What equation do I use to solve for a spin $1$ bosonic field in a curved spacetime background? I was wondering what equation I would use if a wanted to find a solution for a bosonic field with quanta of spin $1$ in a curved spacetime background, with a specified metric, $g_{\mu\nu}$, describing the curvature. I know for a spin $0$ quanta scalar field you can use the curved spacetime Klein-Gordon equation:
$\frac{1}{
\sqrt{(-|g_{\mu\nu}|)}}(\partial_\mu g^{\mu\nu} \sqrt{(-|g_{\mu\nu}|)} \partial_\nu)\psi+m\psi=0$
and for a spin $\frac{1}{2}$ quanta fermionic field you can use the curved spacetime Dirac equation:
$i\gamma^a e^\mu_aD_\mu\Psi-m\Psi=0$
I am not sure though what equation to use for a bosonic field with spin $1$ quanta? A colleague suggested the Joos-Weinberg equation for spin $1$ quanta, but I can't seem to find the curved spacetime version of it. I would appreciate any help.
 A: The minimal coupling prescription says that the "simplest" way to generalize an equation of motion from flat space to curved space is to replace partial derivatives with covariant derivatives, so instead of
\begin{equation}
\partial_\mu F^{\mu\nu} = J^\nu, \ \ F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu
\end{equation}
you would have
\begin{equation}
\nabla_\mu F^{\mu\nu} = J^\nu, \ \ F_{\mu\nu} = \nabla_\mu A_\nu - \nabla_\nu A_\mu
\end{equation}
(although actually you could still write $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ because $F_{\mu\nu}$ is a two-form, or in other words an antisymmetric tensor with all lower indices, but this is a detail).
This set of equations is the typical starting point for studying electromagnetic fields in curved spacetime.
For a massive spin-1 field you would promote covariantize the Proca equation
\begin{equation}
\partial_\mu F^{\mu\nu} + m^2 A^\nu = J^\nu
\end{equation}
into
\begin{equation}
\nabla_\mu F^{\mu\nu} + m^2 A^\nu = J^\nu
\end{equation}
That gives you a starting point for massive spin-1 fields on a curved background.
In principle you can add extra terms proportional to the curvature which would vanish on flat spacetime (and so would recover Maxwell's equations on Earth). Classically, there's no fundamental reason you shouldn't include these kinds of terms. They usually aren't written down because of Occam's Razor. You are apparently just making the theory more complicated without adding any extra predictive power.
Note that there are some important subtleties in defining what "spin" even means on a curved spacetime (since the normal definition is based on the isometries of Minkowski spacetime). The "minimal coupling" prescription is a pragmatic way to bypass that discussion.
