Two boson stars in equilibrium P. Cunha1 C. Herdeiro1 E. Radu1and Ya. Shnir23 1Departamento de Matem atica da Universidade de Aveiro

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Two boson stars in equilibrium

P. Cunha1, C. Herdeiro1, E. Radu1and Ya. Shnir2,3

1Departamento de Matem´atica da Universidade de Aveiro

Center for Research and Development in Mathematics and Applications – CIDMA

Campus de Santiago, 3810-183 Aveiro, Portugal

2BLTP, JINR, Joliot-Curie 6, Dubna 141980, Moscow Region, Russia

3Institute of Physics, University of Oldenburg, Oldenburg D-26111, Germany

September 2022

Abstract

We construct and explore the solution space of two non-spinning, mini-boson stars in equilibrium, in

fully non-linear General Relativity (GR), minimally coupled to a free, massive, complex scalar ﬁeld. The

equilibrium is due to the balance between the (long range) gravitational attraction and the (short-range)

scalar mediated repulsion, the latter enabled by a πrelative phase. Gravity is mandatory; it is shown no

similar solutions exist in ﬂat spacetime, replacing gravity by non-linear scalar interactions. We study the

variation of the proper distance between the stars with their mass (or oscillation frequency), showing it

can be qualitatively captured by a simple analytic model that features the two competing interactions.

Finally, we discuss some physical properties of the solutions, including their gravitational lensing.

arXiv:2210.01833v1 [gr-qc] 4 Oct 2022

Contents

1 Introduction 2

2 The framework 4

2.1 Themodelandtheansatz...................................... 4

2.2 The explicit equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Boundary conditions and physical quantities of interest 6

3.1 Boundary conditions and asymptotic expansion . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2 Quantitiesofinterest ........................................ 7

4 The numerical approach 8

5 DBSs solutions 9

5.1 Domain and illustrative solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5.2 Comparison with the simple eﬀective model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.3 The role of gravity: no ﬂat spacetime static Q-balls in equilibrium . . . . . . . . . . . . . . . 14

6 Gravitational Lensing 16

7 Further remarks and conclusions 17

References 20

1 Introduction

The two-body equilibrium is an old and far reaching problem in General Relativity (GR). As a non-linear

theory, there is no superposition principle. Thus, the putative existence of any two-body equilibrium solution

must be anchored on a balance between diﬀerent interactions.

It was therefore surprising when Bach and Weyl, following Weyl’s formalism for static, axi-symmetric

solutions in GR [1], found a two-body equilibrium solution in fully non-linear vacuum GR. This fact intrigued

Einstein, who in 1936 addressed the problem with Rosen [2], realizing there is indeed an extra (non-vacuum)

ingredient that allows the equilibrium: a conical singularity. The latter can be either interpreted as a strut in

between the two bodies, or two strings connecting each body to inﬁnity. Whichever the chosen interpretation,

the solution is geodesically incomplete, by virtue of the (naked) conical singularity.

A non-singular (on and outside the event horizon) two-body (or in fact N-body) solution of GR was

eventually found, independently, in 1947 [3] by Majumdar and in 1948 by Papapetrou [4] in electro-vacuum.

They found that, under an appropriate ansatz, the Einstein-Maxwell equations fully linearise into a Laplace

equation in ﬂat Euclidean 3-space, which admits a multi-centre harmonic solution. Each centre (in the full

geometry) was later interpreted by Hartle and Hawking [5] as the horizon of an extremal Reissner-Nordstr¨om

black hole (BH).

The existence of the two-centre (or multi-centre) Majumdar-Papapetrou solution can be interpreted as

due to a balance between the gravitational attraction and the electric repulsion. In classical mechanics, two

point masses of magnitude M, each electrically charged with magnitude Q, at a distance r, interact via a

potential energy U=Ug+Ue, where

Ug=−GM2

r, Ue=1

4π0

r.(1.1)

In geometrized units, for which Newton’s constant Gand the vacuum electric permittivity 0obey, G=

1 = 4π0, extremal objects obeying M=Qcan be in equilibrium at any r,i.e. F=−∇U= 0. This is

precisely what occurs for the Majumdar-Papapetrou solution, wherein the extremal BHs can be placed at

any location. Thus, this heuristic classical mechanics argument provides an intuition for the Majumdar-

Papapetrou solution. This argument also suggests the equilibrium is not stable against small perturbations

– there is a ﬂat potential – which in fact is also the case in the full GR problem [6, 7]. Small perturbations

lead to an interesting scattering problem which can be dealt, for small velocities, in the moduli space

approximation - see also [8].

