Trapping Photons by a Line Singularity

Abstract

We present cylindrically symmetric, static solutions of the Einstein field equations around a line singularity such that the energy momentum tensor corresponds to infinitely thin photonic shells. Positivity of the energy density of the thin shell and the line singularity is discussed. It is also shown that thick shells containing mostly radiation are possible in a numerical solution.

Static cylindrical matter shells

“…This is also true for a shell with particles moving only in the axial direction. Similar behavior has been observed for a static cylindrical shell with counter moving photons in z direction [12]. This might be an expected result since for these cases there is no pressure in the fluid against collapse (p φ = 0) and we need an repelling force to keep these shells static.…”

Higher dimensional cylindrical or Kasner type electrovacuum solutions

“…Such particle can also follow axial timelike (null) geodesics if the parameters satisfy the inequality (equality) 2σ(2σ − 2 + p) − p + (p 2 + q 2 )/2 > 0(= 0). These are in accordance with four dimensional Levi-Civita solution [8,[60][61][62].…”

“…Note that if one choses the coordinate x 4 as the radial coordinate and the remaining coordinates as usual cylindrical coordinates in (67) with properly chosen signature, then one obtains the cylindrical static vacuum Levi-Civita solution [8] in its Kasner form. There is a simple transformation between these two different forms of cylindrical vacuum solutions (See for example [26,62]). Actually, Kasner type cosmological [65][66][67] or cylindrical vacuum solutions [26,27] were generalized to higher dimensions.…”

“…In this special case the coordinate r must be extended to include −∞ < r < r 0 . The interesting point of this value of parameters σ = 1/2, p i = 0 is that it is the higher dimensional generalization of four dimensional case where the Levi-Civita metric describes an infinite plane geometry [15,60,62] rather than a cylindrical one. Thus, for this case we have a charged infinite plane at r = r 0 where r in this case should be considered as a Cartesian coordinate.…”

Cylindrically symmetric Brans–Dicke–Maxwell solutions

“…For this metric, the parameter d is related to the energy density of the source of the metric and the parameter W 0 is related to the global topology (conicity) of space-time [22][23][24][25] and gains significance in the context of the cosmic strings. In order to find a plausible interpretation for these parameters, several interior concentric cylinders [22][23][24][25][26][27] and thin shells [28][29][30][31][32] have been constructed as the sources for this particular vacuum solution. For d = 0 this metric is identical to that of a straight cosmic string metric [33].…”