Rotating Boson Stars with Large Self-interaction in (2+1) dimensions

Solutions for rotating boson stars in (2+1) dimensional gravity with a negative cosmological constant are obtained numerically. The mass, particle number, and radius of the (2 + 1) dimensional rotating boson star are shown. Consequently we find the region where the stable boson star can exist. PACS number(s): 04.25.Dm, 04.40.Dg Typeset using REVTEX e-mail: [email protected], [email protected] e-mail: [email protected] 1 Self-gravitating systems have been investigated in various situations. Boson stars ( [1] [2] for reviews) have a very simple constituent, a complex scalar field which is bound by gravitational attraction. Thus the boson star provides us with the simplest model of relativistic stars. The solutions for relativistic boson stars are only numerically obtained in four dimensions. In (2 + 1) dimensions, static equilibrium configurations have been argued [3] in Einstein gravity with a negative cosmological constant. In the previous paper [4], we obtained an exact solution for nonrotating boson star in (2 + 1) dimensional gravity with a negative cosmological constant. We consider that the scalar field has a strong self-interaction. An infinitely large self-interaction term in the model leads to much simplications as in the (3+1) dimensional case [5]. In the present paper, we obtain numerical solutions for a rotating boson star in (2 + 1) dimensional gravity with a negative cosmological constant. We assume that the scalar field has a strong self-interaction as in the previous paper [4]. The rotating boson star in the (3+1) dimensional case has been studied numerically by F. D. Ryan [6]. We wish to study the similarity and the difference, between the (2+1) dimensional model and the model in other dimensions. The study of the rotating boson star will lead to a new aspect of gravitating systems and clarify the similarity and/or the difference among the other dimensional cases. We consider a complex scalar field with mass m and a quartic self-coupling constant λ. The action for the scalar field coupled to gravity can be written down as S = ∫ dx √−g [ 1 16πG (R + 2C)− |∇μφ| −m |φ| − λ 2 |φ| ] , (1) where R is the scalar curvature and the positive constant C stands for the (negative) cosmological constant. G is the Newton constant. Varying the action (1) with respect to the scalar field and the metric yields equations of motion. The equation of motion for the scalar field is ∇φ−mφ− λ |φ| φ = 0, (2) while the Einstein equation is 2 Rμν − 1 2 gμνR = 8πG [ 2Re ( ∇μφ∇νφ− 1 2 |∇φ| gμν ) −m |φ| gμν − λ 2 |φ| gμν ]