Such a field possesses energy because of its spatial gradients and time derivatives and this energy gravitates holding the star together. Less clear is what supports the star against the force of gravity. Its constituent scalar field obeys a Klein–Gordon wave equation which tends to disperse fields. This is the same dispersion which underlies the Heisenberg uncertainty principle. Indeed, Kaup’s original work [126] found energy eigenstates for a semi-classical, complex scalar field, discovering that gravitational collapse was not inevitable. Ruffini and Bonazzola [188] followed up on this work by quantizing a real scalar field representing some number of bosons and they found the same field equations.
None of this guarantees that such solutions balancing dispersion against gravitational attraction exist. In fact, a widely known theorem, Derrick’s theorem [68] (see also [186]), uses a clever scaling argument to show that no regular, static, nontopological localized scalar field solutions are stable in three (spatial) dimensional flat space. This constraint is avoided by adopting a harmonic ansatz for the complex scalar field
and by working with gravity. Although the field is no longer static, as shown in Section 2 the spacetime remains static. The star itself is a stationary, soliton-like solution as demonstrated in Figure 1 There are, of course, many other soliton and soliton-like solutions in three dimensions finding a variety
of ways to evade Derrick’s theorem. For example, the field-theory monopole of ’t Hooft and Polyakov is
a localized solution of a properly gauged triplet scalar field. Such a solution is a topological
soliton because the monopole possesses false vacuum energy which is topologically trapped.
The monopole is one among a number of different topological defects that requires an infinite
amount of energy to “unwind” the potential energy trapped within (see [218] for a general
introduction to defects and the introduction of [189] for a discussion of relevant classical field theory
concepts).
In Section 2, we present the underlying equations and mathematical solutions, but here we are
concerned with the physical nature of these boson stars. When searching for an actual boson star, we look
not for a quantized wave function, or even a semiclassical one. Instead, we search for a fundamental scalar,
say the long-sought Higgs boson. The Large Hadron Collider (LHC) hopes to determine the existence and
nature of the Higgs, with evidence at the time of writing suggesting a Higgs boson with mass
[184]. If the Higgs does not ultimately appear, there are other candidates such as an axion
particle. Boson stars are then either a collection of stable fundamental bosonic particles bound
by gravity, or else a collection of unstable particles that, with the gravitational binding, have
an inverse process efficient enough to reach an equilibrium. They can thus be considered a
Bose–Einstein condensate (BEC), although boson stars can also exist in an excited state as
well.
Indeed, applying the uncertainty principle to a boson star by assuming it to be a macroscopic quantum state results in an excellent estimate for the maximum mass of a BS. One begins with the Heisenberg uncertainty principle of quantum mechanics
and assumes the BS is confined within some radius
http://www.livingreviews.org/lrr-2012-6 |
Living Rev. Relativity 15, (2012), 6
![]() This work is licensed under a Creative Commons License. E-mail us: |