The simplest extension of vacuum GR, apart from electro-vacuum, is argueably scalar-vacuum. Allowing

the scalar ﬁeld to be massive and complex, but still free and minimally coupled to gravity, yields a novel

feature: horizonless self-gravitating solitons exist, describing localized energy lumps, dubbed boson stars

(BSs). These solutions were ﬁrst described in 1968 by Kaup [9], in 1969 by Ruﬃni and Bonazzala [10] and

(in a less known work) in 1968 by Feinblum and McKinley [11]. Single BSs rely on a dispersive scalar ﬁeld

– due to an oscillating amplitude with a harmonic phase ω, but with a time independent energy-momentum

tensor – being conﬁned by gravity. The harmonic phase ωcan neither be too large, with an upper bound set

by the ﬁeld’s mass µ, nor too small, with a minimum frequency ωmin. For ωmin < ω < µ, self-gravitating,

everywhere regular, asymptotically ﬂat spherical solitons exist, for one or more values of the ADM mass (in

units of the ﬁeld’s mass) Mµ, deﬁning one or more branches - see e.g. [12].1Some of these solutions are

dynamically robust – see [15, 16, 17] for reviews – and one may ask if one can set two such boson stars in

equilibrium in this fully non-linear theory.

Non-linear BS solutions that are axi-symmetric, static, asymptotically ﬂat and describe two symmetric

lumps of scalar ﬁeld energy were ﬁrst reported in [18] (see also [19, 20] for a discussion in the Newtonian

limit). More recently, more complex conﬁgurations of many scalar lumps in equilibrium were reported [21]2,

therein dubbed multipolar BSs. Although these are non-linear conﬁgurations, and therefore not a simple

superposition of two BSs, it is natural to interpret the two-centre dipolar BSs (DBSs) as an equilibrium

solutions of two (equal mass) BSs [18].

A key ﬁnding in the works described in the previous paragraph is that DBSs solutions exist only for

a parity odd scalar ﬁeld, with respect to the symmetry plane in between the two individual BSs. This is

equivalent to saying the two BSs have a phase diﬀerence of π. Such a phase diﬀerence sustains a repulsive

scalar interaction, as conﬁrmed (say) by performing head-on collisions of BSs, using numerical relativity

techniques [24]. Since the scalar ﬁeld has mass µ, one may expect to capture the leading interaction between

the two BSs by an interaction potential including both gravity and a meson Yukawa term U=Ug+Us, with

Ug=−GM2

r, Us=gQ2

re−αµr ,(1.2)

where Qdeﬁnes the scalar strength of each BS – its Noether charge – and g, α are constants. This heuristic

model suggests that: 1) For each pair of BSs with given M(and Q=Q(M)) there is a speciﬁc equilibrium

distance L=L[M, Q(M)]; 2) This equilibrium is stable, at least in the point particle approximation. Below

we shall discuss how this simple model indeed captures the distance dependence of the fully non-linear DBSs

and why the stability problem of DBSs is more complex than what this model suggests.

In this paper we shall construct the domain of solutions of DBSs and explore its physical properties,

within the perspective that it is the simplest multi-BSs conﬁguration. This serves both as a test ground for

the properties of equilibrium multi-BSs and as a bridge towards the dynamical problem of collisions of two

BSs. We shall further comment on both these perspectives in the ﬁnal discussion.

