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struggeling to define the metric...

PhysicsTom 5
August 21 2016
0 0

@ecterrab 

Thanks,

I tried along those lines. Instead of the full metric up to 4th order in X, I tried to play with simpler metric to familiarize myself with the workflow, but I already failed with the simple examples below. Strangely, the only thing that worked was the assignment g = - η (but only with a minus sign).

 


> 

restart; with(Physics); Setup(mathematicalnotation = true, coordinates = Y)

`* Partial match of 'coordinates' against keyword 'coordinatesystems'`

  

`Default differentiation variables for d_, D_ and dAlembertian are: `*{Y = (y1, y2, y3, y4)}

  

`Systems of spacetime Coordinates are: `*{Y = (y1, y2, y3, y4)}

  

[coordinatesystems = {Y}, mathematicalnotation = true]

 (1)
> 

Physics:-Define(eta[mu, nu] = rhs(g_[]));

`Defined objects with tensor properties`

  

{Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-d_[mu], eta[mu, nu], Physics:-g_[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](Y)}

 (2)
> 

Physics:-Define(A[mu, nu, alpha, beta], symmetric = {{mu, nu}})

`Defined objects with tensor properties`

  

{A[mu, nu, alpha, beta], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-d_[mu], eta[mu, nu], Physics:-g_[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](Y)}

 (3)
> 

Physics:-Define(B[mu, nu], symmetric = {{mu, nu}})

`Defined objects with tensor properties`

  

{A[mu, nu, alpha, beta], B[mu, nu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-d_[mu], eta[mu, nu], Physics:-g_[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](Y)}

 (4)
> 

Define(g_[mu, nu] = A[mu, nu, alpha, beta]*Y[`~alpha`]*Y[`~beta`])

Error, (in Physics:-Define) expected a spacetime symmetric metric; received: 0 = y1^2*A[1, 2, 1, 1]+y1*y2*A[1, 2, 1, 2]+y1*y2*A[1, 2, 2, 1]+y1*y3*A[1, 2, 1, 3]+y1*y3*A[1, 2, 3, 1]+y1*y4*A[1, 2, 1, 4]+y1*y4*A[1, 2, 4, 1]+y2^2*A[1, 2, 2, 2]+y2*y3*A[1, 2, 2, 3]+y2*y3*A[1, 2, 3, 2]+y2*y4*A[1, 2, 2, 4]+y2*y4*A[1, 2, 4, 2]+y3^2*A[1, 2, 3, 3]+y3*y4*A[1, 2, 3, 4]+y3*y4*A[1, 2, 4, 3]+y4^2*A[1, 2, 4, 4]

  
> 

Define(g_[mu, nu] = B[mu, nu])

Error, (in Physics:-Define) invalid input: Physics:-Setup expects value for keyword parameter metric to be of type Or(truefalse, identical(Euclidean, Minkowski, arbitrary, "not given"), set(`=`), `+`, `*`, `^`, Or(symbol, string, [posint, posint, posint]), list(`=`), Matrix, Array, table), but received B[mu, nu]

  
> 

Define(g_[mu, nu] = eta[mu, nu])

Error, (in Physics:-Define) invalid input: Physics:-Setup expects value for keyword parameter metric to be of type Or(truefalse, identical(Euclidean, Minkowski, arbitrary, "not given"), set(`=`), `+`, `*`, `^`, Or(symbol, string, [posint, posint, posint]), list(`=`), Matrix, Array, table), but received eta[mu, nu]

  
> 

Physics:-Define(g_[mu, nu] = -eta[mu, nu])

`Defined objects with tensor properties`

  

{A[mu, nu, alpha, beta], B[mu, nu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-d_[mu], eta[mu, nu], Physics:-g_[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](Y)}

 (5)
> 

g_[]

g[mu, nu] = (Matrix(4, 4, {(1, 1) = 1, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 2) = 1, (2, 3) = 0, (2, 4) = 0, (3, 3) = 1, (3, 4) = 0, (4, 4) = -1}, storage = triangular[upper], shape = [symmetric]))

 (6)
> 

``


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