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

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.