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Cylindrically symmetric line element canonical form

  1. May 4, 2014 #1

    What is the most general cylindrically symmetric line element in the canonical form?

    Best regards.
  2. jcsd
  3. May 4, 2014 #2


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    It's not clear what you're asking for. Do you mean just axially symmetric, or whole cylinder symmetry? Rotating or nonrotating? Time-dependent or independent?

    A canonical form for stationary axially symmetric solutions was introduced in 1932 by Lewis:

    ds2 = f(dt - ω dφ)2 - f -1 ρ22 - e(dρ2 + dz2)

    where (t, φ, ρ, z) are like cylindrical coordinates (e.g. φ has period 2π and the spacetime is flat where ρ, z → ∞), and f, ω, Γ are functions of ρ and z alone.
  4. May 4, 2014 #3
    Thanks for your answer. Well, I want a a line element time dependent, and where the plane ( t,ρ) is orthogonal to the ( ø,z) plane . Probably the line element will be of the form of:

    [tex]ds^2= A(t,ρ) ( dt^2-dρ^2)-...[/tex] I have some doubts on the rest of the expression, as well on the determination of the function A. Some papers gives de value [tex]A=e^{2Ω} [/tex] where Ω is a function of t and ρ. This solution admits a symmetry axis and is invariant under both rotations about the axis and translations parallel to it. It is a rotating solution, so it admits cross terms [tex]dx^2 dx^3[/tex].

    With best regards.
  5. May 5, 2014 #4


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    Are you aware of the Van Stockum rotating cylinder? The type of solution you're talking about would be a time-dependent generalization of that.
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