### 5.7 Boson and fermion stars

Spherically symmetric static solutions of Einstein’s field equation coupled to a scalar field may be
interpreted as (uncharged, non-rotating) boson stars if they are free of singularities. Because of the
latter condition, the Wyman–Newman–Janis metric (see Section 5.6) does not describe a boson
star. The theoretical concept of boson stars goes back to [178, 290]. The analogous idea of a
fermion star, with the scalar field replaced by a spin 1/2 (neutrino) field, is even older [217].
Until today there is no observational evidence for the existence of either a boson or a fermion
star. However, they are considered, e.g., as hypothetical candidates for supermassive objects
at the center of galaxies (see [300, 323] for the boson and [334, 324] for the fermion case).
For the supermassive object at the center of our own galaxy, evidence points towards a black
hole, but the possibility that it is a boson or fermion star cannot be completely excluded so
far.
Exact solutions that describe boson or fermion stars have been found only numerically (in 3 + 1
dimensions). For this reason there is no boson star model for which the lightlike geodesics could
be studied analytically. Numerical studies of lensing have been carried out by Dabrowski and
Schunck [70] for a transparent spherically symmetric static maximal boson star, and by BiliÄ‡,
NikoliÄ‡, and Viollier [30] for a transparent spherically symmetric static maximal fermion star.
For the case of a fermion-fermion star (two components) see [170]. In all three articles the
authors use the “almost exact lens map” of Virbhadra and Ellis (see Section 4.3) which is
valid for observer and light source in the asymptotic region and almost aligned. Dabrowski and
Schunck [70] also discuss how the alignment assumption can be dropped. The lensing features found
in [70] for the boson star and in [30] for the fermion star have several similarities. In both
cases, there is a tangential caustic and a radial caustic (recall Figure 8 for terminology). A
(point) source on the tangential caustic (i.e., on the axis of symmetry through the observer) is
seen as a (1-dimenional) Einstein ring plus a (point) image in the center. If the (point) source
is moved away from the axis the Einstein ring breaks into two (point) images, so there are
three images altogether. Two of them merge and vanish if the radial caustic is crossed. So the
qualitative lensing features are quite different from a Schwarzschild black hole with (theoretically)
infinitely many images (see Section 5.1). The essential difference is that in the case of a boson or
fermion star there are no circular lightlike geodesics towards which light rays could asymptotically
spiral.