Question

What is the charge of a scalar field in the context of black hole hair?

Answer 1

The electric charge is given by integrating the Hodge dual of the electric current

∫V⋆j=∫Vd⋆F=∫∂V⋆F∫V⋆j=∫Vd⋆F=∫∂V⋆F

.

I was told that the charge of a sclar field ϕϕ is obtained integrating the gradient of ϕϕ over a surface

∫dSnμ∂μϕ∫dSnμ∂μϕ

where nμnμ is a normal vector to the surface. But why is this last formula correct.

I guess that the scalar current is j=dϕj=dϕ, but I'm confused because this is obtained by the equations of motion, not gauge symmetry.

Answer 2

hey

here is answer

first what is scalar fields

Scalar fields are a common theme in modern cosmology. They play a
central role in inflation, and they have frequently been used to describe
dark energy in place of the cosmological constant.

Classical scalar fields
have also been proposed as possible candidates for the dark matter
component of the universe. One of the SFDM models proposes that
galactic haloes are formed by a Bose-Einstein condensation of a scalar
field (Matos 2000, Arbey 2001).
Given the fact that super-massive black holes seem to exist at the centre
of most galaxies, a scalar field configuration should be stable in the
presence of a central black hole, or at least be able to survive for
cosmological time-scales.
Dynamics of scalar fields around black holes.
A simple proposal is to consider that dark matter is made of a massive
scalar field, whose properties could explain the behaviour of dark matter
at all scales. The scenario of galactic formation, for instance, is as
follows:
a sea of scalar field particles fills the Universe and forms
localized primordial fluctuations that could collapse to form stable
objects, which are interpreted as the dark matter halos of galaxies.
Juan Carlos Degollado (UA) Dynamics of scalar fields around black holes.
Scalar fields as cosmological background:
It is assumed that the scalar particles decouple after inflation in the early
universe, after which the field has a simple quadratic potential with no
interactions. Ultra-light particles form a pure ground state condensate
with a high critical temperature behaving like cold dark matter today.
The condition for the formation of BEC in an expanding Universe can be
derived in a simple way if it is assumed that the scalar field particles
reached thermal equilibrium with other particles at early times, then the
thermal evolution of scalar field particles can be described by the
behaviour of the scalar number density of particles nφ during an
adiabatic process.
Juan Carlos Degollado (UA) Dynamics of scalar fields around black holes

hope it helps

thanks