text:
\section{Introduction}
\label{secIntro}

There exist three formulations of general relativity: {\em purely metric}, {\em metric--affine} and {\em purely affine}~\cite{FK3}.
In the purely metric (Einstein--Hilbert) formulation, the metric tensor is a variable (gravitational potential) and the field equations are derived by varying the total action with respect to the metric.
The affine connection is the Christoffel--Levi-Civita connection associated with the metric and the Lagrangian density for the gravitational field is linear in the symmetric part of the Ricci tensor.
In the metric--affine (Einstein--Palatini) formulation, both the metric tensor and a torsionless connection are independent variables (gravitational potentials) and the field equations are derived by varying the action with respect to these quantities.
The Lagrangian for the gravitational field is linear in the symmetric part of the Ricci tensor.
In the purely affine (Einstein--Eddington) formulation, a Lagrangian density depends on a torsionless affine connection and the symmetric part of the Ricci tensor.
This formulation constructs the (symmetric) metric tensor as the Hamilton derivative of the Lagrangian density with respect to the symmetrized Ricci tensor, obtaining an algebraic relation between these two tensors.
The field equations are derived by varying the action with respect to the connection, which gives a differential relation between the connection and the metric tensor.
This relation yields a differential equation for the metric.

Ferraris and Kijowski showed that all three formulations of general relativity are dynamically equivalent and the relation between them is analogous to the Legendre relation between Lagrangian and Hamiltonian dynamics~\cite{FK3}.
The relation between the purely affine and metric--affine picture of general relativity shows that the metric--affine Lagrangian density for the gravitational field is a Legendre term (with respect to the symmetrized Ricci tensor) corresponding to the scalar product of the velocities and momenta in classical mechanics~\cite{Kij}.
The purely metric and metric--affine picture of general relativity are related by a Legendre transformation with respect to the connection, and the Lagrangian density for the gravitational field in these pictures automatically turns out to be linear in the curvature tensor~\cite{FK3}.
The equivalence of the three formulations can also be shown for theories of gravitation with purely affine Lagrangians that depend on the full Ricci tensor and the tensor of homothetic curvature~\cite{univ,nonsym}, and a general connection with torsion~\cite{nonsym,Smal}.
There also exist formulations of gravity in which the dynamical variables are: metric and torsion (Einstein--Cartan theory)~\cite{Hehl,SG}, metric and nonsymmetric connection~\cite{HK,HLS}, tetrad~\cite{Uti}, tetrad and spin connection (Einstein--Cartan--Kibble--Sciama theory)~\cite{KS}, pure spin connection~\cite{sc}, and spinors~\cite{sp}.

Both the purely affine and metric--affine picture yield the same field equation from varying the action with respect to the connection.
The difference between these two formulations is the role of the metric.
In the purely affine picture the metric is defined from the Lagrangian density as a gravitational momentum, while in the metric--affine picture the metric is a variable (configuration)~\cite{FK3}.
Accordingly, the algebraic relation between the Ricci tensor and the metric tensor follows from a Hamilton-derivative definition in the purely affine picture and from a variational principle in the metric--affine picture.
In this paper we examine only the field equations obtained from varying the action with respect to the connection.
Therefore our results are valid for both the purely affine and metric--affine picture.

In general relativity, the electromagnetic field and its sources are considered to be on the side of the matter tensor in the field equations, i.e. they act as sources of the gravitational field.
In unified field theory, the electromagnetic field obtains the same geometric status as the gravitational field~\cite{Goe}.
A general affine connection, not restricted to be metric-compatible and symmetric, has enough degrees of freedom to make it possible to describe the classical gravitational and electromagnetic fields.
Both the purely affine and metric--affine formulation of gravity allow an elegant unification of these fields.
Ferraris and Kijowski showed that while the gravitational field in the purely affine formulation is represented by the symmetric part of the Ricci tensor, the electromagnetic field can be represented by the tensor of homothetic curvature~\cite{FK2}.
The purely affine Lagrangian for the electromagnetic field, that has the form of the Maxwell Lagrangian in which the metric tensor is replaced by the symmetrized Ricci tensor and the electromagnetic field tensor by the tensor of homothetic curvature, is dynamically equivalent to the sourceless Einstein--Maxwell Lagrangian in the metric--affine and metric formulation~\cite{FK2,FK1}.
Such a construction can be generalized to sources~\cite{unif}.
Ponomarev and Obukhov introduced the same construction in the metric--affine formulation of gravity~\cite{PO}.

Affine (metric--affine and purely affine) Lagrangians that depend on the affine connection not only through the symmetric part of the Ricci tensor but also through matter fields, are subject to an unphysical constraint on the source density~\cite{HK,HLS,San}.
This constraint is related to the fact that the symmetrized Ricci tensor is projectively invariant~\cite{HK,HLS,San,Scho} while the matter part of the Lagrangian is generally not.
The inclusion of the tensor of homothetic curvature (the electromagnetic field), which is also projectively non-invariant, in an affine Lagrangian replaces this constraint with the Maxwell equations and restores projective invariance of the total action without constraining the connection~\cite{unif}.

In this paper we consider the {\em metric--affine} formulation of gravity.
In Sec.~\ref{secField} we show, following Ponomarev and Obukhov~\cite{PO}, that the simplest metric--affine Lagrangian which depends on the tensor of homothetic curvature (this Lagrangian has the form of the Maxwell Lagrangian in which the electromagnetic field tensor is replaced by the tensor of homothetic curvature) generates the Einstein--Maxwell equations for the {\em massless} Weyl vector~\cite{HK,HLS} (corresponding to the photon).
Therefore the degrees of freedom contained in the affine connection provide an elegant way of unifying the classical gravitational and electromagnetic fields also in the metric--affine formulation of gravity.
The above Lagrangian was already mentioned, though without further investigation, by Hehl, Lord and Smalley~\cite{HLS} to remedy projective invariance of the gravitational action, and further studied by Vollick using a symmetric connection~\cite{Vol}.
A similar Lagrangian, with the electromagnetic field tensor represented by the curl of the torsion vector, was studied by Hammond~\cite{Ham}.
In Sec.~\ref{secSol} we present the general solution for the affine connection in the presence of matter fields that depend on the connection~\cite{PO}.

In Sec.~\ref{secMas} we use the Lagrange-multiplier method of Hehl and Kerlick~\cite{HK} to show that algebraic constraints on the torsion that break projective invariance of the connection and impose projective invariance on the tensor of homothetic curvature replace the massless Weyl vector with the {\em massive} Weyl vector related to the corresponding Lagrange multiplier.
The mass of this vector is given by the coupling of the affine electromagnetic Lagrangian to the gravitational field.
We examine several algebraic constraints on the torsion tensor.
We observe that the metric--affine formulation of gravity allows for a mechanism that {\em generates masses} of vectors, as it happens for electroweak gauge bosons via spontaneous symmetry breaking of the $SU(2)\times U(1)$ group~\cite{Wei}.
A geometrical mechanism of mass generation was already noted by Ponomarev and Obukhov~\cite{PO} who included the square of the torsion tensor in the gravitational Lagrangian, Hammond~\cite{Ham} who added the square of the torsion vector, and Vollick~\cite{Vol} and Hejna~\cite{Hej} who used a symmetric connection.
In this paper we treat this mechanism systematically using a Lagrange-multiplier formalism~\cite{HK} that allows us to study arbitrary constraints on the affine connection.
We briefly discuss and summarize the results in Sec.~\ref{secSum}.

\section{Field equations in metric--affine gravity}
\label{secField}

A general metric--affine Lagrangian density ${\cal L}$ depends on the affine connection $\Gamma^{\,\,\rho}_{\mu\,\nu}$ and the curvature tensor, $R^\rho_{\phantom{\rho}\mu\sigma\nu}=\Gamma^{\,\,\rho}_{\mu\,\nu,\sigma}-\Gamma^{\,\,\rho}_{\mu\,\sigma,\nu}+\Gamma^{\,\,\kappa}_{\mu\,\nu}\Gamma^{\,\,\rho}_{\kappa\,\sigma}-\Gamma^{\,\,\kappa}_{\mu\,\sigma}\Gamma^{\,\,\rho}_{\kappa\,\nu}$, as well as the symmetric metric tensor $g_{\mu\nu}$ of the Lorentzian signature $(+,-,-,-)$.
The antisymmetric part of the affine connection is the Cartan torsion tensor, $S^\rho_{\phantom{\rho}\mu\nu}=\Gamma^{\,\,\,\,\rho}_{[\mu\,\nu]}$, while its symmetric part can be split into the Christoffel--Levi-Civita connection, $\{^{\,\,\rho}_{\mu\,\nu}\}_g=\frac{1}{2}g^{\rho\lambda}(g_{\nu\lambda,\mu}+g_{\mu\lambda,\nu}-g_{\mu\nu,\lambda})$, and terms constructed from the torsion tensor and the nonmetricity tensor, $N_{\mu\nu\rho}=g_{\mu\nu;\rho}$~\cite{HK,HLS}:
\begin{equation}
\Gamma^{\,\,\rho}_{\mu\,\nu}=\{^{\,\,\rho}_{\mu\,\nu}\}_g+S^\rho_{\phantom{\rho}\mu\nu}+2S_{(\mu\nu)}^{\phantom{(\mu\nu)}\rho}+\frac{1}{2}N_{\mu\nu}^{\phantom{\mu\nu}\rho}-N^\rho_{\phantom{\rho}(\mu\nu)},
\label{split}
\end{equation}
where the semicolon denotes the covariant differentiation with respect to $\Gamma^{\,\,\rho}_{\mu\,\nu}$ (we use the notation of~\cite{Schr}).
The {\em Weyl vector} is defined as~\cite{HK,HLS}:
\begin{equation}
W_\nu=\frac{1}{2}(\Gamma^{\,\,\rho}_{\rho\,\nu}-\{^{\,\,\rho}_{\rho\,\nu}\}_g)=-\frac{1}{4}N^\rho_{\phantom{\rho}\rho\nu}.
\label{Weylvec1}
\end{equation}

We assume that the dependence of ${\cal L}$ on the curvature is restricted to the contracted curvature tensors~\cite{nonsym}, of which there exist three: the symmetric $P_{\mu\nu}=R_{(\mu\nu)}$ and antisymmetric $R_{[\mu\nu]}$ part of the Ricci tensor, $R_{\mu\nu}=R^\rho_{\phantom{\rho}\mu\rho\nu}$, and the antisymmetric {\em tensor of homothetic curvature}:
\begin{equation}
Q_{\mu\nu}=R^\rho_{\phantom{\rho}\rho\mu\nu}=\Gamma^{\,\,\rho}_{\rho\,\nu,\mu}-\Gamma^{\,\,\rho}_{\rho\,\mu,\nu},
\label{homoth}
\end{equation}
which has the form of a curl~\cite{Scho,Schr}.
The tensor of homothetic curvature is proportional to the curl of the Weyl vector:
\begin{equation}
Q_{\mu\nu}=2(W_{\nu,\mu}-W_{\mu,\nu}).
\label{Weylvec2}
\end{equation}
In addition, we assume that ${\cal L}$ does not depend on $R_{[\mu\nu]}$.
The simplest metric--affine Lagrangian density that depends on $P_{\mu\nu}$ is the Einstein--Palatini Lagrangian density of the gravitational field~\cite{FK3}:
\begin{equation}
{\cal L}_g=-\frac{1}{2\kappa}P_{\mu\nu}{\sf g}^{\mu\nu},
\label{LagGrav}
\end{equation}
where $\kappa=8\pi G$ ($c=1$), ${\sf g}^{\mu\nu}=\sqrt{-g}g^{\mu\nu}$ and $g=\mbox{det}(g_{\mu\nu})$.
The total Lagrangian density for the gravitational field and matter is given by ${\cal L}_g+{\cal L}_m$, where ${\cal L}_m$ is the Lagrangian density for matter that depends in general on both the metric and connection.
The variations of ${\cal L}_m$ with respect to the metric and connection define, respectively, the dynamical energy-momentum tensor $T_{\mu\nu}$:
\begin{equation}
T_{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta{\cal L}_m}{\delta g^{\mu\nu}},
\label{EMT}
\end{equation}
and the density conjugate to the connection (called hypermomentum in~\cite{HK,HLS}):
\begin{equation}
\Pi_{\phantom{\mu}\rho\phantom{\nu}}^{\mu\phantom{\rho}\nu}=-2\kappa\frac{\delta{\cal L}_m}{\delta \Gamma^{\,\,\rho}_{\mu\,\nu}},
\label{Pi}
\end{equation}
which has the same dimension as the connection.\footnote{
The variational derivative of a function ${\cal L}(\phi,\phi_{,\mu})$ with respect to a variable $\phi$ is defined as $\frac{\delta{\cal L}}{\delta\phi}=\frac{\partial{\cal L}}{\partial\phi}-(\frac{\partial{\cal L}}{\partial\phi_{,\mu}})_{,\mu}$.
}

