Cylindrically symmetric line element canonical form

1. May 4, 2014

btphysics

Hello,

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

Best regards.

2. May 4, 2014

Bill_K

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.

3. May 4, 2014

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.

4. May 5, 2014

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.