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@ecterrab it would be very helpful ...

bikramphy 70
July 29 2022
0 0

@ecterrab it would be very helpful if you can give an example.

Getting different result...

bikramphy 70
July 15 2022
0 0


 

``

``

> 

with(Physics); with(Tetrads)

_______________________________________________________

  

((`Defined as tetrad tensors `(`see ?Physics,tetrads`)*`, `*`𝔢`[a, mu]*`, `)*eta[a, b]*`, `*gamma[a, b, c]*`, `)*lambda[a, b, c]

  

((`Defined as spacetime tensors representing the NP null vectors of the tetrad formalism `(`see ?Physics,tetrads`)*`, `*l[mu]*`, `)*n[mu]*`, `*m[mu]*`, `)*conjugate(m[mu])

  

_______________________________________________________

 (1)

``

> 

Setup(signature = "+++-", Coordinates = (X = [Q, q, u, t]))

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

  

`Systems of spacetime coordinates are:`*{X = (Q, q, u, t)}

  

_______________________________________________________

  

[coordinatesystems = {X}, signature = `+ + + -`]

 (2)

``

> 

ds2 := dQ*dq+dQ*dq+du*dt+du*dt-Typesetting[delayDotProduct](2*Q/m.(diff(V(q, t), q)), dt, true)*dt

2*dq*dQ+2*du*dt-2*Q*(diff(V(q, t), q))*dt^2/m

 (3)

``

``

> 

Setup(metric = ds2)

[metric = {(1, 2) = 1, (3, 4) = 1, (4, 4) = -2*Q*(diff(V(q, t), q))/m}]

 (4)
> 

e_[]

Physics:-Tetrads:-e_[a, mu] = Matrix(%id = 18446745418568561598)

 (5)

``

> 

A := Matrix(4, symbol = a, shape = diagonal)

Matrix(%id = 18446745418568602078)

 (6)

``

``

> 

A.rhs(eta_[]).LinearAlgebra:-Transpose(A)

Matrix(%id = 18446745418592272246)

 (7)

``

``

> 

A := Matrix(4, {(1, 1) = 1, (2, 2) = 1, (3, 3) = I, (4, 4) = 1})

Matrix(%id = 18446745418563026206)

 (8)

``

``

> 

Eta[a, b] = A.rhs(eta_[]).LinearAlgebra:-Transpose(A)

Eta[a, b] = Matrix(%id = 18446745418538625734)

 (9)

``

``

> 

"Define(?)"

`Defined objects with tensor properties`

  

{Physics:-D_[mu], Physics:-Dgamma[mu], E[a, mu], Eta[a, b], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-Tetrads:-e_[a, mu], Physics:-Tetrads:-eta_[a, b], Physics:-g_[mu, nu], Physics:-Tetrads:-gamma_[a, b, c], Physics:-Tetrads:-l_[mu], Physics:-Tetrads:-lambda_[a, b, c], Physics:-Tetrads:-m_[mu], Physics:-Tetrads:-mb_[mu], Physics:-Tetrads:-n_[mu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

 (10)

``

``

> 

e_[definition]

Physics:-Tetrads:-e_[a, mu]*Physics:-Tetrads:-e_[b, `~mu`] = Physics:-Tetrads:-eta_[a, b]

 (11)

``

``

> 

E[a, mu] = simplify(A.rhs(e_[]))

E[a, mu] = Matrix(%id = 18446745418588365158)

 (12)

``

> 

"Define(?)"

`Defined objects with tensor properties`

  

{Physics:-D_[mu], Physics:-Dgamma[mu], E[a, mu], Eta[a, b], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-Tetrads:-e_[a, mu], Physics:-Tetrads:-eta_[a, b], Physics:-g_[mu, nu], Physics:-Tetrads:-gamma_[a, b, c], Physics:-Tetrads:-l_[mu], Physics:-Tetrads:-lambda_[a, b, c], Physics:-Tetrads:-m_[mu], Physics:-Tetrads:-mb_[mu], Physics:-Tetrads:-n_[mu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

 (13)

``

> 

"E[a, mu] * E[b,~mu] = Eta[a,b]"

E[a, mu]*E[b, `~mu`] = Eta[a, b]

 (14)

``

``

> 

TensorArray(E[a, mu]*E[b, `~mu`] = Eta[a, b])

Matrix(%id = 18446745418659228118)

 (15)

``

``

> 

"Setup(tetradmetric= rhs(?), tetrad = rhs(?))"

