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Question: Showing a tensor is identically equal to zero (need to know what equations to input).

Question:Showing a tensor is identically equal to zero (need to know what equations to input).

guru kido 60 Maple
tensor simplification physics
March 24 2019
1


 

> 

with(Physics)
> 

Setup(mathematicalnotation = true)
> 

Setup(Coordinatesystem = (X = [x1, x2, x3, x4]), metric = -dx3^2*u22+2*dx3*dx4*u12-dx4^2*u11+dx1*dx3+dx2*dx4)

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

  

`Default differentiation variables for d_, D_ and dAlembertian are: `*{X = (x1, x2, x3, x4)}

  

`Systems of spacetime Coordinates are: `*{X = (x1, x2, x3, x4)}

  

[coordinatesystems = {X}, metric = {(1, 3) = 1/2, (2, 4) = 1/2, (3, 3) = -(diff(diff(u(X), x2), x2)), (3, 4) = Physics:-diff(diff(u(X), x1), x2), (4, 4) = -(diff(diff(u(X), x1), x1))}]

 (1)
> 

Setup(spacetimeindices = lowercaselatin)
> 

u22 := diff(u(x1, x2, x3, x4), x2, x2)

Physics:-diff(Physics:-diff(u(X), x2), x2)

 (2)
> 

u12 := diff(u(x1, x2, x3, x4), x1, x2)

Physics:-diff(Physics:-diff(u(X), x1), x2)

 (3)
> 

u11 := diff(u(x1, x2, x3, x4), x1, x1)

Physics:-diff(Physics:-diff(u(X), x1), x1)

 (4)
> 

u24 := diff(u(x1, x2, x3, x4), x2, x4)

Physics:-diff(Physics:-diff(u(X), x2), x4)

 (5)
> 

u13 := diff(u(x1, x2, x3, x4), x1, x3)

Physics:-diff(Physics:-diff(u(X), x1), x3)

 (6)
> 

g_[]

Physics:-g_[a, b] = Matrix(%id = 18446745064427858870)

 (7)
> 

Define(W, quiet)
> 

"W[~i,j,k,l]=1/(2)sqrt('g_[determinant]')g_[~i,~a]g_[~b,~c]LeviCivita[a,j,b,d]Weyl[~d,c,k,l]"

W[`~i`, j, k, l] = (1/2)*Physics:-g_[determinant]^(1/2)*Physics:-g_[`~a`, `~i`]*Physics:-g_[`~b`, `~c`]*Physics:-LeviCivita[a, b, d, j]*Physics:-Weyl[`~d`, c, k, l]

 (8)
> 

Define(W[`~i`, j, k, l] = (1/2)*Physics[g_][determinant]^(1/2)*Physics[g_][`~a`, `~i`]*Physics[g_][`~b`, `~c`]*Physics[LeviCivita][a, b, d, j]*Physics[Weyl][`~d`, c, k, l])

`Defined objects with tensor properties`

  

{Physics:-D_[a], Physics:-Dgamma[a], Physics:-Psigma[a], Physics:-Ricci[a, b], Physics:-Riemann[a, b, c, d], W[`~i`, j, k, l], Physics:-Weyl[a, b, c, d], Physics:-d_[a], Physics:-g_[a, b], Physics:-Christoffel[a, b, c], Physics:-Einstein[a, b], Physics:-KroneckerDelta[a, b], Physics:-LeviCivita[a, b, c, d], Physics:-SpaceTimeVector[a](X)}

 (9)
> 

W[definition]

W[`~e`, f, g, h] = (1/32)*16^(1/2)*Physics:-g_[`~a`, `~e`]*Physics:-g_[`~b`, `~c`]*Physics:-LeviCivita[a, b, d, f]*Physics:-Weyl[`~d`, c, g, h]

 (10)
> 

u11*u22-u12^2+u13+u24 = 0

diff(diff(u(X), x1), x3)+diff(diff(u(X), x2), x4)+(diff(diff(u(X), x2), x2))*(diff(diff(u(X), x1), x1))-(diff(diff(u(X), x1), x2))^2 = 0

 (11)
> 

u113 := diff(-u11*u22+u12^2-u24, x1)

-(diff(diff(diff(u(X), x1), x2), x4))-(diff(diff(u(X), x1), x1))*(diff(diff(diff(u(X), x1), x2), x2))-(diff(diff(u(X), x2), x2))*(diff(diff(diff(u(X), x1), x1), x1))+2*(diff(diff(u(X), x1), x2))*(diff(diff(diff(u(X), x1), x1), x2))

