A general affine connection has enough degrees of freedom to describe theclassical gravitational and electromagnetic fields in the metric-affineformulation of gravity. The gravitational field is represented in theLagrangian by the symmetric part of the Ricci tensor and the classicalelectromagnetic field can be represented by the tensor of homothetic curvature.The simplest metric-affine Lagrangian that depends on the tensor of homotheticcurvature generates the Einstein-Maxwell equations for a massless vector.Metric-affine Lagrangians with matter fields depending on the connection aresubject to an unphysical constraint because the symmetrized Ricci tensor isprojectively invariant while matter fields are not. We show that the appearanceof the tensor of homothetic curvature, which is not projectively invariant, inthe Lagrangian replaces this constraint with the Maxwell equations and restoresprojective invariance of the total action. We also examine several constraintson the torsion tensor to show that algebraic constraints on the torsion thatbreak projective invariance of the connection and impose projective invarianceon the tensor of homothetic curvature replace the massless vector with amassive vector. We conclude that the metric-affine formulation of gravityallows for a mechanism that generates masses of vectors, as it happens forelectroweak gauge bosons via spontaneous symmetry breaking.
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