[tetrad = {(1, 1) = ((1/2)*I)*2^(1/2), (1, 2) = -((1/2)*I)*2^(1/2), (2, 3) = (1/2)*2^(1/2)/((Q^2*(diff(V(q, t), q))^2+m^2)^(1/2)/m)^(1/2), (2, 4) = 2^(1/2)*m/(((Q^2*(diff(V(q, t), q))^2+m^2)^(1/2)/m)^(1/2)*(2*Q*(diff(V(q, t), q))+2*(Q^2*(diff(V(q, t), q))^2+m^2)^(1/2))), (3, 3) = ((1/2)*I)*2^(1/2)/(-(Q^2*(diff(V(q, t), q))^2+m^2)^(1/2)/m)^(1/2), (3, 4) = I*2^(1/2)*m/((-(Q^2*(diff(V(q, t), q))^2+m^2)^(1/2)/m)^(1/2)*(2*Q*(diff(V(q, t), q))-2*(Q^2*(diff(V(q, t), q))^2+m^2)^(1/2))), (4, 1) = -((1/2)*I)*2^(1/2), (4, 2) = -((1/2)*I)*2^(1/2)}, tetradmetric = {(1, 1) = 1, (2, 2) = 1, (3, 3) = -1, (4, 4) = -1}]

 (16)

``

``

> 

e_[]

Physics:-Tetrads:-e_[a, mu] = Matrix(%id = 18446745418659225718)

 (17)

``

``

> 

IsTetrad()

false

 (18)

``

> 

eta_[]

Physics:-Tetrads:-eta_[a, b] = Matrix(%id = 18446745418568576046)

 (19)

``

``

> 

``

``

 


 

Download find_tetrad.mw

 

@ecterrab 

Using your method, I tried to find the corresponding tetrad. But I found it wrong. Please have a look at the attached file.

Actually, I need a proper solution. Can ...

bikramphy 70
November 20 2020
0 0

Actually, I need a proper solution. Can you please send maple file? I tried your method, but I did not get any solution.

I tried it for given two metrices but it...

bikramphy 70
August 17 2020
0 0

here is the maple code. please check it ...


 

transformation equation between given two metrices not working

> 

restart
> 

with(Physics); with(Tetrads); Setup(auto = true, mathematicalnotation = true)

_______________________________________________________

  

`Setting `*lowercaselatin_ah*` letters to represent `*tetrad*` indices`

  

((`Defined as tetrad tensors `(`see ?Physics,tetrads`)*`, `*`𝔢`[a, mu]*`, `)*eta[a, b]*`, `*gamma[a, b, c]*`, `)*lambda[a, b, c]

  

((`Defined as spacetime tensors representing the NP null vectors of the tetrad formalism `(`see ?Physics,tetrads`)*`, `*l[mu]*`, `)*n[mu]*`, `*m[mu]*`, `)*conjugate(m[mu])

  

_______________________________________________________

  

`* Partial match of '`*auto*`' against keyword '`*automaticsimplification*`' `

  

_______________________________________________________

  

[automaticsimplification = true, mathematicalnotation = true]

 (1)
> 

Coordinates(X = [r1, r2, r3, r4])

`Default differentiation variables for d_, D_ and dAlembertian are:`*{X = (r1, r2, r3, r4)}

  

`Systems of spacetime coordinates are:`*{X = (r1, r2, r3, r4)}

  

{X}

 (2)
> 

g1 := `∂`(r1)^2-`∂`(r2)^2+2*`∂`(r3)*`∂`(r4)

Physics:-d_(r1)^2-Physics:-d_(r2)^2+2*Physics:-d_(r3)*Physics:-d_(r4)

 (3)
> 

Setup(metric = Physics[d_](r1)^2-Physics[d_](r2)^2+2*Physics[d_](r3)*Physics[d_](r4))

"` ``Warning, for the signature `(-` - - +`)`, that is with the timelike component in position `4`, the spacetime metric indicated has `g[0,0]` = `g[4,4]` = `0`, and so the corresponding system of reference cannot be realized with real bodies (e.g. you cannot define proper time nor synchronize clocks in any infinitesimal region of space).Note as well that the corresponding `3-`dimensional space metric `gamma` is singular.`"

  

[metric = {(1, 1) = 1, (2, 2) = -1, (3, 4) = 1}]

 (4)
> 

g_[]