This paper is organized as follows. In Section 2 we introduce the model, the ansatz and the explicit

equations of motion to construct DBSs. In Section 3 we discuss the boundary conditions under which

the equations of motion are solved and introduce some relevant physical quantities for the analysis of the

solutions. In Section 4 we present the numerical approach to solve the equations of motion. In Section 5 we

discuss the solution space and illustrate speciﬁc solutions. Moreover, we discuss how a simple analytic model

based on (1.2) captures (to some extent) the behaviour of the fully non-linear solutions. We also provide

1Here we are referring to the fundamental solutions, for which the scalar ﬁeld has no nodes. Excited, spherical solutions

with nodes also exist - see e.g. [13]. Moreover, spinning, axially-symmetric solutions also exist, see e.g. [14], which can be seen

as a diﬀerent sort of excitation, with angular (rather than radial) nodes.

2See also [22, 23] for chains of non-spinning and spinning BSs.

an argument for the absence of two static Q-balls in equilibrium on Minkowski spacetime, regardless of the

speciﬁc self-interactions. In Section 6 we describe the lensing properties of the DBSs and we conclude with

a discussion and ﬁnal remarks in Section 7.

2 The framework

2.1 The model and the ansatz

We consider the action for Einstein’s gravity minimally coupled to a complex massive scalar ﬁeld Φ

S=1

16πG Zd4x√−gR−1

2gαβ Φ∗

, αΦ, β + Φ∗

, βΦ, α−µ2Φ∗Φ,(2.3)

where Gis Newton’s constant, gαβ the spacetime metric, with determinant gand Ricci scalar R, “*” denotes

complex conjugation and µis the scalar ﬁeld mass.

The resulting ﬁeld equations are:

Eαβ ≡Rαβ −1

2gαβR−8πG Tαβ = 0 ,(2.4)

Φ = µ2Φ,(2.5)

where

Tαβ = 2Φ∗

,(αΦ,β)−gαβ[Φ∗

,ν Φ,ν +µ2Φ∗Φ] ,(2.6)

is the energy-momentum tensor of the scalar ﬁeld.

The action (2.3) is invariant under the global U(1) transformation Φ →eiαΦ, where αis constant. This

implies the existence of a conserved current, jν=−i(Φ∗∂νΦ−Φ∂νΦ∗), with jν

;ν= 0. It follows that

integrating the timelike component of this 4-current in a spacelike slice Σ yields a conserved quantity – the

Noether charge:

Q=ZΣ

jt.(2.7)

At a microscopic level, this Noether charge counts the number of scalar particles.

The line-element for the solutions to be constructed in this work possesses two commuting Killing vector

ﬁelds, ξand η, with

ξ=∂t, η =∂ϕ,(2.8)

in a system of adapted coordinates. The analytical study of GR solutions with these symmetries is usually

considered within a metric ansatz of the form

ds2=−e−2U(ρ,z)dt2+e2U(ρ,z)he2k(ρ,z)(dρ2+dz2) + P(ρ, z)2dϕ2i,(2.9)

where (ρ, z) correspond asymptotically to the usual cylindrical coordinates. The corresponding expression

for the scalar ﬁeld is

Φ = φ(ρ, z)e−iωt ,(2.10)

where the real function φis the ﬁeld amplitude and ωis the scalar ﬁeld frequency, which we take to be positive,

without loss of generality. The Einstein–Klein-Gordon (EKG) equations take the following compact form3:

∇2k+ (∇U)2+ 8πGh(∇φ)2+e2k+2U(µ2−2e2Uω2)φ2i= 0 ,

∇2U+1

P(∇U)·(∇P)−8πGe2k+2U(µ2−2e2Uω2)φ2= 0 ,(2.11)

∇2P+ 16πGe2k+2U(µ2−2e2Uω2)φ2= 0 ,

∇2φ+1

P(∇P)·(∇φ)−e2k+2U(µ2−e2Uω2)φ= 0 ,

3There are two further constraint equations which, however, we do not display here.

where we deﬁne

∇2A=∂2A

∂ρ2+∂2A

∂z2,(∇A)·(∇B) = ∂A

∂ρ

∂B

∂ρ +∂A

∂z

∂B

∂z .