The theory based on the Lagrangian density~(\ref{LagGrav}) without any constraints, i.e. with fully independent metric and connection, does not determine the connection uniquely because ${\cal L}_g$ is invariant under projective transformations of the connection~\cite{Hehl,HK,HLS,PO}:
\begin{equation}
\Gamma^{\,\,\rho}_{\mu\,\nu}\rightarrow\Gamma^{\,\,\rho}_{\mu\,\nu}+\delta^\rho_\mu\Lambda_\nu,
\label{proj}
\end{equation}
where $\Lambda_\nu$ is a vector function of the coordinates.
The same problem occurs if ${\cal L}_m$ does not depend on the connection, e.g., for electromagnetic fields or ideal fluids.
Therefore at least four degrees of freedom must be constrained to make such a theory consistent from a physical point of view~\cite{HLS}.
If, however, ${\cal L}_m$ does depend on the connection, e.g., for spinor fields or viscous fluids, the projective invariance of ${\cal L}_g$ imposes four algebraic constraints on the source density~(\ref{Pi}) and restricts forms of matter that can be described by the metric--affine formulation of gravity~\cite{HK,HLS,San}.
Consequently, we must change the form of the Lagrangian for the gravitational field so that it is not projectively invariant.

To break projective invariance of the field part of the Lagrangian we add a term that depends on the tensor of homothetic curvature $Q_{\mu\nu}$~\cite{HLS}.
The tensor $Q_{\mu\nu}$ is invariant only with respect to special projective transformations, i.e. projective transformations~(\ref{proj}) with $\Lambda_\nu=\lambda_{,\nu}$, where $\lambda$ is a scalar function of the coordinates~\cite{Kun}.
The simplest metric--affine Lagrangian density that depends on $Q_{\mu\nu}$, proposed by Hehl, Lord and Smalley~\cite{HLS}, Ponomarev and Obukhov~\cite{PO}, and Vollick~\cite{Vol}, has the form of the Maxwell Lagrangian for the electromagnetic field:
\begin{equation}
{\cal L}_Q=-\frac{\alpha}{4}\sqrt{-g}Q_{\mu\nu}Q^{\mu\nu},
\label{LagEM}
\end{equation}
where $\alpha$ is a constant.
Consequently, the total action is $S=\int d^4x({\cal L}_g+{\cal L}_Q+{\cal L}_m)$.
Varying $S$ with respect to the metric tensor and applying the principle of least action $\delta S=0$ for arbitrary variations of $g^{\mu\nu}$ yields the Einstein equations:
\begin{equation}
P_{\mu\nu}-\frac{1}{2}Pg_{\mu\nu}=\kappa\tilde{T}_{\mu\nu}=\kappa T_{\mu\nu}+\kappa\alpha\Bigl(\frac{1}{4}Q_{\rho\sigma}Q^{\rho\sigma}g_{\mu\nu}-Q_{\mu\rho}Q_\nu^{\phantom{\nu}\rho}\Bigr),
\label{Ein1}
\end{equation}
where $P=P_{\mu\nu}g^{\mu\nu}$ is the Ricci scalar.

The variation of $S$ with respect to the affine connection is $\delta S=-\frac{1}{2\kappa}\int d^4x({\sf g}^{\mu\nu}\delta P_{\mu\nu}+{\sf h}^{\mu\nu}\delta Q_{\mu\nu}+\Pi_{\phantom{\mu}\rho\phantom{\nu}}^{\mu\phantom{\rho}\nu}\delta\Gamma^{\,\,\rho}_{\mu\,\nu})$, where the antisymmetric tensor density ${\sf h}^{\mu\nu}$ is defined as~\cite{unif}:
\begin{equation}
{\sf h}^{\mu\nu}=-2\kappa\frac{\delta{\cal L}_Q}{\delta Q_{\mu\nu}}.
\label{as1}
\end{equation}
For the Lagrangian density~(\ref{LagEM}), the tensor density ${\sf h}^{\mu\nu}$ is linear in the tensor of homothetic curvature:
\begin{equation}
{\sf h}^{\mu\nu}=\kappa\alpha\sqrt{-g}Q^{\mu\nu}.
\label{as2}
\end{equation}
If we do not restrict the connection $\Gamma^{\,\,\rho}_{\mu\,\nu}$ to be symmetric, unlike in~\cite{Vol}, the variation of the Ricci tensor is given by the Palatini formula~\cite{Hehl,Scho,Schr}:
\begin{equation}
\delta R_{\mu\nu}=\delta\Gamma^{\,\,\rho}_{\mu\,\nu;\rho}-\delta\Gamma^{\,\,\rho}_{\mu\,\rho;\nu}-2S^\sigma_{\phantom{\sigma}\rho\nu}\delta\Gamma^{\,\,\rho}_{\mu\,\sigma}.
\label{Palat}
\end{equation}
Using the identity $\int d^4x({\sf V}^\mu)_{;\mu}=2\int d^4x S_\mu{\sf V}^\mu$~\cite{Schr}, where ${\sf V}^\mu$ is an arbitrary vector density and $S_\mu=S^\nu_{\phantom{\nu}\mu\nu}$ is the torsion vector, and applying the principle of least action $\delta S=0$ for arbitrary variations of $\Gamma^{\,\,\rho}_{\mu\,\nu}$, we obtain
\begin{equation}
{\sf g}^{\mu\nu}_{\phantom{\mu\nu};\rho}-{\sf g}^{\mu\sigma}_{\phantom{\mu\sigma};\sigma}\delta^\nu_\rho-2{\sf g}^{\mu\nu}S_\rho+2{\sf g}^{\mu\sigma}S_\sigma\delta^\nu_\rho+2{\sf g}^{\mu\sigma}S^\nu_{\phantom{\nu}\rho\sigma}=\Pi_{\phantom{\mu}\rho\phantom{\nu}}^{\mu\phantom{\rho}\nu}+2{\sf h}^{\nu\sigma}_{\phantom{\nu\sigma},\sigma}\delta^\mu_\rho.
\label{field1}
\end{equation}
This equation is equivalent to
\begin{equation}
{\sf g}^{\mu\nu}_{\phantom{\mu\nu},\rho}+\,^\ast\Gamma^{\,\,\mu}_{\sigma\,\rho}{\sf g}^{\sigma\nu}+\,^\ast\Gamma^{\,\,\nu}_{\rho\,\sigma}{\sf g}^{\mu\sigma}-\,^\ast\Gamma^{\,\,\sigma}_{\sigma\,\rho}{\sf g}^{\mu\nu}=\Pi_{\phantom{\mu}\rho\phantom{\nu}}^{\mu\phantom{\rho}\nu}-\frac{1}{3}\Pi_{\phantom{\mu}\sigma\phantom{\sigma}}^{\mu\phantom{\sigma}\sigma}\delta^\nu_\rho+2{\sf h}^{\nu\sigma}_{\phantom{\nu\sigma},\sigma}\delta^\mu_\rho-\frac{2}{3}{\sf h}^{\mu\sigma}_{\phantom{\mu\sigma},\sigma}\delta^\nu_\rho,
\label{field2}
\end{equation}
where $^\ast\Gamma^{\,\,\rho}_{\mu\,\nu}=\Gamma^{\,\,\rho}_{\mu\,\nu}+\frac{2}{3}\delta^\rho_\mu S_\nu$ is the projectively invariant connection (Schr\"{o}dinger's star-affinity)~\cite{Schr}.

Contracting the indices $\mu$ and $\rho$ in Eq.~(\ref{field1}) yields
\begin{equation}
{\sf h}^{\sigma\nu}_{\phantom{\sigma\nu},\sigma}={\sf j}^\nu,
\label{Max1}
\end{equation}
where
\begin{equation}
{\sf j}^\nu=\frac{1}{8}\Pi_{\phantom{\sigma}\sigma\phantom{\nu}}^{\sigma\phantom{\sigma}\nu}.
\label{Max2}
\end{equation}
Equation~(\ref{Max1}) has the form of the Maxwell equation for the electromagnetic field.
The density $\Pi_{\phantom{\mu}\rho\phantom{\nu}}^{\mu\phantom{\rho}\nu}$, in addition to $T_{\mu\nu}$, represents the {\em source} for the metric--affine field equations.
Since the tensor density ${\sf h}^{\mu\nu}$ is antisymmetric, the current vector density ${\sf j}^\mu$ must be conserved: ${\sf j}^\mu_{\phantom{\mu},\mu}=0$, which constrains how the connection $\Gamma^{\,\,\rho}_{\mu\,\nu}$ can enter a metric--affine Lagrangian density ${\cal L}$: $\Pi_{\phantom{\sigma}\sigma\phantom{\nu},\nu}^{\sigma\phantom{\sigma}\nu}=0$.
We note that if ${\cal L}$ does not depend on $Q_{\mu\nu}$, the field equation~(\ref{Max2}) becomes a stronger, algebraic constraint on how the Lagrangian depends on the connection: $\Pi_{\phantom{\sigma}\sigma\phantom{\nu}}^{\sigma\phantom{\sigma}\nu}=0$, which restricts forms of matter that can be described by the metric--affine formulation of gravity~\cite{HK,HLS,San}.
The dependence of a metric--affine Lagrangian on the tensor of homothetic curvature $Q_{\mu\nu}$, which will be associated with the electromagnetic field, {\em replaces} this unphysical constraint with a field equation for the tensor density ${\sf h}^{\mu\nu}$ (the Maxwell equations if this dependence is given by Eq.~(\ref{LagEM})).\footnote{
One way to overcome the constraint $\Pi_{\phantom{\sigma}\sigma\phantom{\nu}}^{\sigma\phantom{\sigma}\nu}=0$ is to restrict the torsion tensor to be traceless: $S_\mu=0$~\cite{San}.
This condition enters the Lagrangian density as a Lagrange multiplier term $\sqrt{-g}B^\mu S_\mu$, where the Lagrange multiplier $B^\mu$ is a vector.
Consequently, there is an extra term $-2\kappa\sqrt{-g}B^{[\mu}\delta^{\nu]}_\rho$ on the right-hand side of Eq.~(\ref{field1}) and Eq.~(\ref{Max2}) becomes ${\sf h}^{\sigma\nu}_{\phantom{\sigma\nu},\sigma}={\sf j}^\nu+\frac{3\kappa\sqrt{-g}}{8}B^\nu$.
Setting $B^\nu=-\frac{8}{3\kappa\sqrt{-g}}{\sf j}^\nu$ removes this constraint if ${\cal L}$ does not depend on $Q_{\mu\nu}$, or yields the wave equation ${\sf h}^{\sigma\nu}_{\phantom{\sigma\nu},\sigma}=0$ if ${\cal L}$ depends on $Q_{\mu\nu}$.
In both cases, the vector density ${\sf j}^\mu$ does not need to be conserved.
Therefore introducing the dependence of ${\cal L}$ on $Q_{\mu\nu}$ is more suitable than imposing $S_\mu=0$ if we want to unify gravitation and electromagnetism in the metric--affine formalism.
Another way to overcome the constraint $\Pi_{\phantom{\sigma}\sigma\phantom{\nu}}^{\sigma\phantom{\sigma}\nu}=0$ is to restrict the Weyl vector to vanish: $W_\mu=0$~\cite{HLS}.
\label{foot}
}

