"Frame-Field" calculation of Ricci and Weyl and Ricci and Weyl invariants
for the mixmaster metric with constant basis inner product.
> restart:
> grtw():
> qload(mix1);
![`x `^a = vector([Theta, Phi, Psi, T])](images/mix110.gif){kind=small}
![eta^`(a)`*``^`(b)` = matrix([[1, 0, 0, 0], [0, 1, 0...](images/mix113.gif)
![omega1[a] = vector([-exp(a(T))*sqrt(1-Psi^2)/(sqrt(...](images/mix116.gif)
![omega2[a] = vector([-exp(b(T))*Psi/(sqrt(1-Theta^2)...](images/mix118.gif)
![omega3[a] = vector([0, Theta*exp(c(T)), exp(c(T))/(...](images/mix120.gif)
![omega4[a] = vector([0, 0, 0, exp(a(T)+b(T)+c(T))])](images/mix122.gif){kind=small}
> grcalcalter(R(bdn,bdn),8);
Created definition for rot(bdn,bup,bdn)
Created a definition for e(bdn,dn,pdn)
Simplification will be applied during calculation.
Applying routine normal to object g(up,up)
Applying routine normal to object e(bdn,up)
Applying routine normal to object e(bdn,dn,pdn)
Applying routine normal to object lambda(bdn,bdn,bdn)
Applying routine normal to object rot(bdn,bdn,bdn)
Applying routine normal to object rot(bdn,bup,bdn)
Applying routine normal to object R(bdn,bdn,bdn,bdn)
Applying routine normal to object R(bdn,bdn)
> gralter(_,6,7);
Component simplification of a GRTensorII object:
Applying routine expand to object R(bdn,bdn)
Applying routine factor to object R(bdn,bdn)
> grmap(_,autoAlias,`x`);
Applying routine autoAlias to R(bdn,bdn)
Warning, alias or macro a[T] defined in terms of a
Warning, alias or macro b[T] defined in terms of b
Warning, alias or macro c[T] defined in terms of c
> grdisplay(_);
![R[`(a)`]*``[`(b)`] = matrix([[1/2*(2*exp(c)^2*exp(b...](images/mix127.gif)
>
### WARNING: calls to `C` for generating C code should be replaced by codegen[C]
grcalcalter(C(bdn,bdn,bdn,bdn),8);
Simplification will be applied during calculation.
Applying routine normal to object bRicciscalar
Applying routine normal to object C(bdn,bdn,bdn,bdn)
> gralter(_,6,7);
Component simplification of a GRTensorII object:
Applying routine expand to object C(bdn,bdn,bdn,bdn)
Applying routine factor to object C(bdn,bdn,bdn,bdn)
> grmap(_,autoAlias,`x`);
Applying routine autoAlias to C(bdn,bdn,bdn,bdn)
> grdisplay(_);
![C[`(1) (2) (1) (2)`] = 1/6*(-2*exp(a)^2*exp(b)^2+ex...](images/mix132.gif){kind=small}
![C[`(1) (2) (1) (2)`] = 1/6*(-2*exp(a)^2*exp(b)^2+ex...](images/mix133.gif)
![C[`(1) (2) (3) (4)`] = 1/2*(-2*exp(c)^2*c[T]+exp(b)...](images/mix134.gif)
![C[`(1) (3) (1) (3)`] = 1/6*(exp(a)^2*exp(b)^2+exp(c...](images/mix135.gif){kind=small}
![C[`(1) (3) (1) (3)`] = 1/6*(exp(a)^2*exp(b)^2+exp(c...](images/mix136.gif)
![C[`(1) (3) (2) (4)`] = -1/2*(-2*exp(b)^2*b[T]+exp(c...](images/mix137.gif)
![C[`(1) (4) (1) (4)`] = 1/6*(-exp(a)^2*exp(b)^2+2*ex...](images/mix138.gif){kind=small}
![C[`(1) (4) (1) (4)`] = 1/6*(-exp(a)^2*exp(b)^2+2*ex...](images/mix139.gif)
![C[`(1) (4) (2) (3)`] = -1/2*(2*exp(a)^2*a[T]-exp(b)...](images/mix140.gif)
![C[`(2) (3) (2) (3)`] = -1/6*(-exp(a)^2*exp(b)^2+2*e...](images/mix141.gif){kind=small}
![C[`(2) (3) (2) (3)`] = -1/6*(-exp(a)^2*exp(b)^2+2*e...](images/mix142.gif)
![C[`(2) (4) (2) (4)`] = -1/6*(exp(a)^2*exp(b)^2+exp(...](images/mix143.gif){kind=small}
![C[`(2) (4) (2) (4)`] = -1/6*(exp(a)^2*exp(b)^2+exp(...](images/mix144.gif)
![C[`(3) (4) (3) (4)`] = -1/6*(-2*exp(a)^2*exp(b)^2+e...](images/mix145.gif){kind=small}
![C[`(3) (4) (3) (4)`] = -1/6*(-2*exp(a)^2*exp(b)^2+e...](images/mix146.gif)
> grcalc(Rinvars);
Scalar invariant library.
Last modified 25 March 1997.
Created definition for S(bup,bdn)
> gralter(_,6,7);
Component simplification of a GRTensorII object:
Applying routine expand to object Ricciscalar
Applying routine expand to object R1
Applying routine expand to object R2
Applying routine expand to object R3
Applying routine factor to object Ricciscalar
Applying routine factor to object R1
Applying routine factor to object R2
Applying routine factor to object R3
To save space here we display only R and R2.Increase grOptionDisplayLimit to display the other invariants.
> grdisplay(Ricciscalar,R2);
![`R2 ` = -3/512*(-4*b[T,T]+2*exp(a)^2*exp(b)^2-2*exp...](images/mix154.gif)
![`R2 ` = -3/512*(-4*b[T,T]+2*exp(a)^2*exp(b)^2-2*exp...](images/mix155.gif){kind=small}
![`R2 ` = -3/512*(-4*b[T,T]+2*exp(a)^2*exp(b)^2-2*exp...](images/mix156.gif)
![`R2 ` = -3/512*(-4*b[T,T]+2*exp(a)^2*exp(b)^2-2*exp...](images/mix157.gif){kind=small}
> grcalc(Winvars);
Created definition for C(bdn,bdn,bup,bup)
Created definition for C(bup,bup,bup,bup)
> gralter(_,6,7);
Component simplification of a GRTensorII object:
Applying routine expand to object W1R
Applying routine expand to object W1I
Applying routine expand to object W2R
Applying routine expand to object W2I
Applying routine factor to object W1R
Applying routine factor to object W1I
Applying routine factor to object W2R
Applying routine factor to object W2I
To save space here we display only W1R and W1I. Increase grOptionDisplayLimit to display the other invariants.
> grdisplay(W1);
>