-v(theta)*ln(t)+1/4*exp(P[infinity](theta))^2*diff(Q[infinity](theta),`$`(theta,2))^2*t^2*ln(t)^2+V[1](theta)*t^2+(exp(2*P[infinity](theta))*psi[Q](theta)*diff(Q[infinity...](limitsixzw1.gif)
-v(theta)*ln(t)+1/4*exp(P[infinity](theta))^2*diff(Q[infinity](theta),`$`(theta,2))^2*t^2*ln(t)^2+V[1](theta)*t^2+(exp(2*P[infinity](theta))*psi[Q](theta)*diff(Q[infinity...](limitsixzw2.gif)
limitsixzw.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),`$`(theta,2)))*t^2*ln(t)+t^4*Z(t,theta),Q(t,theta) = Q[infinity](theta)+psi[Q](theta)*t^(2*v(theta))+1/2*diff(Q[infinity](theta),`$`(theta,2))*t^(2*v(theta))*ln(t)+t^4*W(t,theta); |
| > | 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$2))/4; |
 := 3/8*exp(P[infinity](theta))^2*diff(Q[infinity](theta),`$`(theta,2))^2-exp(2*P[infinity](theta))*psi[Q](theta)*diff(Q[infinity](theta),`$`(theta,2))-1/4*diff(Q[infinity](theta),`$`(theta,...](limitsixzw3.gif)
| > | 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); |
![`x `^a = vector([t, theta, x1, x2])](limitsixzw14.gif){kind=small}
| > | 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)+t^4*Z(t,theta),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)+t^4*W(t,theta),`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$1)=0,`x`); |
Applying routine subs to WeylSq
| > | grmap(_,subs,v(theta)=1,`x`); |
Applying routine subs to WeylSq
| > | gralter(_,6,7); |
Component simplification of a GRTensorII object:
Applying routine expand to object WeylSq
Applying routine factor to object WeylSq
| > | core1:=factor(simplify(subs(t=0,factor(grcomponent(WeylSq,[])/(t*exp(gamma(t,theta))))))); |
^2+3*diff(P[infinity](theta),theta)^4+4*diff(P[infinity](theta),`$`(theta,2))^2+6*diff(P[infinity](theta),`$`(theta,2))*diff(P[infinity](theta),theta)^2-8*diff(P[infinity](theta)...](limitsixzw22.gif)
^2+3*diff(P[infinity](theta),theta)^4+4*diff(P[infinity](theta),`$`(theta,2))^2+6*diff(P[infinity](theta),`$`(theta,2))*diff(P[infinity](theta),theta)^2-8*diff(P[infinity](theta)...](limitsixzw23.gif)
core:=factor(simplify(limit(factor(grcomponent(WeylSq,[])/(t*exp(gamma(t,theta)))),t=0)));
| > | core2:=factor(expand(subs(V[1](theta)=(exp(P[infinity](theta))^2*diff(Q[infinity](theta),`$`(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$3)=0,%))); |
,`$`(theta,2))+2*exp(P[infinity](theta))*diff(psi[Q](theta),theta)+4*exp(P[infinity](theta))*psi[Q](theta)*diff(P[infinity](theta),theta)+diff(P[infinity](theta),the...](limitsixzw24.gif)
latex(%);
3\, \left( {\frac {d^{2}}{d{\theta}^{2}}}P_{{\infty }} \left( \theta
\right) \right) ^{2}+3\, \left( {\frac {d}{d\theta}}P_{{\infty }}
\left( \theta \right) \right) ^{4}+6\, \left( {\frac {d^{2}}{d{
\theta}^{2}}}P_{{\infty }} \left( \theta \right) \right) \left( {
\frac {d}{d\theta}}P_{{\infty }} \left( \theta \right) \right) ^{2}+
16\, \left( Z \left( 0,\theta \right) \right) ^{2}-48\, \left( {e^{P_
{{\infty }} \left( \theta \right) }} \right) ^{2} \left( {\frac {d}{d
\theta}}P_{{\infty }} \left( \theta \right) \right) ^{2} \left( \psi_
{{Q}} \left( \theta \right) \right) ^{2}-12\, \left( {e^{P_{{\infty }
} \left( \theta \right) }} \right) ^{2} \left( {\frac {d}{d\theta}}
\psi_{{Q}} \left( \theta \right) \right) ^{2}-48\, \left( {e^{P_{{
\infty }} \left( \theta \right) }} \right) ^{2} \left( {\frac {d}{d
\theta}}P_{{\infty }} \left( \theta \right) \right) \psi_{{Q}}
\left( \theta \right) {\frac {d}{d\theta}}\psi_{{Q}} \left( \theta
\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$2))/4); |
 = exp(P[infinity](theta))^2*psi[Q](theta)^2+1/4*diff(P[infinity](theta),`$`(theta,2))](limitsixzw26.gif)
| > | 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)
| > |