The tensor $P_{\mu\nu}$ is invariant under an infinitesimal projective transformation~(\ref{proj}): $\delta\Gamma^{\,\,\rho}_{\mu\,\nu}=\delta^\rho_\mu\delta V_\nu$.\footnote{
Equation~(\ref{Palat}) yields $\delta P_{\mu\nu}=0$ for $\delta\Gamma^{\,\,\rho}_{\mu\,\nu}=\delta^\rho_\mu\delta V_\nu$.
}
Under the same transformation, the tensor $Q_{\mu\nu}$ changes according to $Q_{\mu\nu}\rightarrow Q_{\mu\nu}+4(\delta V_{\nu,\mu}-\delta V_{\mu,\nu})$.
Consequently, the action changes according to $\delta S=-\frac{1}{2\kappa}\int d^4x(\Pi_{\phantom{\mu}\rho\phantom{\nu}}^{\mu\phantom{\rho}\nu}\delta\Gamma^{\,\,\rho}_{\mu\,\nu}+{\sf h}^{\mu\nu}\delta Q_{\mu\nu})=-\frac{1}{2\kappa}\int d^4x(\Pi_{\phantom{\sigma}\sigma\phantom{\mu}}^{\sigma\phantom{\sigma}\mu}+8{\sf h}^{\mu\nu}_{\phantom{\mu\nu},\nu})\delta V_\mu$.
This expression is identically zero due to the field equation~(\ref{Max1}) so, although ${\cal L}_m$ and ${\cal L}_Q$ are not projectively invariant, the total action is.
The formal similarity between the tensor of homothetic curvature $Q_{\mu\nu}$ and the electromagnetic field tensor $F_{\mu\nu}=A_{\nu,\mu}-A_{\mu,\nu}$ (both tensors are curls) suggests that they are proportional to one another~\cite{Goe,FK2}.
If $\hbar=1$ then the constant of proportionality has the dimension of electric charge~\cite{FK2} and, without loss of generality, can be taken equal to the charge of the electron $e$:
\begin{equation}
F_{\mu\nu}=eQ_{\mu\nu}.
\label{prop}
\end{equation}
Accordingly, the Weyl vector is proportional to the electromagnetic four-potential $A_\mu$:
\begin{equation}
W_\mu=\frac{1}{2e}A_\mu.
\label{prop2}
\end{equation}

The degrees of freedom contained in the affine connection provide an elegant way of unifying the classical gravitational and electromagnetic fields in the metric--affine formulation of gravity.
In addition, the Maxwell equations constrain the desired four degrees of freedom so that the metric--affine formulation of gravity is consistent from a physical point of view~\cite{HLS}.
Finally, we can interpret the electromagnetic field in a metric--affine gravity as a field whose role is to {\em restore} projective invariance of a metric--affine Lagrangian broken by matter terms that depend explicitly on the affine connection, without constraining {\em ad hoc} the connection~\cite{unif}.

\section{Solution of field equations}
\label{secSol}

Substituting Eq.~(\ref{Max2}) to~(\ref{field2}) and symmetrizing the indices $\mu$ and $\nu$ yield
\begin{equation}
{\sf g}^{\mu\nu}_{\phantom{\mu\nu},\rho}+\,^\ast\Gamma^{\,\,\,\,\mu}_{(\sigma\,\rho)}{\sf g}^{\sigma\nu}+\,^\ast\Gamma^{\,\,\,\,\nu}_{(\rho\,\sigma)}{\sf g}^{\mu\sigma}-\,^\ast\Gamma^{\,\,\,\,\sigma}_{(\sigma\,\rho)}{\sf g}^{\mu\nu}=\Sigma_{\phantom{\mu}\rho\phantom{\nu}}^{\mu\phantom{\rho}\nu},
\label{field3}
\end{equation}
where
\begin{equation}
\Sigma_{\phantom{\mu}\rho\phantom{\nu}}^{\mu\phantom{\rho}\nu}=\Pi_{\phantom{(\mu}\rho\phantom{\nu)}}^{(\mu\phantom{\rho}\nu)}-\frac{1}{3}\delta^{(\mu}_\rho\Pi_{\phantom{\nu)}\sigma\phantom{\sigma}}^{\nu)\phantom{\sigma}\sigma}-\frac{1}{6}\Pi_{\phantom{\sigma}\sigma\phantom{(\mu}}^{\sigma\phantom{\sigma}(\mu}\delta^{\nu)}_\rho.
\label{sigma}
\end{equation}
Equation~(\ref{field3}) is algebraic and linear in $^\ast\Gamma^{\,\,\,\,\rho}_{(\mu\,\nu)}$ as a function of the metric tensor, its first derivatives and the density $\Pi_{\phantom{\mu}\rho\phantom{\nu}}^{\mu\phantom{\rho}\nu}$.
We decompose the connection $^\ast\Gamma^{\,\,\rho}_{\mu\,\nu}$ as~\cite{unif}:
\begin{equation}
^\ast\Gamma^{\,\,\rho}_{\mu\,\nu}=\{^{\,\,\rho}_{\mu\,\nu}\}_g+V^\rho_{\phantom{\rho}\mu\nu},
\label{sol1}
\end{equation}
where $V^\rho_{\phantom{\rho}\mu\nu}=S^\rho_{\phantom{\rho}\mu\nu}+2S_{(\mu\nu)}^{\phantom{(\mu\nu)}\rho}+\frac{1}{2}N_{\mu\nu}^{\phantom{\mu\nu}\rho}-N^\rho_{\phantom{\rho}(\mu\nu)}+\frac{2}{3}\delta^\rho_\mu S_\nu$ is a projectively invariant deflection tensor.
Consequently, the Ricci tensor $R_{\mu\nu}$ associated with the affine connection $\Gamma^{\,\,\rho}_{\mu\,\nu}$ is given by~\cite{Scho}:
\begin{equation}
R_{\mu\nu}=R_{\mu\nu}^{(g)}-\frac{2}{3}(S_{\nu:\mu}-S_{\mu:\nu})+V^\rho_{\phantom{\rho}\mu\nu:\rho}-V^\rho_{\phantom{\rho}\mu\rho:\nu}+V^\sigma_{\phantom{\sigma}\mu\nu}V^\rho_{\phantom{\rho}\sigma\rho}-V^\sigma_{\phantom{\sigma}\mu\rho}V^\rho_{\phantom{\rho}\sigma\nu},
\label{sol2}
\end{equation}
where $R_{\mu\nu}^{(g)}$ is the Riemannian Ricci tensor constructed from the metric tensor $g_{\mu\nu}$ and the colon denotes the covariant differentiation with respect to $\{^{\,\,\rho}_{\mu\,\nu}\}_g$.
Equation~(\ref{Ein1}) and symmetrized Eq.~(\ref{sol2}) give the Einstein equations in the purely metric formulation:
\begin{equation}
R_{\mu\nu}^{(g)}-\frac{1}{2}R^{(g)}g_{\mu\nu}=\kappa(\tilde{T}_{\mu\nu}+\Theta_{\mu\nu}),
\label{Ein2}
\end{equation}
where $\Theta_{\mu\nu}$ is the effective energy-momentum tensor generated by the affine connection:
\begin{equation}
\Theta_{\mu\nu}=\frac{1}{\kappa}(-V^\rho_{\phantom{\rho}(\mu\nu):\rho}+V^\rho_{\phantom{\rho}(\mu|\rho|:\nu)}-V^\sigma_{\phantom{\sigma}(\mu\nu)}V^\rho_{\phantom{\rho}\sigma\rho}+V^\sigma_{\phantom{\sigma}(\mu|\rho}V^\rho_{\phantom{\rho}\sigma|\nu)})-\frac{1}{2\kappa}g_{\mu\nu}(2V^{\rho\sigma}_{\phantom{\rho\sigma}[\rho:\sigma]}-V^{\sigma\tau}_{\phantom{\sigma\tau}\tau}V^\rho_{\phantom{\rho}\sigma\rho}+V^{\sigma\tau}_{\phantom{\sigma\tau}\rho}V^\rho_{\phantom{\rho}\sigma\tau}).
\label{Ein3}
\end{equation}
The Bianchi identity $(R^{\mu\nu(g)}-\frac{1}{2}R^{(g)}g^{\mu\nu})_{:\nu}=0$ yields the covariant conservation of the total energy--momentum tensor: $(\tilde{T}^{\mu\nu}+\Theta^{\mu\nu})_{:\nu}=0$.

Substituting Eq.~(\ref{sol1}) to~(\ref{field3}) gives
\begin{equation}
V^\mu_{\phantom{\mu}(\sigma\rho)}{\sf g}^{\sigma\nu}+V^\nu_{\phantom{\nu}(\rho\sigma)}{\sf g}^{\mu\sigma}-V^\sigma_{\phantom{\sigma}(\sigma\rho)}{\sf g}^{\mu\nu}=\Sigma_{\phantom{\mu}\rho\phantom{\nu}}^{\mu\phantom{\rho}\nu}.
\label{sol3}
\end{equation}
Its solution is:
\begin{equation}
V^\rho_{\phantom{\rho}(\mu\nu)}=\frac{1}{2\sqrt{-g}}(\Delta_{\phantom{\rho}\nu\phantom{\sigma}}^{\rho\phantom{\nu}\sigma}g_{\mu\sigma}+\Delta_{\phantom{\rho}\mu\phantom{\sigma}}^{\rho\phantom{\mu}\sigma}g_{\nu\sigma}-\Delta_{\phantom{\alpha}\gamma\phantom{\beta}}^{\alpha\phantom{\gamma}\beta}g_{\mu\alpha}g_{\nu\beta}g^{\rho\gamma}),
\label{sol4}
\end{equation}
where
\begin{equation}
\Delta_{\phantom{\mu}\rho\phantom{\nu}}^{\mu\phantom{\rho}\nu}=\Sigma_{\phantom{\mu}\rho\phantom{\nu}}^{\mu\phantom{\rho}\nu}-\frac{1}{2}\Sigma_{\phantom{\alpha}\rho\phantom{\beta}}^{\alpha\phantom{\rho}\beta}g_{\alpha\beta}g^{\mu\nu}.
\label{delta}
\end{equation}
Substituting Eq.~(\ref{Max2}) to~(\ref{field2}) and antisymmetrizing the indices $\mu$ and $\nu$ yield
\begin{equation}
V^\mu_{\phantom{\mu}[\sigma\rho]}{\sf g}^{\nu\sigma}-V^\nu_{\phantom{\nu}[\sigma\rho]}{\sf g}^{\mu\sigma}=\Omega_\rho^{\phantom{\rho}\mu\nu},
\label{field4}
\end{equation}
where
\begin{equation}
\Omega_\rho^{\phantom{\rho}\mu\nu}=\Pi_{\phantom{[\mu}\rho\phantom{\nu]}}^{[\mu\phantom{\rho}\nu]}-\frac{1}{3}\Pi_{\phantom{[\sigma}\sigma\phantom{\nu]}}^{[\sigma\phantom{\sigma}\nu]}\delta^\mu_\rho+\frac{1}{3}\Pi_{\phantom{[\sigma}\sigma\phantom{\mu]}}^{[\sigma\phantom{\sigma}\mu]}\delta^\nu_\rho
\label{omega}
\end{equation}
is a traceless tensor density.
Consequently, we find
\begin{equation}
V^\rho_{\phantom{\rho}[\mu\nu]}=\frac{1}{2\sqrt{-g}}(\Omega_\nu^{\phantom{\nu}\rho\sigma}g_{\mu\sigma}-\Omega_\mu^{\phantom{\mu}\rho\sigma}g_{\nu\sigma}-\Omega_\gamma^{\phantom{\gamma}\alpha\beta}g_{\mu\alpha}g_{\nu\beta}g^{\rho\gamma}).
\label{sol5}
\end{equation}
Equations~(\ref{sol4}) and~(\ref{sol5}) give the tensor $V^\rho_{\phantom{\rho}\mu\nu}$ which is linear in $\Pi_{\phantom{\mu}\rho\phantom{\nu}}^{\mu\phantom{\rho}\nu}$~\cite{unif}.
A general solution for the affine connection in the presence of matter fields that depend on the connection is also given in~\cite{PO}.

