-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...](limitfiveq31.gif)
limitfiveq3.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); |
| > | 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])](limitfiveq312.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),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)=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(factor(grcomponent(WeylSq,[])/(t*log(t)^2*exp(gamma(t,theta)))),t=0)); |
![core := 3*v[2]^2](limitfiveq320.gif){kind=small}
| > | factor(subs(v[2]=diff(v(theta),theta$2),Q[3]=diff(Q[infinity](theta),theta$3),core)*t*log(t)^2*exp(gamma(t,theta))); |
| > | latex(%); |
3\, \left( {\frac {d^{2}}{d{\theta}^{2}}}v \left( \theta \right)
\right) ^{2}t \left( \ln \left( t \right) \right) ^{2}{e^{\gamma
\left( t,\theta \right) }}
| > | kernelopts(cputime); |
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