guru kido

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These are questions asked by guru kido

Setup(mathematicalnotation = true):
Setup(dimension = 3, coordinates = (X = [x, y, t]), metric = -ut^2*(dx^2)+ut^2*(dy^2)+(-ux^2-uy^2+1)*(dt^2)+2*ux*ut*dx*dt+2*uy*ut*dy*dt):
              ut := Physics:-diff(u(X), t)
ux := diff(u(x, y, t), x);
                  ux := Physics:-diff(u(X), x)
uy := diff(u(x, y, t), y);
                  uy := Physics:-diff(u(X), y)
Define(S[mu, nu] = 1/(d-2)*(Ricci[mu, nu]-1/(2*d-2)*Ricci[`~alpha`, alpha]*g_[mu, nu]));
             Defined objects with tensor properties
Define(C[mu, nu, rho] = D_[nu](S[rho, mu])-D_[rho](S[nu, mu]));
             Defined objects with tensor properties
Physics:-g_[mu, nu] = 

C[mu, nu, rho, nonzero];
          [Length of output exceeds limit of 1000000]
utt := diff(u(x, y, t), t, t);
               utt := Physics:-diff(u(X), t $ 2)
uxx := diff(u(x, y, t), x, x);
               uxx := Physics:-diff(u(X), x $ 2)
uyy := diff(u(x, y, t), y, y);
               uyy := Physics:-diff(u(X), y $ 2)
uxt := diff(u(x, y, t), x, t);
                uxt := Physics:-diff(u(X), t, x)
uyt := diff(u(x, y, t), y, t);
                uyt := Physics:-diff(u(X), t, y)
utt = ut*ut*(1/(-ux^2-uy^2+1))*(uxx+uyy)-2*ut*(1/(-ux^2-uy^2+1))*(ux*uxt+uy*uyt);
Basically here the cotton tensor exceeds the output limit in maple, however I am supposed to be showing how the cotton tensor vanishes, the last part of the code is me trying to input a relation of u_tt to various other derivatives of u in the hope that the cotton tensor can be simplified from this equation but nothing has changed as a result, can anyone help me? 

Hi, fairly simple question,
I've been using the physics package to calculate various different tensors but I can't see that I can calculate a cotton tensor for a given metric, is this included in the physics package? If so what is the calling sequence as I cannot find it. I understand I could use the differential geometry package but I do not want to as I don't know how to set up an arbitrary metric, thanks in advance.


Setup(mathematicalnotation = true);
Setup(Coordinatesystem = (X = [x, y, z, t]), metric = 4*dx*dt-dy^2-dz^2+4*u*(dt^2));
    * Partial match of  'Coordinatesystem' against keyword 

    Systems of spacetime Coordinates are: (Xequals(x,y,z,t))
[coordinatesystems = {X}, 

  metric = {(1, 4) = 2, (2, 2) = -1, (3, 3) = -1, (4, 4) = 4 u}]
g_[];      (here g is diplayed in matrix form)

Weyl[alpha, beta, mu, nu, nonzero];
            Physics:-Weyl[alpha, beta, mu, nu] = {}

So basically i'm trying to find the nonzero components of the weyl tensor which returns the empty set, i.e the weyl tensor apparently equals zero but I know this is not the case, can anyone see if I have input something wrong? Thanks in advance

g_[] in matrix form is 
0 0 0 2
0 -1 0 0
0 0 -1 0
2 0 0 4u

UPDATE: u is a function of all varibles and I have not considered this which might be causing the problem

I want to know how to program a metric g_[ ]  so that entries are zero apart from the diagonal.
Basically I am using the physics package and can set it as arbitrary or can set it to be specific values but I just want arbitrary values across the diagonal. e.g

Setup(mathematicalnotation = true);
                 [mathematicalnotation = true]

Setup(metric = arbitrary);
 [metric = {(1, 1) = _F1(X), (1, 2) = _F2(X), (1, 3) = _F3(X), (1, 4) = _F4(X), (2, 2) = _F5(X), (2, 3) = _F6(X),  (2, 4) = _F7(X),

(3, 3) = _F8(X), (3, 4) = _F9(X),  (4, 4) = _F10(X)}]

SO here I want to keep F1 F5 F8 and F10, thanks in advance!



Setup(mathematicalnotation = true);
Setup(Coordinatesystem = (X = [x1, x2, x3, x4]), metric = f(dx1^2+dx2^2+dx3^2+dx4^2));
    * Partial match of  'Coordinatesystem' against keyword 


  Default differentiation variables for d_, D_ and dAlembertian 

   are: (Xequals(x1,x2,x3,x4))
  Systems of spacetime Coordinates are: (Xequals(x1,x2,x3,x4))
Error, (in Physics:-Setup) expected definition of a metric as a tensorial algebraic expression with two free indices; received one with free indices {}


So I currently have:
with(DifferentialGeometry); with(Tensor);
DGsetup([r, theta], pol);
g1 := evalDG(drdr+r^2d(theta)d(theta))
C1 := Christoffel(g1);
However its coming back saying that g1 is not of metric form, am i missing something? Thanks

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