \left( \theta

 \right)  \right) ^{2}+2\, \left( {\frac {d}{d\theta}}\psi_{{Q}}

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

 \right) }}+4\,\psi_{{Q}} \left( \theta \right)  \left( {\frac {d}{d

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

 \left( \theta \right) }} \right)  \left( -{\frac {d^{2}}{d{\theta}^{2

}}}P_{{\infty }} \left( \theta \right) - \left( {\frac {d}{d\theta}}P_

{{\infty }} \left( \theta \right)  \right) ^{2}+2\, \left( {\frac {d}{

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

 \left( \theta \right) }}+4\,\psi_{{Q}} \left( \theta \right)  \left( 

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

_{{\infty }} \left( \theta \right) }} \right) 

>	kernelopts(cputime);

>

subs(diff(Q[infinity](theta),theta$2)=0,diff(Q[infinity](theta),theta$3)=0,V[1](theta)=(exp(P[infinity](theta))^2*diff(Q[infinity](theta),`$`(theta,2))^2/4)*(3/2)-exp(2*P[infinity](theta))*psi[Q](theta)*diff(Q[infinity](theta),`$`(theta,2))-1/4*diff(Q[infinity](theta),`$`(theta,2))^2-1/4*diff(v(theta),`$`(theta,2))+exp(P[infinity](theta))^2*(psi[Q](theta)+diff(Q[infinity](theta),`$`(theta,2))/4)^2+(diff(P[infinity](theta),theta$2)+diff(v(theta),theta$2))/4);

>

latex(%);

V_{{1}} \left( \theta \right) = \left( {e^{P_{{\infty }} \left( \theta

 \right) }} \right) ^{2} \left( \psi_{{Q}} \left( \theta \right) 

 \right) ^{2}+1/4\,{\frac {d^{2}}{d{\theta}^{2}}}P_{{\infty }} \left( 

\theta \right) 

>

>

>

>