limitfourteen.mws

limitfourteen.mws

The special case v(theta)=-1 , with constraints,

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

>    restart:

the following substitution is 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);

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...

>    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 ` = .32e-1

>    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

Keep all derivatives of Q[infinity](theta)

>    grmap(_,subs,diff(Q[infinity](theta),theta$5)=Q[5],`x`); 

Applying routine subs to WeylSq

>    grmap(_,subs,diff(Q[infinity](theta),theta$4)=0,`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)=Q[1],`x`); 

Applying routine subs to WeylSq

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

Applying routine subs to WeylSq

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

Keep all derivatives of psi[Q](theta)

>    grmap(_,subs,diff(psi[Q](theta),`$`(theta,3))=psi[Q3],`x`); 

Applying routine subs to WeylSq

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

Applying routine subs to WeylSq

>   

>    grmap(_,subs,diff(psi[Q](theta),`$`(theta,4))=0,`x`); 

Applying routine subs to WeylSq

Apply constraints

>    grmap(_,subs,diff(psi[Q](theta),theta)=0,`x`): 

Applying routine subs to WeylSq

>    grmap(_,subs,psi[Q](theta)=0,`x`): 

Applying routine subs to WeylSq

>    grmap(_,subs,v(theta)=-1,`x`): 

Applying routine subs to WeylSq

>    denom(grcomponent(WeylSq,[]));

16384*t^9*exp(-1/2*gamma(t,theta))^2

>    factor(subs(t=0,numer(grcomponent(WeylSq,[]))));

0

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

>    Ww:=factor(t*exp(gamma(t,theta))*core);

Ww := -t*exp(gamma(t,theta))*(-16*V[1](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*d...

>    latex(%); 

-t{e^{\gamma \left( t,\theta \right) }} \left( -16\, \left( V_{{1}}

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

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

2}}{d{\theta}^{2}}}P_{{\infty }} \left( \theta \right)  \right) ^{2}+6

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

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

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

 \left( \theta \right)  \right) V_{{1}} \left( \theta \right) +12\,{e^

{2\,P_{{\infty }} \left( \theta \right) }}{Q_{{1}}}^{2} \right)

>    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;

V[1](theta) := 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,...

>    subs(psi[Q](theta)=0,diff(Q[infinity](theta),theta$2)=0,V[1](theta));

1/4*diff(P[infinity](theta),`$`(theta,2))

>    latex(1/4*diff(P[infinity](theta),`$`(theta,2))); 

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

 \right)

>    factor(subs(Q[1]=diff(Q[infinity](theta),theta),V[1](theta)=1/4*diff(P[infinity](theta),`$`(theta,2)),Ww));

3*t*exp(gamma(t,theta))*(diff(P[infinity](theta),`$`(theta,2))^2+diff(P[infinity](theta),theta)^4-2*diff(P[infinity](theta),theta)^2*diff(P[infinity](theta),`$`(theta,2))-4*exp(2*P[infinity](theta))*di...

>    latex(%); 

3\,t{e^{\gamma \left( t,\theta \right) }} \left(  \left( {\frac {d^{2}

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

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

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

\theta \right)  \right) ^{2}{\frac {d^{2}}{d{\theta}^{2}}}P_{{\infty }

} \left( \theta \right) -4\,{e^{2\,P_{{\infty }} \left( \theta

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

 \right)  \right) ^{2} \right)

>   

>    dsolve(diff(P[infinity](theta),`$`(theta,2))^2+diff(P[infinity](theta),theta)^4-4*exp(2*P[infinity](theta))*diff(Q[infinity](theta),theta)^2-2*diff(P[infinity](theta),`$`(theta,2))*diff(P[infinity](theta),theta)^2=0,P[infinity](theta));

P[infinity](theta) = -ln(-2*theta*Q[infinity](theta)+_C1*theta+2*Int(theta*diff(Q[infinity](theta),theta),theta)-_C2), P[infinity](theta) = -ln(2*theta*Q[infinity](theta)+_C1*theta-2*Int(theta*diff(Q[i...

>    latex(P[infinity](theta) = -ln(-2*theta*Q[infinity](theta)+_C1*theta+2*Int(theta*diff(Q[infinity](theta),theta),theta)-_C2)); 

P_{{\infty }} \left( \theta \right) =-\ln  \left( -2\,\theta\,Q_{{

\infty }} \left( \theta \right) +{\it \_C1}\,\theta+2\,\int \!\theta\,

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

\it \_C2} \right)

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