5.3 Reissner–Nordström spacetime

The Reissner–Nordström metric
( 2) 2 ( ) g = − 1 − 2m--+ e- dt2 + ----dr-----2+ r2 d𝜗2 + sin2𝜗d φ2 (110 ) r r2 1 − 2mr-+ er2
is the unique spherically symmetric and asymptotically flat solution of the Einstein–Maxwell equations. It has the form (69View Equation) with
2f(r) − 1 2m e2 r e = S (r) = 1 − ----+ -2, R (r) = ----2m---e2. (111 ) r r 1 − r + r2
It describes the field around an isolated spherical object with mass m and charge e. The Reissner–Nordström metric was found independently by Reissner [286], Weyl [347], and Nordström [242]. A fairly complete list of the pre-1979 literature on Reissner–Nordström geodesics can be found in [305Jump To The Next Citation Point]. A detailed account of Reissner–Nordström geodesics is given in [54Jump To The Next Citation Point]. (The Reissner–Nordström spacetime can be modified by introducing a cosmological constant. This generalized Reissner–Nordström spacetime, whose global structure is investigated in [202], will not be considered here.)

We assume m > 0 and ignore the region r < 0 for which the singularity at r = 0 is naked, for any value of e. Two cases are to be distinguished:

  1. 2 2 0 ≤ e ≤ m; in this case the staticity condition f(r) e > 0 is satisfied on the regions √ -------- 0 < r < m − m2 − e2 and √ -------- m + m2 − e2 < r < ∞, i.e., there are two horizons.
  2. m2 < e2; then the staticity condition ef(r) > 0 is satisfied on the entire region 0 < r < ∞, i.e., there is no horizon and the singularity at r = 0 is naked.

As the net charge of all known celestial bodies is close to zero, the naked-singularity case 2 is usually thought to be of little astrophysical relevance.

By switching to isotropic coordinates, one can describe light propagation in the Reissner–Nordström metric by an index of refraction (see, e.g., [104]). The resulting Fermat geometry (optical geometry) is discussed, in terms of embedding diagrams for the black-hole case and for the naked-singularity case, in [1903] (cf. [159Jump To The Next Citation Point]). The visual appearance of a background, as distorted by a Reissner–Nordström black hole, is calculated in [223]. Lensing by a charged neutron star, whose exterior is modeled by the Reissner–Nordström metric, is the subject of [6869]. The lensing properties of a Reissner–Nordström black hole are qualitatively (though not quantitatively) the same as that of a Schwarzschild black hole. The reason is the following. For a Reissner–Nordström black hole, the metric coefficient R (r) has one local minimum and no other extremum between horizon and infinity, just as in the Schwarzschild case (recall Figure 9View Image). The minimum of R (r) indicates an unstable light sphere towards which light rays can spiral asymptotically. The existence of this minimum, and of no other extremum, was responsible for all qualitative features of Schwarzschild lensing. Correspondingly, Figures 15View Image, 16View Image, and 17View Image also qualitatively illustrate lensing by a Reissner–Nordström black hole. In particular, there is an infinite sequence of images for each light source, corresponding to an infinite sequence of light rays whose limit curve asymptotically spirals towards the light sphere. One can consider the “strong-field limit” [39Jump To The Next Citation Point37Jump To The Next Citation Point] of lensing for a Reissner–Nordström black hole, in analogy to the Schwarzschild case which is indicated by the asymptotic straight line in the middle graph of Figure 15View Image. Bozza [37Jump To The Next Citation Point] investigates whether quantitative features of the “strong-field limit”, e.g., the slope of the asymptotic straight line, can be used to distinguish between different black holes. For the Reissner–Nordström black hole, image positions and magnifications have been calculated in [96], and travel times have been calculated in [201]. In both cases, the authors use the “almost exact lens map” of Virbhadra and Ellis [336Jump To The Next Citation Point] (recall Section 4.3) and analytical methods of Bozza et al. [3937Jump To The Next Citation Point40].

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