### 3.2 Newtonian boson stars

The Newtonian limit of the Einstein–Klein–Gordon Eqs. (9 – 11) can be derived by assuming that the
spacetime metric in the weak field approximation can be written as
where is the Newtonian gravitational potential. In this limit, the Einstein equations reduce to the
Poisson equation
Conversely, by assuming that
in addition to the weak limit of Eq. (55), the Klein–Gordon equation reduces to
which is just the Schrödinger equation with . Therefore, the EKG system is reduced in the
Newtonian limit to the Schrödinger–Poisson (SP) system [98].
The initial data is obtained by solving an eigenvalue problem very similar to the one for boson stars,
with similar assumptions and boundary conditions. The solutions also share similar features and display a
similar behavior. A nice property of the Newtonian limit is that all the solutions can be obtained by
rescaling from one known solution [98],

where is the Newtonian number of particles.
The possibility of including self-interaction terms in the potential was considered in [106], studying also
the gravitational cooling (i.e., the relaxation and virialization through the emission of scalar field bursts) of
spherical perturbations. Non-spherical perturbations were further studied in [27], showing that the final
state is a spherically symmetric configuration. Single Newtonian boson stars were studied in [98], either
when they are boosted with/without an external central potential. Rotating stars were first successfully
constructed in [203] within the Newtonian approach. Numerical evolutions of binary boson stars in
Newtonian gravity are discussed in Section 4.2.

Recent work by Chavanis with Newtonian gravity solves the Gross–Pitaevskii equation, a variant of
Eq. (58) which involves a pseudo-potential for a Bose–Einstein condensate, to model either dark matter or
compact alternatives to neutron stars [46, 45, 47].

Much recent work considers boson stars from a quantum perspective as a Bose-Einstein condensate
involving some number, , of scalar fields. Ref. [160] studies the collapse of boson stars
mathematically in the mean field limit in which . Ref. [130] argues for the existence of bosonic
atoms instead of stars. Ref. [16] uses numerical methods to study the mean field dynamics of
BSs.