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 [178290]. 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 [300323] for the boson and [334324] 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 [70Jump To The Next Citation Point] for a transparent spherically symmetric static maximal boson star, and by Bilić, Nikolić, and Viollier [30Jump To The Next Citation Point] 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 [70Jump To The Next Citation Point] 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 8View Image 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.

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