Equation~(\ref{sol1}) gives the following expression for the tensor of homothetic curvature~\cite{Scho}:
\begin{equation}
Q_{\mu\nu}=-\frac{8}{3}(S_{\nu,\mu}-S_{\mu,\nu})+V^\rho_{\phantom{\rho}\rho\nu,\mu}-V^\rho_{\phantom{\rho}\rho\mu,\nu}.
\label{q1}
\end{equation}
Accordingly, the Weyl vector is:
\begin{equation}
W_\mu=-\frac{4}{3}S_\mu+\frac{1}{2}V^\rho_{\phantom{\rho}\rho\mu}.
\label{Wey}
\end{equation}
The torsion vector is related to the electromagnetic potential $A_\mu$ via Eq.~(\ref{q1}) and the correspondence relation~(\ref{prop}):\footnote{
The torsion vector in~\cite{Ham} is proportional to the electromagnetic potential.
}
\begin{equation}
S_\nu=\frac{3}{8}\Bigl(-\frac{A_\nu}{e}+V^\rho_{\phantom{\rho}\rho\nu}\Bigr).
\label{tor}
\end{equation}
Gauge transformations $A_\nu\rightarrow A_\nu+\lambda_{,\nu}$ (equivalent to special projective transformations) do not affect Eq.~(\ref{q1}) and the total action.
Equations~(\ref{as2}) and~(\ref{Max1}) reproduce the Maxwell equation with source, $\frac{1}{\sqrt{-g}}(\sqrt{-g}F_{\alpha\beta}g^{\nu\alpha}g^{\mu\beta})_{,\nu}=j^\mu$, where $j^\mu$ is the electromagnetic current four-vector~\cite{LL2}, if we identify:
\begin{equation}
{\sf j}^\mu=\frac{\kappa\alpha}{e}\sqrt{-g}j^\mu.
\label{iden}
\end{equation}

If there are no sources, $\Pi_{\phantom{\mu}\rho\phantom{\nu}}^{\mu\phantom{\rho}\nu}=0$, the connection $\Gamma^{\,\,\rho}_{\mu\,\nu}$ depends only on the metric tensor $g_{\mu\nu}$ representing the gravitational field and the torsion vector $S_\mu$ proportional to the electromagnetic potential~\cite{FK2,Ham} and thus to the Weyl vector.
If we impose the Lorentz condition $W^\mu_{\phantom{\mu}:\mu}=0$ (by applying a suitable special projective transformation), Eqs.~(\ref{as2}),~(\ref{Max1}) and~(\ref{q1}) yield the wave equation for the massless Weyl vector:
\begin{equation}
\triangle_g W^\mu-R^\mu_{\phantom{\mu}\nu(g)}W^\nu=0,
\label{q2}
\end{equation}
where $\triangle_g$ denotes the covariant d'Alembert operator with respect to $\{^{\,\,\rho}_{\mu\,\nu}\}_g$.

Lastly, we consider the case when the source density~(\ref{Pi}) is antisymmetric (which we study in the next section):
\begin{equation}
\Pi_{\phantom{\mu}\rho\phantom{\nu}}^{\mu\phantom{\rho}\nu}=\Pi_{\phantom{[\mu}\rho\phantom{\nu]}}^{[\mu\phantom{\rho}\nu]}.
\label{Pi2}
\end{equation}
Equations~(\ref{sigma}),~(\ref{sol4}) and~(\ref{delta}) give $V^\rho_{\phantom{\rho}(\rho\nu)}=-\frac{2}{3\sqrt{-g}}g_{\mu\nu}{\sf j}^\mu$, while Eqs.~(\ref{omega}) and~(\ref{sol5}) give $V^\rho_{\phantom{\rho}[\rho\nu]}=0$.
Therefore we find
\begin{equation}
V^\rho_{\phantom{\rho}\rho\nu}=-\frac{2\kappa\alpha}{3e}j_\nu,
\label{sol6}
\end{equation}
and the tensor of homothetic curvature is:
\begin{equation}
Q_{\mu\nu}=-\frac{8}{3}(S_{\nu,\mu}-S_{\mu,\nu})-\frac{2\kappa\alpha}{3e}(j_{\nu,\mu}-j_{\mu,\nu}).
\label{q3}
\end{equation}
The torsion vector turns out to be, due to Eq.~(\ref{tor}), a linear combination of the electromagnetic potential and the electromagnetic current:
\begin{equation}
S_\nu=-\frac{3}{8e}A_\nu-\frac{\kappa\alpha}{4e}j_\nu.
\label{q4}
\end{equation}
The Weyl vector, proportional to the electromagnetic potential due to Eq.~(\ref{prop2}), satisfies now
\begin{equation}
\triangle_g W^\mu-R^\mu_{\phantom{\mu}\nu(g)}W^\nu=\frac{j^\mu}{2e},
\label{q5}
\end{equation}
playing the role of the {\em massless} vector (photon) generated by the tensor of homothetic curvature in a metric--affine Lagrangian in the presence of electromagnetic sources.

\section{Generation of mass from algebraic constraints on torsion}
\label{secMas}

Combining Eqs.~(\ref{as2}),~(\ref{Max1}),~(\ref{iden}) and~(\ref{q3}) gives
\begin{equation}
-\Bigl(\frac{8e}{3}(S^{\nu,\mu}-S^{\mu,\nu})+\frac{2\kappa\alpha}{3}(j^{\nu,\mu}-j^{\mu,\nu})\Bigr)_{:\mu}=j^\nu.
\label{Max3}
\end{equation}
This equation formally looks like a dynamical equation for the vector field $j^\mu$.
However, the vector $j^\mu$ is not a dynamical field but constitutes the source for the dynamical field $W_\mu$ (or $A_\mu$) since Eq.~(\ref{Max3}) reads
\begin{equation}
2e(W^{\nu,\mu}-W^{\mu,\nu})_{:\mu}=j^\nu,
\label{Max4}
\end{equation}
or Eq.~(\ref{q5}).
We can ask whether it is possible to turn the vector $j^\mu$ into a dynamical field and thus modify Eq.~(\ref{Max4}).
The answer is yes and the mechanism to dynamize the source $j^\mu$ is related to constraining the torsion tensor.
To constrain the torsion we use the Lagrange-multiplier method of Hehl and Kerlick~\cite{HK} that allows to study arbitrary constraints on the affine connection.

\subsection{Torsionless connection}
\label{TC}

Let us consider the strongest constraint on the torsion tensor:
\begin{equation}
S^\rho_{\phantom{\rho}\mu\nu}=0.
\label{cond1}
\end{equation}
This condition enters the Lagrangian density as the Lagrange-multiplier term, $\sqrt{-g}\beta_\rho^{\phantom{\rho}\mu\nu}S^\rho_{\phantom{\rho}\mu\nu}$, where the Lagrange multiplier $\beta_\rho^{\phantom{\rho}\mu\nu}$ is a tensor field antisymmetric in the indices $\mu$ and $\nu$:
\begin{equation}
{\cal L}=-\frac{1}{2\kappa}P_{\mu\nu}{\sf g}^{\mu\nu}-\frac{\alpha}{4}\sqrt{-g}Q_{\mu\nu}Q^{\mu\nu}+\sqrt{-g}\beta_\rho^{\phantom{\rho}\mu\nu}S^\rho_{\phantom{\rho}\mu\nu}+{\cal L}_m.
\label{mas1}
\end{equation}
Varying the corresponding action with respect to the field $\beta_\rho^{\phantom{\rho}\mu\nu}$ gives Eq.~(\ref{cond1}) and thus $S_\mu=0$.
Let us also assume, for simplicity, the absence of matter fields, ${\cal L}_m=0$.
To solve the field equation related to the variation of the connection, we treat the Lagrange-multiplier term as the effective Lagrangian density for matter.\footnote{
In the following text, $j^\nu$ denotes not the electromagnetic current (which is zero in the absence of matter fields) but the effective current four-vector associated with a Lagrange-multiplier constraint on the connection.
}
Consequently, we find
\begin{equation}
\Pi_{\phantom{\mu}\rho\phantom{\nu}}^{\mu\phantom{\rho}\nu}=-2\kappa\sqrt{-g}\beta_\rho^{\phantom{\rho}\mu\nu}
\label{mas2}
\end{equation}
and, because of Eq.~(\ref{iden}),
\begin{equation}
j^\nu=-\frac{2e}{\alpha}\beta_\rho^{\phantom{\rho}\rho\nu}.
\label{mas3}
\end{equation}
Accordingly, the Weyl vector is given by Eqs.~(\ref{Wey}) and~(\ref{sol6}):
\begin{equation}
W_\nu=-\frac{\kappa\alpha}{3e}j_\nu.
\label{sol7}
\end{equation}

The source density~(\ref{mas2}) is antisymmetric so we can apply the results of the preceding section and Eq.~({\ref{Max4}), obtaining
\begin{equation}
(W^{\nu,\mu}-W^{\mu,\nu})_{:\mu}=-\frac{3}{2\kappa\alpha}W^\nu.
\label{Max5}
\end{equation}
The Lorentz condition $W^\mu_{\phantom{\mu}:\mu}=0$ is satisfied automatically, yielding the Einstein--Proca equation for the {\em massive} Weyl vector:\footnote{
The massive Weyl vector loses the meaning of the quantity representing the electromagnetic potential.
}
\begin{equation}
\triangle_g W^\mu-R^\mu_{\phantom{\mu}\nu(g)}W^\nu+m_W^2W^\mu=0,
\label{Proc}
\end{equation}
with mass
\begin{equation}
m_W=\sqrt{\frac{3}{2\kappa\alpha}}.
\label{mas}
\end{equation}
The mass of the vector $W^\mu$ is determined by the coupling constant $\alpha$.
The same result was observed by Vollick who used the symmetric connection in a variational principle~\cite{Vol}.

\subsection{Traceless torsion}
\label{TT}

We now consider a weaker constraint on the torsion tensor, imposing its trace to vanish:
\begin{equation}
S_\mu=0.
\label{cond2}
\end{equation}
This condition enters the Lagrangian density as the Lagrange-multiplier term: $\sqrt{-g}B^\mu S_\mu$ (cf. footnote \ref{foot}), so the total Lagrangian density is:
\begin{equation}
{\cal L}=-\frac{1}{2\kappa}P_{\mu\nu}{\sf g}^{\mu\nu}-\frac{\alpha}{4}\sqrt{-g}Q_{\mu\nu}Q^{\mu\nu}+\sqrt{-g}B^\mu S_\mu,
\label{mas4}
\end{equation}
since varying the corresponding action with respect to the field $B^\mu$ gives Eq.~(\ref{cond2}).
We again treat the Lagrange-multiplier term as the matter part of the Lagrangian, obtaining
\begin{equation}
\Pi_{\phantom{\mu}\rho\phantom{\nu}}^{\mu\phantom{\rho}\nu}=-2\kappa\sqrt{-g}B^{[\mu}\delta^{\nu]}_\rho
\label{mas5}
\end{equation}
and
\begin{equation}
j^\nu=\frac{3e}{8\alpha}B^\nu.
\label{mas6}
\end{equation}
The source density~(\ref{mas5}) is again antisymmetric so we can apply the results of the preceding section and Eqs.~(\ref{Max4}) and~({\ref{sol7}), obtaining the Einstein--Proca equations~(\ref{Max5}) and~({\ref{Proc}) for the {\em massive} Weyl vector of mass~(\ref{mas}).

We obtain the same results if the torsion vector is a gradient of a scalar:
\begin{equation}
S_\mu=\lambda_{,\mu},
\label{cond4}
\end{equation}
which is weaker than the constraint~(\ref{cond2}).
Therefore it is the constraint $S_\mu=\lambda_{,\mu}$, or equivalently $S_{\nu,\mu}-S_{\mu,\nu}=0$, that gives mass to the Weyl vector in our Lagrangian.
From Eq.~(\ref{q1}) it follows that this condition causes the tensor of homothetic curvature to depend only on the projectively invariant tensor $V^\rho_{\phantom{\rho}\mu\nu}$, i.e. {\em imposing projective invariance on the tensor of homothetic curvature gives mass to the vector degree of freedom} of our Lagrangian.

