Coordinate calculation of the Ricci and Weyl tensors from a basis
> restart:
> grtw();
We load the bases.
> qload(mix1);
![`x `^a = vector([Theta, Phi, Psi, T])](images/mix1c11.gif){kind=small}
![eta^`(a)`*``^`(b)` = matrix([[1, 0, 0, 0], [0, 1, 0...](images/mix1c14.gif)
![omega1[a] = vector([-exp(a(T))*sqrt(1-Psi^2)/(sqrt(...](images/mix1c17.gif)
![omega2[a] = vector([-exp(b(T))*Psi/(sqrt(1-Theta^2)...](images/mix1c19.gif)
![omega3[a] = vector([0, Theta*exp(c(T)), exp(c(T))/(...](images/mix1c21.gif)
![omega4[a] = vector([0, 0, 0, exp(a(T)+b(T)+c(T))])](images/mix1c23.gif){kind=small}
From the bases we calculate the metric components.
> grcalc(g(dn,dn,pdn),g(up,up),detg);
> gralter(_,6,7);
Component simplification of a GRTensorII object:
Applying routine expand to object g(dn,dn,pdn)
Applying routine expand to object g(up,up)
Applying routine expand to object detg
Applying routine factor to object g(dn,dn,pdn)
Applying routine factor to object g(up,up)
Applying routine factor to object detg
Though not needed for the following calculations, it is sometimes convenient to see the line element.
> grcalc(ds);
> grdisplay(_);
We now proceed to calculate the coordinate components of the Ricci and Weyl tensors.
> grcalc(Chr(dn,dn,up));
> gralter(_,6,7);
Component simplification of a GRTensorII object:
Applying routine expand to object Chr(dn,dn,up)
Applying routine factor to object Chr(dn,dn,up)
> grcalcalter(R(dn,dn),6);
Simplification will be applied during calculation.
Applying routine expand to object R(dn,dn)
> gralter(_,6,7);
Component simplification of a GRTensorII object:
Applying routine expand to object R(dn,dn)
Applying routine factor to object R(dn,dn)
> grmap(_,autoAlias,`x`);
Applying routine autoAlias to R(dn,dn)
Warning, alias or macro b[T] defined in terms of b
Warning, alias or macro a[T] defined in terms of a
Warning, alias or macro c[T] defined in terms of c
> grdisplay(_);
![`R `[Theta*Theta] = 1/2*(exp(a)^2*exp(c)^4+exp(b)^2...](images/mix1c40.gif)
![`R `[Theta*Theta] = 1/2*(exp(a)^2*exp(c)^4+exp(b)^2...](images/mix1c41.gif)
![`R `[Theta*Phi] = -1/2*Psi*(Psi-1)^2*(Psi+1)^2*(exp...](images/mix1c42.gif){kind=small}
![`R `[Theta*Phi] = -1/2*Psi*(Psi-1)^2*(Psi+1)^2*(exp...](images/mix1c43.gif)
![`R `[Phi*Phi] = -1/2*(-exp(b)^2*Theta^2*exp(c)^4+2*...](images/mix1c44.gif)
![`R `[Phi*Phi] = -1/2*(-exp(b)^2*Theta^2*exp(c)^4+2*...](images/mix1c45.gif){kind=small}
![`R `[Phi*Phi] = -1/2*(-exp(b)^2*Theta^2*exp(c)^4+2*...](images/mix1c46.gif){kind=small}
![`R `[Phi*Phi] = -1/2*(-exp(b)^2*Theta^2*exp(c)^4+2*...](images/mix1c47.gif){kind=small}
![`R `[Phi*Phi] = -1/2*(-exp(b)^2*Theta^2*exp(c)^4+2*...](images/mix1c48.gif)
![`R `[Phi*Psi] = 1/2*Theta*(Psi-1)*(Psi+1)*(exp(b)^4...](images/mix1c49.gif)
![`R `[Psi*Psi] = 1/2*(exp(b)^4-exp(c)^4+exp(a)^4-2*c...](images/mix1c50.gif)
![`R `[T*T] = -a[T,T]-b[T,T]-c[T,T]+2*a[T]*b[T]+2*c[T...](images/mix1c51.gif){kind=small}
> grcalcalter(Ricciscalar,6);
Simplification will be applied during calculation.
