bikramphy

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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 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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