[Paper Review] Nonstatic cylindrically symmetric vacuum spacetimes in generalized Kasner form

Özgür Delice|arXiv (Cornell University)|Nov 1, 2004

Advanced Differential Geometry Research3 citations

TL;DR

This paper investigates nonstatic, cylindrically symmetric vacuum spacetimes by assuming separability of metric functions, deriving two nonstatic solutions for each parameter value of the static Levi-Civita case. The key result is a generalized solution that extends prior time-dependent vacuum solutions, highlighting the absence of a Birkhoff-type theorem in cylindrical symmetry and confirming that gravitational waves can be emitted during cylindrical collapse.

ABSTRACT

Cylindrically symmetric nonstatic vacuum space-times are studied assuming the separability of the metric functions. We see that for any value of the parameter of the static case, there are two corresponding nonstatic solutions in general. Some of the physical properties of the solutions are discussed and compared with the static Levi-Civita spacetime. According to Birkhoff’s theorem all spherically symmetric vacuum spacetimes can be represented by a static solution, namely, the Schwarzschild solution. Due to this fact, there is no gravitational wave emitted during spherical collapse. However, the situation drastically changes when we consider cylindrically symmetric systems since there is no analogue of this theorem in cylindrical symmetry. During gravitational collapse of a cylindrically symmetric system, gravitational waves are emitted and the exterior region of a collapsing cylindrical body is not static [1]. Time-dependent solutions of Einstein equations were studied for different cylindrical systems. An expanding cylindrical radiation filled universe [2], a radiation universe with heat flow [3], nonstatic cosmic strings with a time dependent vacuum exterior [4, 5, 6], nonstatic global strings [7] are some examples of such solutions. It is important to have time dependent vacuum solutions if one wants to match a cylindrical time dependent source with vacuum exterior region. Thus in this paper, we will study cylindrically symmetric time dependent vacuum solutions. It turns out that our solution will be generalization of a solution presented in some of the above papers. If ∂z and ∂φ are the axial and the angular killing vectors desribing cylindrical symmetry, then the most general cylindrically symmetric metric in diagonal form can be written as [8]: ds 2 = e 2(K−U) (−dt 2 + dr 2) + e 2U dz 2 + e −2U W 2 dφ 2, (1)

Motivation & Objective

• To investigate nonstatic cylindrically symmetric vacuum solutions in the absence of a Birkhoff-type theorem for cylindrical symmetry.
• To determine whether time-dependent vacuum solutions exist that generalize known static and dynamic solutions.
• To compare physical properties of the derived nonstatic solutions with the static Levi-Civita spacetime.
• To establish a framework for matching time-dependent cylindrical sources with vacuum exteriors in general relativity.
• To explore the implications of separability in metric functions for constructing exact solutions in cylindrical symmetry.

Proposed method

• Assumes separability of the metric functions in the general cylindrically symmetric diagonal metric form.
• Uses the metric ansatz: ds² = e²⁽ᴷ⁻ᵁ⁾(−dt² + dr²) + e²ᵁ dz² + e⁻²ᵁ W² dφ², with K, U, and W as functions of r and t.
• Applies Einstein’s vacuum field equations to the separable metric, leading to a system of partial differential equations.
• Solves the resulting equations under the assumption of separability, yielding two nonstatic solutions per static parameter value.
• Compares the derived solutions with the static Levi-Civita spacetime and known time-dependent solutions from prior literature.
• Analyzes the physical properties of the solutions, including curvature and causal structure, in the context of gravitational wave emission.

Experimental results

Research questions

• RQ1 Can nonstatic, cylindrically symmetric vacuum solutions be derived under the assumption of separability of metric functions?
• RQ2 How do these nonstatic solutions compare physically with the static Levi-Civita spacetime?
• RQ3 What is the role of the absence of a Birkhoff-type theorem in cylindrical symmetry for gravitational wave emission?
• RQ4 How do the derived solutions generalize previously known time-dependent vacuum solutions for cylindrical systems?
• RQ5 What are the implications of these solutions for matching time-dependent cylindrical sources to vacuum exteriors?

Key findings

• For every parameter value of the static Levi-Civita solution, two corresponding nonstatic vacuum solutions are found, indicating a richer solution space in the time-dependent case.
• The derived solutions generalize existing time-dependent vacuum solutions for cylindrical systems, such as those involving nonstatic cosmic strings or global strings.
• The absence of a Birkhoff-type theorem in cylindrical symmetry implies that gravitational waves can be emitted during the collapse of a cylindrically symmetric body.
• The solutions exhibit explicit time dependence in the metric functions, confirming that the exterior spacetime of a collapsing cylindrical source is not static.
• The separability assumption leads to a tractable system of equations that yields exact, physically meaningful vacuum solutions.
• The physical properties of the nonstatic solutions differ significantly from the static case, particularly in their curvature and causal structure, supporting the emission of gravitational radiation.