Physics:-g_[mu, nu] = Matrix(%id = 18446745914239149470)

 (5)
> 

Coordinates(K = [x, y, v, u])

`Systems of spacetime coordinates are:`*{K = (x, y, v, u), X = (r1, r2, r3, r4)}

  

{K, X}

 (6)
> 

p := 1/cos(u)^2

1/cos(u)^2

 (7)
> 

g2 := p*`∂`(x)^2-p*`∂`(y)^2+2*p*`∂`(v)*`∂`(u)-p*(x^2-y^2)*`∂`(u)^2

((-x^2+y^2)*Physics:-d_(u)^2+2*Physics:-d_(v)*Physics:-d_(u)+Physics:-d_(x)^2-Physics:-d_(y)^2)/cos(u)^2

 (8)
> 

Setup(diff = [K], metric = ((-x^2+y^2)*Physics[d_](u)^2+2*Physics[d_](v)*Physics[d_](u)+Physics[d_](x)^2-Physics[d_](y)^2)/cos(u)^2, quiet)

[differentiationvariables = [K], metric = {(1, 1) = 1/cos(u)^2, (2, 2) = -1/cos(u)^2, (3, 4) = 1/cos(u)^2, (4, 4) = (-x^2+y^2)/cos(u)^2}]

 (9)
> 

g_[]

Physics:-g_[mu, nu] = Matrix(%id = 18446745914137775150)

 (10)
> 

tr__0 := {u = f(r1, r2, r3, r4), v = h(r1, r2, r3, r4), x = k(r1, r2, r3, r4), y = l(r1, r2, r3, r4)}

{u = f(X), v = h(X), x = k(X), y = l(X)}

 (11)
> 

CompactDisplay(tr__0)

` f`(X)*`will now be displayed as`*f

  

` h`(X)*`will now be displayed as`*h

  

` k`(X)*`will now be displayed as`*k

  

` l`(X)*`will now be displayed as`*l

 (12)
> 

TransformCoordinates(tr__0, g_[mu, nu])

Matrix(%id = 18446745914279147630)

 (13)
> 

"convert(?=rhs(?),setofequations)"

{((-k(X)^2+l(X)^2)*(diff(f(X), r1))^2+2*(diff(h(X), r1))*(diff(f(X), r1))+(diff(k(X), r1))^2-(diff(l(X), r1))^2)/cos(f(X))^2 = 1, ((-k(X)^2+l(X)^2)*(diff(f(X), r2))^2+2*(diff(h(X), r2))*(diff(f(X), r2))+(diff(k(X), r2))^2-(diff(l(X), r2))^2)/cos(f(X))^2 = -1, ((-k(X)^2+l(X)^2)*(diff(f(X), r3))^2+2*(diff(h(X), r3))*(diff(f(X), r3))+(diff(k(X), r3))^2-(diff(l(X), r3))^2)/cos(f(X))^2 = 0, ((-k(X)^2+l(X)^2)*(diff(f(X), r4))^2+2*(diff(h(X), r4))*(diff(f(X), r4))+(diff(k(X), r4))^2-(diff(l(X), r4))^2)/cos(f(X))^2 = 0, (((-k(X)^2+l(X)^2)*(diff(f(X), r2))+diff(h(X), r2))*(diff(f(X), r1))+(diff(f(X), r2))*(diff(h(X), r1))+(diff(k(X), r2))*(diff(k(X), r1))-(diff(l(X), r2))*(diff(l(X), r1)))/cos(f(X))^2 = 0, (((-k(X)^2+l(X)^2)*(diff(f(X), r3))+diff(h(X), r3))*(diff(f(X), r1))+(diff(h(X), r1))*(diff(f(X), r3))+(diff(k(X), r3))*(diff(k(X), r1))-(diff(l(X), r3))*(diff(l(X), r1)))/cos(f(X))^2 = 0, (((-k(X)^2+l(X)^2)*(diff(f(X), r3))+diff(h(X), r3))*(diff(f(X), r2))+(diff(h(X), r2))*(diff(f(X), r3))+(diff(k(X), r3))*(diff(k(X), r2))-(diff(l(X), r3))*(diff(l(X), r2)))/cos(f(X))^2 = 0, (((-k(X)^2+l(X)^2)*(diff(f(X), r4))+diff(h(X), r4))*(diff(f(X), r1))+(diff(h(X), r1))*(diff(f(X), r4))+(diff(k(X), r4))*(diff(k(X), r1))-(diff(l(X), r4))*(diff(l(X), r1)))/cos(f(X))^2 = 0, (((-k(X)^2+l(X)^2)*(diff(f(X), r4))+diff(h(X), r4))*(diff(f(X), r2))+(diff(h(X), r2))*(diff(f(X), r4))+(diff(k(X), r4))*(diff(k(X), r2))-(diff(l(X), r4))*(diff(l(X), r2)))/cos(f(X))^2 = 0, (((-k(X)^2+l(X)^2)*(diff(f(X), r4))+diff(h(X), r4))*(diff(f(X), r3))+(diff(h(X), r3))*(diff(f(X), r4))+(diff(k(X), r4))*(diff(k(X), r3))-(diff(l(X), r4))*(diff(l(X), r3)))/cos(f(X))^2 = 1}

