# 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.