Dynamics of Rotating Cylindrical Shells in General Relativity

Abstract

Cylindrical spacetimes with rotation are studied using the Newmann-Penrose formulas. By studying null geodesic deviations the physical meaning of each component of the Riemann tensor is given. These spacetimes are further extended to include rotating dynamic shells, and the general expression of the surface energy-momentum tensor of the shells is given in terms of the discontinuation of the first derivatives of the metric coefficients. As an application of the developed formulas, a stationary shell that generates the Lewis solutions, which represent the most general vacuum cylindrical solutions of the Einstein field equations with rotation, is studied by assuming that the spacetime inside the shell is flat. It is shown that the shell can satisfy all the energy conditions by properly choosing the parameters appearing in the model, provided that $ 0 \le \sigma \le 1$, where $\sigma$ is related to the mass per unit length of the shell.Comment: Typed in Revtex, including three figures. To appear in General Relativity and Gravit

Cylindrical systems in general relativity

“…Pereira and Wang studied the polarizations of the rotating GWs, and found that the geodesic deviation equation (9.8) is replaced by [259], D 2 η µ Dλ 2 = −R µ ναβ l ν l β η α = Φ 00 e µν o + Ψ 0 e µν + +i (Ψ 1 + Φ 01 ) e µν 03 + i (Ψ 1 + Φ 01 ) e µν 13 ] η ν , (9.19) where Ψ 1 ≡ −C µνλρ l µ n ν l λ m ρ , and 2Φ 01 ≡ (R µν − Rg µν /4) l µ m ν [259], A = e γ−ψ , e µν 03 ≡ e µ 0 e ν 3 + e µ 3 e ν 0 , e µν 13 ≡ e µ 1 e ν 3 + e µ 3 e ν 1 , 20) and e 2 , e 3 and e µν + are as defined above. It is remarkable to note that now the Ψ 0 wave has only one polarization, the "+" mode, along e 2 direction, and the "×" mode now is absent.…”

“…A spherical ball consisting of photons cuts SO in the circle S with the point O as its center. The image of the ball at the point P is turned into a spheroid with the main major axis along a line at 45 0 with respect to e1 in the plane spanned by e1 and e3 because of the interaction of Ψ1 and Φ01, while the rays are left undeflected in the e2-direction[259].performing a rotation in the plane spanned by e µ 1 α = π/4, we have e µν 13 = e…”

Rotating metrics admitting non-perfect fluids

“…To study the energy conditions of the energy-momentum tensor, we consider a time-like observer with its four-velocity vector U a [26,27] U a =αu a +βv a +γw a +δz a , (4.5) whereα,β,γ andδ are arbitrary constants. The four-velocity vector U a is subjected to the condition that…”

Trapping Photons by a Line Singularity

“…Here we present a series of graphs showing the components of the Einstein tensor as a function of r for five different values of b (Figs. [5][6][7][8][9]. We consider the energy momentum tensor of this thick shell in the form (35).…”