Seeing relativity-II: Revisiting and visualizing the Reissner–Nordström metric

Abstract

In this paper we study some features of the Reissner-Nordström metric both from an analytic and a visual point of view. We perform an accurate ray tracing and study of null geodesics in various situations. Among the issues we focus on are (i) the comparison with the Schwarzschild case, (ii) the naked singularity case, where, if the electric charge is not too large, some dark shell appears on images despite there is no horizon in the metric, and (iii) the wormhole crossing case, i.e., a visual exploration of the maximal analytic extension of the metric.For any non zero value of M or Q, the region r = 0 is a curvature singularity. Consequently, one has a black hole when horizon(s) surround the singularity, i.e., when M ≥ |Q|. In the opposite case (M < |Q|), one has a naked singularity, whose visual aspect shall be studied in the next sections together with the black hole case. B. Classifying null geodesicsBeing spherically symmetric, any test particle of four-velocity or four-momentum experiences a planar geodesic motion in the metric, so that we can reduce our analysis to the case where the particle is confined within the plane θ = π/2. Also, the metric being static, one can extract two constants of motion for a massive test particle of four-velocity u µ or massless particle of four-wavevector k µ . Those are • The particle total energy per unit of mass or its total energy E := g µt u µ , or E := g µt k µ• The particle projected angular momentum per unit of mass, or its projected total angular momentum, L := −g µϕ u µ , or L := −g µϕ k µ .

A black hole solution of higher-dimensional Weyl–Yang–Kaluza–Klein theory

“…Now, if we convert geometrized units to SI units for the third black hole parameter charge (see, e.g. [132]), we, however, obtain the length-normalized charge-like term as Z ≈ 10 44 C, which is inconsistent with the upper limits of those mentioned above.…”

“…= 0 and the value of constant f 0 as f 0 = 1 but also make the problematic term, 𝕗 1 r −1 , of the 𝕗(r) in (131), which turns into a logarithmic term, 𝕗 1 ln r, in f (r), surprisingly vanish. Finally, the metric function f (r) (128) and warping function (132) reduce to two simple sets as follows:…”