Cylindrically symmetric line element canonical form

In summary, the conversation discusses the most general cylindrically symmetric line element in the canonical form, specifically in terms of its time-dependency and symmetry properties. The canonical form for stationary axially symmetric solutions, introduced by Lewis, is mentioned as well as the Van Stockum rotating cylinder. The conversation also touches on the function A and its determination in the line element.
  • #1
btphysics
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0
Hello,

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

Best regards.
 
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  • #2
btphysics said:
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 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
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


I would like to clarify that the most general cylindrically symmetric line element in canonical form is the metric tensor, which describes the geometry of a space in terms of distances and angles. In cylindrical coordinates, the metric tensor takes the form of ds^2 = dr^2 + r^2dθ^2 + dz^2, where r is the distance from the axis of symmetry, θ is the angular coordinate, and z is the distance along the axis of symmetry. This form is considered canonical because it is the most simple and concise way to express the cylindrically symmetric line element. However, there are other forms that can also describe cylindrically symmetric spaces, such as the Weyl-Lewis-Papapetrou line element. I hope this clarifies your question. Best regards.
 

1. What is a cylindrically symmetric line element canonical form?

A cylindrically symmetric line element canonical form is a mathematical expression used in the study of General Relativity. It describes the geometry of a spacetime that is cylindrically symmetric, meaning that it has the same properties in all directions around a central axis.

2. How is a cylindrically symmetric line element canonical form different from other forms?

A cylindrically symmetric line element canonical form is unique in that it is specifically used to describe spacetimes that have cylindrical symmetry. Other forms, such as the Schwarzschild or Kerr solutions, describe different types of symmetries.

3. What are the variables in a cylindrically symmetric line element canonical form?

The variables in a cylindrically symmetric line element canonical form are typically the coordinates of the spacetime (t, r, θ, φ) and a set of constants that represent the mass and angular momentum of the system.

4. How is a cylindrically symmetric line element canonical form used in physics?

Cylindrically symmetric line element canonical forms are used in the study of General Relativity, specifically in the analysis of spacetimes that have cylindrical symmetry. They are used to describe the geometry of these spacetimes and to make predictions about the behavior of particles and light in these environments.

5. What are some real-world applications of cylindrically symmetric line element canonical forms?

Cylindrically symmetric line element canonical forms have been used to study the gravitational effects of rotating cylinders, such as neutron stars or black holes. They have also been used in the study of cosmic strings, which are hypothetical objects that have cylindrical symmetry and could have formed in the early universe.

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