limittwelve.mws

limittwelve.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 ` = .93e-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 ` = .31e-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)=v[2],`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,2))=psi[Q2],`x`); 

Applying routine subs to WeylSq

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

>    gralter(_,6,7); 

Component simplification of a GRTensorII object:

Applying routine expand to object WeylSq

Applying routine factor to object WeylSq

`CPU Time ` = .140

>    denom(grcomponent(WeylSq,[]));

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

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

65536*exp(P[infinity](theta))^2*exp(0)^2*psi[Q2]^2

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

core := 4*exp(2*P[infinity](theta))*psi[Q2]^2

>    Ww:=core*exp(gamma(t,theta))/t;

Ww := 4*exp(2*P[infinity](theta))*psi[Q2]^2*exp(gamma(t,theta))/t

>    subs(psi[Q2]=diff(psi[Q](theta),`$`(theta,2)),Ww);

4*exp(2*P[infinity](theta))*diff(psi[Q](theta),`$`(theta,2))^2*exp(gamma(t,theta))/t

>    latex(%); 

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

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

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

>