\subsection{Vector torsion}
\label{VT}

We now consider a different constraint on the torsion tensor:
\begin{equation}
S^\rho_{\phantom{\rho}\mu\nu}=\frac{2}{3}S_{[\mu}\delta_{\nu]}^\rho,
\label{cond3}
\end{equation}
i.e. the torsion tensor depends completely on the torsion vector.\footnote{
Contracting the indices $\mu$ and $\rho$ yields the identity.
}
This condition enters the Lagrangian density as the Lagrange-multiplier term: $\sqrt{-g}\gamma_\rho^{\phantom{\rho}\mu\nu}(S^\rho_{\phantom{\rho}\mu\nu}-\frac{2}{3}S_{[\mu}\delta_{\nu]}^\rho)$, where the Lagrange multiplier $\gamma_\rho^{\phantom{\rho}\mu\nu}$ is a tensor field antisymmetric in the indices $\mu$ and $\nu$:
\begin{equation}
{\cal L}=-\frac{1}{2\kappa}P_{\mu\nu}{\sf g}^{\mu\nu}-\frac{\alpha}{4}\sqrt{-g}Q_{\mu\nu}Q^{\mu\nu}+\sqrt{-g}\gamma_\rho^{\phantom{\rho}\mu\nu}\Bigl(S^\rho_{\phantom{\rho}\mu\nu}-\frac{2}{3}S_{[\mu}\delta_{\nu]}^\rho\Bigr).
\label{mas7}
\end{equation}
Varying the corresponding action with respect to the field $\gamma_\rho^{\phantom{\rho}\mu\nu}$ reproduces Eq.~(\ref{cond3}).
We again treat the Lagrange-multiplier term as the matter part of the Lagrangian, obtaining
\begin{equation}
\Pi_{\phantom{\mu}\rho\phantom{\nu}}^{\mu\phantom{\rho}\nu}=-2\kappa\sqrt{-g}\Bigl(\gamma_\rho^{\phantom{\rho}\mu\nu}+\frac{2}{3}\gamma_\sigma^{\phantom{\sigma}\sigma[\mu}\delta^{\nu]}_\rho\Bigr).
\label{mas8}
\end{equation}
The effective current vanishes identically:
\begin{equation}
j^\nu=0.
\label{mas9}
\end{equation}
Accordingly, the Weyl vector is given by Eqs.~(\ref{Wey}) and~(\ref{sol6}):
\begin{equation}
W_\nu=-\frac{4}{3}S_\nu,
\label{sol8}
\end{equation}
and is {\em massless}, satisfying
\begin{equation}
\triangle_g W^\mu-R^\mu_{\phantom{\mu}\nu(g)}W^\nu=0.
\label{q6}
\end{equation}

The condition~(\ref{cond3}) is the only algebraic constraint on the connection that is projectively invariant and contains only the torsion tensor.
Therefore we can state that {\em constraints that break projective invariance generate mass of the vector} associated with the dynamical connection (the Weyl vector), while {\em projectively invariant algebraic constraints on the connection do not generate mass} of this vector.
From Eq.~(\ref{q1}) it follows that this condition causes the tensor of homothetic curvature to depend on the torsion vector which is not projectively invariant.
Since we do not impose projective invariance on the tensor of homothetic curvature, the vector degree of freedom remains massless (cf. Subsec.~\ref{TT}).

\subsection{Nonlinear constraints on torsion}
\label{ST}

We examined three simplest (linear) constraints on the the torsion.
Let us consider the simplest quadratic constraint:
\begin{equation}
S^\mu S_\mu=0,
\label{cond5}
\end{equation}
introducing it through the Lagrange-multiplier term, $\sqrt{-g}C S^\mu S_\mu$.
Varying the corresponding action with respect to the field $C$ reproduces Eq.~(\ref{cond5}).
Following the previous subsections, we find
\begin{equation}
\Pi_{\phantom{\mu}\rho\phantom{\nu}}^{\mu\phantom{\rho}\nu}=-4\kappa\sqrt{-g}CS^{[\mu}\delta^{\nu]}_\rho
\label{qu1}
\end{equation}
and
\begin{equation}
j^\nu=\frac{3e}{4\alpha}CS^\nu.
\label{qu2}
\end{equation}
Accordingly, the Weyl vector is given by
\begin{equation}
W_\nu=-\Bigl(\frac{4}{3}+\frac{\kappa C}{4}\Bigr)S_\nu
\label{qu3}
\end{equation}
and satisfies
\begin{equation}
W^\mu W_\mu=0.
\label{nul}
\end{equation}
Equation~(\ref{q5}) reads
\begin{equation}
\triangle_g W^\mu-R^\mu_{\phantom{\mu}\nu(g)}W^\nu=-\frac{9C}{32\alpha(1+\frac{3\kappa C}{16})}W^\mu.
\label{q7}
\end{equation}

Equation~(\ref{q7}) has the form of the Einstein--Proca equation~(\ref{Proc}) with an effective mass
\begin{equation}
m_W=\sqrt{\frac{9C}{32\alpha(1+\frac{3\kappa C}{16})}},
\label{run}
\end{equation}
which depends on the scalar field $C$.
The field $C$ replaces the degree of freedom that is constrained by Eq.~(\ref{nul}).
The effective mass of the Weyl vector ranges from $m_W=0$ for $C=0$ to $m_W=\sqrt{\frac{3}{2\kappa\alpha}}$, as in Eq.~(\ref{mas}), for $C\rightarrow\infty$.
The dependence of the effective mass on the scalar degree of freedom is a consequence of the non-zero torsion vector $S_\mu$.
Nonlinear constraints, unlike those in Eqs.~(\ref{cond1}) and~(\ref{cond2}), do not impose vanishing of this vector so the Weyl vector is a linear combination of $S_\mu$ and the effective current.
If a nonlinear constraint on the torsion has the form of a scalar, the corresponding Lagrange multiplier is a scalar density and the resulting degree of freedom that appears in place of this constraint is a scalar which enters the effective current and effective mass.
Since the field equations arising from nonlinear constraints on the torsion are not simple Einstein--Maxwell or Einstein--Proca equations, we conclude that these constraints do not correspond to physical conditions (vector particles with constant masses).
Similar conclusions can be made for nonlinear constraints on the connection that have the form of tensors and for constraints that contain covariant derivatives.

\section{Discussion and summary}
\label{secSum}

The degrees of freedom contained in the affine connection not only provide an elegant way of unifying the classical gravitational and electromagnetic fields within the metric--affine formulation of gravity, but also can incorporate massive vectors.
The simplest metric--affine Lagrangian that depends on the tensor of homothetic curvature generates the Einstein--Maxwell equations for the electromagnetic potential corresponding to the massless photon.
In the previous section we examined three linear constraints on the antisymmetric part of the affine connection (the torsion tensor).
We found that constraints that break projective invariance generate the Einstein--Proca equations for a massive vector which appears in place of the photon.
Therefore projective-invariance breaking generates masses of vectors in metric--affine gravity that incorporates the tensor of homothetic curvature.
We also examined one simple quadratic constraint on the torsion and concluded that nonlinear constraints generate the Einstein--Proca equations with running masses that do not correspond to physical vector particles.

Although we only considered algebraic constraints on the torsion, the Lagrange-multiplier formalism allows to study arbitrary constraints on the affine connection.
We can, e.g., require the connection to be metric-compatible~\cite{HK}: $g_{\mu\nu;\rho}=0$, which implies $\Gamma^{\,\,\rho}_{\rho\,\nu}=\{^{\,\,\rho}_{\rho\,\nu}\}_g$ and thus $Q_{\mu\nu}=0$.
Therefore, under this condition, the electromagnetic field loses its association with the affine connection since imposing the metric-compatibility on the connection means that we are dealing with the purely metric formulation of gravity rather than metric--affine.
We can also introduce differential constraints on the torsion, such as $S_{\mu;\nu}=0$.
These constraints, however, impose {\em ad hoc} differential equations on the connection which should result from a variational principle with respect to a geometrical quantity rather than from Lagrange multipliers.
Therefore the only physical constraints on the connection in the metric--affine formulation of gravity should have the form of linear algebraic equations for the torsion tensor.

The presented paper is a mathematical exercise that can give insights into possible geometrical integration of the bosonic part of the Glashow--Weinberg--Salam electroweak model~\cite{Wei,EW} with gravitation.
In the electroweak model, masses of gauge bosons arise naturally from spontaneous symmetry breaking (the Higgs mechanism) of the $SU(2)\times U(1)$ group~\cite{Wei,Hig}.
Gauge-boson masses have also been shown to arise when gauge invariance is spontaneously broken in higher-dimensional theories of gravity~\cite{GEW}.
In this paper we showed that projective-symmetry breaking can generate massive gauge bosons in four dimensions.
To include four gauge bosons instead of one in the metric--affine formulation of gravity (and generate masses for three of them) we should introduce the covariant derivative acting on the electroweak spinor doublet (the fermionic part)~\cite{Wei}.
Since the metric tensor and connection cannot describe spinors in curved spacetime, we should use the Einstein--Cartan--Kibble--Sciama formulation of gravity with the tetrad and spin connection as dynamical variables~\cite{KS} and impose constraints on the Fock--Ivanenko coefficients that appear in the covariant derivative of a spinor~\cite{Lord}.

redpajama_set_name: RedPajamaArXiv


text:
›HEALTH
›Johnson & Johnson's COVID-19 vaccine trial paused over unexplained illness
Johnson & Johnson's COVID-19 vaccine trial paused over unexplained illness
A late-stage study of Johnson & Johnson's COVID-19 vaccine candidate has been paused while the company investigates whether a study participant's "unexplained illness" is related to the shot.
The company said in a statement Monday evening that illnesses, accidents and other so-called adverse events "are an expected part of any clinical study, especially large studies," but that its physicians and a safety monitoring panel would try to determine what might have caused the illness.
The pause is at least the second such hold to occur among several vaccines that have reached large-scale final tests in the U.S.
The company declined to reveal any more details about the illness, citing the participant's privacy.
Temporary stoppages of large medical studies are relatively common. Few are made public in typical drug trials, but the work to make a coronavirus vaccine has raised the stakes on these kinds of complications.
so-called
study...

redpajama_set_name: RedPajamaCommonCrawl


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text:
FInal Homes and Models Released for Sale at Verana in Coastal North San Diego County
Tuesday, April 26, 2016 / Pardee Homes
SAN DIEGO (April 26, 2016) Pardee Homes is pleased to announce that the model homes and final phase release at Verana are now available for preview and purchase. Showcasing Pardee's exclusive GenSmart Suite™, Verana, located just east of Carmel Valley in Pacific Highlands Ranch, is one of the company's most successful new home neighborhoods to be built in the past year in this coveted coastal-close location.
"This unique opportunity to purchase an upgraded, designer appointed model home could be a dream come true for three motivated buyers," said Liz Ramirez, regional director of marketing for Pardee Homes. "Spacious and highly upgraded, the model homes are filled with fabulous designer touches and two of the homes include Pardee's exclusive GenSmart Suite™ offering a separate and private space for living, sleeping and eating that easily accommodates family and friends for short visits or extended living arrangements. You'll also find that the models have fabulous extended patio covers and outdoor rooms, upgraded flooring and kitchen appliances, and exquisite landscaping.
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capture of fugitive Iraqi dictator Saddam Hussein, and interrogated him to boot.
Rep. Mark Green, elected in November to represent Tennessee's 7th Congressional District, is an Army special operations veteran as well as a physician and businessman.
One of his most memorable missions in the military, Green recalls in an interview with The Daily Signal, was helping to capture Saddam on Dec. 13, 2003.
Green had graduated in 1999 from Boonshoft School of Medicine at Wright State University in Dayton, Ohio, did his residency in emergency medicine at Fort Hood, Texas, and was subsequently chosen to serve as a flight surgeon for the 160th Special Operations Aviation Regiment.
Also called the Night Stalkers, the Army regiment provides helicopter support for both general purpose and special operations forces.
Green previously commanded an infantry company in the 82nd Airborne Division, based at Fort Bragg, North Carolina.
As it turned out, the U.S. military found Saddam hiding in a 6- to 8-foot-deep "spider hole," 9 miles outside Tikrit, his hometown in north central Iraq.
Anytime you climb into a special operations helicopter with a bunch of tier one special operators, it's a rush to say the least, right? And then to be in the helicopter listening to them describe how they're going over the target, and they have a special way to do that and code words that they use and things like this.
And the task force commander, who I believe was Admiral [William] McRaven, came on the line and said, 'Do you have Big Jackpot or do you have Little Jackpot?' Because supposedly he had a cook on that target as well.
The Pats won the Super Bowl this year, right? Somebody's going to win it next year. But we're not capturing another totalitarian dictator next year, I don't think.
His superiors asked Green to spend six hours in a detention facility with Saddam the night of his capture.
It was amazing. He of course was very charming, which is what I would anticipate [of] anyone who has a cult of personality, which he did. Of course, he was a murderous dictator, too. But I mean, obviously, you don't rise in the ranks to become Saddam Hussein without some ability to have a cult of personality.
The post This Freshman Lawmaker Interrogated Saddam as Part of Iraqi Dictator's Capture appeared first on The Daily Signal.


