3.2 Newtonian boson stars

The Newtonian limit of the Einstein–Klein–Gordon Eqs. (9View Equation11View Equation) can be derived by assuming that the spacetime metric in the weak field approximation can be written as
g00 = − (1 + 2 V) , gii = 1 + 2 V , gij = 0 for i ⁄= j , (55 )
where V is the Newtonian gravitational potential. In this limit, the Einstein equations reduce to the Poisson equation
∇2V = 4π GT 00 = 4π Gm2 ϕ ¯ϕ. (56 )
Conversely, by assuming that
ϕ(x,t) ≡ Φ (x,t)eimt, (57 )
in addition to the weak limit of Eq. (55View Equation), the Klein–Gordon equation reduces to
i∂ Φ = − --1-∇2 Φ + m V Φ , (58 ) t 2 m
which is just the Schrödinger equation with ℏ = 1. 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],

(N )2 ( N )2 ( N ) ϕ2 = ϕ1 --2 , ω2 = ω1 --2 , r2 = r1 --1 , (59 ) N1 N1 N2
where ∫ N ≡ m dx3ϕ ¯ϕ 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. (58View Equation) 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, N, of scalar fields. Ref. [160] studies the collapse of boson stars mathematically in the mean field limit in which N → ∞. 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.

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