In the (electro-)vacuum case, it is always possible to set P≡ρ, such that only two independent metric

functions appear in the equations, and (ρ, z) become the canonical Weyl coordinates, the system being

integrable [25].

However, setting P≡ρis not possible for the case of interest here and no exact solutions appear to

exist for nonzero (ω, µ), in which case the problem is solved numerically. Then, it is convenient to use

‘quasi-isotropic’ spherical coordinates (r, θ) instead of (ρ, z), with the usual transformation

ρ=rsin θ, z =rcos θ , (2.12)

and the usual coordinate ranges, 0 ≤r < ∞, 0 ≤θ≤π. Also, in order to make contact with our previous

work [26], we redeﬁne U=−F0,k=F1+F0and P=eF2+F0in (2.9). The line-element and scalar ﬁeld

become

ds2=−e2F0(r,θ)dt2+e2F1(r,θ)(dr2+r2dθ2) + e2F2(r,θ)r2sin2θdϕ2,Φ = φ(r, θ)e−iωt ,(2.13)

which is the Ansatz used in this work in the numerical treatment of the problem. The Minkowski spacetime

background is approached for r→ ∞, with F0=F1=F2= 0. One also remarks that the symmetry axis of

the spacetime is given by η= 0, which corresponds to the z-axis, with θ= 0, π.

2.2 The explicit equations of motion

Given the Ansatz (2.13), the explicit form of the Klein-Gordon (KG) equation (2.5) reads

φ,rr +φ,θθ

r2+F0,r +F2,r +2

rφ,r +1

r2(F0,θ +F2,θ + cot θ)φ,θ +e−2F0+2F1ω2−µ2e2F1φ= 0 .(2.14)

The metric functions Fi(i= 1,2,3) satisfy the following second order partial diﬀerential equations (PDEs):

F0,rr +F0,θθ

r2+F0,r +F2,r +2

rF0,r +1

r2(F0,θ +F2,θ + cot θ)F0,θ (2.15)

+ 8πGe2F1µ2−2e−2F0ω2φ2= 0 ,

F1,rr +F1,θθ

r2−F2,r +1

rF0,r +F1,r

r−1

r2(F2,θ + cot θ)F0,θ (2.16)

+ 8πGe2F1 φ2

,r +φ2

,θ

r2+e−2F0+2F1ω2φ2!= 0 ,

F2,rr +F2,θθ

r2+F0,r

r+F0,r +F2,r +3

rF2,r +1

r2h(F0,θ +F2,θ)F2,θ + (F0,θ + 2F2,θ) cot θi(2.17)

+ 8πGe2F1µ2φ2= 0 .

In addition, there are also two constraint equations:

F0,rr +F2,rr + (F0,r −2F1,r)F0,r −(2F1,r −F2,r)F2,r −1

r(F0,r + 2F1,r −F2,r)−1

r2hF0,θθ +F2,θθ

+ (F0,θ −2F0,θ)F0,θ + (F2,θ −2F1,θ)F2,θ −2 cot θ(F1,θ −F2,θ )i+ 16πGφ2

0,r −φ2

,θ

r2= 0 ,(2.18)

F0,rθ +F2,rθ +F0,rF0,θ +F2,rF2,θ −(F1,r F2,θ +F2,rF1,θ)−(F1,rF0,θ +F0,rF1,θ )

−1

r(F0,θ +F1,θ)−cot θ(F1,r −F2,r) + 16πGφ,rφ,θ = 0 ,(2.19)

which are not solved directly, being used to check the numerical accuracy of the results

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TwobosonstarsinequilibriumP.Cunha1,C.Herdeiro1,E.Radu1andYa.Shnir2;31DepartamentodeMatematicadaUniversidadedeAveiroCenterforResearchandDevelopmentinMathematicsandApplications{CIDMACampusdeSantiago,3810-183Aveiro,Portugal2BLTP,JINR,Joliot-Curie6,Dubna141980,MoscowRegion,Russia3InstituteofPhysics,Uni...

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Two boson stars in equilibrium P. Cunha1 C. Herdeiro1 E. Radu1and Ya. Shnir23 1Departamento de Matem atica da Universidade de Aveiro

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