Applying routine expand to object tRicciscalar
> gralter(_,6,7);
Component simplification of a GRTensorII object:
Applying routine expand to object Ricciscalar
Applying routine factor to object Ricciscalar
> grcalcalter(C(dn,dn,dn,dn),6);
Simplification will be applied during calculation.
Applying routine expand to object R(dn,dn,dn,dn)
Applying routine expand to object tRicciscalar
Applying routine expand to object C(dn,dn,dn,dn)
> gralter(_,6,7);
Component simplification of a GRTensorII object:
Applying routine expand to object C(dn,dn,dn,dn)
Applying routine factor to object C(dn,dn,dn,dn)
> grmap(_,autoAlias,`x`);
Applying routine autoAlias to C(dn,dn,dn,dn)
> grdisplay(_);
![`C `[Theta*Phi*Theta*Phi] = 1/6*(-4*exp(a)^2*a[T]*e...](images/mix1c58.gif)
![`C `[Theta*Phi*Theta*Phi] = 1/6*(-4*exp(a)^2*a[T]*e...](images/mix1c59.gif){kind=small}
![`C `[Theta*Phi*Theta*Phi] = 1/6*(-4*exp(a)^2*a[T]*e...](images/mix1c60.gif){kind=small}
![`C `[Theta*Phi*Theta*Phi] = 1/6*(-4*exp(a)^2*a[T]*e...](images/mix1c61.gif){kind=small}
![`C `[Theta*Phi*Theta*Phi] = 1/6*(-4*exp(a)^2*a[T]*e...](images/mix1c62.gif){kind=small}
![`C `[Theta*Phi*Theta*Phi] = 1/6*(-4*exp(a)^2*a[T]*e...](images/mix1c63.gif){kind=small}
![`C `[Theta*Phi*Theta*Phi] = 1/6*(-4*exp(a)^2*a[T]*e...](images/mix1c64.gif){kind=small}
![`C `[Theta*Phi*Theta*Phi] = 1/6*(-4*exp(a)^2*a[T]*e...](images/mix1c65.gif){kind=small}
![`C `[Theta*Phi*Theta*Phi] = 1/6*(-4*exp(a)^2*a[T]*e...](images/mix1c66.gif){kind=small}
![`C `[Theta*Phi*Theta*Phi] = 1/6*(-4*exp(a)^2*a[T]*e...](images/mix1c67.gif)
![`C `[Theta*Phi*Theta*Phi] = 1/6*(-4*exp(a)^2*a[T]*e...](images/mix1c68.gif){kind=small}
![`C `[Theta*Phi*Theta*Psi] = -1/6*(Psi-1)*(Psi+1)*(-...](images/mix1c69.gif)
![`C `[Theta*Phi*Theta*Psi] = -1/6*(Psi-1)*(Psi+1)*(-...](images/mix1c70.gif){kind=small}
![`C `[Theta*Phi*Theta*Psi] = -1/6*(Psi-1)*(Psi+1)*(-...](images/mix1c71.gif){kind=small}
![`C `[Theta*Phi*Theta*Psi] = -1/6*(Psi-1)*(Psi+1)*(-...](images/mix1c72.gif){kind=small}
![`C `[Theta*Phi*Theta*Psi] = -1/6*(Psi-1)*(Psi+1)*(-...](images/mix1c73.gif)
![`C `[Theta*Phi*Theta*Psi] = -1/6*(Psi-1)*(Psi+1)*(-...](images/mix1c74.gif)