text:
class RH_NRF24 : public RHNRFSPIDriver
{
public:

    /// \brief Defines convenient values for setting data rates in setRF()
    typedef enum
    {
	DataRate1Mbps = 0,   ///< 1 Mbps
	DataRate2Mbps,       ///< 2 Mbps
	DataRate250kbps      ///< 250 kbps
    } DataRate;

    /// \brief Convenient values for setting transmitter power in setRF()
    /// These are designed to agree with the values for RF_PWR in RH_NRF24_REG_06_RF_SETUP
    /// To be passed to setRF();
    typedef enum
    {
	// Add 20dBm for nRF24L01p with PA and LNA modules
	TransmitPowerm18dBm = 0,        ///< On nRF24, -18 dBm
	TransmitPowerm12dBm,            ///< On nRF24, -12 dBm
	TransmitPowerm6dBm,             ///< On nRF24, -6 dBm
	TransmitPower0dBm,              ///< On nRF24, 0 dBm
	// Sigh, different power levels for the same bit patterns on RFM73:
	// On RFM73P-S, there is a Tx power amp, so expect higher power levels, up to 20dBm. Alas
	// there is no clear documentation on the power for different settings :-(
	RFM73TransmitPowerm10dBm = 0,   ///< On RFM73, -10 dBm
	RFM73TransmitPowerm5dBm,        ///< On RFM73, -5 dBm
	RFM73TransmitPowerm0dBm,        ///< On RFM73, 0 dBm
	RFM73TransmitPower5dBm          ///< On RFM73, 5 dBm. 20dBm on RFM73P-S2 ?

    } TransmitPower;

    /// Constructor. You can have multiple instances, but each instance must have its own
    /// chip enable and slave select pin.
    /// After constructing, you must call init() to initialise the interface
    /// and the radio module
    /// \param[in] chipEnablePin the Arduino pin to use to enable the chip for transmit/receive
    /// \param[in] slaveSelectPin the Arduino pin number of the output to use to select the NRF24 before
    /// accessing it. Defaults to the normal SS pin for your Arduino (D10 for Diecimila, Uno etc, D53 for Mega,
    /// D10 for Maple)
    /// \param[in] spi Pointer to the SPI interface object to use.
    ///                Defaults to the standard Arduino hardware SPI interface
    RH_NRF24(uint8_t chipEnablePin = 8, uint8_t slaveSelectPin = SS, RHGenericSPI& spi = hardware_spi);

    /// Initialises this instance and the radio module connected to it.
    /// The following steps are taken:g
    /// - Set the chip enable and chip select pins to output LOW, HIGH respectively.
    /// - Initialise the SPI output pins
    /// - Initialise the SPI interface library to 8MHz (Hint, if you want to lower
    /// the SPI frequency (perhaps where you have other SPI shields, low voltages etc),
    /// call SPI.setClockDivider() after init()).
    /// -Flush the receiver and transmitter buffers
    /// - Set the radio to receive with powerUpRx();
    /// \return  true if everything was successful
    bool        init();

    /// Reads a single register from the NRF24
    /// \param[in] reg Register number, one of RH_NRF24_REG_*
    /// \return The value of the register
    uint8_t        spiReadRegister(uint8_t reg);

    /// Writes a single byte to the NRF24, and at the same time reads the current STATUS register
    /// \param[in] reg Register number, one of RH_NRF24_REG_*
    /// \param[in] val The value to write
    /// \return the current STATUS (read while the command is sent)
    uint8_t        spiWriteRegister(uint8_t reg, uint8_t val);

    /// Reads a number of consecutive registers from the NRF24 using burst read mode
    /// \param[in] reg Register number of the first register, one of RH_NRF24_REG_*
    /// \param[in] dest Array to write the register values to. Must be at least len bytes
    /// \param[in] len Number of bytes to read
    /// \return the current STATUS (read while the command is sent)
    uint8_t           spiBurstReadRegister(uint8_t reg, uint8_t* dest, uint8_t len);

    /// Write a number of consecutive registers using burst write mode
    /// \param[in] reg Register number of the first register, one of RH_NRF24_REG_*
    /// \param[in] src Array of new register values to write. Must be at least len bytes
    /// \param[in] len Number of bytes to write
    /// \return the current STATUS (read while the command is sent)
    uint8_t        spiBurstWriteRegister(uint8_t reg, uint8_t* src, uint8_t len);

    /// Reads and returns the device status register NRF24_REG_02_DEVICE_STATUS
    /// \return The value of the device status register
    uint8_t        statusRead();

    /// Sets the transmit and receive channel number.
    /// The frequency used is (2400 + channel) MHz
    /// \return true on success
    bool setChannel(uint8_t channel);

    /// Sets the chip configuration that will be used to set
    /// the NRF24 NRF24_REG_00_CONFIG register when in Idle mode. This allows you to change some
    /// chip configuration for compatibility with libraries other than this one.
    /// You should not normally need to call this.
    /// Defaults to NRF24_EN_CRC| RH_NRF24_CRCO, which is the standard configuration for this library
    /// (2 byte CRC enabled).
    /// \param[in] mode The chip configuration to be used whe in Idle mode.
    /// \return true on success
    bool setOpMode(uint8_t mode);

    /// Sets the Network address.
    /// Only nodes with the same network address can communicate with each other. You
    /// can set different network addresses in different sets of nodes to isolate them from each other.
    /// Internally, this sets the nRF24 TX_ADDR and RX_ADDR_P0 to be the given network address.
    /// The default network address is 0xE7E7E7E7E7
    /// \param[in] address The new network address. Must match the network address of any receiving node(s).
    /// \param[in] len Number of bytes of address to set (3 to 5).
    /// \return true on success, false if len is not in the range 3-5 inclusive.
    bool setNetworkAddress(uint8_t* address, uint8_t len);

    /// Sets the data rate and transmitter power to use. Note that the nRF24 and the RFM73 have different
    /// available power levels, and for convenience, 2 different sets of values are available in the
    /// RH_NRF24::TransmitPower enum. The ones with the RFM73 only have meaning on the RFM73 and compatible
    /// devces. The others are for the nRF24.
    /// \param [in] data_rate The data rate to use for all packets transmitted and received. One of RH_NRF24::DataRate.
    /// \param [in] power Transmitter power. One of RH_NRF24::TransmitPower.
    /// \return true on success
    bool setRF(DataRate data_rate, TransmitPower power);

    /// Sets the radio in power down mode, with the configuration set to the
    /// last value from setOpMode().
    /// Sets chip enable to LOW.
    void setModeIdle();

    /// Sets the radio in RX mode.
    /// Sets chip enable to HIGH to enable the chip in RX mode.
    void setModeRx();

    /// Sets the radio in TX mode.
    /// Pulses the chip enable LOW then HIGH to enable the chip in TX mode.
    void setModeTx();

    /// Sends data to the address set by setTransmitAddress()
    /// Sets the radio to TX mode
    /// \param [in] data Data bytes to send.
    /// \param [in] len Number of data bytes to send
    /// \return true on success (which does not necessarily mean the receiver got the message, only that the message was
    /// successfully transmitted).
    bool send(const uint8_t* data, uint8_t len);

    /// Blocks until the current message (if any)
    /// has been transmitted
    /// \return true on success, false if the chip is not in transmit mode or other transmit failure
    virtual bool waitPacketSent();

    /// Indicates if the chip is in transmit mode and
    /// there is a packet currently being transmitted
    /// \return true if the chip is in transmit mode and there is a transmission in progress
    bool isSending();

    /// Prints the value of all chip registers
    /// to the Serial device if RH_HAVE_SERIAL is defined for the current platform
    /// For debugging purposes only.
    /// \return true on success
    bool printRegisters();

    /// Checks whether a received message is available.
    /// This can be called multiple times in a timeout loop
    /// \return true if a complete, valid message has been received and is able to be retrieved by
    /// recv()
    bool        available();

    /// Turns the receiver on if it not already on.
    /// If there is a valid message available, copy it to buf and return true
    /// else return false.
    /// If a message is copied, *len is set to the length (Caution, 0 length messages are permitted).
    /// You should be sure to call this function frequently enough to not miss any messages
    /// It is recommended that you call it in your main loop.
    /// \param[in] buf Location to copy the received message
    /// \param[in,out] len Pointer to available space in buf. Set to the actual number of octets copied.
    /// \return true if a valid message was copied to buf
    bool        recv(uint8_t* buf, uint8_t* len);

    /// The maximum message length supported by this driver
    /// \return The maximum message length supported by this driver
    uint8_t maxMessageLength();

    /// Sets the radio into Power Down mode.
    /// If successful, the radio will stay in Power Down mode until woken by
    /// changing mode it idle, transmit or receive (eg by calling send(), recv(), available() etc)
    /// Caution: there is a time penalty as the radio takes a finite time to wake from sleep mode.
    /// \return true if sleep mode was successfully entered.
    virtual bool    sleep();

protected:
    /// Flush the TX FIFOs
    /// \return the value of the device status register
    uint8_t flushTx();

    /// Flush the RX FIFOs
    /// \return the value of the device status register
    uint8_t flushRx();

    /// Examine the receive buffer to determine whether the message is for this node
    void validateRxBuf();

    /// Clear our local receive buffer
    void clearRxBuf();

private:
    /// This idle mode chip configuration
    uint8_t             _configuration;

    /// the number of the chip enable pin
    uint8_t             _chipEnablePin;

    /// Number of octets in the buffer
    uint8_t             _bufLen;

    /// The receiver/transmitter buffer
    uint8_t             _buf[RH_NRF24_MAX_PAYLOAD_LEN];

    /// True when there is a valid message in the buffer
    bool                _rxBufValid;
};

/// @example nrf24_client.pde
/// @example nrf24_server.pde
/// @example nrf24_reliable_datagram_client.pde
/// @example nrf24_reliable_datagram_server.pde
/// @example RasPiRH.cpp

#endif

redpajama_set_name: RedPajamaGithub


text:
Q: fast cumulative operation in python? I have several operators f(a, b) like: "a*2+b*3", "a^b" ...
I would like to do something like:
a = a0
for _b in X:
    a = f(a, _b)

a0 will be N-byte value, X will be M*N byte buffer. In most case N <4, and a will not overflow. python reduce() will do the job, but for custom function it seems slow.
Can numexpr or other package speed up this?

A: Here are some options:

*

*You may want to try functools.reduce solution on PyPy (just reduce if you're on Python 2). In addition you can use array instead of list.

*Use numpy.ufunc.reduce if your lists are big enough. It is written in C/Cython, but it takes longer to create small numpy array than a python stdlib's collection of the same size.

Update:

*Use Cython to implement f(a, b), reduce or both.

