[Paper Review] On Spherically Symmetric Non-Static Space-Times Admitting Homothetic Motions

Ragab M. Gad|arXiv (Cornell University)|May 12, 2010

Cosmology and Gravitation Theories4 citations

TL;DR

This paper derives a new self-similar, spherically symmetric solution to Einstein's field equations by imposing a homothetic Killing vector field (HKVF) either orthogonal or parallel to the 4-velocity. The solution, obtained under the assumption of a perfect fluid with equation of state ρ = p, features zero expansion, non-zero acceleration and shear, and exhibits specific tidal force behavior—no radial stretching but transverse compression. The HKVF takes the form η = Φr∂r + Φν∂ν, leading to a metric with explicit ν and r dependence, and the solution is shown to be physically viable with naked singularity conditions analyzable via null geodesics.

ABSTRACT

Spherically symmetric solutions admitting a homothetic Killing vector field (HKVF) either orthogonal, $\eta_{\bot}$, or parallel,$\eta_{||}$, to the 4-velocity vector field, $u^a$, are studied. New self-similar solution of Einstein's field equation is found in the case when HKVF is in a general form. Some physical properties of the obtained solution are examined.

Motivation & Objective

• To derive exact solutions for non-static spherically symmetric space-times admitting homothetic Killing vector fields (HKVFs).
• To examine the physical properties of such solutions, particularly focusing on self-similarity and fluid dynamics.
• To determine the form of HKVF when it is either orthogonal or parallel to the 4-velocity vector field.
• To analyze the conditions under which singularities in the solution may be naked, using transverse radial null geodesics.
• To explore the implications of the equation of state ρ = p in the context of homothetic symmetry and tidal forces.

Proposed method

• The study uses the general form of the spherically symmetric line element ds² = αdν² + 2βdνdr − r²(dθ² + sin²θ dφ²), with α and β as functions of ν and r.
• Homothetic Killing vector fields are derived by solving the Lie derivative condition £ηgab = 2Φgab, leading to two cases: η⊥ (orthogonal to 4-velocity) and η|| (parallel to 4-velocity).
• For the orthogonal case, η⊥ = Φr∂r is derived; for the parallel case, η|| = Φν∂ν is obtained, leading to the combined form η = Φr∂r + Φν∂ν.
• The metric is then specialized using the HKVF condition and the perfect fluid assumption, yielding α = ½(r/ν)² and β = r/ν.
• Kinematic quantities (acceleration, expansion, shear, rotation) are computed from the 4-velocity and covariant derivatives.
• The Einstein field equations are solved under the perfect fluid assumption, leading to explicit expressions for energy density ρ and pressure p.

Experimental results

Research questions

• RQ1 What is the general form of the homothetic Killing vector field when it is orthogonal or parallel to the 4-velocity in non-static spherically symmetric space-times?
• RQ2 Under what conditions can singularities in such solutions be globally naked, as determined by the behavior of radial null geodesics?
• RQ3 What physical properties emerge in a self-similar solution admitting a general HKVF, particularly regarding expansion, shear, and tidal forces?
• RQ4 How do the energy density and pressure relate in a perfect fluid solution with homothetic symmetry, and what equation of state is implied?
• RQ5 What is the behavior of tidal forces (via geodesic deviation) in the radial and transverse directions for this solution?

Key findings

• The homothetic Killing vector field is derived in two cases: orthogonal to 4-velocity as η⊥ = Φr∂r, and parallel as η|| = Φν∂ν.
• A new self-similar solution is obtained with the metric ds² = ½(r/ν)²dν² + 2(r/ν)dνdr − r²(dθ² + sin²θ dφ²).
• The solution has zero expansion scalar (Θ = 0), indicating no volume change in the fluid flow.
• The fluid has non-zero acceleration (˙ua = −1/(2r)δ₁ᵃ) and non-zero shear (σ₁₁ = −2/√2 r, σ² = 1/r²).
• The energy density and pressure are equal: ρ = p = 1/(2κr²), implying a stiff equation of state.
• Tidal forces do not stretch observers in the radial direction (R₁₀₁₀ = 0), but induce transverse compression (R₂₀₂₀ = R₃₀₃₀ = 1/(4ν²)).