Multipolar boson stars: Macroscopic Bose-Einstein condensates akin to hydrogen orbitals

C.A.R Herdeiro,J Kunz,I Perapechka,E Radu,Ya Shnir

doi:10.1016/j.physletb.2020.136027

Abstract

Boson stars are often described as macroscopic Bose-Einstein condensates. By accommodating large numbers of bosons in the same quantum state, they materialize macroscopically the intangible probability density cloud of a single particle in the quantum world. We take this interpretation of boson stars one step further. We show, by explicitly constructing the fully non-linear solutions, that static (in terms of their spacetime metric, gμν) boson stars, composed of a single complex scalar field, Φ, can have a non-trivial multipolar structure, yielding the same morphologies for their energy density as those that elementary hydrogen atomic orbitals have for their probability density. This provides a close analogy between the elementary solutions of the non-linear Einstein–Klein-Gordon theory, denoted Φ(N,ℓ,m), which could be realized in the macrocosmos, and those of the linear Schrödinger equation in a Coulomb potential, denoted Ψ(N,ℓ,m), that describe the microcosmos. In both cases, the solutions are classified by a triplet of quantum numbers (N,ℓ,m). In the gravitational theory, multipolar boson stars can be interpreted as individual bosonic lumps in equilibrium; remarkably, the (generic) solutions with m≠0 describe gravitating solitons [gμν,Φ(N,ℓ,m)] without any continuous symmetries. Multipolar boson stars analogue to hybrid orbitals is also constructed.

Highlights

Atomic orbitals are solutions of the linear, non-relativistic Schrödinger equation in an appropriate electromagnetic potential
By explicitly constructing the fully non-linear solutions, that static boson stars, composed of a single complex scalar field, Φ, can have a non-trivial multipolar structure, yielding the same morphologies for their energy density as those that elementary hydrogen atomic orbitals have for their probability density
This provides a close analogy between the elementary solutions of the non-linear Einstein– Klein-Gordon theory, denoted Φ(N,m), which could be realized in the macrocosmos, and those of the linear Schrödinger equation in a Coulomb potential, denoted Ψ(N,m), that describe the microcosmos

Summary

Introduction

Atomic orbitals are solutions of the linear, non-relativistic Schrödinger equation in an appropriate electromagnetic potential. By explicitly constructing the fully non-linear solutions, that static (in terms of their spacetime metric, gμν ) boson stars, composed of a single complex scalar field, Φ, can have a non-trivial multipolar structure, yielding the same morphologies for their energy density as those that elementary hydrogen atomic orbitals have for their probability density. Multipolar boson stars can be interpreted as individual bosonic lumps in equilibrium; remarkably, the (generic) solutions with m = 0 describe gravitating solitons [gμν , Φ(N, ,m)] without any continuous symmetries.

Results

Conclusion