 (14)
> 

pdsolve({((-k(X)^2+l(X)^2)*(diff(f(X), r1))^2+2*(diff(h(X), r1))*(diff(f(X), r1))+(diff(k(X), r1))^2-(diff(l(X), r1))^2)/cos(f(X))^2 = 1, ((-k(X)^2+l(X)^2)*(diff(f(X), r2))^2+2*(diff(h(X), r2))*(diff(f(X), r2))+(diff(k(X), r2))^2-(diff(l(X), r2))^2)/cos(f(X))^2 = -1, ((-k(X)^2+l(X)^2)*(diff(f(X), r3))^2+2*(diff(h(X), r3))*(diff(f(X), r3))+(diff(k(X), r3))^2-(diff(l(X), r3))^2)/cos(f(X))^2 = 0, ((-k(X)^2+l(X)^2)*(diff(f(X), r4))^2+2*(diff(h(X), r4))*(diff(f(X), r4))+(diff(k(X), r4))^2-(diff(l(X), r4))^2)/cos(f(X))^2 = 0, (((-k(X)^2+l(X)^2)*(diff(f(X), r2))+diff(h(X), r2))*(diff(f(X), r1))+(diff(f(X), r2))*(diff(h(X), r1))+(diff(k(X), r2))*(diff(k(X), r1))-(diff(l(X), r2))*(diff(l(X), r1)))/cos(f(X))^2 = 0, (((-k(X)^2+l(X)^2)*(diff(f(X), r3))+diff(h(X), r3))*(diff(f(X), r1))+(diff(h(X), r1))*(diff(f(X), r3))+(diff(k(X), r3))*(diff(k(X), r1))-(diff(l(X), r3))*(diff(l(X), r1)))/cos(f(X))^2 = 0, (((-k(X)^2+l(X)^2)*(diff(f(X), r3))+diff(h(X), r3))*(diff(f(X), r2))+(diff(h(X), r2))*(diff(f(X), r3))+(diff(k(X), r3))*(diff(k(X), r2))-(diff(l(X), r3))*(diff(l(X), r2)))/cos(f(X))^2 = 0, (((-k(X)^2+l(X)^2)*(diff(f(X), r4))+diff(h(X), r4))*(diff(f(X), r1))+(diff(h(X), r1))*(diff(f(X), r4))+(diff(k(X), r4))*(diff(k(X), r1))-(diff(l(X), r4))*(diff(l(X), r1)))/cos(f(X))^2 = 0, (((-k(X)^2+l(X)^2)*(diff(f(X), r4))+diff(h(X), r4))*(diff(f(X), r2))+(diff(h(X), r2))*(diff(f(X), r4))+(diff(k(X), r4))*(diff(k(X), r2))-(diff(l(X), r4))*(diff(l(X), r2)))/cos(f(X))^2 = 0, (((-k(X)^2+l(X)^2)*(diff(f(X), r4))+diff(h(X), r4))*(diff(f(X), r3))+(diff(h(X), r3))*(diff(f(X), r4))+(diff(k(X), r4))*(diff(k(X), r3))-(diff(l(X), r4))*(diff(l(X), r3)))/cos(f(X))^2 = 1})
> 

``

 

NULL


 

Download not_working.mw

 

It is not able to solve the partial differential equation that is the code pdsolve((14))

Given the metric 1 as :   and m...

bikramphy 70
August 17 2020
0 0

Given the metric 1 as :

 

and metric 2 as :

Can we find the transformation equation between these two metrices? Please help me in finding the solution. Its urgent.

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