Pros: you can achieve almost C performance when it comes to dealing with numbers. You don't have to know Python C API.
Cons: almost C performance comes with writing almost C code. Use pointers, do some malloc, declare types explicitly, get a compatible C compiler. Before starting I would suggest to use cprofile in order to determine the critical code.

redpajama_set_name: RedPajamaStackExchange


With election season in full swing it is time for the governor hand out some money. So in time for the election, but not too soon to yield any data, NY is rolling out Start Up NY. This program was sold to the public as a way to create new jobs by helping start up companies. These companies who partner with a local universities and get a 10 year grace period for taxes for them and their employees. Recently the Start Up board announced some winners in the Rochester Area as three location were approved for the U of R. Two are at the old Kodak Park site and the other is in Henrietta. Even if this program made sense the board fails to understand that neither of these are near the U of R. Of course these applicants for this space are not start up companies either and at least one is just relocating to these tax free zones. While this may seem like a misuse of the program the web site for Start Up NY encourages this sort of behavior. The problems with this program are many, but there is an underlying premise to this program that makes some sense. Start up companies have a lot of disadvantages, it is almost impossible for them to raise capital and they can not afford to hire the workers they need. Thus the state providing incentives would make some sense as it would decrease the failure rate of a start up company. The problem is that this program is not focusing on start up companies nor is it trying to help students at major university turn their ideas into businesses nor does it even see proximity as a requirement. It is being used like every other corporate welfare program the state has ever tried, just a way to give some businesses huge tax breaks. The granting of tax breaks to big business has a long history but has few results. Despite the good intentions of these programs they tend to go companies already in the state who use relocation to lower their costs. All these programs seem to specialize in helping the well connected and frequently use local tax breaks to fund these state projects. In effect they are unfunded mandates. COMIDA is a perfect example this uses local tax breaks to fund this state program thus costing local towns money with little positive return. Supporters claim that the jobs created put millions into the local economy through wages which offsets the lost taxes but this is not how they really work. After all giving tax breaks does not lower the amount of money needed by government it only means the rest of the people have to pay more. With property taxes this means rents go up and property values stagnate so everyone has to pay more for housing. As some of this housing is funded by government money that means taxes need to go up more and a cycle of rising costs is created. The affect is everyone has less money to spend. As our economy is fueled by consumer spending this means less growth everywhere but for the favored few with tax breaks. Which usually means no real job growth. This is held up by the numbers as Ny state job growth during the last decade has mirrored population growth for the state. Finally our economy is based upon competition. When tax breaks are granted to one company that company has a competitive edge allowing them to make a higher profit or lower prices than their competitors. Even when this results in job growth it is often at the expense of other firms without tax breaks. This is particularly evident when tax breaks are granted to commercial businesses. Just because a new restaurant opens that does not mean that more people will pay to go out to eat. Most patrons who attend the new restaurant would have gone to some other place and usually spent roughly the same amount of money. AS a result their is no real growth in our economy. Despite a long history of failure in the job creation market our leaders seem to have no other tool than tax breaks for economic development. Yet investments in education, infrastructure, and health care all have proven track records for economic growth. Our government could also start investing in the future. They could make start up money available, require banks to loan money in local areas, invest in new technologies, or even try new deal style work programs. Despite these options it seems that neither Astorino nor Cuomo have any plan other than tax breaks. Thus barring a Howie Hawkins miracle it looks like 4 more years of austerity for the rest of us.


text:
Mick Pointer, né Michael Pointer, est un batteur britannique né le  à Brill (Angleterre), officiant au sein du groupe Arena.

Biographie
Après une enfance sans histoire de son village natal de Brill, dans le Buckinghamshire, Mick Pointer part rapidement pour Londres. Il fait alors la connaissance du bassiste Doug Irvine, avec qui il fonde rapidement un groupe en 1979 : Silmarillion. Après avoir recruté le claviériste Brian Jelliman, le guitariste Steve Rothery et le chanteur Fish, le nom du groupe est changé en Marillion.

Après avoir participé à trois singles du groupe, au premier album Script for a Jester's Tear et au live The Recital of the Script, les musiciens du groupe lassés de ses faibles performances choisissent de le remplacer, en 1983.

Après un long retrait de la scène, il réapparaît en 1995 au côté de Clive Nolan, au sein du groupe Arena.

À l'occasion des 25 ans du premier album de Marillion, il joue aussi au sein du Mick Pointer Band.

Discographie

Marillion
 Market Square Heroes (single - 1982)
 Script for a Jester's Tear (1983)
 He Knows You Know (single - 1983)
 Garden Party (single - 1983)

Arena
 Voir: discographie d'Arena.

Notes et références

Liens externes

Batteur de rock progressif
Naissance en juillet 1956
Naissance dans le Buckinghamshire
Batteur anglais de rock

redpajama_set_name: RedPajamaWikipedia


text:
Dave Toke's green energy blog
How the centralised generators are trying to strangle the decentralised energy revolution in the UK
Just as the UK Government has stopped onshore renewables (mainly wind power and solar pv) from getting all-important long term power purchase agreements (PPAs) through the feed in tariff system (the big one being now reserved for Hinkley C), so government agencies are moving to make sure that the rules of the electricity market favour centralised generators over decentralised ones. The Government says that no subsidies will be available for onshore wind and solar pv. Yet it is busy doling out subsidies and altering rules to favour big power stations over decentralised renewables.
In setting the regulations, the Government and the agencies, including OFGEM and the National Grid (NG) clearly seem to favour big power plant over other decentralised options for balancing electricity supply and demand including battery storage and demand side response (DSR). Really these technologies should now be routinely combined with renewable energy schemes to create 'virtual power plant' to deliver energy services for consumers. Yet despite the celebrated Clayhill 'subsidy-free' solar pv-battery project, progress is very slow. The revolution is being held back by the dead hand of the centralised power regime.
The Government's preferred solution of course is lots more large gas fired power plant - and nuclear power plants of course. Yet these technologies are falling behind the newer decentralised ones whose costs and information based technologies are becoming more and more economic. But instead of helping decentralised energy, the Government is pushing more and more subsidies towards the old, centralised, solutions.
This action to roll back the revolution is taking place in the 'boileroom' of the electricity system, with its the arcane and often impenetrable rules and language of the electricity market, well away from the understanding of the wider public. However various trade and academic reports are flagging how the centralised generators are trying to hold back the decentralised energy revolution by whatever means are possible.
How is this happening? Essentially there are two strands. First there is the way that the so-called 'capacity market' is oriented to favour the interests of centralised power plant, and second is the way that the regulatory incentives are being geared to penalise smaller and more innovative players and to favour the big ones.
Capacity Market (CM)
Matthew Lockwood, in a recent working paper, tells the story of how the CM has largely been shaped to be a riverstream of income for the existing gas and coal and nuclear power plant. First came the decision to reward all existing generators for providing capacity, providing a subsidy for plants that have been built a long time ago. A much cheaper option would have been to operate a 'strategic reserve' that would fund a dedicated set of assets to be brought in to balance supply and demand. But that. of course, would not help the centralised power plant. Of course the mere term 'capacity' is biased against the decentralised solutions which include DSR and battery storage.
Then has come a series of decisions that have given centralised power plant an inbuilt advantage over decentralised options for balancing demand with supply. Cornwall Energy as well as Matthew Lockwood has written about some of these decisions and how they adversely affect the decentralised players.
First, DSR and battery storage are given much inferior terms compared to the big power stations in the CM. Their contribution is deliberately de-rated, subject to expensive monitoring and accorded much shorter contracts compared to the big power plant operators.
Second, OFGEM has issued rules which slash payments earned by distributed generators, that is small generators, through the TRIAD system. This is a mechanism whereby the system rewards companies which can reduce peak power requirements.
Third, the rules seem to favour the big operators even when it comes to providing storage solutions in what is called 'frequency response' services. This is a mechanism that incentivises those who can produce instant remedies to keep national electricity frequencies with a prescribed margin. Yet the decisions of the National Grid in awarding the contracts seem to favour the big boys.
Clearly the dinosaurs are thrashing about to great effect in an effort to delay the decentralised revolution. They will not win the war, but at the moment they are managing to delay the onward march of decentralised energy.
Without doubt they are winning the propaganda war. Any incentives given to renewables are deemed subsidies, whilst the reality is that these subsidies have been eliminated whilst the effective bank of subsidies given to big power station operators is growing rapidly.
For further info, read:
https://theenergyst.com/ofgem-outlines-deep-cuts-to-small-generators-triad-payments/
http://projects.exeter.ac.uk/igov/wp-content/uploads/2017/10/WP-1702-Capacity-Market.pdf
https://www.cornwall-insight.com/newsroom/all-news/latest-frequency-tender-results-exacerbate-battery-challenges
https://www.carbonbrief.org/in-depth-how-smart-flexible-grid-could-save-uk-40-billion
Posted by Dr David Toke at 11:04 No comments:
Three cheers for Sturgeon as she announces pro-renewables Scottish energy company
In what must come as a welcome boost to the flagging hopes of renewable energy workers and supporters Nicola Sturgeon has announced the Government's intention to establish a publicly owned energy company that will be fuelled specifically from renewable energy.
https://www.energyvoice.com/other-news/152781/breaking-sturgeon-proposes-publicly-owned-not-profit-energy-company/
Credit, of course, should also go to the Scottish Greens upon whose votes the SNP depend for a majority and who have been very influential in pushing forward the green energy agenda.
Of course a lot of detail remains to be worked out, but if what's in the can matches the label then this should be a big opportunity for an industry that has been laid low by Westminster's refusal to fund any further land based wind or solar projects.
The Scottish Government's emphasis is on keeping costs down, but that is not a problem for onshore renewable energy whose costs have been declining rapidly in recent times. What they lack at the moment is long term guarantees about income to be earned for energy generation. Nicola Sturgeon's proposal seems likely to plug this gap.
The Scottish Government (SG) could carry out its mission by various means, provided it achieves the central necessity of issuing long term agreements on levels of payment per unit generated from renewable energy projects. It would also be popular if priority could be given to schemes that are community based, that is owned whole or in part by ordinary people. This is what myself and others were arguing in The Scotsman this morning:
See.....
https://www.pressreader.com/uk/the-scotsman/20171010/282050507282301
and......
http://www.scotsman.com/news/opinion/ilona-amos-green-power-schemes-need-a-boost-1-4582308
Long term power purchase arrangements are needed if renewable energy projects are to get cheap financing deals with banks and investors. Among the options there are two routes to progress in how the Scottish company could buy energy and give long term guaranteed incomes flows to solar, wind, and micro-hydro projects. One is that the company could conduct auctions for the right to be given long term power purchase agreements (PPAs), with companies competing to offer the lowest price per MWh to supply a given tranche of contracts. A second, perhaps more suitable for community renewable schemes, is to, in effect, offer them a standard rate for their power, perhaps linked to the wholesale power price (as argued in the letter to The Scotsman).
Recently a report published by Scottish Renewables suggested that 1 GW of wind power was available for no more than £49 per MWh. Yet renewable energy costs (including the costs of solar pv as well as wind) have continued to fall. See https://www.scottishrenewables.com/news/most-competitive-onshore-wind-projects-baringa/
Solar pv costs have been plunging, and if the Scottish Government can offer long term PPAs (for 15 or preferably even 20 years) then they may be able to entice cutting edge solar pv (and battery?) projects up North as has been developed in the Clayhill project in the South of England.
In recent times the wholesale electricity price has been £45 per MWh. Yet with production of the cheaper gas supplies from the British and Norweigian parts of the North Sea under decline and with our other major supplier (The Netherlands) now restricting future exploitation of gas fields the Scottish Government looks like it will be a winner if it signs up wind and solar projects. They may be competing with electricity from gas power plant fuelled by increasingly expensive supplies from Qatar or other places.
Scotland's proposal for a state owned energy company stands in stark contrast to the nationalisation proposed by Labour which is tinged with support for nuclear power. If Labour's planned nuclear expansion goes ahead it will result in heavy state losses, whilst Scotland's renewable expansion will result in cost savings.
How the Scottish Government could implement a 'subsidy free' scheme for community renewables
The Scottish Government (SG) should open discussions to establish a mechanism to enable 'subsidy free' community renewable schemes. It is great news to see a decline in costs of offshore wind schemes, but as good as they are, they are not involving ordinary people. Community renewable schemes, using a definition given to me by Jon Halle of sharenergy are:
'Community renewable schemes are majority-owned and run by members of the public, including both people local to the scheme and supporters from further afield. They are open and democratically controlled, ensuring wide popular ownership of the energy system and maximising the sharing of both benefits and responsibility.'
This could apply to wind power, solar pv or micro-hydro. A 'subsidy free' scheme should be done on a pilot scheme basis to begin with.
Because of the decline in wind and solar power costs it seems likely that some renewable energy projects in Scotland could be established assuming current levels of power prices that generators can receive on wholesale power markets. The projects would certainly count as 'subsidy free'. But they need long term assurances about income streams, something that the Scottish Government could provide at minimal risk to the public purse.
A scheme could be established by the SG to set up a back-up loan facility to give 'top-up' payments for community renewable generators. This could ensure that the generators received at least the income that they would do if wholesale power prices were at the current level of, say, £45 per MWh. Any loans paid would be paid back when power prices rose above the £45 per MWh level. This arrangement could be guaranteed for 20 years and could be enshrined in agreements issued by the SG to specific schemes. This would give schemes long term financial confidence that could allow them to raise money from banks and investors.
A pilot basis would consist of the scheme being restricted, for an initial proving phase, of no more than 100 MW of capacity.
A recent auction for community wind power projects in Germany saw the projects winning long term power purchase agreements for under £40 per MWh, and this is in a country with much lower windspeeds than are available in Scotland. See https://www.bloomberg.com/news/articles/2017-08-15/german-onshore-wind-power-costs-plummet-in-second-auction
The Clayhill solar pv scheme has recently opened on a 'subsidy free' basis, and this is helped by being able to earn payments for electricity system services by co-locating the solar farm with batteries. See https://www.solarpowerportal.co.uk/blogs/inside_clay_hill_the_uks_first_subsidy_free_solar_farm
ARUP has published an analysis of market possibilities for 'subsidy-free' wind power suggested that schemes costing less than £50-£55 per MWh (in 2012 prices) would count as 'subsidy free'. file:///C:/Users/Toke/Downloads/Enabling%20Investment%20in%20Established%20Low%20Carbon%20Electricity%20Generation%20(2).pdf
While the Westminster Government is reluctant to offer sufficiently good financial conditions to promote the development of much onshore wind, and solar pv, Scotland is in a good position to continue take a lead in promoting community renewable. Although £45 per MWh is not a large amount of money, it does at least ensure that the scheme is 'subsidy-free' and means that the Scottish Government, which does not have the power to levy charges on electricity bills (unlike Westminster), will not suffer large financial losses.
There are some harbingers that suggest that average wholesale electricity prices will rise in future years. These prices are dominated by natural gas prices. The UK is able to access a declining proportion of its gas from cheap British sources. The balance cones increasingly from Dutch and Norwegian fields, yet both sets of supplies appear to have peaked already and there is every indication that they may decline. This will leave the UK increasingly dependent on supplies of LNG from Qatar and other sources which are much more expensive than those from the North Sea or The Netherlands.
Brief financial risk analysis
Let us assume that the scheme was limited to 100 MW in order to demonstrate test the concept, and that this 100 MW consisted of wind power projects. Even if the long term income available from power markets was £10 per MWh less than the level of £45 per MWh that the SG would guarantee paid to wind power generators (eg £35 per MWh compared to £45 per MWh) then the annual 'loss' payable by the Government would be less than £3 million (under this pilot scheme). Recently the wholesale power price has been around the £45 per MWh level. Of course if the average income stream is higher than a 'strike price' of eg £45 per MWh then there would be a surplus of income.
OFGEM publishes recent average monthly wholesale power prices at https://www.ofgem.gov.uk/data-portal/electricity-prices-day-ahead-baseload-contracts-monthly-average-gb
Posted by Dr David Toke at 04:23 1 comment:
How the centralised generators are trying to stran...
Three cheers for Sturgeon as she announces pro-ren...
How the Scottish Government could implement a 'sub...
Dr David Toke
Dr David Toke is Reader in Energy Politics in the Department of Politics and International Relations in the University of Aberdeen. You can see his profile at http://www.abdn.ac.uk/socsci/people/profiles/d.toke He has been campaigning on energy issues for around 30 years, and in 1990 his book 'Green Energy' was an influential argument in the UK for a non-nuclear approach to dealing with global warming. He was a key player in the campaign to establish feed-in tariffs for small renewable projects in the UK, achieved in 2008. He has consistently argued that the UK's proposed nuclear power programme is not only uneconomic compared to renewable energy, but that it is undeliverable short of one or more governments signing what amounts to a 'blank cheque' to pay for the nuclear power plant. His latest book, published by Routledge is called 'Low Carbon Politics'. He has published many papers in leading political science journals on environmental, especially energy (and renewable energy) issues and he is also a frequent and well cited contributor to the journal 'Energy Policy' published by Elsevier. His twitter address is @DaveToke


