Reissner–Nordstrom spacetime

5.3 Reissner–Nordström spacetime

The Reissner–Nordström metric

is the unique spherically symmetric and asymptotically flat solution of the Einstein–Maxwell equations. It has the form (69

) with

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 [305

]. A detailed account of Reissner–Nordström geodesics is given in [54

]. (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:

$2 2 0 ≤ e ≤ m$ ; 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.
$m2 < e2$ ; 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 [190 , 3 ] (cf. [159 ]). 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 [68 , 69 ]. 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 9 ). 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 15 , 16 , and 17 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” [39 , 37 ] 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 15 . Bozza [37 ] 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 [336 ] (recall Section 4.3) and analytical methods of Bozza et al. [39 , 37 , 40 ].