![`C `[Theta*Phi*Theta*T] = 1/2*Psi*Theta*(Psi-1)*(Ps...](images/mix1c75.gif){kind=small}
![`C `[Theta*Phi*Theta*T] = 1/2*Psi*Theta*(Psi-1)*(Ps...](images/mix1c76.gif)
![`C `[Theta*Phi*Phi*Psi] = -1/6*(-3*exp(a)^2*exp(b)^...](images/mix1c77.gif)
![`C `[Theta*Phi*Phi*Psi] = -1/6*(-3*exp(a)^2*exp(b)^...](images/mix1c78.gif){kind=small}
![`C `[Theta*Phi*Phi*Psi] = -1/6*(-3*exp(a)^2*exp(b)^...](images/mix1c79.gif)
![`C `[Theta*Phi*Phi*T] = -1/2*Theta*(-3*exp(b)^2*Psi...](images/mix1c80.gif)
![`C `[Theta*Phi*Phi*T] = -1/2*Theta*(-3*exp(b)^2*Psi...](images/mix1c81.gif){kind=small}
![`C `[Theta*Phi*Psi*T] = -1/2*(-exp(b)^2*a[T]-b[T]*e...](images/mix1c82.gif)
![`C `[Theta*Psi*Theta*Psi] = -1/6*(-exp(a)^2*exp(b)^...](images/mix1c83.gif)
![`C `[Theta*Psi*Theta*Psi] = -1/6*(-exp(a)^2*exp(b)^...](images/mix1c84.gif){kind=small}
![`C `[Theta*Psi*Theta*Psi] = -1/6*(-exp(a)^2*exp(b)^...](images/mix1c85.gif){kind=small}
![`C `[Theta*Psi*Theta*Psi] = -1/6*(-exp(a)^2*exp(b)^...](images/mix1c86.gif){kind=small}
![`C `[Theta*Psi*Theta*Psi] = -1/6*(-exp(a)^2*exp(b)^...](images/mix1c87.gif)
![`C `[Theta*Psi*Theta*T] = -1/2*Psi*(-3*exp(b)^2*b[T...](images/mix1c88.gif)
![`C `[Theta*Psi*Phi*Psi] = 1/6*(Psi-1)*(Psi+1)*(-3*e...](images/mix1c89.gif)
![`C `[Theta*Psi*Phi*Psi] = 1/6*(Psi-1)*(Psi+1)*(-3*e...](images/mix1c90.gif){kind=small}
![`C `[Theta*Psi*Phi*Psi] = 1/6*(Psi-1)*(Psi+1)*(-3*e...](images/mix1c91.gif)
![`C `[Theta*Psi*Phi*T] = -1/2*(Psi^2*b[T]*exp(c)^2+2...](images/mix1c92.gif)
![`C `[Theta*Psi*Phi*T] = -1/2*(Psi^2*b[T]*exp(c)^2+2...](images/mix1c93.gif){kind=small}
![`C `[Theta*T*Theta*T] = 1/6*(-2*exp(a)^2*exp(b)^2*e...](images/mix1c94.gif)
![`C `[Theta*T*Theta*T] = 1/6*(-2*exp(a)^2*exp(b)^2*e...](images/mix1c95.gif){kind=small}
![`C `[Theta*T*Theta*T] = 1/6*(-2*exp(a)^2*exp(b)^2*e...](images/mix1c96.gif){kind=small}
![`C `[Theta*T*Theta*T] = 1/6*(-2*exp(a)^2*exp(b)^2*e...](images/mix1c97.gif){kind=small}
![`C `[Theta*T*Theta*T] = 1/6*(-2*exp(a)^2*exp(b)^2*e...](images/mix1c98.gif){kind=small}
![`C `[Theta*T*Phi*Psi] = -1/2*(exp(c)^2*c[T]+3*exp(a...](images/mix1c99.gif)
![`C `[Theta*T*Phi*Psi] = -1/2*(exp(c)^2*c[T]+3*exp(a...](images/mix1c100.gif){kind=small}
