Cylindrically symmetric line element canonical form

[btphysics]

Hello,

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

Best regards.

 

[Bill_K]

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

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 ρ² dφ² - e^2Γ(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.

 

[btphysics]

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:

$$ds^2= A(t,ρ) ( dt^2-dρ^2)-...$$ I have some doubts on the rest of the expression, as well on the determination of the function A. Some papers gives de value $$A=e^{2Ω}$$ 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 $$dx^2 dx^3$$.


With best regards.

 

[Bill_K]

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.