limitone.mws

limitone.mws

This worksheet examines WeylSq (=C_{a b c d}*C^{a b c d}) and distinguishes the case v(theta)^2=1.

>    restart:

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

>    P(t,theta)=P[infinity](theta)-v(theta)*log(t)+V(theta)*t^2*(log(t))^2+V[1](theta)*t^2+V[2](theta)*t^2*(log(t)),Q(t,theta)=Q[infinity](theta)+q(theta)*t^(2*(v(theta)))+qq(theta)*t^(2*(v(theta)))*log(t);

P(t,theta) = P[infinity](theta)-v(theta)*ln(t)+V(theta)*t^2*ln(t)^2+V[1](theta)*t^2+V[2](theta)*t^2*ln(t), Q(t,theta) = Q[infinity](theta)+q(theta)*t^(2*v(theta))+qq(theta)*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`

`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...

Note that the predefined invariant WeylSq is used

>    grcalc(WeylSq);

`CPU Time ` = .93e-1

The constraints are applied and WeylSq simplified

>    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)*log(t)+V(theta)*t^2*(log(t))^2+V[1](theta)*t^2+V[2](theta)*t^2*(log(t)),Q(t,theta)=Q[infinity](theta)+q(theta)*t^(2*(v(theta)))+qq(theta)*t^(2*(v(theta)))*log(t),`x`); 

Applying routine subs to WeylSq

>    h:=factor(simplify(subs(t=0,factor((4*t^3*grcomponent(WeylSq,[])/(exp(gamma(t,theta))))))));

h := (v(theta)^2+3)*(v(theta)-1)^2*(v(theta)+1)^2

and so as t->0, WeylSq->

>    simplify(exp(gamma(t,theta))*h/(4*t^3));

1/4*exp(gamma(t,theta))*(v(theta)^2+3)*(v(theta)-1)^2*(v(theta)+1)^2/t^3

The following shows the execution time (in seconds)

>    kernelopts(cputime);

56.654

>    restart:

The following just generates tex

>    latex(P(t,theta)=P[infinity](theta)-v(theta)*log(t)+V(theta)*t^2*(log(t))^2+V[1](theta)*t^2+V[2](theta)*t^2*(log(t))); 

P \left( t,\theta \right) =P_{{\infty }} \left( \theta \right) -v

 \left( \theta \right) \ln  \left( t \right) +V \left( \theta \right) 

{t}^{2} \left( \ln  \left( t \right)  \right) ^{2}+V_{{1}} \left( 

\theta \right) {t}^{2}+V_{{2}} \left( \theta \right) {t}^{2}\ln 

 \left( t \right)

>    latex(Q(t,theta)=Q[infinity](theta)+q(theta)*t^(2*(v(theta)))+qq(theta)*t^(2*(v(theta)))*log(t)); 

Q \left( t,\theta \right) =Q_{{\infty }} \left( \theta \right) +q

 \left( \theta \right) {t}^{2\,v \left( \theta \right) }+{\it qq}

 \left( \theta \right) {t}^{2\,v \left( \theta \right) }\ln  \left( t

 \right)

>    latex(1/4*exp(gamma(t,theta))*(v(theta)^2+3)*(v(theta)-1)^2*(v(theta)+1)^2/t^3); 

1/4\,{\frac {{e^{\gamma \left( t,\theta \right) }} \left(  \left( v

 \left( \theta \right)  \right) ^{2}+3 \right)  \left( v \left( \theta

 \right) -1 \right) ^{2} \left( v \left( \theta \right) +1 \right) ^{2

}}{{t}^{3}}}

>