text:
Tag Archives: taxation
Music Business, Resources, Traveling Musician
Canadian Waivers and Taxation for Foreign Artists
May 28, 2015 IM
Ignoring the taxation requirements of a foreign country can lead to unforeseen complications, as this letter illustrates:
We sent an American ensemble to Canada in 2013. We had obtained an approved R-105 waiver in advance. Consequently, no taxes were withheld. We did not realize that we needed to file a Canadian tax return. When we applied for a waiver last summer, we were surprised to have it rejected. The presenter withheld 15% and we'd like to get it back. What do we do now?
R-105 Waivers are used in Canada to reduce or eliminate the 15% required withholding on services provided in Canada by a nonresident. Even with an approved R105 waiver in Canada, you are still required to file a Canadian income tax return the following year; otherwise, subsequent waiver applications will be denied until your tax filings are up-to-date. This is made clear in the approval letter from Revenue Canada. R105 waivers do not represent the final Canadian tax obligation of a foreign artist: the ultimate tax liability can only be determined after an assessment of a Canadian tax return.
There are two types of R105 waiver applications:
1) Treaty-based waivers—Treaty-based waivers are granted if there is a treaty between Canada and another country. Currently, there are more than 80 tax treaties in force. (Visit http://www.fin.gc.ca/treaties-conventions/in_force-eng.asp for a list.) Generally, where there is a treaty, if an artist earns less than $5,000 a year in Canada, with certain restrictions on time spent in Canada, a waiver will be granted. For American artists the amount is less than $15,000 per year.
2) Income/expense waiver—If you do not qualify for a treaty-based waiver, you can still apply for an income/expense waiver. You submit a summary of your gross Canadian income and claim applicable expenses against that amount. The net income is then assessed for tax liability. A waiver may be granted or the required withholding may be less than the required 15%. Applicable expenses include: professional service fees (managers, agents, etc.); accommodations and/or meals; travel to Canada and between places in Canada; mileage for personally owned or rented vehicles used in Canada; equipment rental other than vehicles; and remuneration paid to other persons providing services in Canada (for example, resident or nonresident employees, or subcontractors).
[Note: Fees paid to nonresidents require the 15% withholding unless you acquire an approved R105 or R102 waiver for them as well].
Whether you received an approved waiver or had monies withheld, you should always file a Canadian tax return. It's a requirement for future waiver approvals and you may receive a refund of monies withheld. Individuals should file a T1 return by April 30 and corporations a T2 return by June 30 of the following calendar year. In addition, if you used subcontractors or employees, you will need to issue T4A-NR slips to each individual, remit withheld monies to the receiver general of Canada by the 15th of the month, following the month in which the payment was made to the nonresident and file a T4A-NR information return (T4A-NR slips and summary form) by the last day of February in the year following the year in which the amounts were paid. Note that there are significant penalties for failing to file tax returns and required forms as indicated.
Nonresident artists need to be aware of waivers and tax filing requirements when coming to Canada to perform.
To read this article in French visit: www.internationalmusician.org/la-fiscalite-canadienne-et-les-dispenses.
bamCanadianmusic businessrobert braidtaxationtaxestraveling musicianwaiversworking musician
United States Taxation for Foreign Artists
May 1, 2015 IM
The issue of United States Taxation for foreign artists came up in a recent letter:
I have a band that is looking to play a corporate event in the US at the end of June. My accountant just informed me of the Central Withholding Agreement that could keep as much as 30% of our income—the final total being left at their discretion. My accountant tells me that it is likely just a matter of incorporating the business, which can be done fairly quickly. Can you help?
Dealing with taxation in any country is complex. The regulations regarding taxation for a foreign artist are customarily contained in a tax treaty between the two countries, and will apply differently to individuals than to businesses. However, it is not simply a matter of incorporating an individual or a performing group as a business because the IRS considers who the "beneficial owner" of the income is and, if any individual is named in the corporation, then the income is considered personal, not corporate.
In the US, the requirement is that anyone paying a foreign artist has to withhold 30% in taxes on US income. Anyone who withholds tax, be it a promoter, a venue, a presenter, a manager, or an agent, is designated by the IRS as a "withholding agent." If the foreign artist fails to file a tax return and pay the required taxes, then the "withholding agent" is liable for any unpaid tax.
There are exceptions to this general principle:
A tax treaty may exempt a business from paying tax in a foreign country if it is recognized as a business by the IRS and has no "fixed establishment" in the US.
A tax treaty may allow a certain level of income for independent personal services to be earned tax-free, if the individual has no "fixed base" in the US and spends no more than a certain number of days in the country. For example, the Canada-US Tax Treaty allows an individual artist to earn $15,000 annually without taxation. However, earning more than $15,000 annually would make the individual taxable for the whole amount.
An individual (not a business) can enter into a Central Withholding Agreement (CWA) with the IRS whereby the IRS will estimate the actual tax liability of the artist and withhold less than the required 30%.
In years past, it was often enough to provide a US employer with a W-8BEN or a Form 8233 to avoid the required 30% tax withholding. Now, this is no longer regarded as sufficient. An employer cannot rely on these forms to avoid enforcement of the IRS requirement.
If it transpires that a foreign artist has been "over-taxed," then a refund can be obtained by filing a tax return. Even if your income is exempt from US income tax, it's a good idea to file annually. Individuals should file a Form 1040NR or Form 1040NR-EZ. Businesses should file a Form 1120-F.
At some point in the future it may become necessary to have your tax filings current before you are granted a work visa for the US. It's a good idea to comply with all requirements for foreign taxation.
—I welcome your questions and concerns. Please write to me at: robert@bairdartists.com. While I cannot answer every question I receive in this column, I will feature as many as I can and I promise to answer every e-mail I receive.
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Family sells father's 5,000 beer can collection to brewery
Updated: 8:42 PM EDT Apr 5, 2017
After selling their father's 5,000 plus beer can collection to a brewery in east Texas, the family stopped by to see its new home. The hobby started as beer brewing and eventually transitioned into beer can collecting. It's been 50 years since the tradition began. "There are about 5,000 cans. Dad had about the same number, about 2,400 on the wall in his basement and the rest of them he had stacked in his storeroom as his trading cans," said Steve Aldering, son of the man who collected the cans. Steve and Peg Aldering sold their father's collection in the beginning of 2016 after he passed away to the owners of the newly opened Fredonia Brewery.Friends and family began sending the Aldering family cans from foreign countries."Seeing it here was certainly and emotional but inspirational. it was something he loved," and for the brewery, it's certainly a conversation starter.
After selling their father's 5,000 plus beer can collection to a brewery in east Texas, the family stopped by to see its new home.
The hobby started as beer brewing and eventually transitioned into beer can collecting. It's been 50 years since the tradition began.
"There are about 5,000 cans. Dad had about the same number, about 2,400 on the wall in his basement and the rest of them he had stacked in his storeroom as his trading cans," said Steve Aldering, son of the man who collected the cans.
Steve and Peg Aldering sold their father's collection in the beginning of 2016 after he passed away to the owners of the newly opened Fredonia Brewery.
Friends and family began sending the Aldering family cans from foreign countries.
"Seeing it here was certainly and emotional but inspirational. it was something he loved," and for the brewery, it's certainly a conversation starter.