 (12)
> 

u123 := diff(-u11*u22+u12^2-u24, x2)

-(diff(diff(diff(u(X), x2), x2), x4))-(diff(diff(u(X), x1), x1))*(diff(diff(diff(u(X), x2), x2), x2))-(diff(diff(u(X), x2), x2))*(diff(diff(diff(u(X), x1), x1), x2))+2*(diff(diff(u(X), x1), x2))*(diff(diff(diff(u(X), x1), x2), x2))

 (13)
> 

u133 := diff(-u11*u22+u12^2-u24, x3)

-(diff(diff(diff(u(X), x2), x3), x4))-(diff(diff(diff(u(X), x2), x2), x3))*(diff(diff(u(X), x1), x1))-(diff(diff(u(X), x2), x2))*(diff(diff(diff(u(X), x1), x1), x3))+2*(diff(diff(u(X), x1), x2))*(diff(diff(diff(u(X), x1), x2), x3))

 (14)
> 

u134 := diff(-u11*u22+u12^2-u24, x4)

-(diff(diff(diff(u(X), x2), x4), x4))-(diff(diff(diff(u(X), x2), x2), x4))*(diff(diff(u(X), x1), x1))-(diff(diff(u(X), x2), x2))*(diff(diff(diff(u(X), x1), x1), x4))+2*(diff(diff(u(X), x1), x2))*(diff(diff(diff(u(X), x1), x2), x4))

 (15)
> 

u234 := diff(u(x1, x2, x3, x4), x2, x3, x4)

Physics:-diff(Physics:-diff(Physics:-diff(u(X), x2), x3), x4)

 (16)
> 

u223 := diff(u(x1, x2, x3, x4), x2, x2, x3)

Physics:-diff(Physics:-diff(Physics:-diff(u(X), x2), x2), x3)

 (17)
> 

u133 = -u11*u223-u113*u22+2*u12*u123-u234

-(diff(diff(diff(u(X), x2), x3), x4))-(diff(diff(diff(u(X), x2), x2), x3))*(diff(diff(u(X), x1), x1))-(diff(diff(u(X), x2), x2))*(diff(diff(diff(u(X), x1), x1), x3))+2*(diff(diff(u(X), x1), x2))*(diff(diff(diff(u(X), x1), x2), x3)) = -(diff(diff(diff(u(X), x2), x3), x4))-(-(diff(diff(diff(u(X), x1), x2), x4))-(diff(diff(u(X), x1), x1))*(diff(diff(diff(u(X), x1), x2), x2))-(diff(diff(u(X), x2), x2))*(diff(diff(diff(u(X), x1), x1), x1))+2*(diff(diff(u(X), x1), x2))*(diff(diff(diff(u(X), x1), x1), x2)))*(diff(diff(u(X), x2), x2))-(diff(diff(diff(u(X), x2), x2), x3))*(diff(diff(u(X), x1), x1))+2*(diff(diff(u(X), x1), x2))*(-(diff(diff(diff(u(X), x2), x2), x4))-(diff(diff(u(X), x1), x1))*(diff(diff(diff(u(X), x2), x2), x2))-(diff(diff(u(X), x2), x2))*(diff(diff(diff(u(X), x1), x1), x2))+2*(diff(diff(u(X), x1), x2))*(diff(diff(diff(u(X), x1), x2), x2)))

 (18)
> 

Weyl[`~i`, j, k, l, nonzero]+W[`~i`, j, k, l, nonzero]
> 

Weyl[`~i`, j, k, l, nonzero]-W[`~i`, j, k, l, nonzero]

``

 (19)
> 

`in`(half*flat*metrics, 4*D)
> 

``
> 

NULL
> 

NULL

 

I require one of these Weyl + W or Weyl - W (where W is Weyl tensor with hodge star operator, the equation may be found in my code) to be identically equal to zero. The computer will not recognise that it is identally equal unless i input the original p.d.e which I have, and I have also added some partial differentials up to the third order, maybe I need to include up to the fourth order? I am unsure what to add can someone please help me make this identically equal to zero! Thanks in advance for any help!

(I have put colon's after the equations Weyl+W and Weyl-W as the output was too long, but rest assured its not looking identically equal to zero!)
 

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