limitfivespecial.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

The subcase diff(v(theta),`$`(theta,2))^2-diff(Q[infinity](theta),`$`(theta,3))^2*exp(2*P[infinity](theta))

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

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restart:

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

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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);

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restart:

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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

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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)=diff(v(theta),theta$2)/exp(P[infinity](theta)),`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)=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

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

>	factor(subs(v[2]=diff(v(theta),theta$2),core)*t*log(t)*exp(gamma(t,theta)));

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latex(%);

-3\, \left( {\frac {d^{2}}{d{\theta}^{2}}}v \left( \theta \right) 

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

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

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

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

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

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

\theta}}P_{{\infty }} \left( \theta \right)  \right) t\ln  \left( t

 \right) {e^{\gamma \left( t,\theta \right) }}

>	kernelopts(cputime);

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