![`C `[Theta*T*Phi*T] = -1/6*Psi*(Psi-1)^2*(Psi+1)^2*...](images/mix1c101.gif)
![`C `[Theta*T*Phi*T] = -1/6*Psi*(Psi-1)^2*(Psi+1)^2*...](images/mix1c102.gif){kind=small}
![`C `[Theta*T*Phi*T] = -1/6*Psi*(Psi-1)^2*(Psi+1)^2*...](images/mix1c103.gif)
![`C `[Phi*Psi*Phi*Psi] = 1/6*(Theta-1)*(Theta+1)*(ex...](images/mix1c104.gif)
![`C `[Phi*Psi*Phi*Psi] = 1/6*(Theta-1)*(Theta+1)*(ex...](images/mix1c105.gif){kind=small}
![`C `[Phi*Psi*Phi*Psi] = 1/6*(Theta-1)*(Theta+1)*(ex...](images/mix1c106.gif){kind=small}
![`C `[Phi*Psi*Phi*Psi] = 1/6*(Theta-1)*(Theta+1)*(ex...](images/mix1c107.gif){kind=small}
![`C `[Phi*Psi*Phi*Psi] = 1/6*(Theta-1)*(Theta+1)*(ex...](images/mix1c108.gif)
![`C `[Phi*Psi*Phi*T] = 1/2*Psi*(Theta-1)*(Theta+1)*(...](images/mix1c109.gif)
![`C `[Phi*T*Phi*T] = -1/6*Theta^2*exp(c)^4*exp(a)^2-...](images/mix1c110.gif)
![`C `[Phi*T*Phi*T] = -1/6*Theta^2*exp(c)^4*exp(a)^2-...](images/mix1c111.gif)
![`C `[Phi*T*Phi*T] = -1/6*Theta^2*exp(c)^4*exp(a)^2-...](images/mix1c112.gif)
![`C `[Phi*T*Phi*T] = -1/6*Theta^2*exp(c)^4*exp(a)^2-...](images/mix1c113.gif)
![`C `[Phi*T*Phi*T] = -1/6*Theta^2*exp(c)^4*exp(a)^2-...](images/mix1c114.gif)
![`C `[Phi*T*Phi*T] = -1/6*Theta^2*exp(c)^4*exp(a)^2-...](images/mix1c115.gif)
![`C `[Phi*T*Phi*T] = -1/6*Theta^2*exp(c)^4*exp(a)^2-...](images/mix1c116.gif)
![`C `[Phi*T*Phi*T] = -1/6*Theta^2*exp(c)^4*exp(a)^2-...](images/mix1c117.gif)
![`C `[Phi*T*Phi*T] = -1/6*Theta^2*exp(c)^4*exp(a)^2-...](images/mix1c118.gif)
![`C `[Phi*T*Phi*T] = -1/6*Theta^2*exp(c)^4*exp(a)^2-...](images/mix1c119.gif)
![`C `[Phi*T*Phi*T] = -1/6*Theta^2*exp(c)^4*exp(a)^2-...](images/mix1c120.gif)
![`C `[Phi*T*Psi*T] = 1/6*exp(c)^2*(Psi-1)*(Psi+1)*(e...](images/mix1c121.gif)
![`C `[Phi*T*Psi*T] = 1/6*exp(c)^2*(Psi-1)*(Psi+1)*(e...](images/mix1c122.gif)
![`C `[Phi*T*Psi*T] = 1/6*exp(c)^2*(Psi-1)*(Psi+1)*(e...](images/mix1c123.gif)
![`C `[Psi*T*Psi*T] = 1/6*exp(c)^2*(exp(b)^4+exp(b)^2...](images/mix1c124.gif){kind=small}
![`C `[Psi*T*Psi*T] = 1/6*exp(c)^2*(exp(b)^4+exp(b)^2...](images/mix1c125.gif){kind=small}
![`C `[Psi*T*Psi*T] = 1/6*exp(c)^2*(exp(b)^4+exp(b)^2...](images/mix1c126.gif){kind=small}
>